Chapter VII Non-linear SSI analysis of Structure-Isolated footings -soil system

Chapter VII 192 7.1. Introduction Chapter VII Non-linear SSI analysis of Structure-Isolated footings -soil system A program NLSSI-F has been developed, using FORTRAN, to conduct non-linear soilstructure interaction analysis of a three dimensional frame resting on isolated footings, by using hypoelasticity model for soil, adopting finite element method. The model and the corresponding computer program developed simulate two extreme states of compatibility of the horizontal displacements between the foundation and soil giving insight into the variation of horizontal displacements and horizontal stresses and their intricacies. These results are compared with the non-interactive analysis and as well as with respect to each other, for evaluation of the influence of using the interface elements and its relevance. 7.2. Problem definition The structure under consideration is shown in figure 7.1 and the geometrical details are given in table 7.1. The load considered for analysis is 31kN/m on beams, which is a service load on the structure. As the soil is semi infinite, the size of the soil mass considered is 153 X 2 X 95m as shown in Fig. 7.1. This size is arrived after ascertaining negligibly small stresses at the boundaries. For this three dimensional structure, the SSI analysis is carried out for both the uncoupled and coupled cases of interface between the foundation and soil. 7.3. Numerical formulation Since the numerical methods are more versatile than analytical methods for dealing problems with irregularities in geometry and materials, the finite element method is adopted to study the complex behavior of the SSI system. Finite element formulation in the SSI analysis of the frame - isolated footings -soil system is shown in figure 7.2. The soil is modeled with 33 X 21 X 7 layers in the 192

Chapter VII 193 longitudinal, transverse and vertical directions respectively resulting in 4851 brick elements. Each footing is modelled by four plate elements. The number of plate elements used is 96. The number of beam elements in the longitudinal direction (X-direction) is 8, in transverse (Z-direction) it is 72 and in vertical (y - direction) is 96. The various components of the system with respective degrees of are shown in figure 7.4 and are modeled as follows: 1. Soil mass is modeled using as eight-nodded brick elements with three translational degrees of freedom per node viz., D 7, D 2, and D 8. 2. Individual footing is modeled using plate elements with five degrees of freedom per node i.e., three translational degrees of freedom (viz D1, D 2, and D 3 ) and two rotational degrees of freedom (viz., D4, D 5 ). 3. Columns and beams are modeled as one-dimensional beam elements with six degrees of freedom per node (three translational and three rotational degrees of freedom). Member numbers, for quarter frame are shown in figure 7.3. 7.4. Degrees of freedom in coupled and uncoupled analysis In the coupled analysis, the translational degrees of freedom D 1 and D 3 of the plate elements are bonded with D 7 and D 8 of the soil elements, i.e., D 1 =D 7 and D 3 =D 8. This ensures no slippage between soil and foundation. Refer to figure 7.4. In the uncoupled analysis, the translational degrees of freedom D1 and D3 of the plate elements are not bonded with D7 and D8 of the soil elements. Thus, compatibility is disrupted in the horizontal directions and complete slip between the foundation and soil is allowed. 7.5. Details of analyses and validation of the computer program. The linear stiffness matrices for structural elements and non-linear stiffness matrices for soil elements are generated and included in a general-purpose FORTRAN program for simulating the multiscale structure. The assembly of stiffness matrix is carried out and stored in skyline form for each load increment of 1kN. With the load vector made available, the 193

Chapter VII 194 displacements were solved from the system equation using the Gauss elimination method. The program is validated for each of the element type by comparing the results of standard structures, such as cantilever plate and three-dimensional frames, with results available elsewhere. The whole program is validated for proposed model by comparing results with work done by King and Chandrashekharan (1974). 7.6. Results and discussion The structure is primarily analyzed with fixed base. These results are taken as the reference for comparing the structural responses obtained for the uncoupled and coupled cases. The stresses and displacements obtained are plotted against X/L in the longitudinal direction and B/Z in the transverse direction. Refer figure 7.2. The displacements are expressed in millimeters along the central longitudinal and transverse sections at various depths. The distances are expressed in terms of nondimensional parameters which are expressed as follows (Refer figure: and 4.6) ( X L / 2) L ( Z B / 2) B where X is the distace from center of mat in longitudin al direction Y is the distance from centre of mat in transverse direction 7.6.1. Effect of SSI on displacements of soil The displacements in uncoupled and coupled analyses have been evaluated for different Y/D values, where Y is depth measured from ground level and D is depth of soil considered in finite element model. For the uncoupled analysis, the maximum horizontal displacement (figure 7.6.) in the longitudinal direction occurs at =.58 at ground level, whereas in the transverse direction maximum horizontal displacement (figure 7.7.) occurs at Z/B=.6. The absolute maximum 194

Chapter VII 195 vertical displacement is 19.49 mm (figure 7.8) with a maximum differential settlement of 6.84 mm. For the coupled analysis, the maximum horizontal displacement (figure 7.9.) in the longitudinal direction occurs at =.58 at ground level, whereas in the transverse direction maximum horizontal displacement (figure 7.1.) occurs at Z/B=.82. The absolute maximum vertical displacement is 19.9mm (figure 7.11.) with a maximum differential settlement of 6.57 mm. The maximum displacement from the coupled analysis is.9794 times the max displacement from the uncoupled analysis. Meanwhile, the maximum differential settlement from the coupled analysis is.965 times the maximum differential settlement from the uncoupled analysis. This clearly shows that the coupling has no effect on the maximum settlement of the structure, as well as on the differential settlement. 7.6.2. Effect of SSI on stresses in soil The effects of the uncoupled and coupled analyses on the stresses of the soil are shown in Figures 7.12 to 7.26 and Figures 7.27 to 7.41 respectively. Sigma-x In uncoupled case the maximum compressive stresses of.129 occur at = +.53 and = +.117. Max tensile stresses of.7594 occur at = +.47 and = +.55. The stress at centre is -.157. Refer figures 7.12 to 7.16. In coupled case the maximum compressive stresses of.641 occur at = +.2 and = +.217. Max tensile stresses of.5121 occur at = +.98 and = +.117. The stress at centre is -.375. Refer figures 7.27 to 7.31. Sigma-z In uncoupled case the maximum compressive stresses of.1373 occur at = +.7 and Z/B= +.55. Max tensile stresses of.5946 occur at = +. and = +1.3. The stress at centre is -.543. Refer figures 7.17 to 7.21. 195

Chapter VII 196 In coupled case the maximum compressive stresses of.1198 occur at = +.13 and Z/B= +.. Maximum tensile stresses of.5121 occur at = +.98 and = +.117. The stress at centre is -.115. Refer figures 7.32 to 7.36. Sigma-y In uncoupled case the maximum compressive stresses of.52 occur at = +.33 and Z/B= +.217. Max tensile stresses of.3734 (.3238) occur at = +.33 and = +1.3. The stress at centre is -.3942. Refer figures 7.22 to 7.26. In coupled case the maximum compressive stresses of.4776(-.5877) occur at = +.33 and = +.217. Maximum tensile stresses of.3728(.3281) occur at = +.98 and = +.117. the stress at centre is -.393(-.3137). Refer figures 7.37 to 7.41. In the uncoupled analysis, the maximum longitudinal stress is 6.554 times the longitudinal stress at the centre of the soil and the maximum stress in the transverse direction is 2.5285 times the transverse stress at the centre of the soil. The maximum vertical stress is 1.2688(1.154) times the vertical stress at the centre of the soil. In the coupled analysis, the maximum longitudinal stress is 1.79(7.5) times the longitudinal stress at the centre of the soil and the maximum stress in the transverse direction is 1.641 (4.369) times the transverse stress at the centre. The maximum vertical stress is 1.2153 (4.68) times the vertical stress at the centre of the soil. The maximum longitudinal stress obtained from the coupled analysis is.6229 times the maximum longitudinal stress obtained from the uncoupled analysis. The maximum transverse stress obtained from the coupled analysis is.8725 times the maximum transverse stress obtained from the uncoupled analysis. The minimum vertical stress obtained from the coupled analysis is.9969 times the minimum vertical stress obtained from the uncoupled analysis. The maximum vertical stress obtained from the coupled analysis is.9548 (1.589) times the maximum vertical stress obtained from the uncoupled analysis. 196

