Design of Members. Rui Simões. Department of Civil Engineering University of Coimbra

Transcription

1 Design o embers Rui Simões Department o Civil Engineering Universit o Coimbra

2 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 Contents Introduction Design o columns Design o beams Design o beam columns

3 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 ITRODUCTIO ain internal orces and combinations Bending+Shear Compression+Bending+Shear Tension/Compression Torsion less common Building master example (Cardington - UK)

4 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 ITRODUCTIO ember design: i) resistance o cross sections; ii) member buckling resistance. Clause 6. o Eurocode 3, part. provides dierent approaches, depending o cross section shape, cross section class and tpe o internal orces (, +V, ++V,.): elastic criteria (clause 6..(5)); G V, RESISTACE OF CROSS SECTIOS Cross section classiication - Class ; Class ; Class 3 and Class , 0, 0, 0, x x

5 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 ITRODUCTIO linear summation o the utiliation ratios class //3 (clause 6..(7)); Rd,, Rd,, Rd nonlinear interaction ormulas class / (clause 6..(6)). Section properties gross section, net section (deduction or holes) or eective section (class 4 or shear lag eects) (clause 6.. o EC3--). EBER BUCKLIG RESISTACE Buckling resistance (clause 6.3 o Eurocode 3, part.) must be checked in all members submitted to compressive stresses, which are: members under axial compression ; members under bending moment ; or under a combination o both (+).

6 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS Column cross sections and applications Rolled open or closed sections, welded sections or built-up sections The objective is to maximie the second moment o area in the relevant buckling plan in order to maximie the buckling resistance.

7 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS Compression resistance (clause 6..4 o EC3--) c, Rd.0 is the design value o the axial compression; c,rd is the design resistance to axial compression, given b the minimum o: A i) Plastic resistance A c, Rd 0 (class, or 3) A c, Rd e 0 (class 4) A e - eective area ii) Buckling resistance b,rd, in general the lexural buckling resistance, which is analsed hereater.

8 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS Column Buckling Flexural buckling is in general the buckling mode, which govern the design o a member in pure compression. For this mode in a pinned column, the elastic critical load cr, deined as the maximum load supported b the column, ree rom an tpe o imperections, is given b the well known Euler s ormula: cr E I d dx 0 L x () (x) 0 Buckling in a bending mode (x) cr L E I E I Bending stiness L Buckling length (L E or other support conditions) In speciic cases (e.g. members with cruciorm cross sections) buckling ma occur in other modes: torsional buckling or lexural-torsional buckling.

9 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS Column Buckling cr L E I E cr A L E I E Critical stress E L E i Slenderness i I A Radius o gration cr E E A cr A Euler s curve.0 on-dimensional slenderness Euler s curve E E L E i.0 Imperections or real columns (geometrical imperections and material imperections).

10 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS Buckling Resistance (clause 6.3. o EC3--) A b. Rd A b. Rd e (Class, or 3) (Class 4) Theoretical behaviour is the reduction actor or the relevant buckling mode but eglect BUCKLIG i: 0. or cr 0. 04

11 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS Buckling Resistance (clause 6.3. o EC3--) Flexural buckling A cr L cr i (Class, or 3) A e cr L i cr A e A (Class 4) E ε 35 Torsional or lexural-torsional buckling T A cr (Class, or 3) T A e cr (Class 4) - buckling in lexural buckling mode about axis

12 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS EXAPLE Saet veriication o a column member o the building represented in the igure. 4 A C E A B C D E F 6.00 m b a m.50 m.00 m 6.00 m 4.00 m 4.50 m 4.50 m 4.00 m Building master example i) The inner column E-3 represented in the igure, at base level, is selected. This member has a length o m and is composed b a section HEB 340 in steel S 355. In this column the bending moments (and the shear orce) ma be neglected; the design axial orce (compression) obtained rom the previous analsis is given b = k.

