Chapter 1: Introduction

Transcription

1 Chapter 1: Introdution 1.1 Pratial olumn base details in steel strutures Pratial olumn base details Every struture must transfer vertial and lateral loads to the supports. In some ases, beams or other members may be supported diretly, though the most ommon system is for olumns to be supported by a onrete foundation. The olumn will be onneted to a baseplate, whih will be attahed to the onrete by some form of so-alled holding down assembly. Typial details are shown in Figure 1.1. The system of olumn, baseplate and holding down assembly is known as a olumn base. This publiation proposes rules to determine the strength and stiffness of suh details. I or H Setion olumn Grout Base plate Foundation Paks Tubular sleeve Conial sleeve Anhor plates Holding down bolts Anhor plates Hook bolts Cast in setion Stub on underside of base to transfer shear Cover plate Oversize hole 'T' bolt Underut anhors Cast in hannel Figure 1.1 Typial olumn base details 1.1

2 Other olumn base details may be adopted, inluding embedding the lower portion of olumn into a poket in the foundation, or the use of baseplates strengthened by additional horizontal steel members. These types of base are not overed in this publiation, whih is limited to unstiffened baseplates for I or H setions. Although no detailed guidane is given, the priniples in this publiation may be applied to the design of bases for RHS or CHS setion olumns. Foundations themselves are supported by the sub-struture. The foundation may be supported diretly on the existing ground, or may be supported by piles, or the foundation may be part of a slab. The influene of the support to the foundation, whih may be onsiderable in ertain ground onditions, is not overed in this doument. Conrete foundations are usually reinfored. The reinforement may be nominal in the ase of pinned bases, but will be signifiant in bases where bending moment is to be transferred. The holding down assembly omprises two, but more ommonly four (or more) holding down bolts. These may be ast in situ, or post-fixed to the ompleted foundation. Cast in situ bolts usually have some form of tubular or onial sleeve, so that the top of the bolts are free to move laterally, to allow the baseplate to be aurately loated. Other forms of anhor are ommonly used, as shown in Figure 1.. Baseplates for ast-in assemblies are usually provided with oversize holes and thik washer plates to permit translation of the olumn base. Post-fixed anhors may be used, being positioned aurately in the ured onrete. Other assemblies involve loose arrangements of bolts and anhor plates, subsequently fixed with ementiious grout or fine onrete. Whilst loose arrangements allow onsiderable translation of the baseplate, the lak of initial fixity an mean that the olumn must be propped or guyed whilst the holding down arrangements are ompleted. Anhor plates or similar embedded arrangements are attahed to the embedded end of the anhor assembly to resist pull-out. The holding down assemblies protrude from the onrete a onsiderable distane, to allow for the grout, the baseplate, the washer, the nut and a further threaded length to allow some vertial tolerane. The projetion from the onrete is typially around 1 mm, with a onsiderable threaded length. Post-fixed assemblies inlude expanding mehanial anhors, hemial anhors, underut anhors and grouted anhors. Various types of anhor are illustrated in Figure

3 a b d e f a, b d e f ast in plae post fixed, underut post fixed hemial or ementiius grout post fixed expanding anhor fixed to grillage and ast in-situ Figure 1. Alternative holding-down anhors The spae between the foundation and the baseplate is used to ensure the baseplate is loated at the orret absolute level. On smaller bases, this may be ahieved by an additional set of nuts on the holding down assemblies, as shown in Figure 1.3. Commonly, the baseplate is loated on a series of thin steel paks as shown in Figure 1.4, whih are usually permanent. Wedges are ommonly used to assist the plumbing of the olumn. Figure 1.3 Baseplate with levelling nuts The remaining void is filled with fine onrete, mortar, or more ommonly, non-shrink ementiious grout, whih is poured under and around the baseplate. Large baseplates 1.3

4 generally have holes to allow any trapped air to esape when the baseplate is grouted. Permanent k Temporar wedge Grout h l Figure 1.4 Baseplate loated on steel paks The plate attahed to the olumn is generally retangular. The dimensions of the plate are as required by design, though pratial requirements may mean the base is larger than neessitated by design. Steel eretors favour at least four bolts, sine this is a more stable detail when the olumn is initially ereted. Four bolts also allow the baseplate to be adjusted to ensure vertiality of the olumn. Bolts may be loated within the profile of the I or H setion, or outside the profile, or both, as shown in Figure 1.1. Closely grouped bolts with tubular or onial sleeves are to be avoided, as the remaining onrete may not be able to support the olumn and superstruture in the temporary ondition. Bases may have stubs or other projetions on the underside whih are designed to transfer horizontal loads to the foundations. However, suh stubs are not appreiated by steelwork eretors and should be avoided if possible. Other solutions may involve loating the base in a shallow reess or anhoring the olumn diretly to, for example, the floor slab of the struture. Columns are generally onneted to the baseplate by welding around part or all of the setion profile. Where orrosion is possible a full profile weld is reommended. 1.4

5 1.1. Pinned base details Pinned bases are assumed in analysis to be free to rotate. In pratie pinned bases are often detailed with four holding down bolts for the reasons given above, and with a baseplate whih is signifiantly larger than the overall dimensions of the olumn setion. A base detailed in this way will have signifiant stiffness and may transfer moment, whih assists eretion. In theory, suh a base should be detailed to provide onsiderable rotational apaity, though in pratie, this is rarely onsidered Fixed base details Fixed (or moment-resisting) bases are assumed in analysis to be entirely rigid. Compared to pinned bases, fixed bases are likely to have a thiker baseplate, and may have a larger number of higher strength holding down assemblies. Oasionally, fixed bases have stiffened baseplates, as those shown in Figure 1.5. The stiffeners may be fabriated from plate, or from steel members suh as hannels. Stiffener Figure 1.5 Typial stiffened olumn base detail 1.5

6 1.1.4 Resistane of olumn bases Euroode 3, Setion 6 and Annex L ontain guidane on the strength of olumn bases. Setion 6 ontains priniples, and Annex L ontains detailed appliation rules, though these are limited bases subjet to axial loads only. The priniples in Setion 6 over the moment resistane of bases, though there are no appliation rules for moment resistane and no priniples or rules overing the stiffness of suh bases. Traditional approahes to the design of moment-resisting bases involve an elasti analysis, based on the assumption that plane setions remain plane. By solving equilibrium equations, the maximum stress in the onrete (based on a triangular distribution of stress), the extent of the stress blok and the tension in the holding down assemblies may be determined. Whilst this proedure has proved satisfatory in servie over many years, the approah ignores the flexibility of the baseplate in bending, the holding down assemblies and the onrete Modelling of olumn bases in analysis Traditionally, olumn bases are modelled as either pinned or as fixed, whilst aknowledging that the reality lies somewhere within the two extremes. The opportunity to either alulate or to model the base stiffness in analysis was not available. Some national appliation standards reommend that the base fixity be allowed for in design. The base fixity has an important effet on the alulated frame behaviour, partiularly on frame defletions. 1. Calulation of olumn base strength and stiffness 1..1 Sope of the publiation In reent years, Wald, Jaspart and others have direted signifiant researh effort to the determination of resistane and stiffness. Based on the results of this researh, reommendations for the design and verifiation of moment-resisting olumn bases ould be drafted. This permits the modelling in analysis of semi-ontinuous bases in addition to the traditional pratie of pinned and fixed bases. Both resistane and stiffness an be determined. This publiation ontains proposals for the alulation of the apaity and of the stiffness of moment-resisting bases, with the intention that these be inluded in Euroode 3. This publiation is foused on I or H olumns with unstiffened baseplates, though the priniples in this publiation may be applied to baseplates for RHS or CHS olumns. Embedded olumn base details are exluded from the reommendations in this publiation 1.6

7 The effet of the onrete-substruture interation on the resistane and stiffness of the olumn base is exluded. 1.. Component method The philosophy adopted in this publiation is known as the omponent method. This approah aords with the approah already followed in EC3 and, in partiular, Annex J, where rules for the determination of beam to olumn strength and stiffness are presented. The omponent approah involves identifying eah of the important features in the base onnetion and determining the strength and stiffness of eah of these omponents. The omponents are then assembled to produe a model of the omplete arrangement. Eah individual omponent and the assembly model are validated against test results. 1.3 Doument struture Setion of this doument ontains details of the omponents in a olumn base onnetion, namely: The ompression side - the onrete in ompression and the flexure of the baseplate. The olumn member. The tension side - the holding-down assemblies in tension and the flexure of the baseplate. The transfer of horizontal shear. Eah sub-setion overs a omponent and follows the following format: A desription of the omponent. A review of existing relevant researh. Details of the proposed model. Results of validation against test data. Setion 3 desribes the proposed assembly model and demonstrates the validity of the proposals ompared to test data. Setion 4 makes reommendations for the pratial use of this doument in analysis of steel frames. Setion 5 makes reommendations for the lassifiation of bases as sway or non-sway, in braed and unbraed frames. 1.7

8 1.8

9 Chapter : Component harateristis.1 Conrete in ompression and base plate in bending.1.1 Desription of the omponent The omponents onrete in ompression and base plate in bending inluding the grout represent the behaviour of the ompressed part of a olumn base with a base plate. The strength of these omponents depend primarily on the bearing resistane of the onrete blok. The grout is influening the olumn base bearing resistane by improving the resistane due to appliation of high quality grout, or by dereasing the resistane due to poor quality of the grout material or due to poor detailing. The deformation of this omponent is relatively small. The desription of the behaviour of this omponent is required for the predition of olumn bases stiffness loaded by normal fore primarily..1. Overview of existing material The tehnial literature onerned with the bearing strength of the onrete blok loaded through a plate may be treated in two broad ategories. Firstly, investigations foused on the bearing stress of rigid plates, most were onerned the prestressed tendons. Seondly, studies were onentrated on flexible plates loaded by the olumn ross setion due to an only portion of the plate. The experimental and analytial models for the omponents onrete in ompression and plate in bending inluded the ratio of onrete strength to plate area, relative onrete depth, the loation of the plate on the onrete foundation and the effets of reinforement. The result of these studies on foundations with punh loading and fully loaded plates offer qualitative information on the behaviour of base plate foundations where the plate is only partially loaded by the olumn. Failure ours when an inverted pyramid forms under the plate. The appliation of limit state analysis on onrete an inlude the three-dimensional behaviour of materials, plastifiation and raking. Experimental studies (Shelson, 1957; Hawkins, 1968, DeWolf, 1978) led to the development of an appropriate model for olumn base bearing stress estimation that was adopted into the urrent odes. The separate hek of the onrete blok itself is neessary to provide to hek the shear resistane of the onrete blok as well as the bending or punhing shear resistane aording to the onrete blok geometry detailing. The influene of a flexible plate was solved by replaing the equivalent rigid plate (Stokwell, 1975). This reasoning is based on reognition that uniform bearing pressure is unrealisti and.1

10 that maximum pressure would logially follow the profile shape. This simple pratial method was modified and heked against the experimental results, (Bijlaard, 198; Murray, 1983). Euroode 3 ( Annex L, 199) adopted this method in onservative form suitable for standardisation using an estimate inluding the dimensions of the onrete blok ross-setion and its height. It was also found (DeWolf and Sarisley, 198; Wald, 1993) that the bearing stress inreases with larger eentriity of normal fore. In this ase is the base plate in larger ontat with the onrete blok due to its bending. In ase, when the distane between the plate edge and the blok edge is fixed and the eentriity is inreased, the ontat area is redued and the value of bearing stress inreases. In ase of the rushing of the onrete surfae under the rigid edge is neessary to apply the theory of damage. These ases are unaeptable from design point of view and are determining the boundaries of above desribed analysis..1.3 Proposed model Strength The proposed design model resistane of the omponents onrete in ompression and base plate in bending is given in Euroode 3 Annex L, 199. The resistane of these omponents is determined with help of an effetive rigid plate onept. The onrete blok size has an effet on the bearing resistane of the onrete under the plate. This effet an be onservatively introdued for the strength design by the onentration fator k = j a b 1 1 a b (.1.1) where the geometry onditions, see Figure.1.1, are introdued by a 1 = a + a 5 a min, a a a h + 5 b 1 r 1 (.1.) b 1 = b + b 5 b min, b b b h + 5 a 1 r 1 (.1.3) This onentration fator is used for evaluation of the design value of the bearing strength as.

11 follows f = j β j k γ j f k (.1.4) where joint oeffiient is taken under typial onditions with grout as β j = / 3. This fator βj represents the fat that the resistane under the plate might be lower due to the quality of the grout layer after filling. F Rd a a 1 a r t h b b 1 b r Figure.1.1 Evaluation of the onrete blok bearing resistane The flexible base plate, of the area Ap an be replaed by an equivalent rigid plate with area A eq, see Fig.1.. The formula for alulation of the effetive bearing area under the flexible base plate around the olumn ross setion an be based on estimation of the effetive width. The predition of this width an be based on the T-stub model. The alulation seures that the yield strength of base plate is not exeeded. Elasti bending moment resistane of the base plate per unit length should be taken as M = 1 t f y 6 (.1.4) and the bending moment per unit length on the base plate ating as a antilever of span is, see Figure.1.3. M = 1 f j (.1.4).3

12 A p A A eq Figure.1. Flexible base plate modelled as a rigid plate of effetive area with effetive width When these moments are equal, the bending moment resistane is reahed and the formula evaluating an be obtained 1 1 f t f j 6 = (.1.4) y as = t 3 f f y j γ M (.1.5) The omponent is loaded by normal fore FSd. The strength, expeting the onstant distribution of the bearing stresses under the effetive area, see Figure.1.3 is possible to evaluate for a omponent by = = + (.1.6) F F A f ( t ) L f sd Rd eq j w j olumn F Sd F Rd L t base plate t w f j Figure.1.3 The T stub in ompression, the effetive width alulation The improvement of effetive area due to the plate behaviour for plates fixed on three or four edges an be based on elasti resistane of plates (Wald, 1995) or more onservatively an be limited by the deformations of plate as is reahed for antilever predition. This improvement is not signifiant for open ross setions, till about 3%. For tubular olumns the plate.4

13 behaviour inrease the strength up to 1% aording to the geometry. The pratial onservative estimation of the onentration fator, see Eq. (.1.1), an be preised by introdution of the effetive area into the alulation; into the proedure Eq. (.1.1) - (.1.3). This leads however to an iterative proedure and is not reommended for pratial purposes. The grout quality and thikness is introdued by the joint oeffiient βj, see SBR (1973). For β j = / 3, it is expeted the grout harateristi strength is not less than, times the harateristi strength of the onrete foundation f.g, f and than the thikness of the grout is not greater than, times the smaller dimension of the base plate t g, min (a ; b). In ases of different quality or high thikness of the grout tg, min (a ; b), it is neessary to hek the grout separately. The bearing distribution under 45 an be expeted in these ases, see Figure.1.4., (Bijlaard, 198). The influene of paking under the steel plate an be negleted for the design (Wald at al, 1993). The influene of the washer under plate used for eretion an be also negleted for design in ase of good grout quality f.g, f. In ase of poor grout quality f.g, f it is neessary to take into aount the anhor bolts and base plate resistane in ompression separately. tg t t g tg washer under base plate h o 45 t g 45 o paking Figure.1.4 The stress distribution in the grout.1.3. Stiffness The elasti stiffness behaviour of the T-stub omponents onrete in ompression and plate in bending exhibit the interation between the onrete and the base plate as demonstrated for the strength behaviour. The initial stiffness an be alulated from the vertial elasti deformation of the omponent. The omplex problem of deformation is influened by the flexibility of the base plate, and by the onrete blok quality and size..5

14 The simplified predition of deformations of a rigid plate supported by an elasti half spae is onsidered first inluding the shape of the retangular plate. In a seond step, an indiation is given how to replae a flexible plate by an equivalent rigid plate. In the last step, assumptions are made about the effet of the size of the blok to the deformations under the plate for pratial base plates. The deformation of a retangular rigid plate in equivalent half spae solved by different authors is given in simplified form by Lambe & Whitman, 1967 as δ r = F α ar, (.1.6) E A r where δr is the deformation under a rigid plate, F the applied ompressed fore, ar the width of the rigid plate, E the Young's modulus of onrete, A r the area of the plate, A r = a r L, L the length of the plate, α a fator dependent on ratio between L and a r. The value of fator α depends on the Poison's ratio of the ompressed material, see in Table.1.1, for onrete (ν,15). The approximation of this values as α,85 L an be read from the following Table.1.1. L / a r Table.1.1 Fator α and its approximation α aording to (Lambe and Whitman, 1967) Approximation as α 85, L / a r 1,9,85 1,5 1,1 1,4 1,5 1, 3 1,47 1,47 5 1,76 1,9 1,17,69 / a r With the approximation for α, the formula for the displaement under the plate an be rewritten.6

15 = 85, F δ r E L a r (.1.7) A flexible plate an be expressed in terms of equivalent rigid plate based on the same deformations. For this purpose, half of a T-stub flange in ompression is modelled as shown in Figure.1.5. δ fl E Ip x Figure.1.5 A flange of a flexible T-stub The flange of a unit width is elastially supported by independent springs. The deformation of the plate is a sine funtion, whih an be expressed as δ (x) = δ sin ( ½π x / fl ) (.1.8) The uniform stress on the plate an then be replaed by the fourth differentiate of the deformation multiplied by E Ip, where E is the Young's modulus of steel and Ip is the moment of inertia per unit length of the steel plate with thikness t (I p = t 3 / 1). σ (x) = E s I p ( ½π / fl ) 4 δ sin (½π x / fl ) = E s t 3 The onrete part should be ompatible with this stress 1 (½π / fl ) 4 δ sin ( ½π x / fl ) (.1.9) δ(x) = σ(x) hef / E (.1.1) where h ef is the equivalent onrete height of the portion under the steel plate. Assume that h ef = ξ fl hene δ(x) = σ(x) ξ fl / E (.1.11) Substitution gives δ sin (½π x / fl ) = E t 3 / 1 (½π / fl ) 4 δ sin ( ½π x / fl ) ξ fl / E (.1.1) This may be expressed as.7

16 = t fl 3 π ( ) 1 4 ξ E E (.1.13) The flexible length fl may be replaed by an equivalent rigid length r suh that uniform deformations under an equivalent rigid plate give the same fore as the non uniform deformation under the flexible plate: r = fl / π (.1.14) The fator ξ represents the ratio between hef and fl. The value of hef an be expressed as α ar. From Tab..1 an be read that fator α for pratial T-stubs is about equal to 1,4. The width ar is equal to tw + r, where tw is equal to the web thikness of the T-stub. As a pratial assumption it is now assumed that t w equals to,5 r whih leads to h ef = 1,4 (,5 + ) r = 1,4,5 fl / π =, fl (.1.15) hene ξ =, For pratial joints an be estimated by E 3 N / mm leads to and E 1 N / mm, whih = t fl π ( ) E ( ) ξ t π 1, 198,. (.1.16) E or r = 1,6 t 1,5 t, (.1.17) whih gives for the effetive width alulated based on elasti deformation a = t +,5 t eq.el w (.1.18) The influene of the finite blok size ompared to the infinite half spae an be negleted in pratial ases. For example the equivalent width ar of the equivalent rigid plate is about tw + r. In ase t w is,5 r and r = 1,5 t the width is a r = 3,1 t. That means, peak stresses are even in the elasti stage spread over a very small area. In general, a onrete blok has dimensions at least equal to the olumn with and olumn depth. Furthermore it is not unusual that the blok height is at least half of the olumn depth. It means, that stresses under the flange of a T-stub, whih represents for instane a plate under a olumn flange, are spread over a relative big area ompared to a r = 3,1 t. If stresses are spread, the strains will be low where stresses are low and therefore these strains will not ontribute signifiantly to the deformations of the onrete just under the plate. Therefore, for.8