Chapter VII 197 7.6.3. Effect of SSI on end actions of structural members Let F x. F y and F z represent the axial force, shear force in the y-direction and shear force in the z-direction in the local coordinates, respectively, and M x, M y and M z represent the moments about the local x-, y-, and z-axes, respectively. The second subscripts 1, 2 and 3 represent the non-interactive case, uncoupled case and coupled case, respectively. The numbers in parenthesis are pertaining to paper published by --------------------- On X-beams (Longitudinal direction) Absolute values of member end actions are given in table 7.4 and 7.5. Table 7.6 and 7.7 indicates normalized values with respect to non-interactive analysis. In Table 7.7, the axial forces on the beams of the structure obtained from the uncoupled analysis are -.6 to 4.18 (1.59 to 8.94) times the axial forces obtained from the non-interactive analysis. Negative sign indicates reversal on stresses which occurred at ground floor level. The axial forces on the beams of the structure obtained from the coupled analysis are, -6.51 to 5.21(2.55 to 9.48) times the forces obtained from the non-interactive analysis. The axial forces obtained from the coupled analysis are.95 to 3.95 times the axial forces obtained from the uncoupled analysis. The shear forces on the beams of the structure in the longitudinal direction obtained from the uncoupled analysis are..55 to 1.51 and coupled analyses are.47 to 1.6 (.64 to 1.39) times the shear forces obtained from the non-interactive analysis. The shear forces in coupled analysis are.85 to 1.6 times the shear forces in uncoupled analysis. From Table 7.7, it can be observed bending moments on the beams of the structure in the longitudinal direction obtained from the uncoupled analysis is -.33 to 3.21 and coupled analyses are -.49 to 3.68 times the bending moments obtained from the non-interactive analysis. The end moments on the beams in the longitudinal direction obtained from coupled analysis are.18 to 2.76 times the end moments from the uncoupled analysis. 197

Chapter VII 198 On Z-beams (Transverse direction) Absolute values of member end actions are given in table 7.7 and 7.8. Table 7.9 and 7.1 indicates normalized values with respect to non-interactive analysis. In Table 7.1, the axial forces on the beams of the structure obtained from the uncoupled analysis are -6.88 to 3.46 (-2.78 to 4.45) times the forces from the non-interactive analysis. The axial forces on the beams obtained from the coupled analysis are -4.69 to 5.1 (-3.31 to 6.13) times the forces from the non-interactive analysis. The axial forces obtained from the coupled analysis are - 7.45 to 6.7 (1.1) times the axial forces obtained from the uncoupled analysis. The shear forces on the beams of the structure in the longitudinal direction obtained from both the uncoupled and coupled analysis are.54 to 1.52 and.47 to 1.61 (.64 to 1.39) times the shear forces from the non-interactive analysis. The shear forces obtained from the coupled analysis are.87 to 1.45 times the shear forces obtained from the uncoupled analysis. The end moments on the beams of the structure obtained from the uncoupled and coupled analysis are.45 to 3.22 and -.522 to 3.67 (-.26 to 2.46) times the forces from the noninteractive analysis. The end moments obtained from the coupled analysis are -.58 to 5.52 (1.5 and.999) times the end moments from the uncoupled analysis. Refer table 7.1. On Columns Absolute values of member end actions are given in table 7.11 and 7.12. Table 7.13 and 7.14 indicates normalized values with respect to non-interactive analysis. From table 6.6 (b), it may be observed that the axial forces in the columns of the structure obtained from both from the uncoupled and coupled analyses are.83 to 1.33 (.86 to 1.28) times the axial forces obtained from the non-interactive analysis. The shear forces in X direction obtained from the uncoupled and coupled analysis are 2.19 to 4.71 and 2.6 to 7.18 (2.4 to 2.6) times the forces from the non-interactive analysis. The shear forces in Z direction obtained from the uncoupled and coupled analysis are 2.1 to 4.89 and 1.92 to 7.46 (1.95 to 2.62) times the forces from the non-interactive analysis. The 198

Chapter VII 199 shear force along X and Z axes obtained from the coupled analysis are.89 to 1.57 and.868 to 1.58 times the shear forces obtained from the uncoupled analysis. The bending moments on the columns of the structure with absolute values more than 1kNm are considered for discussion. In table 7.14, the bending moments about the X-axis are 2.7 to 4.34 (1.29 to 2.72) and 1.77 to 6.12 (1.51 to 2.66) times those of the noninteractive analysis for the uncoupled and coupled analyses, respectively. The bending moments for the uncoupled and coupled analyses about the Z-axis are 2.16 to 4.21 (1.1 to 2.8) and 1.91 to 5.93 (1.16 to 2.72) times the non-interactive analysis respectively. The maximum end moments about the X and Z axes obtained from the coupled analysis are.843 to 1.46 and.839 to 1.429 (1.4 and 1.3) times the end moments obtained from the uncoupled analysis.. 7.7. Conclusions 1. The response of the structure changes significantly in uncoupled and coupled soilstructure-interaction analysis when compared to the non-interactive analysis. 2. The presence or absence of interface elements does not affect significantly the settlements or differential settlements in the nonlinear analysis.. 3. In nonlinear analysis the interface element plays a crucial role when the constitutive relations of the soil depend on the state and increment of the stress and strain. The maximum longitudinal stress obtained from the coupled analysis is.6229 times the maximum longitudinal stress obtained from the uncoupled analysis. The maximum transverse stress obtained from the coupled analysis is.8725 times the maximum transverse stress obtained from the uncoupled analysis. The maximum vertical stress obtained from the coupled analysis is.9548 times the maximum vertical stress obtained from the uncoupled analysis. The stresses in the soil decrease with the coupling of horizontal displacements between the footing and soil in nonlinear analysis, where as the stresses were found to increase in linear coupled analysis. 4. Axial forces in structural members 199

Chapter VII 2 The axial forces in X-beams obtained from the coupled analysis are.95 to 3.95 times the axial forces obtained from the uncoupled analysis. The axial forces in Z-direction obtained from the coupled analysis are -7.45 to 6.7 times the axial forces obtained from the uncoupled analysis. However in the case of columns axial forces in columns are found to vary from.988 to 1.1 times the forces in uncoupled analysis. Therefore the coupling of horizontal displacement affects axial forces in beams. The beams in shorter direction are affected more 5. Shear forces in members The shear forces in X-beams (longitudinal direction) in coupled analysis are.85 to 1.6 times the shear forces in uncoupled analysis. The shear forces in Z-beams (transverse direction) obtained from the coupled analysis are.989 to 1.17 times the shear forces obtained from the uncoupled analysis. Therefore it can be concluded that shear forces in beams in shorter direction are more affected.. The shear force for columns, along X and Z axes obtained, from the coupled analysis are.893 to.983 and.868 to.984 times the shear forces obtained from the uncoupled analysis. Increase Shear forces is found at bottom end of ground floor columns. 6. End moments in members The end moments of the beams, in the longitudinal direction, obtained from coupled analysis are.18 to 2.76 times the end moments from the uncoupled analysis.. The end moments of the beams, in transverse direction, obtained from the coupled analysis are -.58 to 5.52 (1.5 and.999) times the end moments from the uncoupled analysis.. The end moments of columns, about the X and Z axes, obtained from the coupled analysis are.843 to 1.46 and.839 to 1.429 (1.4 and 1.3) times the end moments obtained from the uncoupled analysis. When compared to linear analysis the nonlinear analysis is found to have more effect on coupling of horizontal displacements of soil and footing. Therefore it is imperative to model the soil and as well the interface between foundation and soil to get accurate behaviour of three dimensional structures even under static service. 2