13 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS EXAPLE ii) Cross section classiication section HEB 340 in pure compression. Geometric characteristics: A = 70.9 cm, b = 300 mm, h = 340 mm, t =.5 mm, t w = mm, r = 7 mm, I = cm 4, i = 4.65 cm, I = 9690 cm 4, i =7.53cm. echanical properties o the steel: = 355 Pa and E = 0 GPa. c Web in compression (Table 5. o EC3--) c ( ) t (class ) Flange in compression (Table 5. o EC3--) c t (class ) HEB 340 cross section, steel S 355, in pure compression is class.

14 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS EXAPLE iii) Cross section veriication - class in pure compression k c A , Rd k..0 0 iv) Buckling resistance. Buckling lengths Assuming that the design orces were obtained b a second order structural analsis, the buckling lengths are considered (conservativel) equal to the real lengths (mid-distance between loors), given b: Buckling in the plan x- (around ) - L E m Buckling in the plan x- (around ) - L E m Determination o the slenderness coeicients L E 9.59 i LE i

15 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS EXAPLE Calculation o the reduction actor min h b and t.5mm lexural buckling around curve b( 0.34) lexural buckling around curve c ( 0.49). 00mm HEB 340 As and curve c curve b min

16 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF COLUS EXAPLE min 0.69 v) Saet veriication b, Rd A k λ 0.75 As, 336.0k b, Rd 486.k saet is veriied with the cross section HEB 340 in S 355 steel.

17 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Beam cross sections and applications A beam ma be deined as a member subjected essentiall to bending and shear orce. Castellated beams Hot-rolled sections (IPE, HEA or HEB, RHS,.) Welded sections Welded sections in non-uniorm beams

18 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Cross section resistance Uniaxial bending (clause 6..5 o EC3--) c. Rd.0 Class or Class 3 Class 4 W c. Rd pl 0 W c. Rd el.min 0 W c. Rd e.min 0 Bi-axial bending (clause 6..9 o EC3..), pl,. Rd, pl,. Rd.0 I or H CHS RHS ; 5 n but.66.3 n but 6 pl n, Rd

19 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Cross section resistance Shear (clause 6..6 o EC3--) V V c, Rd.0 PLASTIC RESISTACE V pl.rd ELASTIC RESISTACE V A 0 pl. Rd v 3 V V I t S A v Shear area (obtained rom clause 6..6 (3) o EC3-- or rom tables o proiles). 3 G A v V G e. n. a. Shear stresses - 3 Shear buckling or webs without stieners should be veriied in accordance with EC3--5, i: h t w w 7 35 / h w and t w are the height and thickness o the web and is in accordance with EC3--5.

20 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS,V.Rd Cross section resistance Bending and Shear Interaction (clause 6..8 o EC3--) V V 50% V pl, Rd O REDUCTIO h m r r + V 50% V pl, Rd REDUCED OET t w ( ) (V ) r V V pl. Rd For I and H cross sections o equal langes, with bending about the major axis, the bending moment resistance,v,rd is given b (clause 6..8 o EC3--): A w, V. Rd Wpl,, c, Rd 4 t w 0 A W h w t w

21 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Lateral-Torsional Buckling Instabilit phenomenon characteried b the occurrence o large transversal displacements and rotation about the member axis, under bending moment about the major axis ( axis). This instabilit phenomenon involves lateral bending (about axis) and torsion o cross section.

22 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Lateral-Torsional Buckling In the stud o lateral-torsional buckling o beams, the Elastic Critical oment cr plas a undamental role; this quantit is deined as the maximum value o bending moment supported b a beam, ree rom an tpe o imperections. For a simple supported beam with a double smmetric section, with supports prevent lateral displacements and rotation around member axis (twist rotations), but allowing warping and rotations around cross section axis ( and ), submitted to a uniorm bending moment ( standard case ), the elastic critical moment is given b: A C L a) Elevation x B x E cr L G I T E I E I L G I Which depend mainl o: Loading and support conditions; Length between lateral braced sections (L); Lateral bending stiness (E I ); Torsional stiness (G I T ); Warping stiness (E I w ). W T