17 simpliity it is proposed to make no ompensation for the fat that the onrete blok is not infinite. From the strength proedure the effetive with of a T-stub is alulated as a = t + = t + t eq.str w w 3 f f y j γ M (.1.19) Based on test, see Figure , and FE simulation, see Figure.1.9, it may our that the value of aeq.str is also a suffiient good approximation for the width of the equivalent rigid plate as the expression based on elasti deformation only. If this is the ase, it has a pratial advantage for the appliation by designers. However, in the model a eq.str will beome dependent on strength properties of steel in onrete, whih is not the ase in the elasti stage, On Figure.1.6 is shown the influene of the base plate steel quality for partiular example on the onrete quality - deformation diagram for flexible plate t = tw = mm, L eff = 3 mm, F = 1 kn. From the diagram it an be seen that the differene between a eq.el = t w +,5 t and a eq.str = t w + is limited. Deformation, mm,5, a eq.str for strength, S 35 for strength, S 75 for strength, S 355,15,1 t w = mm t = mm F = 1 kn L = 3 mm a eq.el elasti model, S 35, S 75, S 355 δ,5, Conrete, f, MPa k Figure.1.6 Comparison of the predition of the effetive width on onrete - deformation diagram for partiular example for unlimited onrete blok k j = 5, base plate and web thikness mm, L = 3 mm, fore F = 1 kn The onrete surfae quality is affeting the stiffness of this omponent. Based on the tests Alma and Bijlaard, (198), Sokol and Wald, (1997). The redution of modulus of elastiity.9

18 of the upper layer of onrete of thikness of 3 mm was proposed (Sokol and Wald, 1997) aording to the observation of experiments with onrete surfae only, with poor grout quality and with high grout quality. For analytial predition the redution fator of the surfae quality was observed from 1, till 1,55. For the proposed model the value 1,5 had been proposed, see Figure.1.1 and.1.13, see Eq. (.1.). The simplified proedure to alulate the stiffness of the omponent onrete in ompression and base plate in bending an be summarised in Euroode 3 Annex J form as where k F E aeq.el L E aeq.el L = =, (.1.) δ E 1,5*,85E 1,75 E = a eq.el the equivalent width of the T-stub, a e.el = t w +,5 t, L the length of the T-stub, t the flange thikness of the T-stub, the base plate thikness, the web thikness of the T-stub, the olumn web or flange thikness. t w.1.4 Validation The proposed model is validated against the tests for strength and for stiffness separately. 5 tests in total were examined in this part of study to hek the onrete bearing resistane (DEWOLF, 1978, HAWKINS, 1968). The test speimens onsist of a onrete ube of size from 15 to 33 mm with entri load ating through a steel plate. The size of the onrete blok, the size and thikness of the steel plate and the onrete strength are the main variables..1

19 f j / f d Anal. Exp. e t / e Figure.1.7 Relative bearing resistane-base plate slenderness relationship (experiments DeWolf, 1978, and Hawkins, 1968) Figure.1.7 shows the relationship between the slenderness of the base plate, expressed as a ratio of the base plate thikness to the edge distane and the relative bearing resistane. The design approah given in Euroode 3 is in agreement with the test results, but onservative. The bearing apaity of test speimens at onrete failure is in the range from 1,4 to,5 times the apaity alulated aording to Euroode 3 with an average value of 1,75. N, kn <> <> <> f e d b a t =,76 mm 1,5 mm 3,5 mm 6,35 mm 8,89 mm 5,4 mm Anal. a b d ef Exp. <> f, MPa d a x b = 6 x 6 mm Figure.1.8 Conrete strength - ultimate load apaity relationship (Hawkins, 1968).11

20 The influene of the onrete strength is shown on Figure.1.8, where is shown the validation of the proposal based on proposal t w +. A set of 16 tests with similar geometry and material properties was used in this diagram from the set of tests (Hawkins, 1968). The only variable was the onrete strength of 19, 31 and 4 MPa Fore, kn Calulated strength, Eq. (.1.6) Experiment Conrete and grout Conrete Predition based on loal and global deformation, Eq. (.1.) and elasti deformation of the onrete blok Predition based on loal deformation only, Eq. (.1.),1,,3,4,5,6,7,8,9 Deformation, mm t t w δ F L Figure.1.9 Comparison of the stiffness predition to Test.1, (Alma and Bijlaard, 198), onrete blok 8x4x3 mm, plate thikness t = 3, mm, T stub length L = 3 mm The stiffness predition is ompared to tests Alma and Bijlaard, (198) in Figure.1.9. and.1.1. The tests of flexible plates on onrete foundation are very sensitive to boundary onditions (rigid tests frame) and measurements auray (very high fores and very small deformations). The predited value based on eq. (.1.7) is the loal deformation only. The elasti global and loal deformation of the whole onrete blok is shown separately. Considering the spread in test results and the auray ahievable in pratie, the omparison shows a suffiiently good auray of predition..1

21 16 Fore, kn F Calulated strength, Eq. (.1.6) Experiment Conrete and grout Conrete t t w δ L 6 4 Predition based on loal and global deformation, Eq. (.1.) and elasti deformation of the onrete blok Predition based on loal deformation only, Eq. (.1.),1,,3,4,5,6,7,8,9 Deformation, mm Figure.1.1 Comparison of the stiffness predition to Test., (Alma and Bijlaard, 198), onrete blok 8x4x3 mm, plate thikness t = 19 mm, T stub length L = 3 mm Fore, kn Predition, Eq. (.1.) 6 Exluding onrete surfae quality fator 5 4 Experiment W97-15 t F t w L 3 1,1,,3,4,5,6 Parallel line to predition Deformation, mm δ Figure.1.11 Comparison of the stiffness predition to Test W97-15, repeated loading, leaned onrete surfae without grout only (Sokol and Wald, 1997), onrete blok 55 x 55 x 5 mm, plate thikness t = 1 mm, T stub length L = 335 mm.13

22 Fore, kn 6 Exluding onrete surfae quality fator Predition, Eq. (.1.) Experiment W97-1 Parallel lines t to predition d Deformation, mm F t w L,5 1 1,5,5 Figure.1.1 Comparison of the stiffness predition to Test W97-15 repeated and inreasing loading, leaned onrete surfae low quality grout (Sokol and Wald, 1997), onrete blok 55 x 55 x 5 mm, plate thikness t = 1 mm, T stub length L = 335 mm The omparison of loal and global deformations an be shown on Finite Element (FE) simulation. In Figure.1.11 the predition of elasti deformation of rigid plate 1 x 1 mm on onrete blok 5 x 5 x 5 mm is ompared to alulation using the FE model. Vertial deformation at the surfae, mm Vertial deformation along the blok height,,1 F top of the onrete blok elasti deformation of the whole blok }loal deformation under plate predited value eq. (.1.) elasti deformation δ glob deformation at the edge δ edge deformation at the axis δ axis edge axis foot of the onrete blok Vertial deformation, mm,1 Figure.1.13 Calulated vertial deformations of a onrete blok,5 x,5 x,5 m loaded to a defletion of,1 mm under a rigid plate,1 x,1 m; in the figure on the right, the deformations along the vertial axis of symmetry δ axis are given and the alulated deformations at the edge δedge, inluded are the global elasti deformations aording to δglob = F h /(E A ), where A is full the area of the onrete blok Based on these omparisons, the reommendation is given that for pratial design, besides the loal effet of deformation under a flexible plate, the global deformation of the supporting onrete struture must be taken into onsideration..14

23 . Column flange and web in ompression..1 Desription of the omponent In this setion, the mehanial harateristis of the olumn flange and web in ompression omponent are presented and disussed. This omponent, as its name learly indiates, is subjeted to tension fores resulting from the applies bending moment and axial fore in the olumn (Figure..1). The proposed rules for resistane and stiffness evaluation given hereunder are similar to those inluded in revised Annex J of Euroode 3 for the beam flange and web in ompression omponent in beam-to-olumn joints and beam splies. Risk of yielding or instability Figure..1 Component olumn flange and web in ompression.. Resistane When a bending moment M and a axial fore N are arried over from the olumn to the onrete blok, a ompression zone develops in the olumn, lose to the olumn base; it inludes the olumn flange and a part of the olumn web in ompression. The ompressive fore F arried over by the joint may, as indiated in Figure.., is quite higher than the ompressive fore F in the olumn flange resulting from the resolution, at some distane of the joint, of the same bending moment M and axial fore N. In Figure.., the fores F and F are applied to the entroï d of the olumn flange in ompression..15

24 This assumption is usually made for sake of simpliity but does not orrespond to the reality as the ompression zone is not only limited to the olumn flange. The fore F, quite loalized, may lead to the instability of the ompressive zone of the olumn ross-setion and has therefore to be limited to a design value whih is here defined in a similar way than in Annex J for beam flange and web in ompression : where : F = M /( h t ) (..1). fb. Rd. Rd f M.Rd h b t fb is the design moment resistane of the olumn ross-setion redued, when neessary, by the shear fores; M.Rd takes into onsideration by itself the potential risk of instability in the olumn flange or web in ompression; is the whole depth of the olumn ross-setion; is the thikness of the olumn flange. h - tf F M and N applied as in Figure..1 F = N/ + M/(h b -t fb ) F = N/ + M/z F < F It has to be pointed out that Formula (..1) limits the maximum fore whih an be arried over in the ompressive zone of the joint beause of the risk of loss of resistane or instability in the possibly overloaded ompressive zone of the olumn loated lose to the joint. It therefore does not replae at all the lassial verifiation of the resistane of the olumn rossz F Figure.. Loalized ompressive fore in the olumn ross-setion loated lose to the olumn base.16

25 setion. It has also to be noted that formula (..1) applies whatever is the type of onnetion and the type of loading ating on the olumn base. It is also referred to in the preliminary draft of Euroode 4 Annex J in the ase of omposite onstrution and applies also to beam-to-olumn joints and beam splies where the beams are subjeted to ombined moments and shear and axial ompressive or tensile fores. It is therefore naturally extended here to olumn bases. The design resistane given by Formula (..1) has to be ompared to the ompressive fore F (see Figure..) whih results from the distribution of internal fores in the joint and whih is also assumed to be applied at the entroï d of the olumn flange in ompression. It integrates the resistane of the olumn flange and of a part of the olumn web; it also overs the potential risk of loal plate instability in both flange and web...3 Stiffness The deformation of the olumn flange and web in ompression is assumed not to ontribute to the joint flexibility. No stiffness oeffiient is therefore needed..3 Base plate in bending and anhor bolt in tension When the anhor bolts are ativated in tension, the base plate is subjeted to tensile fores and deforms in bending while the anhor bolts elongate. The failure of the tensile zone may result from the yielding of the plate, from the failure of the anhor bolts, or from a ombination of both phenomena. Two main approahes respetively termed "plate model" and "T-stub model" are referred to in the literature for the evaluation of the resistane of suh plated omponents subjeted to transverse bolt fores. The first one, the "plate model", onsiders the omponent as it is - i.e. as a plate - and formulae for resistane evaluation are derived aordingly. The atual geometry of the omponent, whih varies from one omponent to another, has to be taken into onsideration in an appropriate way; this leads to the following onlusions : the formulae for resistane varies from one plate omponent to another; the omplexity of the plate theories are suh that the formulae are rather ompliated and therefore not suitable for pratial appliations..17

26 The T-stub idealisation, on the other hand, onsists of substituting to the tensile part of the joint T-stub setions of appropriate effetive length l eff, onneted by their flange onto a presumably infinitely rigid foundation and subjet to a uniformly distributed fore ating in the web plate, see Figure.3.1. web e m F l eff t flange Figure.3.1 T-stub on rigid foundation In omparison with the plate approah, the T-stub one is easy to use and allows to over all the plated omponents with the same set of formulae. Furthermore, the T-stub onept may also be referred to for stiffness alulations as shown in (Jaspart, 1991) and Yee, Melhers, 1986). This explains why the T-stub onept appears now as the standard approah for plated omponents and is followed in all the modern haraterisation proedures for omponents, and in partiular in Euroode 3 revised Annex J (1998) for beam-to-olumn joints and olumn bases. In the next pages, the evaluation of the resistane and stiffness properties of the T-stub are disussed in the partiular ontext of olumn bases and proposals for inlusion in forthoming European regulations are made..3.1 Design resistane of plated omponents Basi formulae of Euroode 3 The T-stub approah for resistane, as it is desribed in Euroode 3, has been first introdued by Zoetemeijer (1974) for unstiffened olumn flanges. It has been then improved (Zoetemeijer, 1985) so as to over other plate onfigurations suh as stiffened olumn flanges and end-plates. In Jaspart (1991), it is also shown how to apply the onept to flange leats in bending..18

27 In plated omponents, three different failure modes may be identified : a) Bolt frature without prying fores, as a result of a very large stiffness of the plate (Mode 3) b) Onset of a yield lines mehanism in the plate before the strength of the bolts is exhausted (Mode 1) ) Mixed failure involving yield lines - but not a full plasti mehanism - in the plate and exhaustion of the bolt strength (Mode ). Similar failure modes may be observed in the atual plated omponents (olumn flange, end plates, ) and in the flange of the orresponding idealised T-stub. As soon as the effetive length l eff of the idealised T-stubs is hosen suh that the failure modes and loads of the atual plate and the T-stub flange are similar, the T-stub alulation an therefore be substituted to that of the atual plate. In Euroode 3, the design resistane of a T-stub flange of effetive length l eff is derived as follows for eah failure mode : Mode 3: bolt frature (Figure.3..a) F =Σ B (.3.1) Rd, 3 t. Rd Mode 1: plasti mehanism (Figure.3..b) F Rd, 1 4 = l eff m m pl, Rd (.3.) Mode : mixed failure (Figure.3..) F Rd, l = m + ΣB n eff pl, Rd t. Rd m+ n (.3.3).19

28 e n m Mode 3 Mode 1 Mode FRd.3 F B t.rd B t.rd Rd.1 B B B t.rd F Rd. B t.rd Q Q Q Q Figure.3. Failure modes in a T-stub In these expressions : m pl,rd is the plasti moment of the T-stub flange per unit length ( t f y / γ M ) with t = flange thikness, f y = yield stress of the flange, γ M = partial safety fator) m and e are geometrial harateristis defined in Figure.3.. Σ Bt.Rd n l eff is the sum of the design resistanes Bt.Rd of the bolts onneting the T-stub to the foundation (B t.rd =,9 A s f ub / γ Mb where A s is the tensile stress area of the bolts, fub the ultimate stress of the bolts and γmb a partial safety fator) designates the plae where the prying fore Q is assumed to be applied, as shown in Figure.3. (n = e, but its value is limited to 1,5 m). is derived at the smallest value of the effetive lengths orresponding to all the possible yield lines mehanisms in the speifi T-stub flange being onsidered. The design strength F Rd of the T-stub is derived as the smallest value got from expressions (.3.1) to (.3.3) : FRd = min( FRd, 1, FRd,, FRd, 3 ) (.3.4) In Jaspart (1991), the non-signifiative influene of the possible shear-axial-bending stress interations in the yield lines on the design apaity of T-stub flanges has been shown. In Annex J, the influene on Mode 1 failure of baking plates aimed at strengthening the olumn flanges in beam-to-olumn bolted joints is also onsidered. A similar influene may result, in 1 4.

29 olumn bases, from the use of washer plates. The effet of the latter on the base plate resistane will be taken into onsideration in a similar way than it is done in Annex J for baking plates. This alulation proedure reommended first by Zoetemeijer has been refined when revising the Annex J of Euroode 3. Euroode 3 distinguishes now between so-alled irular and non-irular yield lines mehanisms in T-stub flanges (see Figure.3.3.a). These differ by their shape and lead to speifi values of T-stub effetive lengths noted respetively l eff,p and l eff,np. But the main differene between irular and non-irular patterns is linked to the development or not of prying fores between the T-stub flange and the rigid foundation : irular patterns form without any development of prying fores Q, and the reverse happens for non-irular ones. The diret impat on the different possible failure modes is as follows : Mode 1 : Mode : Mode 3 : the presene or not of prying fores do not alter the failure mode whih is linked in both ases to the development of a omplete yield mehanism in the plate. Formula (.3.) applies therefore to irular and non-irular yield patterns. the bolt frature learly results here from the over-loading of the bolts in tension beause of prying effets; therefore Mode only ours in the ase of non-irular yield lines patterns. this mode does not involve any yielding in the flange and applies therefore to any T-stub. As a onlusion, the alulation proedure differs aording to the yield line mehanisms developing in the T-stub flange (Figure.3.3.b) : F = min( F ; F ) Rd Rd, 1 Rd, 3 F = min( F ; F ; F ) Rd Rd, 1 Rd, Rd, 3 for irular patterns (.3.5.a) for non-irular patterns (.3.5.b) In Annex J, the proedure is expressed in a more general way. All the possible yield line patterns are onsidered through reommended values of effetive lengths grouped into two ategories : irular and non-irular ones. The minimum values of the effetive lengths - respetively termed l eff,p and l eff,np - are therefore seleted for ategory. The failure load is then derived, by means of Formula (.3.4), by onsidering suessively all the three possible failure modes, but with speifi values of the effetive length :.1

30 Mode 1 : l eff, 1 = min( l eff, p ; l eff, n ) (.3.6.a) Mode : leff, = leff, n (.3.6.b) Mode 3 : - (.3.6.) Cirular pattern (l eff,p ) Non-irular patterns (l eff,n ) (a) Different yield line patterns 1,8 F/ Σ Bt.Rd Mode Mode 3,6 Mode 1 Mode 1*,4, 4 leff M pl.rd / Σ B t.rd,5 1 1,5,5 (b) Design resistane Figure.3.3 T-stub resistane aording revised Annex J.3.1. Alternative approah for Mode 1 failure The auray of the T-stub approah is quite good when the resistane is governed by failure modes and 3. The formulae for failure mode 1, on the other hand, has been seen quite onservative, and sometimes too onservative, when a plasti mehanism forms in the T-stub flange (Jaspart, 1991)..

31 Therefore the question raised whether refinements ould be brought to the T-stub model of Euroode 3 with the result that the amended model would provide a higher resistane for failure mode 1 without altering signifiantly the auray regarding both failure modes and 3. In (Jaspart, 1991), an attempt has been made in this respet. In the Zoetemeijer s approah, the fores in the bolts are idealised as point loads. Thus, it is never expliitly aounted for the atual sizes of the bolts and washers. If this is done, the following resistane may be expressed for Mode 1 (Jaspart, 1991): F Rd, 1 ( 8n e ) l 1m = w eff, pl, Rd [ mn e ( m+ n) ] w (.3.7) with ew =.5 dw (dw designates the diameter of the bolt/srew or of the washer if any. Of ourse, Equation (.3.7) onfines itself to Zoetemeijer's formulae (.3.) when distane e w is vanishing. During the reent revision of Annex J of Euroode 3, formula (.3.7) whih desribes the Mode 1 failure as dependent on the atual bolt dimensions has been agreed for inlusion as an alternative to formula (.3.)..3

32 .3. Initial stiffness of plated omponents.3..1 Appliation of the T-stub approah For plated omponents, it is also referred to the T-stub onept, see Figure.3.4. F p * a B Q,75n m 4/5 a B F B Q,75n m Q l Figure.3.4 T-stub elasti deformability In a T-stub, the tensile stiffness results from the elasti deformation of the T-stub flange in bending and of the bolts in tension (the role of the latter is plaid by the anhor bolts in setion.3.3 dealing expliitly with olumn bases). When evaluating the stiffness of the T-stub, the ompatibility between the respetive deformabilities of the T-stub flange and of the bolts has to be ensured : * p = b (.3.8) where p* b is the deformation of the end-plate at the level of the bolts; is the elongation of the bolts. In Jaspart (1991), expressions providing the elasti initial stiffness of the T-stub are proposed; they allow the oupling effet between the T-stubs to be taken into onsideration. These expressions slightly differ from those given in the original publiation of Yee and Melhers (1986). The elongation b of the bolts simply results from the elongation of the bolt shank subjeted to tension:.4

33 B b = Lb (.3.9) E A S where Lb is approximately defined as the length of the bolt shank in Euroode 3. From these onsiderations, the elasti deformation of the two T-stub may be derived : F p = (.3.1) E k i,p where the stiffness oeffiient k i,p is expressed as : k i,p Z ( q ) = α 8 4 (.3.11) Z q = Z α p a 1 l b + A S (.3.1) In these formulae : Z 3 = l / b t 3 l = ( m +,75 n ) α 3 1 = 1,5 α α b is the T-stub length α 3 = 6 α 8 α α =,75 n / l All the geometrial properties are defined in Figure The validity of these formulae has been demonstrated in Jaspart (1991) on the basis of a quite large number of omparisons with test results on joints with end-plate and flange leated onnetions got from the international literature..3.. Simplified stiffness oeffiients for inlusion in Euroode 3 The appliation of the T-stub onept to a simplified stiffness alulation - as that to be inluded in a ode suh Euroode 3 - requires to express the equivalene between the atual omponent and the equivalent T-stub in the elasti range of behaviour and that, in a different way than at ollapse; this is ahieved through the definition of a new effetive length alled l eff, ini whih differs from the l eff value to whih it has been referred to in setion.3.1. In.5