Chapter-IV 21 Member No X-coordinate Table 7.4: Stress resultants in beams(x-direction) in frame-raft-soil interaction analysis Y-coordinate Z-coordinate Position of memeber Non-interactive analsysis Fx1 Fy1 M Z1 Fx2/F Uncoupled Vs Non-interactive x1 Fy2/F y1 MZ2/ MZ1 Fx3/F Coupled Vs Non-nteractive x1 Fy3/F y1 Fx3/F Coupled vs Uncoupled (1) (2) (3) (4) (5) (6) (7) (8) (9) (1) (11) (12) (13) (14) (15) (16) (17) 1 64 23.5 4-6.71 72.47 43.33.18 1.42 2.83-2.64 1.5 3.23-14.3 1.5 1.14 69 23.5 4 6.71 82.53-68.5.18.63 -.9-2.64.56 -.24-14.3.89 2.76 69 23.5 4-6.54 77.92 66.28.17 1.7 1.6-4.58 1.1 1.16-26.51 1.3 1.9 2 74 23.5 4 6.54 77.8-64.16.17.93.65-4.58.9.58-26.51.97.9 3 74 23.5 4-6.38 77.5 64.43 -.6 1..94-5.45 1..95 93.95 1. 1.1 6 64 27 4 -.2 74.96 5.51 48.5 1.36 2.34 186. 1.34 2.26 3.84.99.97 69 27 4.2 8.4-63.21 48.5.66 -.6 186..68 -.1 3.84 1.3.18 69 27 4-1.28 77.2 63.65 1.63 1.7 1.7 3.91 1.7 1.9 2.4 1. 1.1 7 74 27 4 1.28 77.8-65.14 1.63.93.67 3.91.93.66 2.4 1. 1. 8 74 27 4-1.12 77.5 64.72 2.23 1..92 4.88 1..92 2.19 1. 1. 11 64 3.5 4-4.27 76.48 54.91 1.61 1.35 2.26 1.54 1.34 2.21.95.99.98 69 3.5 4 4.27 78.52-6.1 1.61.65 -.11 1.54.67 -.7.95 1.2.58 69 3.5 4-3.34 76.65 61.68 2.13 1.7 1.9 2.7 1.7 1.1.97 1. 1.1 74 3.5 4 3.34 78.35-65.93 2.13.93.66 2.7.93.66.97 1. 1. 13 74 3.5 4-3.22 77.5 64.9 2.2 1..92 2.15 1..92.97 1. 1. 16 64 34 4 18.74 71.58 35.93 2.19 1.24 2.13 2.14 1.23 2.8.98.99.98 69 34 4-18.7 83.42-65.52 2.19.8.32 2.14.81.35.98 1.1 1.9 69 34 4 18.53 77.44 64.69 3.39 1.3.91 3.31 1.3.92.98 1. 1.1 17 74 34 4-18.5 77.56-65.2 3.39.97.76 3.31.97.76.98 1. 1. 18 74 34 4 18.22 77.5 64.56 3.71 1..9 3.63 1..9.98 1. 1. 21 22 Exterior Beams 64 23.5 45-6.71 72.47 43.33. 1.51 3.21-3.42 1.6 3.68 765.33 1.6 1.15 69 23.5 45 6.71 82.53-68.5..55 -.31-3.42.47 -.49 765.33.85 1.6 69 23.5 45-6.54 77.92 66.28.15 1.8 1.8-5.56 1.11 1.19-36.1 1.3 1.1 74 23.5 45 6.54 77.8-64.16.15.91.57-5.56.89.5-36.1.97.88 23 74 23.5 45-6.38 77.5 64.43 -.5 1..92-6.51 1..94 125.94 1. 1.2 26 27 64 27 45 -.2 74.96 5.51 54.5 1.43 2.62 215.5 1.41 2.53 3.95.98.96 69 27 45.2 8.4-63.21 54.5.59 -.28 215.5.61 -.22 3.95 1.4.79 69 27 45-1.28 77.2 63.65 1.49 1.9 1.11 4.16 1.9 1.12 2.79 1. 1.1 74 27 45 1.28 77.8-65.14 1.49.91.59 4.16.91.59 2.79 1. 1.1 28 74 27 45-1.12 77.5 64.72 2.12 1..9 5.21 1..9 2.46 1. 1. 31 32 64 3.5 45-4.27 76.48 54.91 1.67 1.43 2.51 1.58 1.41 2.45.95.99.98 69 3.5 45 4.27 78.52-6.1 1.67.59 -.33 1.58.6 -.28.95 1.3.84 69 3.5 45-3.34 76.65 61.68 2.24 1.9 1.12 2.17 1.9 1.13.97 1. 1. 74 3.5 45 3.34 78.35-65.93 2.24.91.58 2.17.91.59.97 1. 1.1 33 74 3.5 45-3.22 77.5 64.9 2.33 1..9 2.26 1..9.97 1. 1. 36 37 64 34 45 18.74 71.58 35.93 2.38 1.29 2.4 2.33 1.28 2.34.98.99.98 69 34 45-18.7 83.42-65.52 2.38.75.18 2.33.76.21.98 1.1 1.19 69 34 45 18.53 77.44 64.69 3.79 1.4.91 3.69 1.4.92.97 1. 1.1 74 34 45-18.5 77.56-65.2 3.79.96.7 3.69.96.71.97 1. 1.1 38 74 34 45 Interior Beams 18.22 77.5 64.56 4.18 1..88 4.7 1..88.97 1. 1. x2 Fy3/F y2

