Digest 011, December 011, 1519-155 Determination of Base Stresses in Rectangular Footings under Biaial Bending 1 Güna ÖZMEN* ABSTRACT All the footings of buildings in seismic regions are to be designed according to biaial bending moments. The purpose of this paper is to develop a general method for calculating the base pressures of rectangular footings under biaial bending. First the footings which are eposed to large eccentricit are classified according to the shape of the pressure region. Then the formulation given b Löser for the design of rectangular columns subjected to biaial bending are generalized and applied to the calculation of base stresses. Since the position of the neutral ais is not known initiall, a process of successive approimations is developed. The application of the computation procedure is demonstrated on a numerical eample. Kewords: Biaial bending, single footings, base pressures, successive approimations. 1. INTRODUCTION It ma be considered both safe and economic to provide an individual foundation for each column, in cases when the bearing capacit of the foundation soil is sufficientl high or the loads are comparativel low. These individual foundations which are called Footings are generall constructed rectangular in shape, [1], [], [], [4], [5]. As a special case, the schematic elevation of a rectangular footing under the effect of vertical load V and unidirectional bending moment M is shown in Figure 1. Figure 1: Footing under the effect of uni-directional moment * Istanbul Technical Universit, (Retired), Istanbul, Turke - gunozmen@ahoo.com Published in Teknik Dergi Vol., No. 4 October 011, pp: 5659-5674
Determination of Base Stresses in Rectangular Footings under Biaial Bending The dimensions of the footing are B B. The base stresses for this kind of footings are readil computed b using the well known formulae of strength of materials. The eccentricit e of the vertical load is defined as M e. (1) V However, when the vertical load is situated outside of the region which is called the Core, i.e. when e B / 6 () an unstressed zone develops at the base of the footing. This special state is named Large Eccentricit and the corresponding stress distribution is as shown in Figure. Figure : Stress distribution for large eccentricit In this case, maimum base pressure is computed b V 4 V ma. () cb B (B e). FOOTINGS UNDER BIAXIAL BENDING In the Turkish Earthquake Code which is valid since 007, it is required that the interaction of the two orthogonal ecitation directions is to be taken into account, [6]. Hence, in seismic regions it is necessar to design all the columns (and foundations) of the buildings to be under the effect of biaial bending. Computation of the base stresses of the footings under biaial bending moments is a rather comple task ecept certain special cases. This subject is investigated in a rather comprehensive and detailed manner b Köseoğlu and etensive eplanations are presented, 150
Güna ÖZMEN [1]. For triangular and trapezoidal pressure zones eact formulae are given while the formulae for pentagonal zones are approimate. Trupia and Sagun have also developed eact formulae for triangular and trapezoidal pressure zones and presented a design chart for pentagonal zones, [5]. The purpose of this stud is to develop a base stress computation method which is independent of the shape of the pressure zone, i.e. valid for triangular, trapezoidal and pentagonal zone shapes. To begin with, the eplanations given b Köseoğlu will be summarized and discussed. Schematic stress distribution of a rectangular footing under the effect of biaial bending together with vertical load is shown in Figure. Figure : Loading at footing base and stress distribution Compression zone is shown as shaded on the figure. It is assumed that the load V is acting vertical to the paper surface at the centroid of the rectangular area. The sign convention for the load is taken positive in downward direction. In the cases where the related column is situated at a different location than the centroid, the forces acting on the footing can be transferred to the centroid and the formulae given below can be applied without modification. Positive directions for the bending moments M and M are so chosen that the maimum base stress occurs at the lower left corner of the footing base. The coordinates of the application point of the vertical load V are 151