34 view of the determination of the stiffness oeffiient k ip, two problems have to be investigated: the response of a T-stub in the elasti range of behaviour; the determination of l eff, ini. These two points are suessively addressed hereunder. T-stub response The T-stub response in the elasti range of behaviour is overed in setion The orresponding expressions are rather long to apply, but some simplifiations may be introdued: to simplify the formulae: n is onsidered as equal to 1,5 m; to dissoiate the bolt deformability (Figure.3.5.) from that of the T-sub (Figure.3.5.b). The value of q given by expressions (.3.1) may then be simplified to : q 1,5 α α 3 α1 = = (.3.13) 3 α 6 α 8 α as soon as it is assumed, as in Figure.3.5.b, that the bolts are no more deforming in tension ( A s = ). q further simplifies to: q = 1,8 (.3.14) by substituting 1,5 m to,75 n as assumed previously. The stiffness oeffiient given by formula (.3.11) therefore beomes : k i,p 3 193,64 193,64 t l eff, ini = = (.3.15) 3 Z ( 4,5m ) The effetive length l eff,ini has been substituted to b..6

35 Figure.3.5 Elasti deformation of the T-stub Finally : k i,p 3 3 l eff,init l eff, init = 1,63 (.3.16) 3 3 m m In the frame of the assumptions made, it may be shown that the prying effet inreases the bolt fore from,5 F to,63 F (Figure.3.5.). In Euroode 3, the deformation of a bolt in tension is taken as equal to : B L b b = (.3.17) E As By substituting B by,63 F in (.3.17), the stiffness oeffiient of a bolt row with two bolts may be derived : A s k i b, = 1,6 (.3.18) Lb Definition of effetive length l eff,ini In Figure.3.5., the maximum bending moment in the T-stub flange (points A) is expressed as M max =,3 F m. Based on this expression, the maximum elasti load (first plasti hinges in the T-stub at points A) to be applied to the T-stub may be derived : F 4l t f l t f eff,ini y eff, ini y el = = (.3.19) 1,88 m 4γ M 1,88 m γ M.7

36 In Annex J, the ratio between the design resistane and the maximum elasti resistane of eah of the omponents is taken as equal to 3/ so : F l t f = 3 eff, ini y Rd Fe = l,859 m γ (.3.) MO As, in Figure.3.5.b, the T-stub flange is supported at the bolt level, the only possible failure mode of the T-stub is the development of a plasti mehanism in the flange. The assoiated failure load is given by Annex J as: F Rd l t f eff y = FRd = (.3.1) m γ Mo where l eff is the effetive length of the T-stub for strength alulation. By identifiation of expressions (.3.) and (.3.1), l eff, ini may be derived : l = = l l (.3.) eff,ini,859 eff, 85 eff Finally, by introduing equation (.3.) in the expression (.3.18) giving the value of k i,p for any plated omponent : k i, p 3,85 l eff,1 t = (.3.3) m Extension to base plates To evaluate the resistane and stiffness properties of a base plate in bending and anhor bolts in tension, referene is also made to the T-stub idealisation Resistane properties Three failure modes are identified in Setion for equivalent T-stubs of beam endplates and olumn flanges: Mode 1, Mode and Mode 3. Related formulae may be applied to olumn base plates as well. But in the partiular ase of base plates, it may happen that the elongation of the anhor bolts.8

37 in tension is suh, in omparison to the flexural deformability of the base plate, that no prying fores develop at the extremities of the T-stub flange. In this ase, the failure results either from that of the anhor bolts in tension (Mode 3) or from the yielding of the plate in bending (see Figure.3.6) where a two hinges mehanism develops in the T-stub flange. This failure is not likely to appear in beam-to-olumn joints and splies beause of the limited elongation of the bolts in tension. This partiular failure mode is named Mode 1*. F* Rd.1 B B Figure.3.6 Mode 1** failure The orresponding resistane writes : F ** Rd.1 l eff m pl. Rd = (.3.4) m When the Mode 1* mehanism forms, large base plate deformations develop; they may result in ontats between the onrete blok and the extremities of the T-stub flange, i.e. in prying fores. Further loads may therefore be applied to the T-stub until failure is obtained through Mode 1 or Mode. But to reah this level of resistane, large deformations of the T-stub are neessary, what is not aeptable in design onditions. The extra-strength whih separates Mode 1* from Mode 1 or Mode in this ase is therefore disregarded and Formula (.3.4) is applied despite the disrepany whih ould result from omparisons with some experimental tests. As a result, in ases where no prying fores develop, the design resistane of the T-stub is taken as equal to : Rd ** ( FRd,1, FRd,3 ) F = min (.3.5) when FRd,3 is given by formula (.3.1). In other ases, the ommon proedure explained in setion.3.1 is followed. The riterion to distinguish between situations with and without prying fores is disussed in setion As explained in Setion.3.1.1, irular and non-irular yield line patterns have to be.9

38 differentiated when deriving the effetive length l eff of the T-stub : The non-irular patterns referred to in revised Annex J of Euroode 3 over ases where prying fores develop at the extremities of the plated omponent. The irular patterns develop without any prying. Conerning Mode 1* failure, only irular patterns have therefore to be taken into onsideration and the non-irular patterns proposed by Euroode3 have to be disregarded. Mode 1* identifies then exatly to Mode 1 and, in order to ensure that the design resistanes provided by formulae (.3.) and (.3.4) are equal, the effetive lengths for irular patterns defined in revised Annex J have to be multiplied by a fator before being implemented in Formula (.3.4). Besides that, non-irular patterns not involving prying fores in the bolts may our. These ones may be onsidered through Formula (.3.4), but by introduing appropriate effetive length harateristis. The lowest of the effetive lengths between those derived for irular and non-irular patterns respetively is that whih will determines the design resistane of the T-stub. Table.3.1 indiates how to selet the values of l eff for two lassial base plate onfigurations, in ases where prying fores develop and do not develop..3

39 PLATE CONFIGURATIONS PRYING FORCES DEVELOP PRYING FORCES DO NOT DEVELOP e m Figure.3.7 Anhor bolts loated between the flanges l l l 1 eff = = = α π m m min leff, = l1 ( 4 m + 1,5 e ) ( l ; l ) 1 where m and n are represented in Figure.3.7. and α is defined in EC3 Annex J. l l l 1 eff = = 4 = α π m m min leff, = l1 ( 4 m + 1,5 e ) ( l ; l ) 1 where m and n are represented in Figure.3.7. and α is defined in EC3 Annex J. e w e b p Figure.3.8 Anhor bolts loated in the extended part of the base plate e x m x l 1 = 4.mx+1,5 ex l = π mx l 3 =,5.bp l 4 =,5.w+.mx+,65.ex l 5 = e+.mx+,65.ex l 6 = π m x + e l = ( l ) 1 ; l ; l 3 ; l 4 ; l 5 6 ( l ; l ; l ) min l eff,1 ; l eff = min ; l 5 where bp, m, e, mx, ex and w are given in Figure.3.8. l 1 = 4.mx+1,5 ex l = 4π mx l 3 =,5.bp l 4 =,5.w+.mx+,65.ex l 5 = e+.mx+,65.ex l 6 = π m x + 4 e l = ( l ) 1 ; l ; l 3 ; l 4 ; l 5 6 ( l ; l ; l ) min l eff,1 ; l eff = min ; l 5 where bp, m, e, mx, ex and w are given in Figure.3.8. Table.3.1 Values of the T-stub effetive length.31, 1,, 1,

40 It has to be noted that these formulae only apply to base plates where the anhor bolts are not loated outside the beam flanges, as indiated in Figure.3.9. (a) Cases overed (b) Cases not overed Figure.3.9 Limits of validity of the formulae given in Figures.3.7 and Stiffness properties The elasti deformation of a T-stub in tension is disussed in setion.3..1 and aurate formulae for stiffness evaluation are suggested. They have been used in setion.3.. to derive simplified expressions for inlusion in Revised Annex J of Euroode 3. the stiffness oeffiient for the T-stub flange in bending : k p 3,85 l eff t = (.3.6) 3 m.3

41 the stiffness oeffiient for the anhor bolts in tension: A s k b = 1,6 (.3.7) Lb where Lb is the anhor bolt length desribed hereunder. These two expressions relate to situations where prying fores develop at the extremities of the T-stub flange as a result of a limited bolt-axial deformation in omparison with the bending deformation of the flange. In "no prying ases", the deformation p of the base plate is easily derived as : p F = (.3.8.a) E k i, p where the stiffness oeffiient k i,p is expressed as : 3 l eff,ini t k i, p = (.3.9.b) 3 m and that of the bolts (without preloading) : b F = (.3.3.a) E k i,b with : A s k i b, = (.3.31.b) Lb Lb is the effetive free length of the anhor bolts (Figure 3..1). It is defined as the sum of two ontributions L fl and L el. L fl is the free length of the anhor bolts, i.e. the part of the bolt whih is not embedded. L el is the equivalent free length of the embedded part of the anhor bolt; it may be approximated to 8d (Wald, 1993), where d is the nominal diameter of the anhor bolt. Should the embedded length of the bolt be shorter than 8d, then the atual length of the bolt would be onsidered. A justifiation of this definition of Lel is given in setion

42 L fl L el L d Figure.3.1 Effetive length of the anhor bolts If the approximation of the l eff,ini value suggested in setion.3.3. is again onsidered - formula (.3.6) -, the stiffness oeffiient for the base plate in the ase of "no prying" onditions may be finally expressed as : 3 leff t k i, p =,45 (.3.3) 3 m Boundary for prying effets The elasti deformed shape of a T-stub in tension depends on the relative deformability of the flange in bending and the anhor bolts in tension (see setion.3.). In Figure 3..11, the bolt and flange deformations ompensate suh that the ontat fore Q just vanishes. For a higher bolt deformability, no ontat will develop, while ontat fores will appear for a lower bolt deformability. The situation illustrated in Figure.3.11 therefore onstitutes a limit ase to whih a prying boundary may be assoiated. This is expressed as follows : L 7m n A s b.boundary =. (.3.33) 3 l eff t If, as a further assumption, n is defined as equal to 1,5 m (setion.3..), then : L 8,8 m A 3 s b.boundary =. (.3.34) 3 l eff t Θ p F m n b = Θ p n b Q = Q = Figure.3.11 The T-stub deformation when prying fore Q vanishes.34

43 If Lb is higher than Lb,boundary, then the following formulae apply : formulae (.3.3) and (.3.31.b) for stiffness; formulae (.3.4) and (.3.1) for resistane. In the opposite ase, other formulae have to be referred to : formulae (.3.6) and (.3.7) for stiffness; formulae (.3.1), (.3.) or (.3.7) and (.3.3) for resistane Agreement between the simplified and sophistiated stiffness models In the following figures, the validity of the simplifiations brought to the theoretial stiffness model presented in setion.3..1 to derive the so-alled simplified model desribed in setions.3.. and.3.3. is shown on three examples : T-stub with : l,333 mm eff = 458, L b = 15 mm, A s = 48 mm², m = 5 mm (Figure.3.1.a), where the n/m ratio takes the following values:,5; 1,; 1,5 and.. T-stub with : l = 458,333 mm, L eff b = 3 mm, A s = 48 mm², m = 5 mm, (Figure.3.1.b), where the n/m ratio takes the following values :,5; 1,; 1,5 and.. T-stub with : l eff = 458,333 mm, L b = 6 mm, A s = 48 mm², m = 5 mm, (Figure.3.1.), where the n / m ratio takes the following values :,5; 1,; 1,5 and.. The boundary between "prying" and "no prying" fields is omputed by means of formulae (.3.33) and (.3.34) for the theoretial and simplified models respetively..35

44 Stiffness L b = 15 mm 4 3 Simplified model Sophistiated model (n / m =,5) 1 t / m,,4,6,8 1 1, 7 Stiffness Lb = 15 mm Simplified model Sophistiated model (n / m = 1,) t / m,,4,6,8 1 1, Stiffness Lb = 15 mm Simplified model Sophistiated model (n / m = 1,5) t / m,,4,6,8 1 1, 7 Stiffness L b = 15 mm 3 Simplified model Sophisstiated model (n / m =,) 1 t / m,,4,6,8 1 1, (a) Lb = 15 mm.36

45 3,5 Stiffness 3,5 1,5 1,5 Lb = 3 mm Simplified model Sophistiated model (n / m =,5) t / m,,4,6,8 1 1, 3,5 Stiffness 3,5 1,5 1,5 Lb = 3 mm Simplified model Sophistiated model (n / m = 1,) t / m,,4,6,8 1 1, 3,5 Stiffness 3,5 1,5 1,5 L b = 3 mm Simplified model Sophistiated model (n / m = 1,5) t / m,,4,6,8 1 1, 3,5 3,5 1,5 1,5 Lb = 3 mm Simplified model Sophistiated model (n / m =,) t / m,,4,6,8 1 1, (b) Lb = 3 mm.37

46 1,8 1,6 1,4 1, 1,8,6,4 Lb = 6 mm Simplified model Sophistiated model (n / m =,5), t / m,,4,6,8 1 1, 1,8 1,6 1,4 1, 1,8,6,4 Lb = 6 mm Simplified model Sophistiated model (n / m = 1,), t / m,,4,6,8 1 1, 1,8 1,6 1,4 1, 1,8,6,4 Lb = 6 mm Simplified model Sophistiated model (n / m = 1,5), t / m,,4,6,8 1 1, 1,8 1,6 1,4 1, 1,8,6,4, Lb = 6 mm Simplified model Sophistiated model (n / m =,) t / m,,4,6,8 1 1, () L b = 6 mm Figure.3.1 Comparisons between theoretial and simplified models.38

47 .3.4 Anhorage of the bolts in the onrete blok Different types of anhor bolts are used as shown in Figure a) b) ) d) e) f) ast-in-plae (a), underut (b), adhesive (), grouted (d), expansion (e), anhoring to grillage beams (f) Figure.3.13 Basi types of anhoring The anhoring resistane is provided by CEB rules based on the ultimate limit state onept. As already said, this resistane has to be suh that the anhor bolts fail in tension before the anhorage (pull-out of the anhor, failure of the onrete, ) reah its own resistane. For a single anhor, the following failure modes have to be onsidered (CEB, 1994): Pull-out failure (NRd.p ) Conrete one failure (NRd.) Splitting failure of the onrete (NRd.sp) Similar verifiations are required for anhor groups. e p d a 1 h ef a,7 t 1 h 1 t h t 1 Figure.3.14 The geometry of the in-situ-ast headed anhor bolts.39

48 The most eonomial solutions for anhoring are, for instane, hooked bars for light anhoring, ast-in-plae headed anhors and bounded anhors to drilled holes. The more expensive anhoring systems suh as grillage beams embedded in onrete are designed for large frames. Models for the anhoring design resistane ompatible with Euroodes have been published in CEB Guide (CEB, 1997) and by Eligehausen in (Eligehausen, 199) and (Eligehausen, 1991). The alulation of the anhoring design resistane of ast-in-situ headed anhor bolts loaded in tension is presented here below. The pullout failure design resistane may be obtained as: N Rd. p = p k Ah / γ Mp (.3.35) where p k is taken in non-raked onrete as: p k = 11, f k (.3.36) and Ah is the bearing area of the head; for irular head of diameter dh, it writes: A h = π (d h - d ) / 4 (.3.37) The onrete one failure design resistane is given as N Rd. A = N Rd. Ψ s.nψ e.nψre.nψ ur.n, (.3.38) A.N.N where N. = k1 f h / γ (.3.39) Rk,5 k 1,5 ef M is the harateristi resistane of a single fastener. The oeffiient k 1 ould be taken for non raked onrete as: k 1 = 11, [ N / mm ] (.3.4) The geometri effet of spaing p and edge distane e is inluded in alulation of the area of the one, see Figure.3.15, as: A = p (.3.41).N.N r.n ( p + p )( p p ) A = + (.3.4) r.n 1 r.n.4

49 for examples in Figure.3.15.a and.3.15.b or as:.n ( e,5 pr.n ) pr. N A = + (.3.43) for Figure It is possible to onsider approximately: p r.n, e 3, h (.3.44) r.n ef,5 p r.n p r.n p 1 p r.n p r.n p,5 p r.n e,5 p r.n a) b) ) Figure.3.15 An idealised onrete one, individual anhor (a), anhor group (b), single anhor at edge () The disturbane of the stress distribution in the onrete may be introdued through the following parameter: e Ψ s.n =,7 +,3 1 (.3.45) e r.n The parameter Ψe.N takes into aount the group effet. Parameter Ψre.N is used for small embedded depths (h ef 1 mm). The resistane is inreased in non-raked onrete by parameter Ψ ure. N = 14., The splitting failure for the in-situ-ast anhors is prevented if the onrete is reinfored or by limiting: The spaing: p min = 5 d h 5 mm (.3.46) The edge distanes:.41

50 emin = 3 dh 5 mm (.3.47) And the height of the onrete blok: hmin = hef + th +, (.3.48) where: t h is thikness of the anhor bolt head and the required onrete over for reinforement. For fastenings with an edge distane e >,5 h ef pull-out resistane may be omitted. in all diretions, a hek of the harateristi d h ef d h a h t h Figure.3.16 Headed embedded anhor bolt The detailed omplex desription of the evaluation formulae for the design resistane of different types of fastenings in tension is inluded in the CEB Guide (CEB, 1997). When alulating the anhoring resistane, the toleranes for the position of the bolts should be taken into aount aording to Euroode 3, lause (ENV , Part 1.1)..3.5 Definition of the equivalent free length of the anhor bolts The relative displaement between the surfae of a onrete foundation and an embedded bar subjeted to tension fores has been observed experimentally by Salmon (1957), Wald (Wald et al, 1993; Wald, 1995). From these experimental observations, the length Lt on whih the tensile stress in the embedded part of the bar dereases from a σ value (at the onrete surfae) to a zero value is seen to be approximately equal to 4 d. This length may obviously vary during the loading and dramatially hange beause of loal looses of bound resistane between steel and onrete. In the alulations of the stiffness properties of the anhor bolts in tension, a onstant stress in the bar σ is assumed to at on a so-alled equivalent free length L el (see Figure.3.1); is therefore expressed as:.4

51 B L el = (.3.49) E A s If σbx designates the bond stress between the onrete and the embedded bar (see Figure.3.17), the axial stress σ x along the bar writes: x 4σ bx σ x = σ dx (.3.5) d where σ = B/A s is the axial stress in the non-embedded part of the bar at the onrete surfae. L t σ bx B Figure.3.17 Example of bond stress distribution along the embedded bar x In x = L t (Figure.3.17), σ x equals zero and therefore: L t 4σ bx σ = dx. (.3.51) d The strain along the bar writes εx = σx / E ; by onsidering this relation and Equations (.3.5) and (.3.51), the elongation of the bar may be expressed as: L t 4 LL t t = ε x dx = bx dx. d E σ (.3.5) x From Equation (.3.5), is seen to depend on the distribution of σbx stresses along the bar. Hereunder three different assumptions are made for what regards the distribution: onstant, linear and paraboli. If σbx is onstant and independent of x (σbx = σb ), see Figure.3.18: σ b L t s = d E (.3.53).43

52 L t σ b B Figure.3.18 Constant distribution of bond stresses along the embedded bar σ b σ d = 4 L t (.3.54) and: σ L σ dlt σ L b t t = = = (.3.55.a) d E 4d L E E t F b L eqt = (.3.55.b) EA Finally by omparing Equations (.3.55.b) and (.3.49): L = Lt d = (.3.56) el 1 If the bond stress varies linearly, as shown in Figure.3.19: σ b L = t (.3.57) 3d E L t σb B Figure.3.19 Linear distribution of bond stresses along the embedded bar σ b = σ d (.3.58) L t and: L Lt 4d = = = d (.3.59) 3 3 el 8.44