Chapter-IV 22 Member No X-coordinate Table 7.5: Stress resultants in beams(z-direction) in frame-raft-soil interaction analysis Y-coordinate Z-coordinate Position of memeber Non-interactive Analsysis Fx1 Fy1 M Z1 Uncoupled Vs Non-interactive Fx2/Fx1 Fy2/Fy1 MZ2/MZ1 Coupled Vs Non-nteractive Fx3/Fx1 Fy3/Fy1 Fx3/Fx2 Coupled vs Uncoupled Fy3/Fy2 (1) (2) (3) (4) (5) (6) (7) (8) (9) (1) (11) (12) (13) (14) (15) (16) (17) 81 64 23.5 4-6.88 72.52 43.37 3.79.96 2.63 3.69.96 2.96.97 1. 1.13 64 23.5 45 6.88 82.48-68.3 4.18 1. -.2 4.7 1. -.13.97 1. 5.52 82 64 23.5 45-6.94 77.5 65.82.39 1.39.74-2.2 1.45.77-5.71 1.4 1.4 84 64 27 4.17 74.93 5.5.39.66 2.21-2.2.61 2.12-5.71.92.96 64 27 45 -.17 8.7-63.4.74 1. -.4-2.91 1..2-3.93 1. -.58 85 64 27 45-1.16 77.5 64.23-3.59 1.34.76-21.76 1.32.77 6.7.98 1.2 87 64 3.5 4-4.57 76.45 55.4-3.59.68 2.14-21.76.7 2.1 6.7 1.3.98 64 3.5 45 4.57 78.55-6.3.99 1. -.9 3.91 1. -.4 3.94 1..46 88 64 3.5 45-3.94 77.5 63.9 1.66 1.34.76 1.56 1.32.77.94.99 1.1 9 64 34 4 18.94 71.71 36.29 1.66.67 2.4 1.56.69 1.99.94 1.2.98 64 34 45-18.9 83.29-65.3 2.11 1..28 2.1 1..32.95 1. 1.11 91 64 34 45 93 Exterior Beams 19.18 77.5 65.14 2.1 1.24.7 2.5 1.22.72.98.99 1.2 69 23.5 4-4.57 76.45 43.37 2.1.8 3.4 2.5.81 3.46.98 1.1 1.14 69 23.5 45 4.57 78.55-68.3 2.9 1. -.27 2.81 1. -.41.97 1. 1.5 94 69 23.5 45-3.94 77.5 65.82.6 1.48.68-3.2 1.56.72-51.21 1.5 1.5 96 69 27 4 18.94 71.71 5.5.6.58 2.5-3.2.51 2.39-51.21.88.96 69 27 45-18.9 83.29-63.4.55 1. -.28-4.7 1. -.21-7.45 1..76 97 69 27 45 19.18 77.5 64.23-5.18 1.42.71-27.47 1.39.72 5.31.98 1.2 99 69 3.5 4-4.57 76.45 55.4 1.75 1.41 2.22 1.64 1.4 2.22.93.99.999 69 3.5 45 4.57 78.55-6.3 1.75.6 -.17 1.64.61 -.17.93 1.3.986 1 69 3.5 45-3.94 77.5 63.9 2.28 1..74 2.15 1..74.94 1. 1.1 69 34 4 18.94 71.71 36.29 2.31 1.29 2.13 2.25 1.28 2.13.97.99.999 12 69 34 45-18.9 83.29-65.3 2.31.75.23 2.25.76.23.97 1.1 1.7 13 69 34 45 Interior Beams 19.18 77.5 65.14 3.27 1..69 3.17 1..69.97 1. 1.1 74 23.5 4-6.88 72.52 43.37 -.17 1.52 2.3-3.72 1.61 2.37 22.24 1.5 1.3 15 74 23.5 45 6.88 82.48-68.3 -.17.54.2-3.72.47. 22.24.87 -.14 16 74 23.5 45-6.94 77.5 65.82.35 1..68-4.69 1..69-13.51 1. 1.6 18 Interior Beams 74 27 4.17 74.93 5.5-6.88 1.45 2.46-31.18 1.42 2.45 4.53.98.997 74 27 45 -.17 8.7-63.4-6.88.58 -.22-31.18.6 -.21 4.53 1.5.981 19 74 27 45-1.16 77.5 64.23 1.7 1..72 5.1 1..72 4.69 1. 1.1 74 3.5 4-4.57 76.45 55.4 1.81 1.45 2.32 1.68 1.43 2.32.93.99.999 111 74 3.5 45 4.57 78.55-6.3 1.81.56 -.26 1.68.58 -.26.93 1.3.992 112 74 3.5 45-3.94 77.5 63.9 2.39 1..72 2.25 1..72.94 1. 1. 74 34 4 18.94 71.71 36.29 2.42 1.32 2.22 2.35 1.3 2.22.97.99.999 114 74 34 45-18.9 83.29-65.3 2.42.73.16 2.35.74.17.97 1.2 1.8 115 74 34 45 19.18 77.5 65.14 3.46 1..66 3.35 1..66.97 1. 1.1

Chapter-IV 23 Member No Table 7.6: Translational stress resultants in columns in frame-raft-soil interaction analysis X-coordinate Y-coordinate Z-coordinate Position of memeber Non-interactive analsysis F x1 F y1 F z1 Uncoupled Vs Non-interactive Fx2/Fx1 Fy2/Fy1 Fz2/ Fz1 Coupled Vs Non-nteractive Fx3/Fx1 Fy3/Fy1 Fz2 / Fz1 Coupled vs Uncoupled (1) (2) (3) (4) (5) (6) (7) (8) (9) (1) (11) (12) (13) (14) (15) (16) (17) 153 64 2 4 591.1 7.74-7.66 1.33 4.12 3.76 1.34 6.14 5.61 1. 1.49 1.49 154 69 2 4 929.3 -.36-7.66 1.4-57. 4.49 1.4-87.97 6.86 1. 1.54 1.53 155 74 2 4 916.4.12-7.66 1.12 49.8 4.89 1.12 75.58 7.46 1. 1.54 1.52 159 64 23.5 4 446.1 14.45-14.5 1.31 2.29 2.17 1.3 2.6 1.92.99.9.89 16 69 23.5 4 696.3 -.53-14.5 1.4-38.51 2.4 1.4-36.75 2.1 1..95.87 161 74 23.5 4 689.3 -.3-14.5 1.11-146.33 2.5 1.11-143. 2.17.99.98.87 165 64 27 4 296.2 14.47-14.4 1.29 2.36 2.23 1.28 2.32 2.2.99.98.98 166 69 27 4 464.2.73-14.4 1.3 29.49 2.49 1.3 28.42 2.45 1..96.98 167 74 27 4 459.1 -.19-14.4 1.11-25.26 2.61 1.1-25.5 2.57.99.99.98 171 64 3.5 4 143.3 18.74-18.9 1.24 2.19 2.1 1.23 2.14 2.4.99.98.98 172 69 3.5 4 232.6 -.2-18.9 1.3-18.75 2.31 1.3-15.45 2.25 1..97.97 173 74 3.5 4 Exterior columns 226.8 -.31-18.9 1.9-15.48 2.42 1.9-15.42 2.35 1. 1..97 177 64 2 45 929.9 7.74.51 1.2.85-6.4 1.3 4.71-23.8 1. 1.52 1.54 178 69 2 45 1268 -.36.51.86-5.14-9.33.83-65.19-28.5 1. 1.57 1.57 179 74 2 45 1255.12.51.9-4.5-1.6.88 55.75-31.1 1. 1.55 1.58 183 64 23.5 45 697.4 14.45.45 1.2 2.61-28.5 1.3 2.52-32.5 1..89.93 184 69 23.5 45 947.7 -.53.45.86-41.5-35.1.84-46.23-39.8 1.1.95.93 185 74 23.5 45 94.6 -.3.45.9-172. -37.9.89-178.7-43.2 1.1.96.93 189 64 27 45 464.9 14.47 -.87 1.2 2.39 15.3 1.2 2.59 17.41 1..98.95 19 69 27 45 632.9.73 -.87.87 31.7 18.4.85 34.7 2.82 1.1.96.95 191 74 27 45 627.8 -.19 -.87.91-27.2 19.8.89-3.58 22.41 1..98.95 195 64 3.5 45 232.4 18.74 -.24 1.1 2.23 57.5 1.2 2.38 66.21 1..98.96 196 69 3.5 45 321.7 -.2 -.24.89-113 69.6.88-128.5 79.5 1.1.97.96 197 74 3.5 45 Interior Columns 315.9 -.31 -.24.93-16.6 75.2.92-18.77 85.96 1..98.96 Fx3/Fx2 Fy3/Fy2 Fz3/ Fz2

