Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) Suggested Modifications of the Conventional Rigid Method for Mat Foundation Design S. Shihada 1, J. Hamad and M. Alshorafa 3 1 Professor, Civil Engr. Dept, IUG-Gaza Assistant Professor, Civil Engr. Dept, IUG-Gaza- Corresponding Author: 3 M.Sc in Structural Engineering 1 sshihada@iugaza.edu.ps jhamad@iugaza.edu.ps 3 mznshorafa7@gmail.com Abstract:- The conventional rigid method for mat foundation design is characterized by its ease in execution and therefore, suitable for hand calculations and for small-size mats. Nevertheless, the method is impeded by its inability to satisfy the equations of static equilibrium, which makes the evaluation of correct shear forces and bending moments rather impossible. This study aims at satisfying the equilibrium equations, by suggesting three modification procedures of the conventional rigid method, in order to construct correct shear force and bending moment diagrams. Based on the results of this study, it is found that the three proposed modification approaches constitute lower-bound, average and upper-bound solutions to the internal forces, with maximum differences between upper and lower-bound solutions not exceeding 16. Keywords:- Mat; Shear; Moment; Rigid; Modification Factor; Conventional I. INTRODUCTION The structural design of reinforced concrete mat foundations has been for many years one of the least satisfactory areas of design [1]. Mats may be designed and analyzed as either rigid bodies or as flexible plates supported by an elastic foundation. An exact theoretical design of a mat on elastic foundation can be made; however a number of factors reduce the exactness to a combination of approximations. These include difficulty in predicting subgrade responses, variations in soil properties, mat shape, variety of superstructure loads and effect of superstructure stiffness on mat. The analysis and design is carried out using any of the following methods []: Conventional Rigid Method, Approximate Flexible Method, Finite Difference Method and Finite Element Method. For rigid mat design using the conventional rigid method, two approaches have been suggested; the inverted floor system and the combined footing approach [3]. 418 Teng [4] describes the conventional rigid method, where the pressure under the mat follows a planar distribution such that the centroid of the bearing pressure coincides with the line of action of the resultant force of all column loads acting on the mat. Then, the mat is analyzed as a whole in each of the two perpendicular directions and the total shear forces and bending moments at any section cutting across the mat is equal to the arithmetic sum of all forces and reactions on the left, or right, of this section. The stress distribution along this section is a problem of a highly indeterminate nature. ACI committee 336R [] suggests that mats may be designed and analyzed as either rigid bodies or as flexible plates supported by elastic foundation. In case column spacing is less than 1.75 divided by or the mat is very thick and variation of column loads and spacing is not over 0, mat may be designed by treating it as a rigid body and considering strips both ways. These strips are analyzed as combined footings with multiple column loads and loaded with the soil pressure on the strip and column reactions equal to loads obtained from the superstructure analysis. Since a mat transfers load horizontally, any given strip may not satisfy vertical load summation. The effect of column spacing on the behavior of a five story building is studied by Naratajan and Videivelli [5], where they conclude that column spacing has a marginal effect on the contact pressure. Moreover, they state that the increase in mat thickness results in reduced settlement increased bending moments and reduced uniform pressure. Bowels [6] states that the mats may be designed as rigid structures where the mat is sub-divided into a series of continuous beams (strips) centered on the appropriate column lines. For the series of beams, shear and moment diagram may be established using either combined footing analysis or beam moment coefficients. When the soil bearing pressure is low say 5 kn/m he suggests that the mat may be designed as an inverted flat slab, using heavy beams from column to column.
Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) The portion between beams is designed as a conventional one or two way slabs. Furthermore, Bowels [7] requires that the strip loads need to be adjusted so that statics is satisfied since the shear between adjacent strips is not included in the strip free body. For column loads not falling at the center of the strip area, a nonlinear soil pressure diagram is to be used to close the shear and moment diagrams. Later on, it is affirmed that the method is not recommended at present because of the substantial amount of approximations and the wide availability of computer programs that are relatively easy to use [8, 9]. Das [10] presents the conventional rigid method using strips between column lines in both directions. He proposes a method for satisfying static equilibrium of forces resulting from ignoring shear between adjacent strips. This is done through two sets of modification factors, one for column loads and the other for soil pressures at both ends of each of the individual strips. The soil pressure under each strip is taken as the average of the two values at the end of each strip. Furthermore Das [11] proposes that the soil pressure is not to be averaged at the bottom of each strip while adopting the same modification procedure described in Das [10]. In this work, the conventional rigid method, using the strip method, is to be modified using three different approaches that satisfy the equilibrium of forces in the vertical direction as well as the summation of moments at any point along the considered strip. Consequently, correct shear force and bending moment diagrams can be achieved. III. THE CONVENTIONAL RIGID METHOD-CASE STUDY A case-study of mat foundation design is worked out using the conventional rigid method as described in Das [10] to show its shortcomings. See Figure-1 and Table-1 for dimensions and loading. Note that ACI 318-08 load factors are followed [1]. II. IMPORTANCE OF MODIFIED/PROPOSED CONVENTIONAL RIGID METHOD The conventional rigid method is characterized by its simplicity and ease in execution. On the other hand, the resultant of column loads doesn't coincide with the resultant of soil pressure under the individual strips, which leads to violation of the static equilibrium equations. Most prestigious foundation design textbooks shy away from this fact either by selecting symmetrically-loaded strips and using uniform soil pressure to reduce the eccentricity to zero, or by drawing mistaken shear force and bending moment diagrams that do not close [7, 9, 10, 11]. Others analyzed the mat as a whole in each of the two perpendicular directions and evaluated the shears and moments along selected sections [4]. The stress distribution along this section is a problem of a high indeterminacy. Figure 1 Layout of mat foundation (cm) 419
Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) Column D.L (kn) L.L (kn) Table 1 Load calculations Q u (kn) X i (m) Q u X i (kn.m) Y i (m) Q u Y i (kn.m) C1 780 390 1560 0 0.00 1 3760 C 1601 801 30 5 16010 1 6740 C3 1446 73 89 10 890 1 6073 C4 671 336 134 15 0130 1 818 C5 157 786 3144 0 0.00 14 44016 C6 331 1616 646 5 3310 14 90468 C7 953 1477 5906 10 59060 14 8684 C8 1383 69 766 15 41490 14 3874 C9 1338 669 676 0 0.00 7 1873 C10 804 140 5608 5 8040 7 3956 C11 868 1434 5736 10 57360 7 4015 C1 1360 680 70 15 40800 7 19040 C13 603 30 106 0 0.00 0 0.00 C14 175 638 550 5 1750 0 0.00 C15 1316 658 63 10 630 0 0.00 C16 66 313 15 15 18780 0 0.00 Σ 51654 Σ 381970 Σ 561988 Step 1: Evaluate the factored net soil pressure under the mat The eccentricity e x is given as 381970 e x 7.5 0.105 m 51654 The eccentricity e y is given as 561988 e y 10.5 0.38 m 51654 q u, net 5,1654 ( 54350) x 1961 y 358.4 7645.9 14985.9 144.1 ( 7.1) x 13.1 y Step - Draw shear and bending moment diagrams The mat is divided into four strips in the first perpendicular direction and another four in the second direction. Strip BDKM which is.4 m long and 5 m wide is considered here for demonstrable purposes. qc 160.57 ql 131.4 kn / m kn / m The average uniform soil pressure is given by 160.57 131.4 qu,avg 145.9 kn / m Total soil reaction = 145.9 (.4) (5) = 16340.8 kn Total column loads = 178 kn 16340.8 178.5 = 17081.6 kn Average load 17081.6 Column modification factor = 0. 958 178.0 17081.6 Soil pressure modification factor = 1. 045 16340.9 Average modified soil pressures are 167.8 kn/m and 137. kn/m at points C and L respectively. Shear force and bending moment diagrams for strip BDKM are shown in Figures- and 3. 40
Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) 585.8 3190.1 334.6 196. -481.8-483. -3003.4-3040.4 Figure Shear force diagram for strip BDKM (kn) -350.3-94.6-5610.0-3016.9-848.5 05. 875.0 79.6 It is noticed that while the shear force diagram satisfies the equilibrium of forces in the vertical direction, the bending moment diagram fails to do so, yielding a bending moment of 3016.9 kn.m at the end of the strip, instead of zero. Figure 3 Bending moment diagram for strip BDKM (kn.m) IV. PROPOSED MODIFICATIONS OF THE CONVENTIONAL RIGID METHOD: In this section three proposed modifications are applied to the conventional rigid method and shear and bending moment diagrams are drawn for strip BDKM. A. First proposed modification: In this proposed modification, the strip shown in Figure- 4 is treated as a combined footing with the planar soil distribution evaluated for the entire mat being ignored. Therefore, a new soil pressure under mat ends is evaluated based on the strip columns loads from the following equation. Qu Qu ex B/ q 1, ( new ) (1) A I y Figure 4 Strip loads- First proposed modification Using the above equation, the resultants of the soil pressure under the strip and the resultant of the columns loads will have the same line of action. Then, shear force and bending moment diagrams can be easily constructed. Modified loads acting on strip BDKM are shown in Figure- 5. Q u 30 646 5608 505 178KN 30(0.7 ) 646(7.7 ) 5608(14.7 ) 550( 1.7 ) X l 10.65m 178.4 e x 10.65 0.55m A.45 11m 41
Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01).43 4 5 I 4683.1m 1 178 178(0.55)(.4 )(0.5 ) q1 18.56 kn / m 11 46831 178 178(0.55)(.4 )(0.5 ) q 135.68 kn / m 11 46831 The modified soil pressure and column loads for strip BDKM are shown in Figure-5. Figure 5 Loads acting on strip BDKM- First proposed modification Figures 6 and 7 show the shear and bending moment diagrams, respectively for strip BDKM. One can easily observe that the equilibrium equations are satisfied for shear as well as for bending moment. 636.7 3518.6 66 07.8-477. -565.3-943.4-98 Figure 6 Shear force diagram for strip BDKM- First proposed modification (kn) -3449.9-144.5-91 3.1 166.7 3859.8 3048.8 Figure 7 Moment diagram for strip BDKM- First proposed modification (kn.m) B. Second proposed modification: This includes modifying the columns loads on the strip only through modification factors for columns loads based on the planar soil pressure under the entire mat. Two modification factors are employed in order to make the resultant of the modified column loads coincide with the resultant of the soil pressure under the strip. Column loads situated to the left of the resultant are multiplied by a modifying factor F 1 and column loads situated to the right of the resultant are multiplied by a second modifying factor F as follows. Application of the mentioned process on strip BDKM, shown in Figure-8 is outlined next. Equation () is evaluated through application of static equilibrium of forces in the vertical direction. q1 q F1 ( QLeft ) F QRight ) Bi B Equation (3) is evaluated through application of static equilibrium on summation of F ( Q x ) F Q ) 1 Left i Right xi moments. q1 q q q1 Bi B 3q q 1 B )) (3) 4
Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) Figure 8 Strip Loads- Second proposed modification Solving Equations () and (3), the values of F 1 and F can be easily obtained. Therefore, the shear force and bending moment diagrams can be constructed. The column loads on the strip and soil pressure under strip BDKM are shown in Figure-9. Substituting in Eqn. (), one gets 646 F 5608 550 F1 30 160.6 131. (5)(.4) 9664F1 8158 F 163409kN Substituting in Eqn. (3), F 1 30 (0.7) 646(7.7)) F (5608(14.7) 550(1.7)) ( (131.) 160.6.4 16340.9 3(160.6 131.) 51998.8 F 1 13777.6 F 189164.7 Solving equations (-a) and (3-a) gives F 1 = 0.891 and F = 0.948. The modified column loads are as follows: Q 1 mod = F 1 Q 1 = 0.891(30) = 853 kn Q mod = F 1 Q = 0.891(646) = 5757 kn Q 3 mod = F Q 3 = 0.948(5608) = 5314 kn Q 4 mod = F Q 4 = 0.948(550) = 417 kn The soil pressure and modified column loads for strip BDKM are shown in Figure-9. (-a) Figure 9 Loads on strip BDKM based on the second proposed modification Figures 10 and 11 show the shear and bending moment diagrams respectively for strip BDKM. One can easily see that the equilibrium equations are satisfied for shear force, as well as for bending moment. (3-a) 560.4 3134.9 484.1 1955.5-460.9-9.4-6.4-830. Figure 10 Shear force diagram for strip BDKM - Second proposed modification (kn) 43
Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) -311.7-185.1-704.9 196.3 161.1 333.1 3035.3 Figure 11 Moment diagram for strip BDKM - Second proposed modification (kn.m) C. Third proposed modification: This proposed modification involves both of the columns loads on the strip and the applied soil pressure under the mat. The strip is modified by finding the average loads required to make the resultant of column loads equal to and coincide with that of the average loads at mid point between the influence points of column loads and soil reaction. Two sets of modifying factors are applied to make the resultant of the modified column load equal and coincide with that of the average loads. The first factor will be applied to column loads on the left side of the resultant of the modified column loads, while the second factor will be applied to column loads on the right side of the resultant. Then, the shear force and bending moment diagrams can then be constructed. The loads acting on the strip are shown in Figure-1 and the process is detailed as follows. Figure13 Modified loads on strip BDKM-Third proposed modification Qtotal Qi 178 kn 160.6 131. Soil reaction( qavgbi B) *5*.4 16340.9 kn Figure 1 Loads on strip BDKM before application of the third proposed modification Solving Equations (-a) and (3-a), gives F1 and F values. The modified soil pressure and column modified loads for strip BDKM are shown in Figures-13 and 14. 16340.9 178 Average load 17081.5 kn xl 10.65m and x p 10.8m, 10.65 10.8 so,x average 10.74m Substituting in Equations () and (3) gives F 1 = 0.945 and F = 0.975. The modified column loads are as follows: Q 1 mod = F 1 Q 1 = 0.945*30 = 306 kn Q mod = F 1 Q = 0.945*646 = 6106 kn Q 3 mod = F Q 3 = 0.975*5608 = 5465 kn Q 4 mod = F Q 4 = 0.975*550 = 485 kn And ; 1,mod q 133.6,mod q 171.4 kn/ m kn / m 44
Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) Figure 15 and 16 show the shear and bending moment diagrams respectively for strip BDK M. One can easily notice that equilibrium equations are satisfied for shear as well as for bending mo ment Figure 14. Applied load on the strip BDKM- Third proposed modification 598 334.7 556.8 015.4. -469.6-47.5-781. -908.3 Figure 15 Shear force diagram for strip B D K M - Third proposed modification (kn) -384.8-1353.7-813.4 09.5 164.1 3591. 3047.5 Figure 16 Moment diagram for strip B D K M - Third proposed modification (kn.m) V. DISCUSSION OF THE RESULTS From the results obtained from the three modification procedures it is noticed that the first modification procedure represents an upper bound solution of the results, while the second procedure represents a lower bound solution. Moreover, the third proposed modification procedure represents an average solution of the first and second proposed modification procedures. The bending moments obtained from the three procedures are shown in Figure-17. The differences in bending moments are shown in Table-, where the differences for the upper-bound solution obtained from the first proposed modification procedure range from 0.44 to 15.84 compared with the lower-bound solution obtained from the second modification procedure. Similarly, the average solution obtained from the third proposed modification procedure 45 ranges from 0.40 to 7.78 compared with the lowerbound solution. Figure 17 Bending moments obtained from the three modification procedures for BDKM (kn.m)
Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) Table Bending moments for Strip BDKM and percentages of differences among the three solutions Procedure 1st Proced ure nd Proced ure 3rd Proced ure Exte rior + ve.3 1 13.6 5 19.6 3 0 0.9 5 6.7 Exterior Span (t.m) Inter ior - ve 344.9 9 10.51 31.1 7 0 38.4 8 5. Inte rior + ve 385. 98 15.8 4 333. 1 0 359. 1 7.78 Interior Span (t.m) Inte rior - ve 14. 45 10.8 5 18. 51 0 135. 37 5.34 Inte rior + ve 304. 88 0.44 303. 53 0 304. 75 0.40 Ext. - ve 9. 10 7.99 70. 49 0 81. 34 4.01 Exterio r Span (t.m) Ext. + ve 16.67 3.48 16.11 0 16.41 1.86 Figure 18 shows the shear forces obtained from the three modification procedures. The differences in shear forces are shown in Table 3, where the differences for the upperbound solution obtained from the first proposed modification procedure ranges from 3.54 to 13.61 compared with the lower-bound solution obtained from the second modification procedure. Similarly, the average solution obtained from the third proposed modification procedure range from 1.89 to 6.71 compared with the lower-bound solution. Figure 18 Shear forces obtained from the three modification procedures for BDKM (kn) Table-3 Shear forces for Strip BDKM and percentages of differences among the three solutions (kn) Proce dure 1st Proce dure nd Proce dure 3rd Proce dure Column No. Rig Left ht 63.6 56. 7 53 13.6 11.9 1 0 56.0 4 0 59.8 0 6.71 9. 4 0 4. 75 5.89 Column No. 6 Rig Left ht 351. 94. 86 34 1. 1. 4 4 313. 49 0 33. 47 6.05 6. 4 0 78. 1 6.05 Column No. 10 Rig Left ht 6. 98. 6 5.71 5.36 48. 41 0 55. 68.93 VI. CONCLUSIONS 83. 0 0 90. 83.76 Column No. 14 Rig Left ht 07. 47.7 8 5.99 3.54 195. 55 0 01. 54 3.06 46.0 9 0 46.9 6 1.89 The three modification procedures suggested by the authors for mat foundation design have succeeded in solving the main problem associated with the conventional rigid method, which is satisfaction of the equilibrium equations when constructing shear force and bending moment diagrams for the individual strips for the mat. The three obtained solutions represent lower bound, average and upper bound solution for shear forces and bending moments for each individual strip of the mat. Since two-way action is ignored in analyzing the strips, it is recommended that the lower bound solution associated with modifying column loads only be used in evaluating shear forces and bending moments in the strips. The maximum differences in bending moments obtained from the three procedures is less than 16. The maximum differences in shear forces obtained from the three procedures is less than 14. 46