53 If the bond stress varies non-linearly, as shown in Figure.3. (ubi parabola): L t σ b B Figure.3. Non-linear distribution of bond stresses along the embedded bar σ b L t (.3.6) 5dE and: L Lt 4d = = = 4, d. (.3.61) 5 5 el 8 In order to selet the distribution of bound stresses in an appropriate way, FEM simulations orroborated by experiments (Wald et al, 1995; Sokol, Wald, 1997) have been arried out in Prague. An example of suh simulations is shown in Figure.3.1. From these numerial work, the linear distribution of bound stresses has been seleted and, as a result, the equivalent free length of the embedded part of the anhor bolt is onsidered as equal to: L el = 8d (.3.6) as indiated in setion F δ, 1, σ, MPa 5 mm F=16,3 kn for F=53,1 kn Fore, kn 18 16,3 kn 1 experiment FEM 53,1 kn δ, mm,5 1 1,5,5 Figure.3.1 FEM simulation of bolt elongation.45

54 Results of omparisons between experimental tests and Equation (.3.6) are shown in Figures.3. and.3.3. The agreement is seen to be quite good. Fore, kn Experiment Š 11/94 Experiment Š 1/94 Predition φ P6-4 x P1-95 x ,,4,6,8 1 Deformation, mm Figure.3. Comparison to experiments with long bolts (Wald et al, 1995) d = 4 mm, f k = 4,1 MPa Fore, kn Experiment W13/98 Experiment W14/97 Predition φ P6-4 x P1-95 x ,,4,6,8 1 1, 1,4 1,6 Deformation, mm Figure.3.3 Comparison to experiments with long bolts (Sokol and Wald, 1997), d = 4 mm, f k = 33,3 MPa, In the ase of short anhor bolts with an embedded length smaller than 4d, Formula (.3.6) no more applies..46

55 .3.6 Experimental validation of the proposed formulae In the following figures, omparisons between experimental tests on isolated T-stubs in tension and the models for haraterisation desribed in this setion are presented. Three urves are reprodued on eah diagram: The experimental urve B- ; The theoretial urve based on the more sophistiated available models for stiffness and strength predition (the model is said omplex ); The theoretial urve based on the more simple available rules for stiffness and strength predition (the model is said simplified ); The omparisons of these simplified and omplex alulation models give very similar results (exept for experiments W97-1 and W97-). On the other hand, a quite good agreement with the experimental tests is observed. In tests W97-1 and 97-, the resistanes given by the omplex and simplified models are signifiantly different. The explanation is the following: aording to the simplified model, prying fores develop between the T-stub and the onrete foundation while no suh prying fores develop aording to the omplex model. The formulae used for the strength predition are therefore different. Anyway, the level of safety and auray of the simplified urve remains quite good in both tests by omparison with the experimental results..47

56 35 Fore, kn 35 Fore, kn 3 Experiment 3 5 Simplified predition 5 Simplified predition 15 Complex alulation 15 Complex alulation m = W97-1 Deformation, mm W97- Deformation, mm No olapse mode observed Figure.3.4 Load- defletion diagrams for experiments W 97-1 and W 97- (Sokol and Wald, 1997). Bolts M4, plate mm, m = 3 mm, n = 4 mm, onrete blok 55 x 55 x 5 mm. No ollapse was reahed due to loading ell limitation 35 Fore, kn 35 Fore, kn 3 5 Simplified predition Experiment 3 5 Simplified predition Experiment 15 Complex alulation 15 Complex alulation m = W97-3 Deformation, mm W97-4 Deformation, mm Collapse mode Figure.3.5 Load-defletion diagrams for experiments W 97-3 and W 97 4 (Sokol and Wald, 1997). Bolts M4, plate mm, m = 5 mm, n = 4 mm, onrete blok 55 x 55 x 5 mm. Mode ollapse (breaking of bolts and plate mehanism)..48

57 35 Fore, kn 35 Fore, kn Experiment Experiment 15 Predition 15 Predition m = W97-5 Deformation, mm Collapse mode W97-6 Deformation, mm Figure.3.6 Load-defletion diagrams for experiments W 97-5 and W (Sokol and Wald, 1997). Bolts M4, plate 1 mm, m = 3 mm, n = 4 mm, onrete blok 55 x 55 x 5 mm. Mode ollapse (breaking of bolts and plate mehanism). 35 Fore, kn 35 Fore, kn 3 W W Experiment Experiment 15 Complex alulation 15 Complex alulation m = 67 1 Simplified predition 1 Simplified predition 5 5 Deformation, mm Deformation, mm Collapse mode Figure.3.7 Load-defletion diagrams for experiments W and W (Sokol and Wald, 1997). Bolt M4, plate mm, m = 67 mm, n = 4 mm, onrete blok 55 x 55 x 5 mm. Mode ollapse (breaking of bolts and plate mehanisms)..49

58 35 Fore, kn 35 Fore, kn Experiment 15 Experiment m = 5 1 Predition 1 Predition 5 W97-7 Deformation, mm W97-8 Deformation, mm Figure.3.8 Load-defletion diagrams for experiments W 97-7 and W (Sokol and Wald, 1997). Bolts M4, plate 1 mm, m = 5 mm, n = 4 mm, onrete blok 55 x 55 x 5 mm. Mode 1 ollapse and ollaspe in onrete blok (W 97 7) and no ollapse reahed beause of very large deformations (W 97 8). 35 Fore, kn 35 Fore, kn 3 W W m = Experiment 15 Experiment Predition 5 Predition Deformation, mm Deformation, mm Figure.3.9 Load defletion diagram for experiment W 97-9 and W (Sokol and Wald, 1997). Bolts M4, plate 1 mm, m = 65 mm, n = 4 mm, onrete blok 55 x 55 x 5 mm. Collapse in the onrete blok (W 97 9) and no ollapse reahed beause of very large deformations (W 97 1)..5

59 .4 Shear transfer.4.1 General introdution Horizontal shear fore may be resisted by (see Fig..4.1): a) Frition between the base plate, grout and onrete footing, b) Shear and bending of the anhor bolts, ) A speial shear key, whih is for example a blok of I-stub or T-setion or steel pad welded onto the underside of the base plate, d) Diret ontat, whih an be ahieved by reessing the base plate into the onrete footing. (a) (b) () (d) Figure.4.1 Column bases loaded by horizontal shear fore. In most ases, the shear fore an be resisted through frition between the base plate and the grout. The frition depends on the minimum ompressive load and on the oeffiient of frition. Prestressing the anhor bolts will inrease the resistane of the shear fore transfer by frition. Sometimes, for instane in slender buildings, it may happen that due to horizontal fores (wind loading) the normal ompressive fore is absent or is a tension fore. In suh ases, the horizontal shear fore usually annot be transmitted through frition between the base plate and the grout. If no other provisions are installed (e.g. shear studs), the anhor bolts will have to transmit these shear fores. Beause the grout does not have suffiient strength to resist the bearing stresses between the bolt and the grout, onsiderable bending of the anhor bolts may our, as is indiated in Fig..4.. The main failure modes are rupture of the anhor bolts (loal urvature of the bolt exeeds the dutility of the bolt material), rumbling of the grout, failure (splitting) of the onrete footing and pull out of the anhor bolt. Due to the horizontal displaement, not only shear and bending in the bolts will our, but also the tensile fore in the bolts will be inreased due to seond order effets. The horizontal omponent of the inreasing tensile fore gives an extra ontribution to the shear resistane..51

60 The inreasing vertial omponent gives an extra ontribution to the transfer of load by frition. These fators explain the shape of the load deformation diagram in Fig..4.. F h F v F h δ h (e) δ h Figure.4. Column base loaded by shear and tension. The thikness of the grout layer has an important influene on the horizontal deformations. In the tests reported by the Stevin Laboratory (Stevin, 1989), deformations at rupture of the anhor bolts were between about 15 and 3 mm, whilst grout layers had a thikness of 15, 3 and 6 mm. The deformations have to be taken into aount in the hek of the servieability limit state. Beause of the rather large deformations that may our, this hek may govern the design. The size of the holes may have a onsiderable influene on the horizontal deformations, espeially of ourse when oversized holes are applied. It may be useful in suh ases to apply larger washers under the nuts, to be welded onto the base plate after eretion, or to fill the hole by a two omponent resin. For the appliation of suh resin, referene is made to (ECCS, 1994). For the design of fasteners, the CEB has published a Design Guide (CEB, 1996). In setion.4., the CEB design model for shear load transfer is summarised. In the Stevin Laboratory (Stevin, 1989), a model for the load deformation behaviour of base plates loaded by ombinations of tension or ompression and shear has been developed. This model is desribed in setion.4.3. In setion.4.4, the test results obtained in Delft (Stevin, 1989) are ompared with the CEB model and the Stevin Laboratory model. It is noted that in the models as explained here, only the behaviour of the base plate and the anhor bolts is onsidered. For other failure modes (for the onrete) referene is made to the CEB Design Guide..5

61 .4. CEB Design Guide model.4..1 Introdution In the CEB Design Guide (CEB, 1996), the load transfer from a fixture (e.g. base plate) into the onrete is overed. The CEB Design Guide overs many types of anhors and the possible failure modes of the onrete. The design of the fixture (e.g. the base plate) must be performed aording to the appropriate ode of pratie. In ase of steel fixtures a steel onstrution ode is used. Muh bakground information an be found in the CEB state of the art report: "Fastenings to onrete and masonry strutures" (CEB, 1994). It reviews the behaviour of fastenings in onrete and masonry for the entire range of loading types (inluding monotoni, sustained, fatigue, seismi and impat loading), as well as the influene of environmental effets, based on experimental results. Existing theoretial approahes to predition of the behaviour of anhors are desribed. The CEB Design Guide onsists of three parts: Part I: General provisions. 1. Sope.. Terminology. 3. Safety onept. 4. Determination of ation effets. 5. Non raked onrete. 6. General requirements for a method to alulate the design resistane of a fastening. 7. Provisions for ensuring the harateristi resistane of the onrete member. Part II: Charateristi resistane of fastenings with post installed expansion and underut anhors. 8. General. 9. Ultimate limit state of resistane elasti design approah. 1. Ultimate limit state of resistane plasti design approah. 11. Ultimate limit state of fatigue. 1. Servieability limit state. 13. Durability. Part III: Charateristi resistane of fastenings with ast in plae headed anhors. 14. General. 15. Ultimate limit state of resistane elasti design approah. 16. Ultimate limit state of resistane plasti design approah..53

62 17. Ultimate limit state of fatigue. 18. Servieability limit state. 19. Durability. Referenes. The Design Guide makes a distintion between elasti analysis and plasti analysis. The use of elasti analysis is ompulsory when the expeted mode of failure is brittle. A brittle failure may be assumed in the ase of onrete break-out, splitting failure or rupture of steel with insuffiient dutility. The required dutility is determined by the degree of load redistribution assumed in the analysis. For example, in a plasti analysis, the dutility must be suffiient to aommodate yielding of all anhors on the tension side. It is stated that in the ase of dutile behaviour of the anhorage, the elasti design approah is onservative. For the transfer of shear fores, two methods are onsidered, namely: Frition between the fixture (e.g. base plate) and the grout or onrete and Shear/bending of the anhors. These two methods are desribed in hapter 4: "Determination of ation effets"..4.. Frition between base plate and grout/onrete In setion 4.1 of the CEB Guide it is stated that when a bending moment and/or a ompression fore is ating on a fixture, a frition fore may develop, whih for simpliity may onservatively be negleted in the design of the anhorage. If it is to be taken into aount, then the design value of this fri tion fore VRd,f may be taken as: with V Rd,f = V Rk,f / γ Mf = µ C Sd / γ Mf (.4.1) µ = oeffiient of frition C Sd = ompression fore under the fixture due to the design ations γmf = 1.5 (ultimate limit state) γ Mf = 1.3 (limit state of fatigue) γmf = 1. (servieability 1imit state) In general, the oeffiient of frition between a fixture and onrete may be taken as µ =.4. The frition fore V Rd,f should be negleted if the thikness of grout beneath the fixture is thiker than 3 mm (e.g. in ase of levelling nuts), see Fig..4.3(b) and for anhorages lose to an edge, see Fig..4.3()..54

63 In the elasti design approah, the frition fore alulated by equation (.4.1) is usually subtrated from the shear fore ating on the fixture and in the plasti design approah it is added to the design shear resistane of the fastening. For anhorages lose to an edge it is generally assumed that the edge failure starts from the anhors losest to the edge. The resistane of the anhorage is inreased if the frition fore is ating on the side of the fixture farthest away from the edge (Fig..4.3.(d)), but is not influened by a frition fore ating on the failed onrete (Fig..4.3()). Figure.4.3 Frition fore due to a resulting ompression reation on the fixture. In ases (a) and (d), frition fore may be onsidered in the design. In ases (b) and (), frition fore should not be onsidered in the design (Fig. 16 in the CEB Design Guide). In onlusion, it an be stated that for "normal olumn bases", aording to the CEB Guide, load transfer through frition should be negleted beause of the fat that in normal steel onstrutions the thikness of the grout is always more than 3 mm..55

64 .4..3 Shear/bending of anhor bolts (elasti) In setion of the CEB Guide, guidane is given for the design of anhors and onrete when anhors are loaded by shear, if elasti analysis is assumed. Muh attention is paid to the distribution of the shear loads on the anhors. (a) Distribution of shear loads All anhors in a group are assumed to partiipate in arrying shear loads if the following two onditions are met: (i) (ii) The edge distane is suffiiently large to ensure steel failure of the anhor; and The anhors are welded to or threaded into the fixture, or in the ase of anhorages with a learane hole in the fixture, the diameter of the learane hole is d 1.d. The bolt is then assumed to bear against the fixture, see Fig..4.4(a). Figure.4.4 Examples of anhorages with a large edge distane with a learane hole in the fixture where all anhors will ontribute to the transmission of shear fores. In ase (a) the bolt is assumed to bear against fixture. In ase (b) the sleeve is assumed to bear against the fixture (Fig. in the CEB Design Guide). If the edge distane is small, so that onrete edge failure will our and steel failure of the anhors will be preluded independent of the hole learane, or if the hole learane is larger than indiated in the previous paragraph, only those anhors having the lowest alulated resistane are assumed to arry loads. For example, see Figures.4.5 and.4.6. The positioning of slotted holes in the fixture parallel to the diretion of the shear load an be used to prevent partiular anhors in the group from arrying load. This method an be used to relieve anhors lose to an edge, whih would otherwise ause a premature edge failure (see Fig..4.7)..56

65 Figure.4.5 Examples of load distribution for anhorages lose to an edge or orner of the onrete member (Fig. in the CEB Design Guide). Figure.4.6 Examples of load distribution if the hole learane is large (Fig. 3 in the CEB Design Guide). Figure.4.7 Example of load distribution for an anhorage with slotted holes (Fig. 4 in the CEB Design Guide). For the resistane of anhor bolts, two ases are onsidered, namely (b) and (): (b) Shear loads without lever arm Shear loads ating on anhors may be assumed to at without a lever arm if both of the following onditions are fulfilled..57

66 (1) The fixture must be made of metal and in the area of the anhorage be fixed diretly to the onrete without an intermediate layer or with a levelling layer of mortar with a thikness 3 mm. () The fixture must be adjaent to the anhor over its entire thikness. () Shear loads with lever arm If the onditions (1) and () of the preeding setion (b) are not fulfilled, the length l of the lever arm is alulated aording to equation (.4.): l = a 3 + e l (.4.) with e l = distane between shear load and onrete surfae a 3 =.5 d for post-installed and ast-in-plae anhors (Fig..4.8(a)) a3 = if a washer and a nut are diretly lamped to the onrete surfae (Fig..4.8(b)) d = nominal diameter of the anhor bolt or thread diameter (Fig..4.8(a)) Figure.4.8 Lever arm (Fig. 6 in the CEB Design Guide). The design moment ating on the anhor is alulated aording to equation (.4.3): MSd = VSd l α M (.4.3).58

67 The value of αm depends on the degree of restraint of the anhor at the side of the fixture. No restraint (α M = 1.) should be assumed if the fixture an rotate freely (see Fig..4.9(a)). Full restraint (αm =.) may be assumed only if the fixture annot rotate (see Fig..4.9(b)) and the hole in the fixture is smaller than 1.d (the bolt is then assumed to bear against the fixture, see Fig..4.4(a)) or if the fixture is lamped to the anhor by a nut and a washer (see Fig..4.8). If restraint of the anhor is assumed, the fixture and/or the fastened element must be able to take up the restraint moment. Figure.4.9 Examples of fastenings (a) without and (b) with full restrain of the anhor at the side of the fixture (Fig. 7 in the CEB Design Guide) Plasti analysis In setion 4.. of the CEB Guide, plasti analysis is dealt with. In setion 4...1, the field of appliation is given. It is stated that in a plasti analysis it is assumed that signifiant redistribution of anhor tension and shear fores will our in a group. Therefore, plasti analysis is aeptable only when the failure is governed by dutile steel failure of the anhor. It is stated that pull-out or pull-through failure of the anhor may our at large displaements allowing for some redistribution of tension fores. However, there may not be a signifiant redistribution of shear fores. In view of the lak of information on the required behaviour, a plasti analysis should not be used for this type of failure. To ensure a dutile steel failure the CEB Guide gives several onditions that should be met. These onditions onern: (1) The arrangement of the anhors. It may be assumed that base plates meet these onditions..59

68 () The ultimate strength of a fastening as governed by onrete failure, should exeed its strength as governed by steel failure (equation (.4.4)): Rd, 1.5 Rd,s fuk /fyk (.4.4) with R d, = design onrete apaity of the fastening (onrete one, splitting or pull out failure (tension loading) or onrete pry-out or edge failure (shear loading)) R d,s = design steel apaity of the fastening Equation (.4.4) should be heked for tension, shear and ombined tension and shear fores on the anhors. (3) The nominal steel strength of the anhors should not exeed f uk = 8 Mpa. The ratio of nominal steel yield strength to nominal ultimate strength should not exeed fyk / fuk =.8, while the rupture elongation (measured over a length equal to 5d) should be at least 1%. (4) Anhors that inorporate a redued setion (e.g. a threaded part) should satisfy the following onditions: (a) For anhors loaded in tension, the strength N uk of the redued setion should either be greater than 1.1 times the yield strength N yk of the unredued setion, or the stressed length of the redued setion should be 5d (d = anhor diameter outside redued setion). (b) For anhors loaded in shear or whih are to redistribute shear fores, the beginning of the redued setion should either be 5d below the onrete surfae or in the ase of threaded anhors the threaded part should extend d into the onrete. () For anhors loaded in ombined tension and shear, the onditions (a) and (b) above should be met. (5) The steel fixture should be embedded in the onrete or fastened to the onrete without an intermediate layer or with a layer of mortar with a thikness 3 mm. (6) The diameter of the learane hole in the fixture should be 1.d (the bolt is assumed to bear against fixture; see Fig..4.4(a))..6

69 .4..5 Conlusions for the appliability of the CEB Design Guide for base plates From the above onditions, espeially ondition (5), it beomes lear that aording to the CEB Design Guide plasti design is only allowed for base plates without grout layer or with a grout layer not thiker than 3 mm. For usual base plate onstrution this means that aording to the CEB Design Guide, plasti design is not allowed. The above onlusion means that aording to the CEB Design Guide, the only appliable method for base plates with grout layers thiker than 3 mm is elasti design with the model "Shear loads with lever arm", where the influene of the grout is disregarded. This seems logial sine the grout does not have suffiient resistane to prevent high bending moments in the anhors. In ondition () the relation between the required design onrete apaity of the fastening and the design steel apaity of the fastening is given. It appears that for e.g. 8.8 anhors it gives: R d, 1.56 R d,s (.4.5) In this COST publiation it is assumed that failure of the onrete will not our before failure of the base plate or anhor. The above requirement seems adequate to ensure this prerequisite Resistane funtions for shear load In setion of the Design Guide, the following required verifiations in the ase of shear loading are given (elasti design approah): Steel failure, shear load without lever arm (V Rk,s ) Steel failure, shear load with lever arm (V Rk,sm ) Conrete pry-out failure Conrete edge failure For V Rk,s and V Rk,sm the following equations are given: VRk,s = k As fyk (.4.6) VRk,sm = α M M Rk.s l VRk,s (.4.7) with.61