Chapter-IV Non-linear Dynamic analysis of Soil Structure Interaction of Three Dimensional Structure For Varied Soil conditions 24 Table 7.7:: End moments in columns in frame-raft -soil interaction analysis Member No X-coordinate Y-coordinate Z-coordinate Position of memeber Non-interactive analsysis My 1 Mz 1 Uncoupled Vs Non-interactive My2/ My1 Mz2/ Mz1 Coupled Vs Non-nteractive My3/ My1 Mz3/ Mz1 Coupled vs Uncoupled My3/ My2 Mz3/ Mz2 Member No X-coordinate Y-coordinate Z-coordinate Position of memeber Non-interactive analsysis Uncoupled Vs Non-interactive My2/ My1 Mz2/ Mz1 Coupled Vs Non-nteractive My3/ My1 Mz3/ Mz1 Coupled vs Uncoupled My3/ My2 Mz3/ Mz2 (1) (2) (3) (4) (5) (6) (7) (8) (9) (1) (11) (12) (13) (1) (2) (3) (4) (5) (6) (7) (8) (9) (1) (11) (12) (13) 153 64 2 4 9.3 9.18 4.465 4.928 7.44 8.15 1.666 1.654 177 64 2 45 -.57 9.18-29.1 5.685-49.8 9.617 1.71 1.692 153 64 23.5 4 17.77 17.9 3.47 3.79 4.684 5.112 1.375 1.378 177 64 23.5 45-1.23 17.9-21 4.22-3 5.936 1.429 1.47 154 69 2 4 9.3 -.32 5.435-92.4 9.229-155 1.698 1.68 178 69 2 45 -.57 -.32-34.7-15 -61.2-18 1.763 1.719 154 69 23.5 4 17.77 -.92 4.2-45.9 5.656-66.5 1.47 1.448 178 69 23.5 45-1.23 -.92-25.3-52.9-36.9-77.4 1.455 1.463 155 74 2 4 9.3.17 5.977 52.94 1.1 86.71 1.69 1.638 179 74 2 45 -.57.17-37.9 59.53-67 98.82 1.769 1.66 155 74 23.5 4 17.77.26 4.343 44.65 6.119 65.42 1.49 1.465 179 74 23.5 45-1.23.26-27.6 51.15-4.2 75.4 1.457 1.467 159 64 23.5 4 25.6 25.43 2.119 2.243 1.778 1.913.839.853 183 64 23.5 45-1.22 25.43-21 2.464-19.3 2.77.92.843 159 64 27 4 25.27 25.14 2.213 2.346 2.61 2.218.931.945 183 64 27 45 -.37 25.14-69.1 2.583-65.1 2.434.941.942 16 69 23.5 4 25.6-1.3 2.338-27.2 1.921-25.5.822.937 184 69 23.5 45-1.22-1.3-25.9-32.8-23.8-3.7.921.936 16 69 27 4 25.27 -.57 2.459-63.2 2.275-61.4.925.971 184 69 27 45 -.37 -.57-84.3-75.5-79.5-73.2.943.97 161 74 23.5 4 25.6.2 2.427 371 1.973 35.5.813.945 185 74 23.5 45-1.22.2-28.1 455-25.8 424.5.918.933 161 74 27 4 25.27 -.14 2.571-56.6 2.369-57.1.921 1.8 185 74 27 45 -.37 -.14-91.3-68.9-85.9-68.5.941.995 165 64 27 4 25.22 25.37 2.251 2.374 2.226 2.342.989.987 189 64 27 45 1.22 25.37 21.52 2.614 2.45 2.578.95.986 165 64 3.5 4 25.5 25.26 2.216 2.343 2.171 2.296.979.98 189 64 3.5 45 1.83 25.26 14.63 2.577 13.97 2.523.955.979 166 69 27 4 25.22 1.1 2.59 37.3 2.482 35.64.989.963 19 69 27 45 1.22 1.1 25.78 43.64 24.53 41.92.952.961 166 69 3.5 4 25.5 1.55 2.466 24.48 2.414 23.65.979.966 19 69 3.5 45 1.83 1.55 17.44 28.76 16.68 27.72.956.964 167 74 27 4 25.22 -.29 2.634-29 2.62-28.8.988.993 191 74 27 45 1.22 -.29 27.78-35.1 26.39-34.4.95.979 167 74 3.5 4 25.5 -.39 2.586-21.5 2.527-21.3.977.989 191 74 3.5 45 1.83 -.39 18.78-26 17.92-25.4.954.976 171 64 3.5 4 29.99 29.65 2.118 2.213 2.65 2.163.975.978 195 64 3.5 45.97 29.65 28.13 2.417 26.95 2.361.958.977 171 64 34 4 36.29 35.93 2.76 2.167 2.27 2.121.976.979 195 64 34 45 -.13 35.93-218 2.357-28 2.35.955.978 172 69 3.5 4 29.99.12 2.34 39.3 2.279 3.2.974.97 196 69 3.5 45.97.12 33.68 364.9 32.32 353.3.96.968 172 69 34 4 36.29 -.83 2.285-47 2.229-45.5.975.969 196 69 34 45 -.13 -.83-262 -55.6-251 -53.8.957.967 173 74 3.5 4 29.99 -.64 2.45-12.5 2.381-12.5.972.996 197 74 3.5 45.97 -.64 36.37-15.2 34.85-14.9.958.982 173 74 34 4 36.29 -.46 2.391-19.1 2.327-19.1.973.999 197 74 34 45 -.13 -.46-284 -23.1-271 -22.7.956.983

Chapter-IV 25 Table 7.8: Comparison of responses in uncoupled and coupled, non-linear analysis of frame-isolated footing -soil system Response Un-coupled analysis coupled analysis coupled analysis/ uncoupled analysis 1 Maximum displacement - -.9795 Sl No 2 Maximum Differential settlement - -.9737 3 Min/Min to Max/Max (Longitudinal -.641 to -.754 to.498 to.85 stress).51.129 4 Min/Min to Max/Max (Vertical stress) -.4 to.37 -.5 to.37 1 to.8 5 Min/Min to Max/Max (Transverse stress) -.1198 to.5582 -.1373 to.5946.939 to.872 Table 7.9: Comparison of stress resultants in uncoupled and coupled, non-linear analysis of frame-isolated footing -soil system Un-coupled coupled Sl No Response analysis/ Non-interactive analysis analysis/ Non-interactive analysis coupled analysis/ uncoupled analysis 1 Axial forces in X-beams.827 to 4.176.825 to 4.56.967 to 1.9 2 Axial forces in Z-Beams 2.96 to 3.46 2.45 to 3.349.967 to.976 3 Axial forces in columns.827 to 1.334.826 to 1.339.988 to 1.9 4 Shear forces in X-beams.551 to 1.511.471 to 1.63.854 to 1.6 5 Shear forces in Z-Beams.726 to 1.318.739 to 1.33.989 to 1.17 6 Shear forces in columns Y 2.187 to 2.594 2.6 to 2.54.893 to.983 7 Shear forces in columns Z 2.9 to 2.61 1.91 to 2.56.868 to.984 8 Moments in X Beams -.335 to 3.29 -.492 to 3.681.179 to 2.755 9 Moments in Z Beams -.448 to 3.219 -.523 to 3.672 -.578 to 5.521 1 Moments in columns about Y -axis 2.76 to 3.47 1.778 to 4.684.839 to 1.375 11 Moments in columns about Z-axis 2..167 to 4.219 1.913 to 5.936.843 to 1.46

1 Z ( Transverse direction) X (Longitudinal direction) L Plan B 2 Figure7.1. Structure-footing-soil system Y (Vertical direction) Elevation X (Longitudinal direction) 1 2 2 A A B C C B Footing discretization Figure7.2. Details of FEM Model for Frame isolated Footings Soil Interaction Analysis

2 Y 16, 36 17, 37 18, 38 171,196 172,197 173,198 11, 31 12, 32 13, 33 89,13 88,12 195,196,197 171,172,173 Z 166,19 167,191 165,189 6, 26 7, 27 8, 28 X 159,183 16,184 161,185 88,1 85, 97 87, 99 189,19,191 84, 96 165,166,167 159,16,161 1, 21, 2, 22 3, 33 183,184,185 2 1 A B C 153,177 153,18 A 154,18 154,178 155,179 B A 155,18 82, 94 C 1 2 81, 93 177,178,179 153,154,155 Figure 7.3. Member numbers for Quarter frame

3 D 2 D 3 D 4 D 1 Load in KN 2 4 6 8 1 12 Plate element -5 D 6 D 2 D 5 D 4 D 1 D Displacement in mm -1-15 -2 D 8 D 7 D 3-25 At A1 At B1 At C1 At A2 At B2 At C2 element Column Brick element Figure 7.4: Degrees of freedom of Figure 7.5. Load displacement curves for plate load test