Determination of Base Stresses in Rectangular Footings under Biaial Bending v M M v. (4) V V both of which are negative according to the chosen sign conventions. The angle between the neutral ais and ais is denoted b. The neutral ais ma eist in various locations and inclinations depending on the values of V, M, M and the footing dimensions B, B. An Unstressed Zone in a variet of shapes and sizes ma occur according to the position of the neutral ais. It ma be shown that the shape and size of the compression zone depends on the application point of the vertical load V. The base of the footing ma be considered as being separated into 1 regions according to the shape of the compression zone. These regions ma be collected in 5 groups, as shown in Figure 4, giving the same number to the regions of the same character. Figure 4: Footing base regions according to compression zone shapes The compression zone tpes corresponding to these regions are shown in Figure 5. Compression zones are shown as shaded in the figure. Smmetric shapes with respect to and/or aes ma also occur due to the aes of smmetr. In the following, characteristics are summarized for various tpes and stress computation formulae are presented. 15
Güna ÖZMEN Figure 5: Compression zone tpes Tpe 1: When the application point of the vertical load V remains in the rhombus shaped region, i.e. core, which is denoted as 1 in Figure 4, all the footing base is under compression. In this case, which is called Small Eccentricit, stresses at all corners can be computed b the well known formula V M B M B (5) F I I where F, I and I, are the base area and moments of inertia around aes and, respectivel. Tpe : When the application point of vertical load V is within the regions indicated with in Figure 4, Large eccentricit takes place where a trapezoidal shaped unstressed zone occurs. In this case, the compression zone is also trapezoidal as shown in Figure 5 and the maimum corner stress ma be computed b using the formulae given in [1], i.e. b 15
Determination of Base Stresses in Rectangular Footings under Biaial Bending t B B B B v 1 ; tg 1 v (6) v t v 1V B t ma. (7) B tg B 1t Tpe : When the application point of vertical load V is within the regions denoted as in Figure 4, large eccentricit is again the case where trapezoidal shapes prevail for both the compression and unstressed zones, Figure 5. In this case the maimum corner stress ma also be computed b using the formulae given in [1], i.e. b t B B B B v 1 ; tg 1 v (8) v t v 1V B t ma. (9) B tg B 1t Tpe 4: When the application point of vertical load V is within the regions denoted as 4 in Figure 4, large eccentricit takes place as well. In this case the compression and the unstressed zones are pentagonal and triangular, respectivel, Figure 5. Eact computation of the corner stresses proves to be rather comple for this tpe of stress distribution. However, approimate formulae which ma be considered adequate for practical applications are presented in [1] as follows: v v (10) B B V ma 1.9(6 1)(1 )(. ). (11) B B 154
Güna ÖZMEN Tpe 5: The oval region encircled b a dashed line in Figure 4 is called the Secondar Core which contains whole of the regions 1 and 4 and certain parts of the regions and. When the application point of vertical load V is outside of the secondar core, Ecessive large eccentricit occurs. This case where the compression zone is smaller then the unstressed zone should not be preferred in application. It is stated that, this tpe of stress distribution is not allowed b some of the design codes, [1], [7]. The regions shown as 5 remain wholl outside of the secondar core. In these regions, the triangular compression zones are smaller than the pentagonal unstressed zones. Stress computation formulae for this tpe do not eist in the references. In this stud, a method for base stress computation for the case of large eccentricit is developed. The presented method is general, i.e. independent of the shape of the pressure zone.. CROSS SECTION CHARACTERISTICS OF THE PRESSURE ZONE The section characteristics of the pressure zone should be determined prior to stress calculations. A footing base with pentagonal pressure zone is shown in Figure 6. This tpe of pressure zone ma be considered general, i.e. covering all the other tpes as special cases, as will be eplained shortl. Figure 6: Footing base and pressure zone parts Intersection points of the neutral ais with the aes u and v are denoted b P u and P v, respectivel. The respective distances of these points to the lower left point of the base are designated as A and C. These quantities will be used as main variables in stress calculations. In order to obtain the section characteristics, the pressure zone is divided into 155