70 k =.6 (.4.8) A s = stressed ross setion in the shear plane M Rk,s = M Rk,s ( 1- N Sd /N Rd,s ) (.4.9) M Rk,s = harateristi bending resistane of an individual anhor M Rk,s = 1.5 W el f yk (.4.1) N Rd,s = N Rk,s / γ Ms (.4.11) N Rk,s = A s f yk (.4.1) γms = 1. if fuk 8 Mpa and fyk / fuk.8 (.4.13) γ Ms = 1.5 if f uk 8 Mpa or f yk / f uk.8 (.4.14) NSd = applied normal fore α M = fator depending on the support onditions, see Fig..4.9 l = length of lever arm In setion.4.4 of this COST publiation, a omparison of the CEB model with test results obtained in the Stevin Laboratory will be presented, together with a omparison of the Stevin Laboratory model. The Stevin Laboratory model is presented in the next setion..4.3 Stevin Laboratory Model Introdution In normal steel onstrution a grout layer up to 6 mm thikness may be applied under the base plate. In most ases the shear fore an be transmitted via frition between the base plate and the grout. Beause the grout does not have suffiient strength to resist the bearing stresses between the bolt and the grout, onsiderable bending of the anhor bolts may our, as is indiated in Fig..4.. Fig gives one of the test speimens that were tested in the Stevin Laboratory (Stevin, 1989). It shows the bending deformation of the anhor bolts and also the rumbling of the grout and final raking of the onrete. In this setion, the strutural behaviour of anhor bolts loaded by ombinations of shear and tensile fore will be analysed..6

71 Figure.4.1 One of the tested speimens loaded by a ombination of shear fore and tensile fore The analytial model In Fig..4.11, the deformations and some important measures are indiated. Fig..4.1 shows the shematisation of the deformations and the fores, whih are taken into aount in the analytial model. In the Figures, the following symbols are used: A s = tensile stress area of the anhor bolt Ft = applied tensile fore F h = applied shear fore (horizontal fore) Fw = frition fore between base plate and grout N b = normal fore in the grout F a = normal fore in the anhor bolt δa = elongation of the anhor bolt δ b = ompression of the grout layer δh = horizontal displaement of the base plate v = atual thikness of the grout layer vr = thikness of the grout layer in the analytial model: vr = v +.5 db (vr = l as in CEB) d b = bolt diameter.63

72 Figure.4.11 Deformation and some measurements of the tested anhor bolts. Figure.4.1 Shematisation of the deformations and fores in the analytial model..64

73 Due to the horizontal displaement of the base plate, bending of the bolts will our. Also, the tensile fore in the bolt (F a ) will inrease. Due to this, the shear fore F w between grout and base plate will inrease too. Already at rather small horizontal deformations (δh), the tensile fore F a in the bolt reahes the yield fore F yb = A s f yb. This means that the bending moments in the anhor bolts rapidly derease and the horizontal omponent Fah of Fa (Fig..4.1) rapidly inreases. Also the frition fore F w will inrease. Beause of the high tensile fore in the bolt, the bending moments in the bolt will be small. In the analytial model the bending moments in the bolts are not taken into aount. The bearing stresses of the grout-bolt ontat are not taken into aount either, beause they are small ompared to other fores. For the horizontal equilibrium it follows: F h = F a δ h vr + δ a + F w (.4.15) F w = f w ( F vr δb a Ft ) vr + δa (.4.16) with fw = oeffiient of frition grout-base plate. For the deformation it follows: δ h = (vr + δa ) (vr δb ) (.4.17) Beause δ h is small ompared with δa, this an be simplified to: δ h = vr ( δa + δb) (.4.18) For the "elasti" part of the behaviour, δ a and δ b an be written as: F. v a r δ a = (.4.19) E A s (Fa F t) vr δ b = (.4.) EgroutAgrout.65

74 Beause Egrout Agrout is muh greater than EAs, δb will be small ompared with δa. Therefore δb is not taken into aount further. Beause of the geometry it follows: vr + δ a = δh + vr (.4.1) For (.4.15), (.4.16) and (.4.18) an be written with (.4.1): F h = F a δ δ h h + r v + F w (.4.) F w = f w ( F a vr δh + v r Ft ) (.4.3) δh = (.4.4) v r δa For every δh the elongation δa an be alulated via (.4.4), then with (.4.19) the fore Fa and with (.4.) and (.4.3) the horizontal fore F h. The above formulae are valid for F a F ay, where F ay = A s f yb (.4.5) For F a = F ay it follows with (.4.), (.4.3) and (.4.5): F h = f δ y h A s + v r (δ h + f w v r ) f w F t (.4.6) with f y δ h = vr (.4.7) E For the oeffiient of frition fw,d the following design values are proposed: sand-ement mortar f w,d =. speial grout (e.g. Pagel IV) f w,d =.3.66

75 Comparison with one of the tests To demonstrate the analytial model, one of the test results is alulated, namely DT6 in (Stevin, 1989). In this test, a tensile fore F t = 141 kn was kept onstant in the olumn, while the horizontal fore Fh was inreased. Sand-ement mortar Figure.4.13 Test speimen DT6. Data: - anhor bolts M, grade f ub = 176 N/mm (measured value) - fyb = 861 N/mm (assumed as.8 fub) - ε ub = 1% (measured value) - A s = 45 mm - grout = sand-ement mortar - v = 3 mm Calulation: v r = v +.5 d b = = 4 mm For δh aording to (.4.7) it follows:.67

76 δh = vr f yb 861 = 4 E 1 = 3.6 mm For F h aording to (.4.6) it follows ( bolts): F h = ( ) = 1-8 = 94 kn E.g., for δh = 3.6 = 7. mm, it follows: F h = = 13 kn Fig gives the test result together with the result of the analytial model. The value F v.rd = 15 kn is explained in the next setion. Figure.4.14 Comparison of the analytial model with the result of test DT6. It is noted that in Fig a linear relationship is assumed till the value of δh equals δh aording to (.4.7) and F a equals A s f yb. This part of the load deformation urve is alled "elasti". Failure ourred in the anhor bolts (rupture) at the edge of the base plate, due to loal high bending strains..68

77 Ultimate design strength The ultimate strength is a funtion of the strength of the various parts in the olumn base and of the dutility of the anhor bolts. A greater dutility allows a greater horizontal displaement and thus a greater Fah (Fig.4.1) and onsequently a greater Fh. It an also be noted that δ h should be limited, both at servieability and at ultimate limit state. To predit the ultimate strength (if governed by the anhor bolt) a relation must be found between the loal strain in the bolt and the horizontal deformation. Furthermore, the strain apaity (dutility) of the various anhor materials must be known. In the tests it was lear that the applied 4.6 grade anhor bolts were muh more dutile than the 8.8 grade bolts. A differene in dutility an also be found in the requirements in the relevant produt standards. It appeared not easy to establish a reliable model for the strains and to find reliable data for the bending strain apaity of various anhor bolt materials. Therefore, a simplified method is proposed for the strength of 4.6 and 8.8 grade anhor bolts. This simplified method is adopted in the Duth Standard (NEN 677, 199). 4.6 anhors: 8.8 anhors:.375 fub As F v.rd = (.4.8) γ Mb.5 fub A s F v.rd = (.4.9) γ Mb with γ Mb = 1.5. These resistane funtions are similar to the "normal" Euroode 3 funtions for bolts loaded in shear: 4.6 and 8.8 bolts:.6 f ub A s F v.rd = (.4.3) γ Mb After heking the design resistane, the horizontal displaements should be heked for the servieability limit state and for the ultimate limit state..4.4 Comparison with test results Test results and omparison with the Stevin Laboratory model In the test programme (Stevin, 1989), 16 tests were arried out. In Fig the main dimensions of the test speimens are given. Table.4.1 gives a summary of the parameters in the test programme, while in table.4. a summary is given of the main test results and a.69

78 omparison with the proposed resistane funtions, aording to the Stevin Laboratory model (= NEN 677). Figure.4.15 Test speimens; in the tests with 4.6 grade anhor bolts, four bolts were applied and in the tests with 8.8 grade anhors two bolts. Table.4.1 Parameters in the tests. Part of the base plate struture Parameter Variation Grout Type of grout Speial grout Pagel IV Sand - ement mortar :1 No grout Thikness of grout 15 mm 3 mm 6 mm Anhor bolt Strength and dimension 4.6 M 8.8 M Anhoring length 5 mm with a bend at the end of the bar 6 mm with anhor plate Conrete Reinforement With reinforement Without reinforement.7

79 Table.4. Summary of test results and omparison with design values aording to the proposed resistane funtions (Stevin, 1989). Test DT1 DT DT3 DT4 DT5 DT6 DT7 DT8 DT9 DT1 DT11 DT1 DT13 DT14 DT15 DT16 Anhors number Class Yield stress fyb Ultimate stress fub Thikness of grout (v) Tensile fore Ft ultimate shear Measured fore Fh Failure mode *) Mpa MPa Mm kn kn kn Craking Craking Rupture Rupture Rupture Rupture Rupture Rupture Pull-out Rupture Craking Rupture Rupture Craking Craking Craking Design value Fv.Rd **) *) Craking = raking of the onrete Rupture = rupture of the anhor Pull-out = pull-out of the anhor **) These values were alulated with the measured material properties and dimensions. Test / design value Comparison with the CEB model First the appliation of the CEB model is demonstrated for one of the tests. Test DT5 is taken as an example, see also table.4. and.4.3: Bolts M, grade 8.8 fub-measured = 115 Mpa take f yk = = 9 MPa take γms = 1. (for 8.8 bolts).71

80 A s = π d s / 4 = 45 mm d s = mm NRk,s = As fyk = = 6 kn N Rd,s = N Rk,s / γ Ms = 6 / 1. = 188 kn N Sd = 141 / = 7.5 kn We l = d 3 3 π s π = 3 3 = 541 mm 3 M Rk,s = 1.5 W el f yk = = Nmm M Rk,s = M Rk,s ( 1- N Sd /N Rd,s ) = (1 7.5 / 188) = Nmm VRk.s = k As fyk = = 136 kn α M M Rk.s V Rk,sm = = l 3 + / 1-3 = 3.4 kn V Rk.s = 136 kn For the test with bolts it follows 3.4 = 46.8 kn ~ 47 kn, see table.4.3. Table.4.3: Comparison of test results with design values aording to the CEB model and Stevin Laboratory model. Test NRd,s per bolt*) Nsd per bolt Nsd / NRd,s M Rk,s*) MRk,s*) Length l VRk,sm group*) Test / CEB Test / Stevin**) DT1 DT DT3 DT4 DT5 DT6 DT7 DT8 DT9 DT1 DT11 DT1 DT13 DT14 DT15 DT16 kn kn Nm Nm mm kn *) Calulated with the measured material properties and dimensions as given in table.4.. **) The Stevin values are taken from table

81 .4.5 Conlusions Regarding the models The CEB model gives very onservative results, espeially when a large tensile fore is present and / or the length l (thikness of the grout layer) is large. The main reason is that the CEB model does not take aount of the positive influene of the grout layer. The CEB model does not give rules to determine the deformation. The Stevin Laboratory model gives muh more onsistent results. It gives also rules to determine the deformation Regarding the observed behaviour The strength in shear of anhor bolts is onsiderably lower than the shear strength of bolts in bolted onnetions between steel plates. The dutility of the anhor bolts is an important fator for the strength. The lower dutility of 8.8 grade bolts ompared to 4.6 grade bolts is refleted in the lower oeffiient of the resistane funtion. F F v.rd v.rd = =.67 = The influene of a tensile fore Ft in the olumn an be negleted for the determination of the shear resistane. The shear resistane is almost independent of the thikness of the grout layer. The deformations are greatly dependent on the thikness of the grout layer. The type of grout has virtually no influene on the shear resistane. A "better" grout, e.g. "Pagel IV" gave less deformations in the tests. In general it an be stated that the quality of the grout layer has great influene on the strength and espeially the deformations. In the design, not only the shear resistane should be heked, but also the deformations at servieability and ultimate limit state. It is obvious that also other failure modes, like splitting of the onrete blok, et., should be heked. For this hek the CEB Design Guide is probably a good guidane..73

82 .74

83 3.1 Influene of load history Chapter 3: Assembly proedure One the resistane, stiffness and deformation apaity of the joint omponents have been alulated, the next step is to determine the resistane, stiffness and rotational apaity of the whole joint. The proedure to follow in this proess is alled the "assembly proedure". In setion 3. an overview of assembly proedures resulting from the COST C1 projet is given, whereas in setion 3.3 a simplified assembly proedure is desribed for pratial design purposes. Unlike beam-to-olumn joints, an important aspet of assembly proedures and their omparison with test results for olumn bases is the (monotoni) loading history applied to the joint. In ase of beam-to-olumn joints without normal fore in the beam, the rotation of the joints is only dependent on the applied moment. In the ase of base plate joints, however, the moment-rotation harateristi is also dependent on the normal fore in the olumn. Consider the base plate joint of Figure M -N Sd Sd Figure Base plate joint. Assume that the base plate is loaded to its moment resistane M Rd = 16 knm in ombination with a normal ompressive fore N Rd = -66 kn. With help of a design model, design moment-rotation diagrams are alulated as shown in Figure In this figure two lines are indiated, the solid line labelled "proportional" and the dotted line labelled "non-proportional". In the proportional ase, the moment M Sd is applied to the joint together with a normal fore 3.1

84 N Sd equal to N Rd M Sd / M Rd. In other words, when the M Sd is equal to zero, N Sd is equal to zero and when M Sd is equal to M Rd, N Sd is equal to N Rd. In the non-proportional ase, it is assumed that the normal fore N Rd is applied first to the joint and that then the moment M Sd is applied step by step until it reahes M Rd. Non Proportional (N=onst) Moment Proportional Left side anhor bolts in tension, right side onrete in ompression Both sides of the joint in ompression Rotation Figure 3.1. Design moment-rotation diagram for the joint of Figure From Figure 3.1. it an be seen that the initial stiffness under non-proportional loading is muh higher then under proportional loading. This is beause onrete in ompression is muh stiffer than anhor bolts in tension. In the ase of non-proportional loading during the initial loading steps, both sides of the joint are firmly held in ontat by the normal fore N Rd. That means that hardly no deformations our, even if a small moment is applied to the joint. When the moment M Sd exeeds a ertain value, the left side of the joint is no longer in ompression and the anhor bolts are ativated. The joint looses then its high stiffness. In ase of proportional loading, the anhor bolts on the left hand side of the joint are ativated immediately. The initial stiffness is therefore lower than in the ase of non-proportional loading, where the anhor bolts are inative. In Figure 3.1. it an be seen that the design moment resistane of the joint is independent of the load history applied (proportional or non-proportional) to the joint. The normal fore in the olumn has a ritial effet on the design moment resistane. When the normal fore N Sd ating at a joint is a tensile fore, the initial stiffness in ase of non-proportional loading will probably be lower than in ase of proportional loading beause due to the tensile normal fore, all anhor bolts will be held in tension during the initial load 3.

85 steps of non-proportional loading, whih will result in a high flexibility. Load history is important when reviewing the assembly proedures and also when omparing tests with design models. For instane, in many ases, tests are performed with nonproportional loading and design models assume proportional loading. The test results and the design models may in that ase not be ompared diretly with eah other and a orretion needs to be made to take the load history into aount. In design praitie, a proportional load history is normally onsidered. 3. Review of existing models 3..1 Overview Chapter desribes how the load - deformation urve of eah individual omponent of a olumn base joint may be determined. In order to end up with a moment-rotation urve, whih represents the global behaviour of the joint, the omponent harateristis have to be assembled. For this purpose, appropriate analytial or mehanial models are used, for example spring models. The assembly of the omponents of a olumn base joint with an end plate is similar to that of beam-to-olumn joints. However, the assembly is signifiantly affeted by the interation of normal fores and bending moments. Model Mehanial Sophistiate d Two dimensional Complex Model Simplified Stiffness Analytial Simplified Strength Simplified Strength Simplified Strength and stiffness M-φ urve Non-linear Non-linear Stiffness Strength Strength Non-linear Components Non-linear Bi-linear Stiffness Strength Strength Bi-linear desription only only only Effetive Annex L [1] Retangular Retangular Annex L [1] Retangular Web area of an for H setion for H setion negleted equivalent rigid plate Stress to onrete Non-linear springs Elasto-plasti (1 stress patterns) Three patterns only Compatibility Yes Yes Yes Not for strain Referene Guisse et al, Wald, Wald et al., Wald et al., ,8 b Plasti only Plasti only Plasti only Simplified Guisse et al, 1996 Figure 3..1 Comparison of different assembly proedures Simplified Steenhuis,

86 Different predition models are available for alulation, for list of models see Wald, Figure 3..1 gives a brief overview of the different assembly models based on omponent method, see Wald, 1995; Guisse at all., 1996; Wald, 1996; Steenhuis, It an be seen that some models onsider the full non-linear behaviour, while others only deal with the resistane or with the stiffness of the joint. Most of the models fulfil the requirements for ompatibility (i.e. equilibrium of internal fores, deformations, et.), but some of them in a simplified way only. The mehanial model presented in Guisse at all, 1996 is based on a D non-linear spring simulation, see Figure 3... It an take into aount not only the non-linear behaviour of eah omponent, but it an also simulate the influene of hanges of the effetive area under bending. This mehanial model is ertainly useful for researhers performing sientifi bakground studies and omparisons. Test results demonstrate a good auray of the model. However, the model is too omplex for daily design pratie. A more pratial approah is the omponent method to predit the design resistane, stiffness and deformation apaity. N Sd M Sd effetive area anhor bolt olumn web larger ontat area non-linear omponent behaviour Fore, kn Fore, kn larger ontat area for bending Fore, kn plate to onrete ontat Anhor bolt Deformation Base plate Deformation Conrete Deformation Figure 3.. Mehanial model for the assembly proedure of omponents aording to Guisse at all, 1996 As an be seen in Figure 3..1, the various models use different approahes to determine the effetive area. In the models it is assumed that this effetive area represents a rigid plate with a ertain stress distribution in the ompression zone. It an be understood that the shape of this effetive area will strongly influene the omplexity of the assembly models. This is beause the assembly proedure should take the normal fores and the bending moments ating to the joint into aount. Their atual values will influene the position of the neutral axis of stresses, whih also depends on the shape of the effetive area. Most proposals for the effetive area are based on the model given in Annex L of Euroode 3 [1], whih assumes plasti distribution of stresses in the onrete under the base 3.4

87 plate (bearing width ). Furthermore, the models assume a ertain pattern for the ompressive stresses. The simplest approah is a full plasti stress distribution for the resistane alulation. Effetive bearing area Resistane Stiffness Component Modelling Assembling Stress patterns N Sd M Sd N Sd M Sd Fore, kn M, knm E k b Anhor bolt,5 N, kn k k p k b k k E k p E k Base plate,5 Conrete,5 Deformation, δ, mm Stiffness predition M, knm N = konst. Rotation, rad Figure 3..3 The Assembly based on retangular shape of base plate Wald, 1995, the simplifiation to three patterns only for high bending moments and low bending moments respetively, the bi-linear load-deformation urves of the omponents and the moment-normal fore diagram MSd M N Sd k p k b k N Sd Sd k k Fore, kn E k p E k E k b Deformation, Anhor bolt,5 Base plate,5,5 Conrete δ, mm Moment, knm Experiment W7-4.-prop Predition HE 16 B t = h = 5 1 N M Rotation, mrad Figure 3..4 The assembling with an effetive area around the olumn flanges only Steenhuis, For high bending moments and low bending moments respetively, the bilinear load-deformation urves of the omponents and the moment-rotation diagram for the joint in omparison to test no W7-4.-prop are shown, see Wald et al, 1995 The predition of the rotational stiffness has to be based on simplifiations of the effetive 3.5

88 area in order to allow for the development of a proedure with a limited degree of omplexity. The simplified area using a retangular shape enables taking all stress distributions (1 patterns) into aount. A simpler solution, for example with only three ategories of high, medium and low normal fore is also an option, see Figure 3..3 from Wald, 1993, where f y represents yielding in the tension part and f j represents the onrete bearing resistane. The simplest modelling takes the areas under the flanges only into aount, see Figure 3..4 based on Steenhuis, This has the advantage of simple modelling without losing auray under pure bending, when the ontat is orretly predited, but it has the disadvantage of losing the auray under high normal fore and a very small bending moment, espeially for ross setions with a signifiant effetive area under the olumn web. 3.. Simplified Prague model for resistane and stiffness Strength Design The effetive bearing area A eff and bearing strength f j is alulated aording to Euroode 3 Annex L, 199. This area is used for strength alulation Wald (1995). The resistane of the tension part of olumn bases loaded by an axial fore and a bending moment F t.rd, is alulated aording to Euroode 3 Annex J, The equilibrium of internal fores ould be alulated in elasti or plasti stages. The ative part of the effetive area A eff an be alulated from the equilibrium equation, see Figure N Rd = Aeff f j Ft. Rd (3..1) M = F r A f r Rd t. + (3..) Rd b eff j N Sd M Rd A eff ative part r b r F t.rd f j Figure 3..5 Equilibrium of internal fores 3.6