4 L o n g i t u d i n a l D i s p l a c e m e n t s 5 4 Y/D D i s p l a c e m e n t s.75.6-3. 2-2. 7-2. 2-1. 7-1. 2 -. 7 -. 2. 3. 8 1. 3 1. 8 2. 3 2. 8-1 3 2 1-2 - 3-4. 5 6, - 4. 4-5 X / L Figure 7.6. Horizontal displacements in longitudinal directions in uncoupled analysis. T r a n s v e r s e D i s p l a c e m e n t s 6 4 Y/D.75 D i s p l a c e m - 3. 2-2. 7-2. 2-1. 7-1. 2 -. 7 -. 2. 3. 8 1. 3 -. 7 3 1. 8 2. 3 2. 8 2.6-2 - 4. 6, - 4. 8 5-6 Z / B Figure7.7. Horizontal displacements in transverse directions in uncoupled analysis. V e r t i c a l D i s p l a c e m e n t s 5-3. 2-2. 7-2. 2-1. 7-1. 2 -. 7 -. 2. 3. 8 1. 3 1. 8 2. 3 2. 8 D i s p l a c e m e n t s - 5-1 - 1 5. 5 6, - 1 2. 4 3 Y/D.75.6. 4, - 1 9. 9-2 - 2 5 X / L Figure7.8. Vertical displacements in uncoupled analysis..45.3.15.5

5 L o n g i t u d i n a l D i s p l a c e m e n t s 4 3 2 1-3. 2-2. 7-2. 2-1. 7-1. 2 -. 7 -. 2. 3. 8 1. 3 1. 8 2. 3 2. 8 D i s p l a c e m e n t s - 1-2 - 3. 8 2, - 3. 4 7-4 X / L Figure 7.9. Horizontal displacements in (mm) longitudinal directions in coupled analysis. T r a n s v e r s e d i s p l a c e m e n t s 5 4 3 2 1 Y/D.75.6-3. 2-2. 7-2. 2-1. 7-1. 2 -. 7 -. 2. 3. 8 -. 1 6 2. 3 1. 8 2. 3.45 2. 8-1 - 2-3 - 4 1. 3 3 3 3 3 3 3 3, - 4. 7.3.15-5 Z / B Figure7.1. Horizontal displacements in transverse directions in coupled analysis..5 V e r t i c a l D i s p l a c e m e n t s 5-3. 2-2. 7-2. 2-1. 7-1. 2 -. 7 -. 2. 3. 8 1. 3 1. 8 2. 3 2. 8 D i s p l a c e m e n t s - 5-1 - 1 5. 5 6, - 1 2. 6 5 Y/D.75.6-2. 1 6, - 1 9. 4 9.45-2 5 X / L Figure 7.11. Vertical displacements in coupled analysis.3.15.5

6 L o n g itu d in a l s tr e s s. 1 5. 1 2 5. 1. 7 5 Y/D.75.6.5 5,.4 7,. 7 5 9 3 7 3 S tr e s s in N /m m 2-1.3-1.2-1.1-1 -.9 -.8 -.7 -.6 -.5 -.4 -.3. 5. 2 5 `` -.2 -.1 -. 2 5.1.2.3.4.5.6.7-2.9-2.58-2.26-1.94-1.62-1.3 -.98 -.69 -.53 -.47 -.4 -.33 -.27 -.2 -.13 -.7.8.9.7 1.13.2.27.33.4.47.53.69.98 1.3 1.62 1.94 2.26 2.58 2.9 1.1 1.2 1.3 -. 5 -. 7 5.1 1 6 7, -.5 3, -. 1 2 8 5 1 9 5 -. 1 -. 1 2 5 Z/B -. 1 5 X /L.8 1 6 7.5 5.4 5.3 3 3.2 1 6 7.1 1 6 7 Figure7.12. Longitudinal stress (in MPa) at ground level in uncoupled analysis -.65--.6 -.6--.55 -.55--.5 -.5--.45 -.45--.4 -.4--.35 -.35--.3 -.3--.25 -.25--.2 -.2--.15 -.15--.1 -.1--.5 -.5- -.5.5-.1.1-.15.15-.2.2-.25.25-.3.3-.35.35-.4.4-.45.45-.5 Figure 13: Longitudinal Stress (in MPa) contours at section Z/B=. in uncoupled analysis Longitudinal Stress -2.9-2.58-2.26-1.94-1.62-1.3 -.98 -.69 -.53 -.47 -.4 -.33 -.27 -.2 -.13 -.7 X/L.7.13.2.27.33.4.47.53.69.98 1.3 1.62 1.94 2.26 2.58 2.9 -.55--.5 -.5--.45 -.45--.4 -.4--.35 -.35--.3 -.3--.25 -.25--.2 -.2--.15 -.15--.1 -.1--.5 -.5- -.5.5-.1.1-.15.15-.2.2-.25.25-.3.3-.35.35-.4.4-.45.45-.5.5-.55 Figure7.14 : Longitudinal Stress (in MPa) contours at section Z/B=.116 in uncoupled analysis

7 `` -.125--.1 -.1--.75 -.75--.5 -.5--.25 -.25- -.25.25-.5.5-.75.75-.1 Figure7.15. Longitudinal stress (in MPa) contours at ground level in uncoupled analysis Longitudinal stress.75-.1.1.5-.75.75.5.25-.5 2 Stress in N/mm.25 -.25 -.5 -.75 `` -.25 -.25- -.5--.25 -.1 -.125-2.9-2.26-1.62 -.98 -.53 -.4 -.27 -.13.13.27.4 X/L.53.98 1.62 2.26 2.9 -.75--.5 -.1--.75 -.125--.1 Figure7.16. Variation of longitudinal stress (in MPa) at ground level in uncoupled analysis

8 Vertical stress.15.1.5-1.3-1.2-1.1-1 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 -.5 -.47, -.7853519 -.1.1.2.3.4.5.6.7.8.9 1 1.1 1.2 1.3 Stress in N/mm -.15 `` -.47, -.19933998 -.2, -.2393931 -.25 -.3 -.35, -.3942789 -.4 -.45 -.33, -.518361 -.5 -.55 -.6 -.65 X/L.8167.55.45.333.2167.1167 Figure 7.17. Vertical stress (in MPa) at ground level in uncoupled analysis Vertical Stress -2.9-2.58-2.26-1.94-1.62-1.3 -.98 -.69 -.53 -.47 -.4 -.33 -.27 -.2 -.13 -.7 X/L.7.13.2.27.33.4.47.53.69.98 1.3 1.62 1.94 2.26 2.58 2.9 -.44--.43 -.43--.42 -.42--.41 -.41--.4 -.4--.39 -.39--.38 -.38--.37 -.37--.36 -.36--.35 -.35--.34 -.34--.33 -.33--.32 -.32--.31 -.31--.3 -.3--.29 -.29--.28 -.28--.27 -.27--.26 -.26--.25 -.25--.24 -.24--.23 -.23--.22 -.22--.21 -.21--.2 -.2--.19 -.19--.18 -.18--.17 -.17--.16 -.16--.15 -.15--.14 -.14--.13 -.13--.12 -.12--.11 -.11--.1 -.1--.9 -.9--.8 -.8--.7 -.7--.6 -.6--.5 -.5--.4 -.4--.3 -.3--.2 -.2--.1 -.1--6.94E-18-6.94E-18-.1.1-.2.2-.3.3-.4 Figure7.18. Vertical stress (in MPa)contours Vertical Stress at section Z/B=. in uncoupled analysis -2.9-2.58-2.26-1.94-1.62-1.3 -.98 -.69 -.53 -.47 -.4 -.33 -.27 -.2 -.13 -.7 X/L -.51--.5 -.5--.49 -.49--.48 -.48--.47 -.47--.46 -.46--.45 -.45--.44 -.44--.43 -.43--.42 -.42--.41 -.41--.4 -.4--.39 -.39--.38 -.38--.37 -.37--.36 -.36--.35 -.35--.34 -.34--.33 -.33--.32 -.32--.31 -.31--.3 -.3--.29 -.29--.28 -.28--.27 -.27--.26 -.26--.25 -.25--.24 -.24--.23 -.23--.22 -.22--.21 -.21--.2 -.2--.19 -.19--.18 -.18--.17 -.17--.16 -.16--.15 -.15--.14 -.14--.13 -.13--.12 -.12--.11 -.11--.1 -.1--.9 -.9--.8 -.8--.7 -.7--.6 -.6--.5 -.5--.4 -.4--.3 -.3--.2 -.2--.1 -.1- -.1.1-.2.2-.3.3-.4.7.13.2.27.33.4.47.53.69.98 1.3 1.62 1.94 2.26 2.58 2.9 Figure7.19. Vertical stress (in MPa)contours at section Z/B=.116 in uncoupled analysis