Determination of Base Stresses in Rectangular Footings under Biaial Bending three parts consisting of two rectangular and one triangular region. The dimensions a 1, c 1, a, c, a and c can be calculated easil b using the values B, B, A and C. The conditions related to the various pressure zone tpes and the dimensions of the section parts for these tpes are shown in Tables 1 and, respectivel. Table 1: Conditions for pressure zone tpes Tpe Condition B B 1 A B & C B & 1 A C A B C B & A B C B & B B 4 A B & C B & 1 A C A B C B 5 & Table : Dimensions of pressure zone parts Tpe a 1 c 1 a c a c 1 B B 0 0 0 0 (C-B )/tgα B 0 0 A-a 1 B 0 0 B (A-B ) tgα B C-c 4 (C-B )/tgα B B -a 1 (A-B ) tgα B -a 1 c 5 0 0 0 0 A C As can be seen in Table, all dimensions a i and c i are nonzero for onl Tpe 4. Hence this pressure zone tpe can be considered general covering all the other tpes as special cases. For each part, coordinates u g, v g of the individual centroids G 1, G, G with respect to the aes u and v, areas F and moments of inertia I s, I t and I st with respect to the individual aes s,t are shown in Table. 156
Güna ÖZMEN Table : Section Characteristics of pressure zone parts Rectangular part (1) Rectangular part () Triangular part () 1 a u g 1 1 c v g 1 F 1 c 1 1 1 a1 a a1 a 1 c 1 c c a a c a c 1 I s a1c1 1 a c 1 ac 6 I t a 1 1 c 1 a c 1 a c 6 I st 0 0 ac 7 Area F and the coordinates u g and v g of the centroid G for the whole pressure zone are computed b F i i1 F (1) u v g g i1 u g,i F i (1) F i1 v g,i F i. (14) F Moments of inertia with respect to the aes and passing through the centroid G can be epressed b s,i i1 i1 ifi I I F (15) 157
Determination of Base Stresses in Rectangular Footings under Biaial Bending I t,i i1 i1 iei I F (16) I st,i i1 I F e f (17) i1 i i i where e i and f i denote the distances of pressure zone parts centroids to the centroid G, Figure 7. Figure 7: Distances of pressure zone parts centroids 4. COMPUTATION OF STRESSES For man ears, the book known as Löser has been one of the main sources of reference in reinforced concrete for structural engineers, [8]. This book includes etensive and detailed information on the design of RC structural elements according to the elastic theor. Nowadas a considerable portion of the information included in this book is outdated due to the usage of the Ultimate Strength theor in designing RC structural elements. However, it is seen that the formulae included in the section named Rectangular cross sections subjected to biaial bending moments in this book can be used for computation of stresses in footings under biaial bending. As a matter of fact, if the contribution of reinforcement is removed from the said formulae, the remaining parts consist of the computation of concrete stresses and can be used for determining base stresses of footings. Moreover, when the above given epressions for moments of inertia are used in the Löser formulae, stress computation procedure will be generalized including pentagonal pressure zones. The generalized stress computation procedure for footings will be eplained in the following pages. 158
Güna ÖZMEN The coordinates v, v of the application point of the vertical load V were given b Eqs. (4). The coordinates of this point with respect to aes and passing through the centroid of the compression zone are B B v v u g ; v v vg, (18) Figure 8: Auiliar variables for stress computations If the angle between the vector GV and horizontal ais is denoted b β, then v tgβ (19) v and the slope of the neutral ais can be computed b I Itgβ tgα, (0) I tgβ I In Figure 8, [8]. The distances of the points of intersection of the neutral ais with aes and to point G are I I / tgα ; 0 tgα, (1) F 0 0 v 159