23 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS CG G C j g j g T W w cr C C C C E I G I L k I I k k L k E I C s a g A s j I da 0.5 Elastic critical moment cr P C cr, < cr P C P cr, > cr C - applicable to member with smmetric and mono-smmetric cross sections, - include the eects o the loading applied below or above the shear centre; - several degrees o restriction to lateral bending (k ) and warping (k w ); - several shapes o bending moment diagram (C, C and C 3 in the next tables). Lateral-Torsional Buckling

24 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Lateral-Torsional Buckling Elastic critical moment - Publication nº 9 do ECCS (Boissonnade et al. 006). - Beam sotware

25 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Lateral-Torsional Buckling Lateral-torsional buckling resistance (clause 6.3. o EC3--) b. Rd.0 W b. Rd W = W pl. Class and ; W = W el. Class 3; W = W e. Class 4. is the reduction actor or lateral-torsional buckling, which can be calculated b one o two methods, depending o member cross section.

26 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Lateral-Torsional Buckling i) General method W 0. 5 cr Table cr - Elastic critical moment

27 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Lateral-Torsional Buckling ii) Alternative method (rolled sections or equivalent welded sections) W cr 0.5,0.0 Table 6.5 -, (ma be speciied in ational Annexes o Eurocode 3) cr - Elastic critical moment

28 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS Lateral-Torsional Buckling,mod, mod k c.0 eglect B i:,0 cr,0

29 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS EXAPLE Saet check o a beam o the building illustrated in the igure (along line E). The beam is composed b a IPE 600 with 9 m length at the central span; the lateral spans with 6 m length (the governing spans) are composed b a section IPE 400 in steel S 355. For the lateral buckling check, two cases are considered: a) a beam with 6 m length, laterall braced onl at the end support sections; b) a beam with 6 m length, laterall braced at the end support sections and at mid-span section. A C E A B C D E F The geometrical and mechanical properties o the section IPE 400 in S 355 steel are: A = cm, b = 80 mm, h = 400 mm, t = 3.5 mm, t w = 8.6 mm, I = 330 cm 4, i = 6.55 cm, I = 38 cm 4, i = 3.95 cm, b a m 4.50 m.50 m.00 m I T = 5.08 cm 4 ; Iw = 490x0 3 cm 6 ; 6.00 m = 355 Pa and E = 0 GPa m 4.50 m 4.50 m 4.00 m Building plan master example

30 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS EXAPLE a) Beam laterall braced at supports i) The internal orces (neglecting the axial orce) are represented in the igure. The design values are = 4.3 km and V = 75.9 k. V, 70.7 k 75.9 k 40. k 55.7 km 39. k 46.3 km 7.6 k 75. k 93.7 km.4 km 99. km 09.7 km ii) Cross section classiication, Web (an internal part) in bending: 4.3 km 3.6 km c t km Flange (outstand part) in compression: c t (80 8.6) The cross section is class

31 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS EXAPLE iii) Cross section veriication Bending resistance: 4.3 km Wpl, km 6 3 Shear resistance: V h t w w 75.9 k V A 4 3 v pl, Rd k So, it is not necessar to veri the shear buckling resistance. Bending + Shear: V 75.9 k 0.50 V pl, Rd k So, it is not necessar to reduce the bending resistance to account or the shear orce. Cross section resistance is veriied.

32 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS EXAPLE iv) Lateral buckling resistance Assuming the support conditions o the standard case and the loading applied at the upper lange level, the elastic critical moment can be obtained rom the ollowing equation, with L = 6.00 m, k = k w =.0, C.80 and C.60 (Boissonnade et al., 006) and g = 00 mm. 0.5 E I k I W cr C g 3 j g 3 k L kw I E I cr km k L G IT C C C C j A CG m 6 m g =00 mm IPE 400 C 3 m 93.7 km.4 km (Using Beam > cr = km), 4.3 km

33 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS EXAPLE 3 cr 64.7 km; W Wpl, 307cm. 68 W 0. 5 cr General method: Table Rolled cross section IPE 400 with h/b=400/80=.> - Curve b 0.34 b , Rd km km So, the saet is veriied (utiliation ratio = 4.3/9.9=0.88)