9 `` -.55--.5 -.5--.45 -.45--.4 -.4--.35 -.35--.3 -.3--.25 -.25--.2 -.2--.15 -.15--.1 -.1--.5 -.5- -.5 Figure7.2. Vertical stress (in MPa) contours at ground level in uncoupled analysis V e rtic a l S tre s s -. 5.5 -. 5 - -. 1 --. 5 -.5 -.1 -. 1 5 --. 1 S tre s s in N /m m -.15 -.2 -.25 -.3 -.35 `` -. 2 --. 1 5 -. 2 5 --. 2 -. 3 --. 2 5 -.4 -.45 -.5 -.55-2.9-2.26-1.62 -.98 -.53 -.4 -.27 -.13 X /L.13.27.4.53.98 1.62 2.26 2.9 -. 3 5 --. 3 -. 4 --. 3 5 S 16 -. 4 5 --. 4 Z /B -. 5 --. 4 5 S 1 -. 5 5 --. 5 Figure7.21. Variation of vertical stress (in MPa) at ground level in uncoupled analysis

1 Transverse stress.15.125.1.75.5 Stress in N /m m -1.3-1.2-1.1-1 -.9 -.8 -.7 -.6 -.5 -.4 -.3.25 -.2 -.1 -.25.1.2.3.4.5.6.7.8.9 1 1.1 1.2 1.3 -.5 -.75 -.1 -.125 -.15 Z/B.8167.5.47.33.2.13 Figure7.22. Transverse stress (in MPa) at ground level in uncoupled analysis Transverse Stress @ X /L=. -2.9-2.3 6 7-1.8 3 3-1.3 -.8 1 7 -.5 5 -.4 5 -.3 3 3 -.2 1 7 -.1 1 7..1 1 7.2 1 7.3 3 3.4 5.5 5.8 1 7 1.3 1.8 3 3 2.3 6 7 2.9 Z /B -. 9 --. 8 5 -. 8 5 --. 8 -. 8 --. 7 5 -. 7 5 --. 7 -. 7 --. 6 5 -. 6 5 --. 6 -. 6 --. 5 5 -. 5 5 --. 5 -. 5 --. 4 5 -. 4 5 --. 4 -. 4 --. 3 5 -. 3 5 --. 3 -. 3 --. 2 5 -. 2 5 --. 2 -. 2 --. 1 5 -. 1 5 --. 1 -. 1 --. 5 -. 5 - -. 5. 5 -. 1. 1 -. 1 5. 1 5 -. 2. 2 -. 2 5. 2 5 -. 3. 3 -. 3 5. 3 5 -. 4. 4 -. 4 5. 4 5 -. 5. 5 -. 5 5. 5 5 -. 6 Figure7.23. Transverse stress (in MPa) contours at section Z/B=. in uncoupled analysis Transverse S tress @ X /L=.13-2.9-2.3 6 7-1.8 3 3-1.3 -.8 1 7 -.5 5 -.4 5 -.3 3 3 -.2 1 7 -.1 1 7. Z /B -. 1 3 5 --. 1 3 -. 1 3 --. 1 2 5 -. 1 2 5 --. 1 2 -. 1 2 --. 1 1 5 -. 1 1 5 --. 1 1 -. 1 1 --. 1 5 -. 1 5 --. 1 -. 1 --. 9 5 -. 9 5 --. 9 -. 9 --. 8 5 -. 8 5 --. 8 -. 8 --. 7 5 -. 7 5 --. 7 -. 7 --. 6 5 -. 6 5 --. 6 -. 6 --. 5 5 -. 5 5 --. 5 -. 5 --. 4 5 -. 4 5 --. 4 -. 4 --. 3 5 -. 3 5 --. 3 -. 3 --. 2 5 -. 2 5 --. 2 -. 2 --. 1 5 -. 1 5 --. 1 -. 1 --. 5 -. 5 - -. 5. 5 -. 1. 1 -. 1 5. 1 5 -. 2. 2 -. 2 5. 2 5 -. 3. 3 -. 3 5. 3 5 -. 4. 4 -. 4 5. 4 5 -. 5. 5 -. 5 5. 5 5 -. 6 Figure7.24. transverse stress (in MPa)contours at section Z/B=.13 in uncoupled analysis.1 1 7.2 1 7.3 3 3.4 5.5 5.8 1 7 1.3 1.8 3 3 2.3 6 7 2.9

11 `` -.15--.125 -.125--.1 -.1--.75 -.75--.5 -.5--.25 -.25--1.735E-18-1.735E-18-.25.25-.5.5-.75 Figure7.25. Transverse stress (in MPa) contours at ground level in uncoupled analysis T ra n s v e rs e S tre s s. 5 -. 7 5. 7 5. 2 5 -. 5. 5-1.7 3 E -1 8 -. 2 5. 2 5 -. 2 5 --1.7 3 E -1 8 `` -. 2 5 -. 5 S tre s s in N /m m -. 5 --. 2 5 -. 7 5 --. 5 -. 7 5 -. 1 -. 1 --. 7 5 -. 1 2 5 -. 1 2 5 --. 1 2.9 Z /B -2.9 -. 1 5 -. 1 5 --. 1 2 5 Figure7.26. Variation of vertical stress (in MPa) at ground level in uncoupled analysis

12 Longitudinal Stress.15.125.1.1167, -.98,.512127.75.5 Stress in N/mm -1.3-1.2-1.1-1 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2.25 -.1 -.25.1.2.3.4.5.6.7.8.9 1 1.1 1.2 1.3 -.5 -.75 -.1.2167,.2, -.6413929 -.125 -.15 X/L.8167.55.45.333.2167.1167 Longitudinal Stress Figure7.27. Longitudinal stress (in MPa) at ground level in coupled analysis -2.9-2.58-2.26-1.94-1.62-1.3 -.98 -.69 -.53 -.47 -.4 -.33 -.27 -.2 -.13 -.7 X/L.7.13.2.27.33.4.47.53.69.98 1.3 1.62 1.94 2.26 2.58 2.9 -.5--.45 -.45--.4 -.4--.35 -.35--.3 -.3--.25 -.25--.2 -.2--.15 -.15--.1 -.1--.5 -.5--8.674E-19-8.674E-19-.5.5-.1.1-.15.15-.2.2-.25.25-.3.3-.35.35-.4.4-.45.45-.5.5-.55 Figure 7.28. Longitudinal Stress (in MPa) Longitudinal contours Stress at section Z/B=. in coupled analysis -2.9-2.58-2.26-1.94-1.62-1.3 -.98 -.69 -.53 -.47 -.4 -.33 -.27 -.2 -.13 -.7 X/L.7.13.2.27.33.4.47.53.69.98 1.3 1.62 1.94 2.26 2.58 2.9 -.65--.6 -.6--.55 -.55--.5 -.5--.45 -.45--.4 -.4--.35 -.35--.3 -.3--.25 -.25--.2 -.2--.15 -.15--.1 -.1--.5 -.5- -.5.5-.1.1-.15.15-.2.2-.25.25-.3.3-.35.35-.4.4-.45.45-.5 Figure7.29. Longitudinal Stress (in MPa) contours at section Z/B=. in coupled analysis

13 `` -.75--.5 -.5--.25 -.25- -.25.25-.5.5-.75 Figure7.3. Longitudinal stress (in MPa) contours at ground level in coupled analysis Longitudinal Stress.5-.75.75.5.25-.5 Stress in N/mm.25 -.25 `` -.25 -.25- -.5 -.75 -.5--.25-2.9-1.3 -.4 X/L -.7.27.69 2.26 -.75--.5 Figure7.31. Variation of longitudinal stress (in MPa) at ground level in uncoupled analysis