REFERENCES International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) [1 ] Eden, W., McRostie, G., Hall, J., 1973- Measured Contact Pressures Below Raft Supporting A stiff Building, Canadian Geotechnical Journal, Vol. 10, pp. 180-19. [ ] American Concrete Institute (ACI) Committee 336R, Suggested Design Procedures for Combined Footing and Mats (ACI 336.R-88, Reapproved 00), Detroit, Michigan, USA, 00. [3 ] Gupta, S., Mat Foundations Design and Analysis with a Practical Approach, New Age International limited Publishers, New Delhi, 1997. [4 ] Teng, W., Foundation Design, Prentice Hall, Prentice Hall Inc., Englewood Cliffs, 196. [5 ] Natarajan, K., Vidivelli, B., 009- Effect of Column Spacing on the Behavior of Frame-Raft and Soil Systems, Journal of Applied Sciences, Vol. 9, No. 0, pp. 369-3640. [6 ] Bowles, J., Foundation Analysis and Design, nd ed., McGraw-Hill Book Co., New York, USA, 1997. [7 ] Bowles, J., Foundation Analysis and Design, 3rd ed., McGraw-Hill Book Co., New York, USA, 198. [8 ] Bowles, J., Foundation Analysis and Design, 4th ed., McGraw-Hill Book Co., New York, USA, 1996. [9 ] Bowles, J., Foundation Analysis and Design, 5th ed., McGraw-Hill, International Edition, 1997. [10 ] Das, B., Principles of Foundation Engineering, PWS Engineering, Boston, Massachusetts, USA, 1984 [11 ] Das, B., Principles of Foundation Engineering, 4th ed., PWS Engineering, Boston, Massachusetts, USA, 1999.. [1 ] American Concrete Institute (ACI). 008. Building Code Requirements for Structural Concrete (318-08) and Commentary (318 R-08), Farmington Hills, Michigan, USA, 008. NOTATION A = total area of the mat B = length of mat strip B i = width of mat strip between centers of adjacent strips D.L= Column's service dead load L.L = Column's service live load e x, e y = coordinates of the resultant force relative to the center of area of the mat E c = modulus of elasticity of concrete F 1= modification factor for column loads located to the left of the resultant F = modification factor for column loads located to the right of the I = moment of inertia of the strip of width B i I x, I y = moment of inertia of the area of the mat with respect to the x and Q u = factored column loads Q left = summation of column loads located to the left of the resultant Q right = summation of column loads located to the right of the resultant Q total,mod = modified column loads q avg,mod = modified average soil pressure X i = coordinate of column load in x-direction, relative to the point of origin Y i = coordinate of column load in y-direction, relative to the point of origin x,y = coordinate of any given point on the mat with respect to x and y axes passing through the centroid of the mat x l = the distance between Q total and the left edge of the mat strip x p = the distance between the resultant of average soil pressure and the left edge of mat strip xl x p xaverage q u,net = factored net soil pressure q u,avg = average factored net soil pressure resultant K = characteristic coefficient = s Bi 4 4 Ec I k s = coefficient of subgrade reaction 47
Website: www.ijetae.com (ISSN 50-459, Volume, Issue 4, April 01) Samir Shihada is professor in structural engineering at the department of civil engineering in the Islamic University of Gaza. He has extensive experience in teaching and practicing structural concrete design where he has published a refereed book entitled Reinforced Concrete Design. His research interests include structural concrete design codes, seismic design and fire-resistant concrete. Furthermore, he has served on several government committees dealing with building damage evaluation and engineering education. Jehad T. Hamad has a Ph.D. in the field of Geotechnical/Geoenvironmental Engineering - Louisiana State University - December 1990. He worked at Southern Illinois University and Bir-ziet University. Currently, He is a member of the Civil Engineering department at the Islamic University-Faculty of Engineering. He worked as a consultant engineer in a number of private and public engineering firms in USA and Gaza Strip in the field of Geotechnical/Geoenvironment Engineering. Dr Hamad has published many papers in field of analysis and design of landfills, improvement of soils and the impact on environment 48