34 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS EXAPLE b) Beam laterall braced at supports and mid-span A B C i) Cross section veriications are not changed. ii) Lateral buckling check: As the beam is laterall braced at mid span cross section, the critical moment can be evaluated with L = 3.00 m and a conservative hpothesis o k = k w =.0. For the given bending moment shape between lateral braced cross sections, C =.6 (Boissonnade et al., 006). EI C cr k L k k w I W I k L GI T EI C C C g 3 j g 3 j 0. 5 C, 3 m m m 93.7 km.4 k km cr km (Using Beam cr = km)

35 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEAS EXAPLE cr km; W Wpl, 307cm W 0. 5 cr General method: Table Rolled cross section IPE 400 with h/b=400/80=.> - Curve b 0.34 b , Rd km km So, the saet is veriied (utiliation ratio = 4.3/4.9=0.8)

36 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS Cross section resistance (clause 6..9 o EC3--) Class or Uniaxial bending, Rd Double-smmetric I or H sections n 0.5 a,, Rd pl,, Rd but,, Rd pl,, Rd, Rd pl,, Rd, i n a,, Rd n pl. Rd pl,, Rd o reduction i 0.5 pl, Rd a 0.5 hw tw 0 n a a n a A b t A ( axis) i h w t w 0 ( axis) pl HEA Eixo de menor inércia - Bending about minor axis - Eixo de maior inércia - Bending about major axis -.0 pl,, pl,

37 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS Cross section resistance (clause 6..9 o EC3--) Class or Bi-axial bending n,,. Rd pl, Rd,,. Rd.0 I or H ; 5 n Circular hollow sections but Rectangular hollow sections,,.66.3 n 6 Class 3 or 4 x, 0, x, A I, I Bending, shear and axial orce (clause 6..0 o EC3--) Similar to bending and shear interaction.

38 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS ember stabilit embers with high slenderness subjected to bending and compression, ma ail b lexural buckling or lateral-torsional buckling. Flexural buckling and lateral-torsional buckling (doubl-smmetric cross-section): Rk k., Rk, k., Rk,.0 (Eq. 6.6 o EC3--) e, (class 4) Rk k., Rk, k., Rk,.0 (Eq. 6.6 o EC3--) k,k,k and k - interaction actors, which are dependent o instabilit phenomena and plasticit Annex A o EC3-- (ethod ) or Annex B (ethod ).

39 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS ember stabilit i) embers with closed hollow sections or open sections restrained to torsion are not susceptible to torsional deormation. ii) embers with open sections (I or H sections) are susceptible to torsional deormation. embers not susceptible to torsional deormation checking o lexural buckling against -axis and -axis, considering eqs. (6.6) and (6.6) with =.0 and interaction actors k,k,k and k in members not susceptible to torsional deormation. embers susceptible to torsional deormation checking o lateral-torsional buckling, considering eqs (6.6) and (6.6) with according to 6.3. o EC3-- and interaction actors k, k, k and k in members susceptible to torsional deormation.

40 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS ember stabilit ethod (Annex B o EC3--) Interaction actors or members not susceptible to torsional deormations (Table B. o EC3--).

41 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS ember stabilit ethod (Annex B o EC3--) Interaction actors or members susceptible to torsional deormations (Table B. o EC3--).

42 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS ember stabilit ethod (Annex B o EC3--) Equivalent actors o uniorm moment C mi (Table B.3 o EC3--)

43 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS EXAPLE 3 Saet check o a beam-column o the irst store o the building illustrated in the igure. The member, composed b a HEB 30 cross section in steel S 355, has a length o The relevant geometric characteristics o HEB 30 cross section are: A = 6.3 cm ; W pl, = 49 cm 3, I = 3080 cm 4, i = 3.8 cm; I = 939 cm 4, i = 7.57 cm; I T = 5. cm 4 and I W = 069 x 0 3 cm 6. The mechanical characteristics o the material are: = 355 Pa, E = 0 GPa and G = 8 GPa.