89 N = konst. f y M, knm M N M, knm f j N = konst. f y tension pattern f j ompression N, kn simplifiation of three patterns only aording to the normal fore (low, medium, high) Rotation, rad Figure 3..6 Moment - normal fore resistane diagram for olumn base introduing the internal fore distribution in elasti and inelasti region, f y is representing yielding of the tension part, f j is bearing resistane of the onrete, see Wald, Rotational Stiffness The rotational stiffness is determined aording to stiffness model published in Euroode 3 Annex J, S j = E µ i z 1 k i (3..3) where E is the modulus of elastiity of the steel and z is the lever arm. The ratio between the rotational stiffness µ with respet to the bending moment an be alulated as S µ = = j. ini κ M Sd S M j Rd ξ 1 (3..4) In the formula above, ξ is shape parameter of the urve, κ is oeffiient introduing the beginning of non-linear part of the urve and M sd is ating bending moment, see WALD, The extension of the model ompared to Euroode approah is the introdution of the normal fore. It is distinguished between three basi ollapse modes of the base plate joint, see Wald and Sokol, 1995 and Wald and Sokol, They differ by the bearing stress distribution 3.7

90 under the base plate with respet to the tension part, see Figure The onrete bearing stress f j is never reahed, when the olumn base joint is loaded by low axial fore (ompared to the ultimate bearing apaity). The ollapse ours either by yielding of the bolts or by development a plasti mehanism in the base plate (pattern 1). When medium axial fore is applied, the onrete bearing stress f j and the strength of the tension part F t.rd are reahed at the ollapse (pattern ). For a high axial fore, ollapse of the onrete ours while stress in the tension part is not developed (pattern 3). initial distribution pattern 1 low axial fore pattern medium axial fore pattern 3 high axial fore f j f j elasti tension part distribution at the ollapse plasti tension part Figure 3..7 The stress distribution for three patterns of the base plate joint in initial and ollapse stages The axial fore N 1. representing the boundary between low and medium fores is alulated for two anhor bolts, see Wald et al, 1996 and Wald, N 1. = 1 ap bp f j ( z+ ap) k ap bp f j + 4 As ft + k kb k p A f (3..5) s t where f t = FtRd. / As is equivalent stress in the tension part, a p, b p are dimensions of the effetive area of the plate, see Figure 3..8, and z is the lever arm of anhor bolts. 3.8

91 b p a p Figure 3..8 Simplifiation of the effetive area for the stiffness alulations The boundary between medium and high fores N.3 is given by N 3. = ( ) z+ a b f A f p p j s t a b f (3..6) p p j pattern 1 pattern pattern 3 /3 1/3 k p k p k b k k b k k k z z z Figure 3..9 The spring simulation of deformation of the base plate joint 3.9

92 Table 3..1 Values to be onsidered in stiffness alulation for different patterns Wald and Sokol, 1995 Pattern 1 Low axial fore N Sd N 1. Medium axial fore N 1. < N Sd < N.3 Relevant Stiffness oeffiients k i k p, k b, k 1,1 6 k p, k b, k κ ξ Lever arm linear transition ( r ) 1 z = b ap linear transition between z 1 and z 3 1 k 1 k ( for pattern1) ( for pattern3) 3 High axial fore k, k 1,5 8 N.3 N Sd z 3 = a p Simplified Liège model for resistane Introdution Experimental tests have been arried out at the University of Liège (Guisse et al, 1996) on olumn bases with two or four anhor bolts. They have shown that the olumn bases have a very high semi-rigid behaviour, even for so-alled nominally pinned onnetions; this is known to be potentially benefiial when designing building frames. On the basis of the knowledge got from these tests and from the available literature, a simple analytial model aimed at prediting the ultimate and design resistanes has been developed and validated through omparisons with the experiments. The model, whih is briefly desribed in this setion appears as an appliation of the priniples of the omponent method, with some referenes to annexes L and J of Euroode 3 for what regards the haraterization of the individual basi omponents. But nowadays a omplete model requires also the predition of the rotational stiffness of the olumn bases. However, the omplexity of the problem is suh that the development of a simple and reliable stiffness model appeared as quite ontingent. Guisse et al therefore deided to fous on the 3.1

93 development of a sientifi tool, i.e. a mehanial model, allowing, through rather long and iterative alulation proedures, to simulate aurately the non-linear response of the olumn bases from the first loading steps to failure. The interested reader will find details about the simple analytial model and the more sophistiated one in (Guisse et al, 1996). In the next paragraphs, guidelines on how to use the simple analytial model for the evaluation of the resistane of olumn bases are given Sope of the presented model All the figures given here below relate to a "4 anhor bolts" onfiguration but the proedure applies both to " and 4 anhor bolts" onfigurations. The equations presented over ases where a part of the onrete blok is subjeted to ompressive fores (ontat zone with the base plate). The model may easily be extended to ases where no ontat fores develop between the blok and the foundation (fores arried out by the anhor bolts in tension only) Desription of the model PRELIMINARY CALCULATIONS Evaluation of the resistane properties of the omponents - onrete in ompression - anhor bolts in tension - base plate in bending Definition of the rigid equivalent plate - Calulation aording Annex L Euroode 3 (or improved proedure if available) A eff - Definition of a fititious rigid equivalent plate b eff x h eff with an area equal to A eff (Figure 3...1) 3.11

94 h h EC3 rigid equivalent plate b b eff h eff Fititious retangular equivalent plate : h eff = h + Figure 3..1 Equivalent retangular rigid plate b eff = A eff /h eff - As an approximation, b eff may be hosen as equal to,8 b. CALCULATION PROCEDURE At ultimate limit state (olumn based subjeted to may our : - bolts are ativated in tension; - bolts are not ativated in tension. * M Rd and - Two sets of formulae are available to over both situations. N * Rd ), two different situations The user first makes an assumption and deides whether, at first sight, anhor bolts are likely or not to be ativated in tension. Aording to his deision, he follows hereafter one of the two proedures respetively named : - assembly proedure with no anhor bolts in tension; - assembly proedure with anhor bolts in tension. Inside the proedures, a riterion is given to hek the validity of the assumption. If it is not fulfilled, the user selets then the proedure he had first disregarded. The proposed formulae simply results from the expression of the equilibrium between the applied bending moment and axial fore and the resultant fores in the anhor bolts and/or the 3.1

95 ontat fores between the base plate and the onrete foundation (a retangular distribution of ontat stresses is assumed). Assembly proedure with no anhor bolts in tension e N * Rd h d h pr h b eff e e N * Rd e h eff N * Rd Figure Internal and external fores if no anhor bolts in tension h pr, whih is defined as h eff -e, is derived from both following equilibrium equations (Figure 3..11) : M N e = * Rd = * Rd 1 h e eff eff * Rd N b f j (3..7) (3..8) At the same time, the values of The solution is valid as long as : h pr * M Rd and * N Rd are extrated. h + h' d (3..9) eff 3.13

96 Assembly proedure with anhor bolts in tension h eff + h d h pr b eff f j F b(hpr) Figure 3..1 Internal fores if one anhor bolt row in tension h pr is derived from the three following equations (Figure 3..1) h pr = b eff f j N * Rd + F + F t. Rd t. Rd h eff 1 + h' d (3..1) M N * Rd = * Rd e (3..11) M * Rd = b eff + F h t. Rd pr f ( h j eff h eff,5( h h + h' d) h eff pr pr + h' d) h ( eff + h' d) (3..1) The values of * M Rd and The solution is valid as long as : * M Rd are extrated at the same time.,5( h + h' d) < h < ( h + h' d) (3..13) eff pr eff 3.14

97 If,5( h eff + h' d ) > hpr, then the anhor bolts are subjeted to fores equal to their design resistanes in tension and equations (3..1) to (3..1) have to be replaed by the following ones: h pr * Rd N + Ft. Rd = (3..14) b f eff j M N M * Rd = * Rd * Rd + F t. Rd e = b eff h ( eff h pr f j h eff + h' d ) h pr (3..15) (3..16) 3..4 Simplified Delft model for resistane and stiffness This paragraph presents an assembly proedure for determination of strength properties M Rd under normal fore N Sd and rotational stiffness S j for olumn base steel joints negleting the ontribution of the olumn web. The assembly proedure based on Euroode 3 Annex L, but the ontribution of the olumn web is negleted, see Figure N Sd N Sd M Sd M Sd t f ½h - t f ½h - t f t f 1 F ten,l,rd F om,l.rd z ten,l F ten,r,rd z ten,r F om,r,rd m m m z om,l z om,r Figure Base plate joint used as example in assembly proedure 3.15

98 In Figure the ompliated behaviour of the base plate is simplified to a system with four springs: two springs ating in ompression more or less under the flanges of the olumn and two springs ating in tension representing the anhor bolts. A designer needs to deide whih springs will be ativated to resist the loading on the joint (normal fore N Sd and moment M Sd ). This deision an be made as follows. Sine the joint is loaded with a normal ompressive fore and it is loaded with a moment in lokwise diretion, the right side of the joint will be ativated in ompression and not in tension. In the ourse of this paragraph it is assumed that M sd / N Sd > z om,r, so the left anhor bolts will be ativated. The assembly proedure is of ourse not restrited to this loading ase: N Sd may be a tensile fore and M sd may be opposite to lokwise. In Table 3.. the assembly proedure is summarized for N Sd as ompressive fore, M sd lokwise and M sd / N Sd > z om,r. 3.16

99 Table 3..: Summary sheet Assembly proedure negleting the ontribution of the olumn web, part urve left bolts in tension, right onrete in ompression. Presentation for onstant normal fore N Sd Range of validity: N Sd z om,r < M Rd Presentation for proportional loading Range of validity: z om,r < e S < e S = M Sd / N Sd NSd NSd M Sd MSd δ trn,l ϕ k ten,l k om,r δ om,r Ften,l F om,r z z zten, zo zten,l zom,r Strength Strength M Rd = the smallest of: F ten,l,rd z + z om,r N Sd F om,l,rd z - z ten,l N Sd z = z om,r + z ten,l Stiffness S j = ϕ = E z µ Σ 1/k i M Sd - e k N Sd e k = z om,r k om,r - z ten,l k ten,l k om,r + k ten,l µ = (1,5 y),7 but µ 1 y = M Sd + ½ z N Sd M Rd + ½ z N Sd Modelling S j M Rd = the smallest of: F ten,l,rd z 1 - z om,r e S F om,r,rd z 1 + z ten,l e S z = z om,r + z ten,l Stiffness S j = ϕ = e S E z e S - e k M Sd S j µ Σ 1/k i e k = z om,r k om,r - z ten,l k ten,l k om,r + k ten,l µ = (1,5 y),7 but µ 1 y = 1 + ½ z e S M Rd + ½ z M Sd e S Modelling Column Column M Sd M Sd ek NSd S j e k N Sd Sj M Sd M Sd 3.17

100 The proedure is given in two presentations: a presentation for onstant normal fore and a presentation for proportional loading. The differene between both is the loading history. In the presentation for onstant normal fore, in a first step the normal fore N Sd is applied. Then, in a seond step, the moment M Sd is built up, till the moment resistane M Rd has been reahed. In the presentation for proportional loading, during the load history the ratio e S = M Sd / N Sd remains onstant. The means, in the beginning of the loading history, both M Sd and N Sd are small. The moment M Sd and N Sd are built up simultaneously until M Sd reahes M Rd. The first presentation is useful for the alibration of the assembly proedure with tests, where it is ommon first to apply a normal fore and in a seond step the bending moment. The seond presentation has more pratial benefit, beause unlike in the ase of a onstant normal fore, with proportional loading it is during the whole loading history lear whih omponents at in tension or ompression. This results finally in advantages for modelling, see Table 3... The assembly proedure for strength is rather straightforward and an be derived from simple equilibrium as an be seen from Table 3... The assembly proedure for stiffness is similar to the stiffness model of Euroode 3 Annex J. However, there are some differenes. When fousing on the presentation for proportional loading the differenes are as follows: In Eurode 3 Annex J the stiffness formula is S j = E z µ Σ 1 k i when a joint is loaded with pure e S bending moment. For base plates, an additional fator should be taken into aount. e S - e k This fator reflets the fat that due to normal fore, the stiffness of a joint is will be higher or lower. The fator e k is dependent on the stiffness of the omponents in the joints. When N Sd = e S, is equal to 1, so the expression beomes equal to the one in Annex J. e S - e k In Annex J the ratio between the initial stiffness and the seant stiffness µ is equal to (1,5 y),7 but µ 1with y = M Sd / M Rd. However, due to the normal fore ating at the joint, ertain omponent may be brought in an earlier stage to ollapse, whih has an effet on the ratio µ. 1 + ½ z e S This effet is taken into aount be adopting y = M Rd + ½ z. The impliation is that even if M Sd e S M Sd is small, the ratio µ may be larger than 1 due to the normal fore. The assembly proedure is very simple and straightforward to apply by pratitioners. The proedure has no iterative harater, whih makes it suitable for ode implementation. 3.18

101 3.3 Tentative assembly model Simplified model In setion 3. a desription is given of assembly models that desribe the moment-rotation harateristis of base plate joints. For pratial use, these models may be improved by further simplifiation. First simplifiations are desribed in (Steenhuis, 1998). In this COST report, a simplified assembly model is given overing all load ases and all situations. Due to its simpliity, this model will differ from test results in some respets. However, a designer an apply this simplified assembly proedure in a straightforward manner, without iterative proedures. In setion 3.3. the assembly proedure with respet to strength is desribed. Setion desribes the assembly proedure with respet to stiffness. In setion a summary table is given for the whole assembly proedure. A omparison with tests is presented in setion Strength The assembly proedure for strength is desribed based on an example as shown in Figure In Figure the drawing onvention is that the arrow representing the normal fore indiates ompression (N Sd ). A positive applied moment is represented by a lokwise arrow as shown (M Sd > ). The ompliated behaviour of the base plate is simplified to a system with four springs: two springs ating in ompression under the flanges of the olumn and two springs ating in tension representing the anhor bolts. The loading is assumed to at proportionally. This aords with the behaviour of strutural systems, where an inrease of normal fore in the olumn is oupled with an inrease of the bending moment to be transferred. In this model, the ontribution of the resistane of the onrete under the olumn web is negleted. This may lead to some onservatism in ase of slender olumn setions, espeially with respet to strength properties. However, taking into aount the ontribution of the onrete portion under the olumn web will lead to onerous iterative proedures. In this simplified assembly proedure it is proposed to neglet this influene. For the simple ase of a ompressive normal fore without bending moment it is of ourse possible to inlude this ontribution as done in (Euroode 3 Annex L, 199). 3.19

102 -N Sd -N Sd M Sd M Sd t f ½h - t f ½h - t f t f 1 F ten,l,rd F om,l.rd z ten,l F ten,r,rd z ten,r F om,r,rd m m m z om,l z om,r Figure Base plate joint used as example in assembly proedure The symbols in Figure are as follows: F ten,l,rd = the apaity of the left side of the joint in tension determined by the omponents "base plate in bending" and "anhor bolts in tension" see hapter ; F ten,r,rd = the apaity of the right side of the joint in tension determined by the omponents "base plate in bending" and "anhor bolts in tension" see hapter ; F om,l,rd = the apaity of the left side of the joint in ompression determined by the omponents "base plate in bending", "onrete in ompression" and "olumn web and flange in ompression", see hapter ; F om,r,rd = the apaity of the right side of the joint in ompression determined by the omponents "base plate in bending", "onrete in ompression" and "olumn web and flange in ompression", see hapter ; z ten,l = the distane between the enter of the olumn and the tensile reation fore on the left side of the joint, z ten,l = ½ (h - t f,. ) z ten,r = the distane between the enter of the olumn and the tensile reation fore on the right side of the joint, z ten,r =. ½ h - t f, - m z om,l = the distane between the enter of the olumn and the ompressive reation fore on the left side of the joint, z om,l = ½ (h - t f, ). z om,r = the distane between the enter of the olumn and the ompressive reation fore on the right side of the joint, z om,r = ½ (h - t f, ). As a simplifiation, the ompressive reation fores are, in aordane with Revised Annex J, assumed to at at the enter of the olumn flanges. For the ase of non-extending base plates, this simplifiation is not fully in line with the real behaviour of the omponents "onrete in ompression" and "base plate in bending", but this simplifiation is felt to be aurate enough 3.

103 for pratial situations. A designer needs to deide whih springs will be ativated to resist the loading on the joint (normal fore N Sd and moment M Sd ). This deision an be made as follows. Sine the joint is loaded with a normal ompressive fore and it is loaded with a moment in lokwise diretion, the right side of the joint will be ativated in ompression and not in tension. Whether the left side of the joint is ativated in tension or ompression an be determined with the following proedure: Define e S = M sd / N Sd as the eentriity of ations If e S < -z om,r then the anhor bolts on the left hand side of the joint will be ativated. In the other ase, the onrete will be ativated. For this example it is assumed that e S < -z om,r, so the left anhor bolts will be ativated. The resistane for bending moment M Rd under a onstant normal fore N Sd an now be determined based on simple equilibrium as follows: -N Sd M Sd F ten,l,rd z F om,r,rd z ten,l z om,r Figure 3.3. Resulting system of fores Consider Figure 3.3.: M Sd z om,r + z ten,l + N Sd z om,r z om,r + z ten,l < F ten,l,rd (3.3.1) M Sd N Sd z ten,l - < F om,r,rd (3.3.) z om,r + z ten,l z om,r + z ten,l If during the loading history of a olumn base joint the ratio e S remains onstant (proportional loading), the equilibrium equations an be presented as follows: 3.1

104 M Sd < F ten,l,rd z 1 + z (3.3.3) om,r e S M Sd < F om,r,rd z 1 - z (3.3.4) ten,l e S The maximum value for M sd under a normal fore N Sd is defined as M Rd, hene: M Rd = the minimum of: F ten,l,rd z z om,r e S + 1 and -F om,r,rd z z ten,l e S - 1 (3.3.5) where: N Rd = M Rd / e S (3.3.6) z = z om,r + z ten,l (3.3.7) A similar proedure an be used to derive formulae for all other load ases (tensile normal fore and anti-lokwise moments). The resulting formulae are given in setion Stiffness The stiffness model is desribed based on the example of Figure For the omponents "anhor bolts in tension" and "base plate in bending" and "onrete in ompression" and "base plate in bending", elasti stiffness fators are defined in aordane with hapter of this report. The omponent "olumn web and flange in ompression" is not onsidered to ontribute to the flexibility of the joint. The elasti stiffness fator for the left bolt group is equal to k ten,l. and the elasti stiffness fator for the right onrete portion is equal to k om,r. The model of the equivalent springs is shown in Figure

105 -N Sd M Sd δ ten,l ϕ k ten,l z k om,r δ om,,r z ten,l z om,r Figure Spring model It is assumed that the springs at at loations as desribed in setion The elasti deformation of the joint an be alulated as follows: δ ten,l = δ om,r = M Sd + N Sd z om,r z om,r + z ten,l z om,r + z ten,l = M Sd + N Sd z om,r (3.3.8) E k ten,l z E k ten,l M Sd z om,r + z ten,l - E k om,r N Sd z ten,l z om,r + z ten,l The rotation of the joints is as defined as follows: Where: = M Sd - N Sd z ten,l z E k om,r (3.3.9) ϕ = µ (δ ten,l + δ om,r ) / z (3.3.1) µ is a fator equal to 1 or more. In Revised Annex J (Euroode 3, 1998) the fator µ varies between 1 and 3 for end plated joints. For instane, µ = 1 when M Sd is smaller than /3 M Rd and µ = 3 when M Sd is M Rd. The value of µ for M Sd is between /3 M Rd and M Rd is equal to: Hene: µ = (1,5 M Sd M Rd ),7 It is proposed to adopt a similar model for base plate joints. 3.3