14 Vertival Stress.15.1.5-1.3-1.2-1.1-1 -.9 -.8 -.7 -.6 -.5 -.4 -.3 -.2 -.1 -.5 -.47, -.733485 -.1.1.2.3.4.5.6.7.8.9 1 1.1 1.2 1.3 Stress N/mm -.15 -.2 -.47, -.28212298 -.25 -.3 -.35 -.4 -.45, -.22891776 ``, -.33512566 -.33, -.47763923 -.5 -.55 -.6 -.65 X/L Figure7.32. Vertical stress (in MPa) at ground level in coupled analysis Vertical stress @ Z/L=. -2.9-2.58-2.26-1.94-1.62-1.3 -.98 -.69 -.53 -.47 -.4 -.33 -.27 -.2 -.13 -.7 X/L.7.13.2.27.33.4.47.53.69.98 1.3 1.62 1.94 2.26 2.58 2.9 -.43--.42 -.42--.41 -.41--.4 -.4--.39 -.39--.38 -.38--.37 -.37--.36 -.36--.35 -.35--.34 -.34--.33 -.33--.32 -.32--.31 -.31--.3 -.3--.29 -.29--.28 -.28--.27 -.27--.26 -.26--.25 -.25--.24 -.24--.23 -.23--.22 -.22--.21 -.21--.2 -.2--.19 -.19--.18 -.18--.17 -.17--.16 -.16--.15 -.15--.14 -.14--.13 -.13--.12 -.12--.11 -.11--.1 -.1--.9 -.9--.8 -.8--.7 -.7--.6 -.6--.5 -.5--.4 -.4--.3 -.3--.2 -.2--.1 -.1- -.1.1-.2.2-.3 Figure7.33. Vertical stress (in MPa) contours at section Z/B=. in coupled analysis Vertical stress @ Z/L=.2167-2.9-2.58-2.26-1.94-1.62-1.3 -.98 -.69 -.53 -.47 -.4 -.33 -.27 -.2 -.13 -.7 X/L -.48--.47 -.47--.46 -.46--.45 -.45--.44 -.44--.43 -.43--.42 -.42--.41 -.41--.4 -.4--.39 -.39--.38 -.38--.37 -.37--.36 -.36--.35 -.35--.34 -.34--.33 -.33--.32 -.32--.31 -.31--.3 -.3--.29 -.29--.28 -.28--.27 -.27--.26 -.26--.25 -.25--.24 -.24--.23 -.23--.22 -.22--.21 -.21--.2 -.2--.19 -.19--.18 -.18--.17 -.17--.16 -.16--.15 -.15--.14 -.14--.13 -.13--.12 -.12--.11 -.11--.1 -.1--.9 -.9--.8 -.8--.7 -.7--.6 -.6--.5 -.5--.4 -.4--.3 -.3--.2 -.2--.1 -.1- -.1.1-.2.2-.3.3-.4 Figure7.34. Vertical stress (in MPa)contours at section Z/B=.2167 in coupled analysis.7.13.2.27.33.4.47.53.69.98 1.3 1.62 1.94 2.26 2.58 2.9

15 `` -.5--.45 -.45--.4 -.4--.35 -.35--.3 -.3--.25 -.25--.2 -.2--.15 -.15--.1 -.1--.5 -.5- -.5 Figure7.35. Vertical stress (in MPa)contours at ground level in coupled analysis Vertical Stress -.5.5 -.5- -.5 -.1--.5 Stress in N/mm -.1 -.15 -.2 -.25 -.3 `` -.15--.1 -.2--.15 -.25--.2 -.35 -.4 -.3--.25 -.45 -.5-2.9-1.3 -.4 X/L -.7.27.69 2.26-2.9 Z/B 2.9 -.35--.3 -.4--.35 -.45--.4 -.5--.45 Figure7.35. Variation of vertical stress (in MPa)at ground level in coupled analysis

16 Transverse stress. 1 5. 1 2 5. 1. 7 5. 5 S tre s s in N /m -1.3-1.2-1.1-1 -.9 -.8 -.7 -.6 -.5 -.4 -.3. 2 5 -.2 -.1 -. 2 5.1.2.3.4.5.6.7.8.9 1 1.1 1.2 1.3 -. 5 -. 7 5 -. 1 -. 1 2 5 -. 1 5 Z /B.8 1 6 7.5.4 7.3 3.2.1 3 Figure7.37. Transverse stress (in Transverse MPa) at stress ground level in coupled analysis Z /B -2.9-2.3 7-1.8 3-1.3 -.8 2 -.5 5 -.4 5 -.3 3 -.2 2 -.1 2. -. 1 1 5 --. 1 1 -. 1 1 --. 1 5 -. 1 5 --. 1 -. 1 --. 9 5 -. 9 5 --. 9 -. 9 --. 8 5 -. 8 5 --. 8 -. 8 --. 7 5 -. 7 5 --. 7 -. 7 --. 6 5 -. 6 5 --. 6 -. 6 --. 5 5 -. 5 5 --. 5 -. 5 --. 4 5 -. 4 5 --. 4 -. 4 --. 3 5 -. 3 5 --. 3 -. 3 --. 2 5 -. 2 5 --. 2 -. 2 --. 1 5 -. 1 5 --. 1 -. 1 --. 5 -. 5 - -. 5. 5 -. 1. 1 -. 1 5. 1 5 -. 2. 2 -. 2 5. 2 5 -. 3. 3 -. 3 5. 3 5 -. 4. 4 -. 4 5. 4 5 -. 5. 5 -. 5 5. 5 5 -. 6.1 2 Transverse stress Figure7.38. Transverse stress (in MPa) contours at section Z/B=. in coupled analysis.2 2.3 3.4 5.5 5.8 2 1.3 1.8 3 2.3 7 2.9 Z /B -2.9-2.3 7-1.8 3-1.3 -.8 2 -.5 5 -.4 5 -.3 3 -.2 2 -.1 2..1 2.2 2.3 3.4 5.5 5.8 2 1.3 1.8 3 2.3 7 2.9 -. 1 2 --. 1 1 5 -. 1 1 5 --. 1 1 -. 1 1 --. 1 5 -. 1 5 --. 1 -. 1 --. 9 5 -. 9 5 --. 9 -. 9 --. 8 5 -. 8 5 --. 8 -. 8 --. 7 5 -. 7 5 --. 7 -. 7 --. 6 5 -. 6 5 --. 6 -. 6 --. 5 5 -. 5 5 --. 5 -. 5 --. 4 5 -. 4 5 --. 4 -. 4 --. 3 5 -. 3 5 --. 3 -. 3 --. 2 5 -. 2 5 --. 2 -. 2 --. 1 5 -. 1 5 --. 1 -. 1 --. 5 -. 5 - -. 5. 5 -. 1. 1 -. 1 5. 1 5 -. 2. 2 -. 2 5. 2 5 -. 3. 3 -. 3 5. 3 5 -. 4. 4 -. 4 5. 4 5 -. 5. 5 -. 5 5 Figure7.39. Transverse stress (in MPa) contours at section Z/B=.2167 in coupled analysis

`` -.125--.1 -.1--.75 -.75--.5 -.5--.25 -.25- -.25.25-.5.5-.75 Figure7.4. Transverse stress (in MPa) contours at ground level in coupled analysis Transverse stress.5-.75.75.25-.5.5.25 -.25 `` -.25 -.5 Stress in N/mm -.25- -.5--.25 X/L -.7.27.692.26-2.9-1.3-.4 2.9 Z/B -.75 -.1 -.125-2.9 -.75--.5 -.1--.75 -.125--.1 Figure7.41. Variation of vertical stress (in MPa) at ground level in coupled analysis 17