Determination of Base Stresses in Rectangular Footings under Biaial Bending respectivel. The distances of the points of intersection of the neutral ais with aes u and v to the lower left point can be epressed b A u / tgα ; C u tgα, () g 0 g g respectivel, as in Figure 8. Then the base stress at an point with coordinates and is computed b 0 g V tgα σ 1 F 0 () and the maimum base stress at lower left point b AV σma, (4) F [8]. 0 5. COMPUTATION PROCEDURE As eplained in the preceding section, in order to compute the auiliar variables v, v, tg, tg, 0 and 0, the Cross Section Characteristics of the pressure zone are needed. But all the sectional characteristics are dependent on the main variables A and C ecept for the small eccentricit case, i.e. Tpe 1, which can be seen b inspecting Tables and. For the cases of large eccentricit, a successive approimation method should be applied, since the values of A and C are not known initiall. Computation procedure can be outlined as follows: 1. Firstl, all corner stresses are calculated b using Eq. (5). If all the corner stresses are positive (compression) then small eccentricit is the case and stress computation is completed.. If at least one of the corner stresses is found to be negative (tension), than large eccentricit will occur. In this case successive approimation is started b choosing proper initial values for A and C.. The conditions in Table 1 are inspected b using the chosen A and C values and the tpe of the compression zone is determined. 4. The dimensions and cross sectional characteristics of the compression zone parts are computed b using the formulae given in Tables and, respectivel. 5. The cross sectional characteristics for the whole of the compression zone are computed b using Eqs. (1) (17). 150
Güna ÖZMEN 6. Auiliar variables and new values for A and C are computed b using Eqs. (18) (1) and (), respectivel. 7. If the new values of A and C are not sufficientl close to the previous values, steps 6 are repeated. 8. When the new values of A and C are sufficientl close to the previous values, successive approimation procedure is terminated and corner stresses are computed b means of Eq. (). This procedure is quickl convergent and the values of corner stresses are not highl sensitive to the variations in the values of A and C. Numerical applications have shown that, step numbers are not ver much dependent on the chosen initial values of A and C. It can be said that, the most appropriate initial values for A and C are the values obtained b simple proportioning using the initial stress values computed b Eq. (5). 6. NUMERICAL APPLICATION A rectangular footing whose dimensions are B =.50 m and B = 1.50 m is under the effect of a vertical load of V = 400 kn together with bending moments M = 10 knm and M = 150 knm. Corner stresses found b using Eq. (5) are shown in Table 4. Table 4: Initial corner stresses Corner σ (kpa) Lower left 0.7 Upper left 74.7 Upper right 117. Lower right 18.7 Since the stress at upper right corner is negative (tension) large eccentricit is the case; hence successive approimations should be applied. Initial values for A and C are computed b simple proportioning using the above corner stresses ielding A = 4.06 m ; C = 1.98 m. If the conditions in Table 1 are inspected b using these values it is seen that B B A B ; C B ; 1, A C 151
Determination of Base Stresses in Rectangular Footings under Biaial Bending i.e. the compression zone is pentagonal, (Tpe 4). B using the relevant formulae in Tables and, dimensions and cross section characteristics of the compression zone parts are computed. Then moments of inertia of the whole compression zone are obtained as I = 0.57 m 4 I = 1.5499 m 4 I = -0.0 m 4 b using Eqs. (1) (17). The auiliar variables are computed b means of these values and Eqs. (18) (1) which ield the new values for A and C as A =.874 m ; C = 1.796 m through Eqs. (). Since these values are not sufficientl close to the initial values it is necessar to continue successive approimations. The results of successive steps are shown in Table 5. Table 5: Results of successive steps Step A (m) C (m) σ ma (kpa) 1 4.06 1.98 0.7.874 1.796 66.8.807 1.768 7.1 4.804 1.767 7. It is seen that the results of 4 th step are sufficientl close to those of rd step. Hence the successive approimations is terminated and final corner stresses are computed b Eq. () which are shown in Table 6. Table 6: Final corner stresses Corner σ (kpa) Lower left 7. Upper left 56.5 Upper right Lower right 18.0 15