106 ϕ = µ (δ ten,l + δ om,r ) / z = µ ( M Sd + N Sd z om,r + M Sd - N Sd z ten,l ) / z (3.3.11) z E k ten,l z E k om,r ϕ =.. µ z E (M Sd + N Sd z om,r k ten,l + M Sd - N Sd z ten,l k om,r ) (3.3.1) If the moment N Sd e k is defined as the moment M Sd required if ϕ = under a onstant normal fore N Sd, see Figure 3.3.4, then: ϕ = µ z E (N Sd e k + N Sd z om,r k ten,l + N Sd e k - N Sd z ten,l k om,r ) = (3.3.13) Hene: e k + z om,r + e k - z ten,l = (3.3.14) k ten,l k om,r k om,r (e k + z om,r ) +k ten,l ( e k -z ten,l ) = (3.3.15) e k = z ten,l k ten,l z om,r k om,r k om,r + k ten,l (3.3.16) -N Sd original loation δ ten,l = δ om,r ϕ = -e k k ten,l z z t,l z,r Figure Definition of e k under zero rotation Hene, e k is defined as the entroid of the springs with stiffness oeffiients k ten;l and k om;r It has to be noted that e k is a purely theoretial definition. This an be seen in Figure where the left tensile spring (anhor bolts), ats in ompression. By defining S j = M Sd / ϕ, S j an be written: 3.4

107 S j = M Sd E z M Sd + e k N Sd µ Σ 1 k i (3.3.17) hene: S j = e S E z e S + e k µ Σ 1 (3.3.18) k i where: Σ 1 k i = (3.3.19) k ten,l k om,r Thus, when alulating the stiffness of a base plate joint, a modifiation fator equal to is used, whih reflets the ontribution of the normal fore in the proportional loading. e S e S + e k Overview of resistane and stiffness formula for all loadases In setion 3.3. and a desription of the moment rotation relationship was given for the ase when the left side of the joint was in tension and the right side in ompression. In Table an overview is given for all situations. For all loadases, i.e. normal fore N Sd > or N Sd and moment M Sd > or M Sd, formula are given. 3.5

108 Table 3.3.1: Moment resistane MRd (MSd > is lokwise, MSd is anti-lokwise) and rotational stiffness Sj. dependent on the diretion of normal fore NSd (NSd > is tension, NSd is ompression) and es = M Sd NSd = M Rd NRd NSd > e S > zten.l N Sd es - z om.r MRd = min ( F ten,l,rd z Sj = es es + ek zom,r es µ ( 1 kten,l + 1 E z ; -F om,r,rd z zten,l + 1 kom,r es ) - 1 ) N Sd > < es zten.l M Rd = min ( Ften,l,Rd z ; F ten,r,rd z zten,r zten,l es Sj = es es + ek es µ ( 1 ) E z kten,l + 1 kten,r N Sd > - zten.r < es M Rd = max ( Ften,l,Rd z ; F ten,r,rd z zten,r zten,l es ) es ) N Sd > es - zten.r MRd = max ( -F om,l,rd z Sj = es es + ek µ ( zten,r es + 1 E z 1 kom,l + 1 kten,r NSd e S > zom.l ; F ten,r,rd z zom,l - 1 es ) ) NSd < es z om.l M Rd = max ( -Fom,l,Rd z zom,r es Sj = + 1 es es + ek ; -F om,r,rd z zom,l - 1 µ ( es 1 E z kom,l ) + 1 kom,r N Sd - zom.r < e S M Rd = min ( -Fom,l,Rd z ) zom,r es + 1 ; -F om,r,rd z zom,l es - 1 ) ek = z om,r k om,r - z ten,l k ten,l k om,r + k ten,l z = z om,.r + z ten,l ek = z ten,r k ten,r - z ten,l k ten,l k ten,r + k ten,l z = z ten,.r + z ten,l ek = z ten,r k ten,r - z om,l k om,l k ten,r + k om,l z = z ten,.r + z om,l ek = z om,r k om,r - z om,l k om,l k om,r + k om, z = z om,.r + z om,l µ = (1,5 M Sd ),7 but µ 1 M Rd Left side in tension Right side in ompression left side in tension right side in tension left side in ompression right side in tension left side in ompression right side in ompression NSd MSd 3.6

109 3.3.5 Comparison with test data The assembly proedure desribed in this hapter together with the models the strength and stiffness of the omponents as given in hapter have been implemented in a spreadsheet. Using the model of this report, preditions have been made of moment rotation urves of three series of tests arried out by Wald et al, 1995 and Vandegans, The predited moment rotation urves have been ompared with the test urves. In the predition, the real measured properties of the materials have been adopted with partial safety fators equal to 1,. Also work has been done to ompare the model test arried out by others. This work has not been onluded yet. Sine in this report omparison are only made for two series of tests, without further onfirmation, the model annot be reommended for use in pratie. The tests show that the behaviour of a base plate joint is very sensitive to the quality of the onrete or grout immediately under the base plate. In the model presented in this doument it is assumed that the onrete and the grout are of good quality. The results of the omparison of the model with the other test series by Wald et al, 1995 and Vandegans, 1997 are reported in Figure to Figure In the ase of non-proportional loading, first the design moment M Rd and the design normal fore N Rd have been alulated based on the methods given in this hapter. Then, the non-proportional loading in the model is simulated by first applying the design normal fore N Rd to the model. Then the moment M Sd is applied in steps from to M Rd. The value of N Rd is in general somewhat lower then the normal fore atually applied in the test. In general, despite all simplifiations, the model design model shows good agreement with the test data. Only in ase of very high normal fore, e.g. test S-19, see Figure , the model is onservative. This ould be expeted beause the ontribution of the web to the load arrying apaity of the base plate is negleted in the assembly model. Also, it should be noted that the model works equally well in ases where only one bolt row is loated in between the olumn flanges and in ases where two bolt rows are applied outside the olumn flanges. 3.7

110 9 Moment, knm 8 W7-4.-prop Analytial Proportional Experiment Rotation, mrad HE 16 B M P - 3 x 3 x 33 x 5 55 x 55 x 55 Moment, knm 9 8 W7-4.-prop Normal Fore, kn Figure Experimental and predited moment - rotational diagrams for shown normal fore history, experiment W7-4.-prop, (Wald et al, 1995) Moment, knm W Analytial Non Proportional Experiment 1 Rotation, mrad HE 16 B M P - 3 x 3 x 33 x 5 55 x 55 x 55 Moment, knm 6 5 W Normal Fore, kn 1 Figure Experimental and predited moment - rotational diagrams for shown normal fore history, experiment W8-4.-onst, (Wald et al, 1995) Moment, knm Moment, knm Analytial Non Proportional Experiment 1 Rotation, mrad W9-4.- HE 16 B M P - 3 x 3 x 33 x 5 55 x 55 x 55 6 W Normal Fore, kn 1 Figure Experimental and predited moment - rotational diagrams for shown normal fore history, experiment W9-4.-, (Wald et al, 1995) 3.8

111 9 Moment, knm 8 W1-4.-prop Analytial Proportional 3 Experiment 1 Rotation, mrad HE 16 B M P - 3 x 3 x 33 x 5 55 x 55 x 55 Moment, knm W1-4.-prop7 Normal Fore, kn 5 1 Figure Experimental and predited moment - rotational diagrams for shown normal fore history, experiment W1-4.-prop7, (Wald et al, 1995) Moment, knm W W Analytial Non Proportional 1 Experiment 5 Rotation, mrad HE 16 B M P - 3 x 3 x 33 x 5 55 x 55 x 55 Moment, knm W Normal Fore, kn Figure Experimental and predited moment - rotational diagrams for shown normal fore history, experiment W , (Wald et al, 1995) 8 Moment, knm 7 W1-4.-prop Analytial Proportional Experiment Rotation, mrad HE 16 B M P - 3 x 3 x 33 x 5 55 x 55 x 55 Moment, knm 8 W1-4.-prop Normal Fore, kn Figure Experimental and predited moment - rotational diagrams for shown normal fore history, experiment W1-4.-prop1, (Wald et al, 1995) 3.9

112 Moment, knm Moment, knm 7 PC S Analytial Non Proportional Experiment 1 Rotation, mrad HE B M P - 8 x 8 3 x 5 x x 6 x 6 1 S-1 Normal Fore, kn Figure 3.. Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S-1, (Vandegans, 1997) Moment, knm Analytial Non Proportional Experiment S-4 Rotation, mrad Moment, knm 1 9 S HE B M P - 8 x x 5 x 5 1 x 6 x Normal Fore, kn Figure 3..1 Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S-4, (Vandegans, 1997) Moment, knm S-8 6 Analytial Non Proportional 4 Experiment Rotation, mrad Moment, knm 1 1 S-8 HE B - 9 M P - 8 x 8 3 x 5 x 5 1 x 6 x Normal Fore, kn Figure 3.. Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S-8, (Vandegans, 1997) 3.3

113 Moment, knm Moment, knm Analytial Non Proportional Experiment S-15 Rotation, mrad HE B M P - 8 x x 5 x 5 1 x 6 x S-15 Normal Fore, kn Figure 3..3 Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S-15, (Vandegans, 1997) Moment, knm Moment, knm S-19 8 Analytial Non Proportional 6 4 Experiment Rotation, mrad S-19 1 HE B - 9 M P - 8 x 8 3 x 5 x 5 1 x 6 x Normal Fore, kn Figure 3..4 Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S-19, (Vandegans, 1997) Moment, knm Moment, knm S Analytial Non Proportional 15 1 Experiment 5 Rotation, mrad HE 14 B - 9 M P - x 3 x 5 x 5 1 x 6 x S Normal Fore, kn Figure 3..5 Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S14-1, (Vandegans, 1997) 3.31

114 Moment, knm Analytial Non Proportional Experiment S14-4 Rotation, mrad HE 14 B - 9 M P - x x 5 x 5 1 x 6 x Moment, knm S14-4 Normal Fore, kn Figure 3..6 Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S14-4, (Vandegans, 1997) Moment, knm Analytial Non Proportional Experiment S14-75(1) Rotation, mrad HE 14 B - 9 M P - x x 5 x 5 1 x 6 x Moment, knm S14-75(1) Normal Fore, kn Figure 3..7 Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S14-75(1), (Vandegans, 1997) Moment, knm S14-75() 3 Analytial Non Proportional Experiment 1 Rotation, mrad Moment, knm 6 5 HE 14 B M P - x 3 3 x 5 x 5 1 x 6 x 6 1 S14-75() Normal Fore, kn Figure 3..8 Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S14-75(), (Vandegans, 1997) 3.3

115 Moment, knm Analytial Non Proportional Experiment S14-1 Rotation, mrad HE 14 B M P - x 5 3 x 5 x 5 1 x 6 x Moment, knm S14-1 Normal Fore, kn Figure 3..9 Experimental and predited moment - rotational diagrams for shown normal fore history, experiment S14-1, (Vandegans, 1997) 3.33

116 3.34

117 Chapter 4: Modelling and idealisation for frame analysis 4.1 Introdution Analysis of steel frames is ommonly arried out assuming the olumn bases to be either rigid or entirely free to rotate (pinned). However, introduing a rotational stiffness in the analysis model at the olumn bases produes a more realisti predition of the frame behaviour. Compared to analysis with pinned bases, lateral (sway) defletions are redued. Compared to analysis with fixed bases, lateral (sway) defletions are inreased, whih is partiularly important in the design of unbraed frames. Some national design standards give reommendations for the modelling of olumn bases whih attempt to reflet the real olumn base behaviour rather that the extremes of fixed or pinned bases. For example, these reommendation relate the olumn base stiffness to the stiffness of the supported olumn, and do not neessarily reflet the atual olumn base details. The proposal ontained in this publiation allow to alulate the rotational stiffness and the moment resistane of olumn bases. This hapter briefly desribes the proedure for inorporating these alulated harateristis in the frame analysis. 4. Modelling Careful examination of both the test results and the proposed models demonstrates that the stiffness and the resistane of olumn bases are influened by the relationship between the applied moment and the applied normal fore, see Figure 4.1. M low normal fores high normal fores Figure 4.1 Influene of the normal fore on the moment-rotation behaviour Hene, the determination of the joint properties needs some onsideration of the global frame response. On the other side the frame response depends on the joint behaviour if the joint may not be assumed be rigid. This leads to an iterative proedure. This setion is aimed in providing some guidane how to model a olumn base joint in the frame in order to take into aount this interation in a pratial way. In general, the joint behaviour needs to be onsidered in the frame analysis. However, in ertain ases, it is 4.1

118 possible to neglet the joint behaviour by making simplified assumption, i.e. to assume that the joints may be modelled either as ontinuous or as pinned. This approah is used in daily design in most ases. The tool to justify this approah is the lassifiation system whih is disussed more in detail in hapter 5. With the help of the lassifiation system, it is possible to lassify a joint as rigid (in this ase, the deformation of the joint are small ompared to the deformation of the frame and hene, the rotation of the joint may be negleted) or as nominally pinned (in this ase, the joint will not transmit signifiant moments and it is able to rotate like an ideal hinge). Dependent on the lassifiation the olumn bases may be modelled as shown in Figure 4.. Method of global analysis Classifiation of the olumn base elasti nominally pinned rigid semi-rigid rigid-plasti nominally pinned full-strength partial-strength elasti-plasti nominally pinned rigid and full-strength semi-rigid partial strength semi-rigid full-strength rigid partial strength Type of joint model simple ontinuous semi-ontinuous Figure 4. Joint modelling dependent on the lassifiation and on the method of global analysis As above-mentioned atual behaviour of a olumn base joint depends on its atual loading. This may be expressed by means of the atual M Sd /N Sd ratio, where M Sd is the applied moment and N Sd is the applied normal fore. This aspet is disussed more in detail in hapter 3.1. In order to determine the olumn base properties, a good guess of the ating fores (i.e. the ratio M Sd/NSd) is required. The following iterative proedure is reommended to take the interation between frame behaviour on one side and the olumn base behaviour on the other side into aount. This general proedure is illustrated in Figure 4.3 If a joint is modelled either as ontinuous, it is obvious that there is no interation between the joint behaviour and the frame response. Therefore it is reommended to start the proedure with the assumption of rigid and fullstrength joints whih are modelled in the frame as ontinuous, see Figure 4., as a first guess. (Note that the design model presented in the previous hapters assumes that the olumn bases may transfer typially some moments, this is for example due the presene of normal fores. Therefore the simple modelling is of minor interest in this ase and it is reommended to use the ontinuous modelling as a starting point.) With the assumption of rigid olumn bases, the frame analysis an be performed without any knowledge of the olumn base properties. In pratie, this step is made anyway in the pre-design stage in order to get an idea about 4.

119 the setion sizes. As a result, the fores ating on the olumn bases are known. Based in the M/N ratio obtained in this analysis the resistane and the stiffness of the joints may be determined aording to the models desribed in hapters and 3. START Frame modelling with ontinuous olumn bases Global frame analysis Frame modelling with semi-ontinous olumn base Determination of olumn base properties not O.K. hek assumptions for stiffness and resistnae O.K. STOP Figure 4.3 Iterative proedure Of ourse one have to hek now if the assumptions made in the beginning are valid. This is easy beause a lassifiation of the joint will give the answer: If the atual stiffness is higher than a ertain lassifiation boundary, the assumption of rigid olumn was valid. If this is not the ase, a further frame analysis is neessary. However, the atual joint stiffness should now be taken into aount and the olumn bases are modelled as semiontinuous. Beause the joint behaviour influenes the frame response, the resulting data for M and N ating on the joints will differ from the previous alulation and new values for the stiffness and resistane should be derived. The iteration an be ontinued by repeating the two previous steps. The proedure an be esaped if the differene between the olumn base stiffness of the last step and the previous step is less than an An approah to find suh an aeptable value is proposed by Steenhuis et al (1994). 4.3 Idealisation Generally the olumn base behaviour may be represented in the frame by using a non-linear rotational spring. However, the spring should be linear for a simple elasti global analysis. As the general behaviour of a olumn base joint is non-linear a so-alled idealisation of the moment-rotation urve is neessary to derive a linearised urve as shown in figure 4.4. Here, a linear stiffness Sj (seant stiffness) at the level of the applied moment is taken as a lower bound value. If the applied moment is not know, the seant stiffness should be determines at the rotation when the moment -rotation urve reahes the plasti moment resistane (dashed line). 4.3

120 M M j,rd M j,sd S j φ lim for S j φ Figure 4.4 Linearised moment-rotation urve Based on the requirement, that a simplified urve should lead to safe results on one side but should give on the other side most eonomi solutions, the revised Annex J of Euroode 3 uses an empirial approah to determine a more eonomial value model for an idealised stiffness of beam-to-olumn joints. Parameter studies were arried out in order to ompare the frame behaviour when on one side the more aurate model (i.e. the nonlinear moment rotation urve) is used as a referene and one the other side the proposed simplified urve is used to represent the joint behaviour. However suh studies are not yet available for olumn bases and the only safe approah is to take a seant stiffness to the non-linear urve 4.4

121 Chapter 5: Classifiation of olumn base joints 5.1 Sope In Euroode 3 Chapter 6 and revised Annex J, the stiffness lassifiation of beam-to-olumn joints into rigid, semi-rigid and pinned joints is seen to be dependent on the strutural system, aording its braed or unbraed harater. In fat, this riterion should not be the single one to determine the lass of the joints; it is obvious that the sway or non sway harater of the struture is a signifiant parameter as well. So in order to larify the point, without entering in long explanations, it an be stated that the reommendations provided in Euroode 3 for the stiffness lassifiation of beam-to-olumn joints are limited to joints belonging to braed - non sway and unbraed - sway strutures; the other ases, respetively braed - sway and unbraed - non sway being not presently overed. Similar onlusions apply to the proposals made in the present hapter for the stiffness lassifiation of olumn bases. For what regards the strength lassifiation, the same boundaries than those proposed in Euroode 3 revised Annex J for beam-to-olumn joints may be referred to for olumn bases so as to distinguish between full-strength, partial strength and pinned joints. 5. Column bases in braed - non-sway frames A modifiation of the atual moment-rotation harateristi of olumn bases is likely to affet the whole response of braed - non-sway frames, and in partiular the lateral displaements of the beams and the bukling resistane of the olumn. This seond aspet - the bukling resistane of the olumns - is the one for whih the influene is rather important, as seen in Figure 5.1 (Wald and Seifert, 1991), whih shows how the bukling length oeffiient of a olumn pinned at the upper extremity is affeted by the variation of the olumn base rotational stiffness. The bukling length oeffiient K is reported on the vertial axis and is expressed as the ratio between the elasti ritial load (Fr,pin ) of the olumn pinned at both extremities and that of the same olumn but restrained by the olumn base at the lower extremity (F r,res ); it is shown to vary from 1, (pinned - pinned support onditions) to,7 (pinned - fixed support onditions). K = F F r, pin r, res (5.1) 5.1

122 F r pin E, = π L I (5.a) F r res E I ( K L ), = π (5.b) Κ = 1,9,8 F r.pin F r,res S j,ini,pin S j,ini,stif t = 1 mm a1 = b1 = 8 mm a = b = 5 mm h = 1 mm M 4-4 S j,ini,pin = 7 1 knm / rad t = 4 mm a 1 = b1 = 4 mm a = b = 5 mm h = 1 mm M 4-4 S j,ini,stif = 74 8 knm / rad,7,6,1,1 1, 1, log S _ Figure 5.1 Elasti ritial bukling load versus olumn base initial stiffness where E is the modulus of elastiity of steel; L and I are respetively the system length and the moment of inertia of the olumn. In Figure 5.1, the non-dimensional stiffness S of the olumn base is reported in a logarithmi sale on the horizontal axis. S = S j. ini E I L (5.3) S j,ini is the initial elasti stiffness in rotation of the olumn base. The numerial values indiated in Figure 5.1 have been obtained by onsidering the partiular ase of a 4 m length olumn with a HE B ross-setion. The atual initial stiffness of two typial olumn bases is reported in Figure 5.1: olumn base with a base plate and two anhor bolts inside the H ross-setion; this onfiguration is traditionally onsidered as pinned, but possesses an initial stiffness S j,ini,pin equal to 7 1 knm / rad; olumn base with a base plate and four bolts outside the olumn ross-setion; suh a olumn base is usually onsidered as rigid even if its initial stiffness S j,ini,stif = 74 8 knm / rad is not infinite. 5.