44 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS EXAPLE 3 The design internal orces obtained through the structure analsis (second order) or the various load combinations are illustrated in the igure. Two simpliication are assumed or the subsequent design 0 k 53.0 k 65.5 km 7.3 km 9.8 km veriications: i) the shear orce is suicient small so can be neglected; ii) the shape o the bending moment 47 k 4. k 58.6 km 3.0 km diagram is linear. 630 k 4. k 57. km 9.3 km Design values are: = 704 k;, =4.8km 84 k 40. k 55.9 km 7.9 km at the base cross section. 053 k 39.3 k 53.8 km 9.0 km i) Cross section classiication 6 k 43. k 68.6 km 54.5 km As the compression orce is high, the cross section is classiied under compression onl (conservative approach). 496 k 50.5 k 73.5 km 0.6 km As the section HEB 30 is a stock section, even under this load condition, is class. 704 k V, 9.4 k 4.8 km,

45 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS EXAPLE 3 ii) Cross section resistance The design internal orces:, = 4.6 km and = 704 k (compression). pl 4, Rd A k As, 704 k, 576.k, the axial orce resistance is veriied. pl Rd Since, 704 k 0.5 pl, Rd 43.5k, in accordance with clause 6..9.(4) o EC3--, it is necessar to reduce the plastic bending resistance (to,,rd ): 3, W 6 3 pl, pl,, Rd km o.0 a n 0.30,, Rd pl,, Rd km 0.5 a A b t A n pl, Rd As,, 4.8 km,, Rd km, the bending resistance is veriied

46 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS EXAPLE 3 iii) Veriication o the stabilit o the member In this example the ethod is applied. As the member is susceptible to torsional deormations (thin-walled open cross section), it is assumed that lateral-torsional buckling constitutes the relevant instabilit mode. Since, = 0, the ollowing conditions must be veriied:, k.0 Rk, Rk Rk k,, Rk.0 Step : characteristic resistance o the cross section A Rk k 6, Rk Wpl, km

47 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS EXAPLE 3 Step : reduction coeicients due to lexural buckling, and h b and 0.5mm lexural buckling around curve b( 0.34) lexural buckling around curve c ( 0.49). t 00mm HEB 30 Plane x - L E, = m. L E, i Plane x - L E, = m L E, i

48 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS EXAPLE 3 Step 3: calculation o the using the alternative method applicable to rolled or equivalent welded sections (clause o EC3--) The length between braced sections is L = m. The critical moment cr assuming a linear diagram, in this example obtained just b Beam sotware, is given b: 5 km 0.6 km km 4.8 km cr 39 Rolled I or H sections with h b , curve b, and 0.34 Table Taking 0.4 and 0. 75,

49 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS EXAPLE 3 Step 3: calculation o the using the alternative method applicable to rolled or equivalent welded sections (clause o EC3--) The correction actor k c, according to Table 6.6 o EC3--, with = 0.6/(-4.8) = , is given b: k c ( 0.43) k c The modiied lateral-torsional buckling reduction actor is given b:, mod So,,.00 must be adopted. mod

50 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS EXAPLE 3 Step 4: interaction actors k and k. The equivalent actors o uniorm moment C m and C m are obtained based on the bending moment diagram, between braced sections according to the direction in case o C m and laterall in case o C m. Assuming the member braced in direction and laterall just at the base and top cross sections, the actors C m and C m must be calculated based on the bending moment diagram along the total length o the member. Since the bending moment diagram is assumed linear, deined b:,,base = -4.8 km;,,top = 0.4 km, rom Table B.3 o EC3--, is obtained: ,, top,, base 5 km 0.6 km C m C m ( 0.40) km,

51 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 DESIG OF BEA COLUS EXAPLE 3 Because the member is susceptible to torsional deormations, the interaction actors k and k are obtained rom Table B. o EC3--, through the ollowing calculations: k k As As C k m k C C m ; Rk m C m Rk Rk Rk , then k Step 5: Finall, the veriication o equations 6.6 and 6.6 o EC3-- ields: then k 0. 8 Section O.K.

52 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 Free sotware or design o steel members in accordance with EC3--. Beam-columns design Design o cellular beams

53 Eurocodes Design o steel buildings with worked examples Brussels, 6 7 October 04 Thank ou or our attention rads@dec.uc.pt