Güna ÖZMEN It is seen that the eact results are obtained in merel 4 steps, i.e. the proposed method is quickl convergent. On the other hand, b using the approimate formula (11) given b Köseoğlu [1] ields σ ma = 7.1 kpa which has an error of merel 0.6%. Weighted average error for all the corner stresses is computed as ±.7% where the weights are taken as the absolute values of the stresses. Investigations on several numerical eamples have revealed that the average errors for Köseoğlu formulae are in the order of ± 5 %; hence the ma be considered suitable for practical applications. The chart and formulae given b Trupia and Sagun also give results which are sufficientl accurate. 7. CONCLUSIONS In this stud, a method for base stress computation for rectangular footings under the effect of biaial loading is developed. The results obtained ma be summarized as follows: 1. The method which is developed for the case of large eccentricit is independent of the shape of the compression zone. Namel, it is valid for all tpes of compression zones including triangular, trapezoidal and pentagonal shapes.. The proposed successive approimations procedure is quickl convergent i.e. the results are obtained after a few steps.. The proposed method and computation procedure can be easil adapted to computer b using an programming language. 4. Köseoğlu formulae are tested and it is determined that the ma be used successfull in practical applications. Smbols A: Abscissa of the intersection point of the neutral ais with the horizontal ais passing through lower left point, a 1, a, a : Horizontal dimensions of pressure zone parts, B : Dimension of the footing in direction, B : Dimension of the footing in direction, C: Ordinate of the intersection point of the neutral ais with the vertical ais passing through lower left point, c: Distance to the footing s edge in the case of uni-aial bending, c 1, c, c : Vertical dimensions of pressure zone parts, e: Eccentricit in the case of uni-aial bending, 15
Determination of Base Stresses in Rectangular Footings under Biaial Bending e 1, e, e : Horizontal distances of the centroids of pressure zone parts to the general centroid, F: Area of the footing base (pressure zone), f 1, f, f : Vertical distances of the centroids of pressure zone parts to the general centroid, I s, I t, I st : Moments of inertia of pressure zone parts, I : Moment of inertia of footing base (pressure zone) with respect to ais, I : Moment of inertia of footing base (pressure zone) with respect to ais, I : Product moment of inertia of footing base (pressure zone), M : Bending moment about ais, M : Bending moment about ais, t, t : Auiliar variables, u g, v g : Coordinates of the centroid of pressure zone, V: Vertical load, v : Abscissa of the application point of the vertical load, v : Ordinate of the application point of the vertical load, α: Inclination of the neutral ais, β: Angle between the vector passing through the application point of the vertical load and horizontal ais, ε: Auiliar coefficient, σ: Corner stress, σ ma : Maimum corner (edge) stress. References [1] Köseoğlu, S., Foundations Statics and Construction, Matbaa Teknisenleri Printing House, Istanbul, 1986, (In Turkish). [] Erso, U., Reinforced Concrete Floors and Foundations, Evrim Publishing House, Ankara, 1995, (In Turkish). [] Celep, Z., Kumbasar, N., Reinforced Concrete Structures, Sema Printing, Istanbul, 1996, (In Turkish). [4] Aka, İ., Keskinel, F., Çılı, F., Çelik, O. C., Reinforced Concrete, Birsen Publishing House, Istanbul, 001, (In Turkish). [5] Trupia, A., Sagun, A. Reinforced Concrete Shallow Foundations, Nobel Printing & Distribution, Ankara, 009, (In Turkish). 154
Güna ÖZMEN [6] Specification for Buildings to be Constructed in Seismic Regions, Ministr of Public Works and Settlements, Ankara, March 007, (In Turkish). [7] DIN 1054 1976, Foundation Ground Safet Loads of Foundation Ground, Bulletin of Ministr of Public Works and Settlements, No. 81, Ankara, 1984, (In Turkish). [8] Löser, B., Löser Bemessungsverfahren, Verlag von Wilhelm Ernst & Sohn, Berlin, 1948. 155