123 As for beam-to-olumn joints, it may be onluded that stiff olumn bases always deforms slightly in rotation while presumably pinned ones exhibit a non-zero rotational stiffness. Some olumn bases are however so flexible or so rigid that the strutural frame response obtained by onsidering the atual olumn base harateristis in rotation is not signifiantly different from that obtained by modelling respetively the olumn bases as perfetly pinned or rigid. For beam-to-olumn joints, this has led to the onept of stiffness lassifiation into pinned, semi-rigid and rigid joints, see Euroode 3. The stiffness lassifiation in Euroode 3 revised Annex J is ahieved by omparing the initial stiffness of the beam-to-olumn joints to boundary values. For instane, rigid beam-to-olumn joints are haraterised by a stiffness higher than 8 E I b / L b where I b and L b are respetively the moment of inertia and the length of the beam. This rigidity hek is based on a so-alled "5% riterion" whih says that a joint may be onsidered as rigid if the ultimate resistane of the frame in whih it is inorporated is not affeted by more than 5% in omparison with the situation where fully rigid joints are onsidered. A rigid lassifiation boundary for olumn bases may be derived by adopting the same basi priniple. To ahieve it, the single storey - single bay braed - non-sway frame shown in Figure 5..a is onsidered. The study of the sensitivity of the frame to a variation of the olumn base stiffness properties is influened by the beam and olumn harateristis in bending, E Ib / Lb and E I / L respetively. Two limit ases are however obtained when: the beam is rather stiff as shown in Figure 5..b (E I b / L b = ); the beam is rather flexible (or when a pinned joint onnet the beam to the olumn); this situation is illustrated in Figure 5.. (E I b / L b = ). N N, L b E I b, L E I, L EI, L EI a) Portal frame (non-sway) b) Isolated olumn (top-fixed) ) Isolated olumn (top-pinned) Figure 5. Portal frame and isolated olumns for lassifiation study The appliation of the "5% riterion" to the first limit ase (olumn fixed at top extremity) writes as follows: 5.3

124 π π EI ( KL ) EI ( 5, L ) 95, (5.4) For sake of simpliity, it is first applied to the ritial elasti bukling loads and not to the ultimate ones (integrating the effets of plastiity, imperfetions,...); it may be demonstrated that by doing so a safe value of the rigid stiffness boundary is obtained. From Equation (5.4), the minimum value of the bukling length oeffiient may be derived: K, 513 (5.5) Aording to Euroode 3 Annex E on the «effetive bukling length of members in ompression», the K oeffiient is expressed as a funtion of restraint oeffiients (k l, k u ) at both olumn ends and writes in this speifi ase: K 1, 145 ( k k ), 65 k k = l u l u, 364 ( k k ), 47 k k l u l u (5.6) with: k l 4EI / L = at lower extremity (5.7.a) 4EI / L + S j, ini k u = at upper extremity (5.7.b) From Equations (5.5) to (5.7), the minimum value of the elasti initial stiffness Sj,ini that presumably rigid olumn bases have to exhibit may be derived as follows (rounded value) : S / j, ini 48EI L (5.8) A similar approah may be followed for the seond limit ase (Figure 5..) and the following boundary is extrated: S / j, ini 4EI L (5.9) This limit is less restritive than the first one; this shows that: the stiffness requirement is system dependent; the stronger requirement on joint stiffness is obtained in struture where the flexibility at the extremities of the olumn is stritly resulting from that of the olumn base. 5.4

125 This is physially understandable and, as a onsequene, the following simple stiffness lassifiation boundary to distinguish between rigid and semi-rigid olumn bases is suggested: Rigid olumn bases: Semi-rigid olumn bases: S / j, ini 48EI L (5.1.a) S / j, ini < 48EI L (5.1.b) As a further step, a less onservative boundary may be derived by applying the "5 % riterion" to the ultimate resistane F u of the olumn. This one may be approximately expressed as (Massonnet and Save, 1976) : 1 N u 1 1 = + (5.11) N N p r where N p and N r designate respetively the squash load and the ritial elasti bukling load of the olumn. As N r = N p / λ, λ being the redued slenderness of the olumn, Equation (5.11) writes also: 1 N u = N p (5.1) 1+ λ With referene to Figure 5..b, the redued slenderness of the olumn with a fully stiff olumn base at lower end equals : λ rigidolumn base =, 5λ o (5.13.a) while that of the olumn with a semi-rigid olumn base writes : λ semi -rigid olumn base = Kλ o (5.13.b) where = 1). λ o is the redued slenderness of the olumn assumed as pinned at both extremities (K The appliation of the "5 % riterion" to the ultimate olumn resistane therefore gives, N p being onsidered as onstant : 1+ (,5) λ 1+ K λ o o,95 (5.14.a) 5.5

126 or : 1 K, (5.14.b) 5λ o Expression (5.14.b) may be ompared to the (5.5) one. For high λ o values, both expressions are similar. In suh ases, the ultimate resistane N u equals N r and a high boundary value of Sj,ini (see Formula 5.8) is required. For low values of λ o, the ondition (5.14.b) relaxes and, as a onsequene, less severe boundary values of S j,ini are required, the influene of the ross-setion yielding beoming then more predominant than the instability. For λ o =,48, Equation (5.14.b) writes K,7, what means that any olumn base, even a perfetly pinned none, will be onsidered as rigid. By integrating Formulae (5.14.b) in Formulae (5.6) to (5.7.b), the following stiffness boundaries are obtained : If λ o,48 S j, ini (5.15.a) If λ o >,48 S j,ini 1,145,838µ 4EI 1,6µ 1 L (5.15.b) with : 1 µ = 1+ (5.16) 5λ o For pratial appliations, simpler expressions are proposed whih fit rather well, as seen in Figure 5.3, with the exat ones (Formulae 5.15) in the usual range of appliation ( λ o to 3). These are: If λ o,5 Sj,ini (5.17.a) If,5 < λ o < 3,93 S j,ini 7 ( λ o - 1)EI /L (5.17.b) If λ o 3,93 S j,ini 48 EI /L (5.17.) As a safe approximation, Formula (5.8) may obviously be applied for any olumn slenderness. 5.6

127 S / (E I / L ) j,ini Simple stiffness boundary Exat stiffness boundary Redued slenderness (pinned olumn) Figure 5.3 Exat and simple stiffness boundaries Suh an approah allows to lassify the olumn bases aording to the olumn properties only. A more preise boundary dependent on the k u oeffiient (Formulae 5.6) ould obviously be derived but its appliation would be far more ompliated, and this seems not to be in line with the expeted simpliity. A similar stiffness boundary ould be defined, on the same basis, for pinned joints. The value obtained is however so low that few atual olumn bases are likely to exhibit suh a lower initial stiffness. On the other hand, in the present ase, the boundary is an upper value, and even if the atual joint stiffness is higher, nothing may prevent the designer to onsider still the joint as pinned, as it is presently done in design. As a onsequene, no pinned lassifiation boundary is derived and proposed here. 5.3 Column bases in unbraed sway frames The unbraed sway frames are more sensitive than braed non-sway ones to the variation of the rotational properties of olumn bases, mainly beause of their high sensitivity to lateral defletions as well as to hanges of the overall stability onditions when the lateral flexibility inreases. To illustrate this statement, a single-bay single-storey unbraed sway frame is onsidered in Figure 5.4. The diagram indiates the evolution with inreasing values of S of the ratio β s = y S / y P between the lateral defletion y S of the frame with atual olumn base stiffness and the defletion yp of the frame with assumed ideally pinned olumn bases. The nondimensional stiffness S defined by Equation (5.3) is again reported in a logarithmi sale. First order elasti theory is used to ompute the y values. 5.7

128 y / y S P 1,,8 5 kn HE B 115 kn HE B 115 kn y 4 m,6,4, S j,ini,pin S j,ini,stif 5 m,1,1 1 1 log S Figure 5.4 Sensitivity of the sway defletion to a variation of the olumn base stiffness in a portal frame A stiffness lassifiation boundary similar to that expressed in the ase of braed nonsway frames may again be derived here on the basis of a "5% resistane riterion". For unbraed sway frames also it may be demonstrated that the more restritive situation orresponds to the limit ase where the beam flexural stiffness is rather high in omparison with that of the olumns. The derivation of the lassifiation boundary is therefore arried out by referring to the isolated olumn represented in Figure 5.5.b. F N FN, L b E I b, L EI, L EI, L EI a) Portal frame (non-sway) b) Isolated olumn (top-fixed) ) Isolated olumn (top-pinned) Figure 5.5 Unbraed sway portal frame and isolated olumns for lassifiation study Aording to the well-known Merhant-Rankine formula, the elasto-plasti seond order resistane load fator λ u of the whole frame may be expressed as: 1 λ u 1.9 = + (5.18) λ λ r p where λr and λp designate respetively the elasti ritial load fator (resulting from an elasti linear instability analysis) and the plasti load fator (resulting from a rigid plasti first order analysis). 5.8

129 The range of appliation of the formula is defined as follows: λr 4 1 λ p (5.19) From Equations (5.18) and (5.19), it may be easily demonstrated by keeping λ p onstant that the appliation of the "5% resistane riterion" to the load-fator λu results in a possible variation of % of the value of λ r. As a onsequene, the following riteria may be written: π EI ) EI ) ( KL π ( L As a result:,8 (5.) K 1,118 (5.1) For unbraed sway frames, the K - k relationship given by Equation (5.6) has to be replaed by the following one: K = + 1, ( k k ), 4 k k l u l u + + 1, 8 ( k k ), 6 k k l u l u (5.) while Equations (5.7.a) and (5.7.b) remain unhanged. The ombination of these equations leads to the following expression of the stiffness lassifiation boundary: S / j, ini 9EI L (5.3) However the "5% resistane riterion" fully disregards the aspets of lateral frame defletions whih have been pointed out as important. It may be shown (Wald and Sokol, 1997) that the lateral defletion ys of the portal frame illustrated in Figure 5.5.a writes: y S = F L 3 E I ( 3 + S ) + 6 ( 4 + S ) S ( 1 S ) ς ς (5.4) where S = S E I j, ini / L (5.5) 5.9

130 ς = E I E I b / / L L b (5.6) For S, the defletion for the frame with rigid olumn bases may be derived from (5.4): y R = F L 3 E I ς ς, (5.7) In omparison with the ase where rigid olumn bases are used (Formula 5.7), the atual frame - where the olumn bases possesses some degree of flexibility - will experiene a larger defletion (Formula 5.4); this inrease of the lateral displaement may be expressed in terms of perentage w as follows: y y S R = 1 + ω (5.8) As far as lassifiation is onerned, an "ω % resistane riterion" may be suggested with the objetive to limit the inrease of the lateral displaement of the atual frame to ω % of the defletion evaluated in the ase of rigid olumn bases. In Formula (5.8), this means that the sign "=" should be replaed by " ". By ombining expressions (5.4), (5.7) and (5.8), the value of the minimum rotational stiffness that the olumn bases should exhibit to be onsidered as rigid from a displaement point of view is derived: S + ς ς ( + ω) ς ( 4 6 ) ω ς ς (5.9) This ondition is illustrated in Figure 5.6 (Wald and Sokol, 1997). The required stiffness is seen to be rather insensitive to the values of ς for signifiant values of w. Conservatively the values obtained for ς = may be seleted, i.e.: for ω = %, the following stiffness boundary is obtained S for ω = 1% S 3 for ω = 5% S

131 y ς = E E I b I / L b / L F N E I b N,4,35,3,5,,15,1 ω,3,5,1,15,,5 S E I E I S L b L, S Figure 5.6 Displaement lassifiation riteria for olumn bases As a onsequene, the displaement lassifiation riterion is seen to be muh more restritive than the resistane one given by Equation (5.3). The seletion of the value for the boundary is obviously strongly related to the level of auray whih is thought to be neessary for the evaluation of the lateral frame defletion. A value of 1% appears to be quite realisti and the following stiffness lassifiation boundary for presumably rigid olumn bases may be therefore proposed: Rigid olumn bases: S 3 E I L (5.3.a) jini, / Semi-rigid olumn bases: S < 3 E I L (5.3.b) j, ini / For similar reasons than those given for braed non-sway frames no lassifiation boundary for presumably pinned olumn bases is suggested. 5.4 Proposed lassifiation The proposed stiffness lassifiation for olumn bases is based on the sensitivity of the frame response to the flexural harateristi of the olumn. The lassifiation boundaries may be easily evaluated for two strutural systems, respetively the braed non-sway and unbraed sway ones. No reommendation is available for the other ases. The stiffness boundaries are shown in Figure 5.7. They are as follows: 5.11

132 braed non-sway frames Rigid olumn bases: If λ o,5 S j,ini (5.31.a) If,5 < λ o < 3,93 S j,ini 7 ( λ o - 1)EI /L (5.31.b) If λ o 3,93 Sj,ini 48 EI/L (5.31.) Semi-rigid olumn bases: If λ o,5 all joints rigid (5.31.d) If,5 < λ o < 3,93 S j,ini < 7 ( λ o - 1)EI /L (5.31.e) If λ o 3,93 S j,ini < 48 EI /L (5.31.f) unbraed sway frames Rigid olumn bases: S 3 E I L (5.3.a) j, ini / Semi-rigid olumn bases: S < 3 E I L (5.3.b) j, ini / In Figure 5.7, the value of the stiffness boundary for the braed - non sway frames is shown in a partiular ase where the redued slenderness of the olumns is equal to λ o = 1. 36, what leads to a stiffness boundary value of 1 EI /L. Relative moment 1,,8,6,4, rigid olumn-base onnetions S j.ini..n = 3 E I / L S j.ini..s = 1 E I / L semi-rigid olumn-base onnetions the area of hinge behaviour,1,,3,4 λ o = 1,36 Rotation,,5 φ Figure 5.7 Proposed lassifiation system aording to the initial stiffness 5.1

133 Chapter 6: Referenes Alma J.G.J., Bijlaard, F.S.K. (198): Berekening van kolomvoetplaten, Rapport No. BI-8-46/ , IBBC-TNO, Delft. Bouwman, L.P., Gresnigt A.M., Romeijn, A. (1989): Onderzoek naar de bevestiging van stalen voetplaten aan funderingen van beton. (Researh into the onnetion of steel base plates to onrete foundations), Stevin Laboratory report /6, Delft. Bijlaard F.S.K. (198): Rekenregels voor het ontwerpen van kolomvoetplaten en experimentele verifiatie, Rapport No. BI-81-51/ , IBBC-TNO, Delft. CEB (1994): Fastenings to onrete and masonry strutures, State of the art report. Comité Euro-International du Béton. Thomas Telford Publishing, London, ISBN CEB (1996): Design of fastenings in onrete, Design Guide. Comité Euro-International du Béton. Thomas Telford Publishing, London, ISBN: DeWolf J.T. (1978): Axially Loaded Column Base Plates, Journal of the Strutural Division ASCE, Vol. 14, No. ST4, pp DeWolf J.T., Sarisley E. F. (198): Column Base Plates with Axial Loads and Moments, Journal of the Strutural Division ASCE, Vol. 16, No. ST11, pp ECCS (1994): European reommendations for bolted onnetions with injetion bolts. ECCS publiation No. 79, Brussels. Eligehausen R. (199): Design of fastenings in onrete using partial safety fators, in German, Bauingenieur, No. 65, pp Eligehausen R. (1991): Fastenings to Reinfored Conrete and Masonry Strutures, State-ofart Report, Part I, Part II, CEB bulletin N 6, EPF Lausanne. Euroode 3 (199): ENV , Part 1.1, Design of Steel Strutures, European Prenorm, CEN, Brussels. Euroode 3, Revised Annex J (1998): Joints in Building frames, CEN, Doument ENV :199/A, Brussels. Guisse S., Vandegans D., Jaspart J.P. (1996): Appliation of the omponent method to olumn bases - Experimentation and development of a mehanial model for haraterization. Report n MT195, Researh Centre of the Belgian Metalworking Industry, Liège, Belgium. Hawkins N.M. (1968): The bearing strength of onrete loaded through rigid plates, Magazine of Conrete Researh, Vol., No. 63, Marh, pp

134 Hawkins N.M. (1968): The bearing strength of onrete loaded through flexible plates, Magazine of Conrete Researh, Vol., No. 63, June, pp Jaspart J.P. (1991): Etude de la semi-rigidité des noeuds poutre-olonne et son influene sur la résistane et la stabilité des ossatures en aier, Ph.D. Thesis, Department MSM, University of Liège. Lambe T.W., Whitman R.V. (1969): Soil Mehanis, MIT, John Wiley & Sons, In., New York. Massonnet Ch., Save M., (1976), Calul plastique des onstrutions, Volume 1: Strutures dépendant d un paramètre, Third edition, Nelissen, Liège, Belgium. Murray T.M. (1983): Design of Lightly Loaded Steel Column Base Plates, Engineering Journal AISC, Vol., pp NEN 677 (199): Staalonstruties TGB 199, Basiseisen en basisrekenregels voor overwegend statish belaste onstruties (TGB 199 Steel Strutures, Basis requirements and basi rules for alulation of predominantly statially loaded strutures), Delft. Niyogi S.K. (1973): Effet of Side Slopes on Conrete Bearing Strength, Journal of the Strutural Division, ASCE, Vol. 14, No. ST3, pp Salmon C.G., Shenker L., Johnston G. J. (1957): Moment-Rotation Charateristis of Column Anhorages, Transation ASCE, Vol. 1, No. 5, pp SBR (1973): Mortelvoegen in montagebouw, Rapport no. 34, Stihting Bouwreseah, Samson Alphen aan de Rijn. Shelson W. (1957): Bearing Capaity of Conrete, Journal of the Amerian Conrete Institute, Vol. 9, No. 5, Nov., pp Sokol Z., Wald F. (1997): Experiments with T-stubs in Tension and Compression, Researh Report G1496, CVUT, Praha. Steenhuis C.M. (1998): Assembly proedure for base plates, report 98-CON-R447, TNO Building and Constrution Researh, Delft, the Netherlands. Steenhuis C.M., Gresnigt A.M., Weynand K. (1994): Pre-Design of semi-rifig joint in steel frames, in COST C1, Proeeding of the nd state of the art workshop, Prague, pp Stokwell F.W. Jr. (1975): Preliminary Base Plate Seletion, Engineering Journal AISC, Vol. 1, No. 3, pp Vandegans D. (1997): Column Bases: Experimentation and Appliation of Analytial Models, Researh Centre of the Belgian Metalworking Industry, MT 196, p.8, Brussels, Belgium. 6.

135 Wald F. (1993): Column-Base Connetions, A Comprehensive State of the Art Review, CVUT, Praha. Wald F. (1995): Patky Sloupù - Column Bases, CVUT Praha, 1995, p. 137, ISBN Wald F. (1999): Anhor bolts design, Aktion 6p1, CVUT, Praha. Wald F., Obata M., Matsuura S., Goto Y. (1993): Prying of Anhor Bolts, Nagoya University Report, pp Wald F., Obata M., Matsuura S., Goto Y. (1993): Flexible Baseplates Behaviour using FE Strip Analysis, Ata Polytehnia, CVUT Vol. 33, No. 1, Prague, pp Wald F., Seifert J. (1991): The Column-Base Stiffness Classifiation, Proeedings of the Nordi Steel Colloquium, 9-13 September, Odense, pp Wald F., Sokol Z. (1995): The Simple olumn - base stiffness modelling, in Steel Strutures - Eurosteel 95, ed. Kounadis, Balkema, Rotterdam, pp Wald F., Sokol Z. (1997): Column Base Stiffness Classifiation, Proedings of SDSS 97, Colloquium on Stability and Dutility of Steel Strutures, Nagoya, Japan, pp Wald F., Sokol Z. (1997): Model of Column Base Stiffness, in STESSA 97, in International Conferene Behaviour of Steel Strutures in Seismi Areas, Kyoto, Japan, pp , ISSN Wald F., Sokol Z., Šimek I., Barantova Z. (1995): Column Base Semi-Rigid Behaviour, Poddajnost patek oelovýh skelet, in Czeh, Report G 8118, CVUT, Praha. Wald F., Sokol Z., Steenhuis M. (1996): Proposal of the Stiffness Design Model of the Column Bases, in Connetion in Steel Strutures III, Behaviour, Strength, and Design, Proeedings of the Third International Workshop, May 9-31, 1995, Trento, Italy, Pergamon Press, London, 1996, p , ISBN Yee Y.L., Melhers R.E. (1986): Moment rotation urves for bolted onnetions, Journal of the Strutural Division, ASCE, Vol. 11, ST3, pp Zoetemeijer P. (1974): A design method for the tension side of statially loaded bolted beamto-olumn onnetions, Heron, Delft University, Vol., n 1. Zoetemeijer P. (1985): Summary of the researhes on bolted beam-to-olumn onnetions, Report , University of Tehnology, Delft, The Netherlands. 6.3