Case 3: Rigid pile cap with d increased from l m to 3.5m, such that d3 is increased by 43 times
Case 4: Rigid pile cap with v increased from 0.15 to 0.9669, such that 1/ (1 – v2) is increased by 50 times
It is noteworthy that v is usually limited by 0.5, and the exceptionally large value of  0.9669 in Case 4 is only an imaginary case for comparison purpose. The deformed shapes of the pile caps are shown in Fig. 2.19. It can be observed that although the flexural rigidity of the pile caps are increased by almost the same amount for Cases 2, 3 and 4, Cases 2 and 3 behave as a rigid pile cap whereas Case 4 behaves as a flexible pile cap, which allows bending and twisting in addition to rigid body displacement and rotation. This shows that the flexural rigidity obtained by Equation (2.9) cannot be used to distinguish between a flexible cap and a rigid cap.
 
Consider the cross-section 2-2 as shown in Fig. 2.16 for Cases 1, 2 and 3. The maximum shear stress along the 1-direction and that along the 3-direction are tabulated in Table 2.1, and the shear stress distribution for the three cases are plotted in Fig. 2.20. As shown in Table 2.1, the maximum shear stresses for Cases 1 and 2 are of a similar magnitude in both the 1- and 3-directions. For Case 3,however, both the maximum S21 and the maximum S23 are greatly reduced. This shows that increasing the rigidity of the pile cap by increasing the Young’s modulus does not alter the magnitude of the shear stresses along the cross-section 2-2, which is subjected to pure torsional moment; on the contrary, increasing the rigidity of the pile cap by increasing the cap depth can greatly reduce the shear stresses. This is reasonable on the ground that the total shear force along the cross-section 2-2 is calculated by the shear stresses multiplied by the total cross-sectional area. With an increase in the total cross-sectional area for Case 3, the shear stresses should be reduced.

The shear stress distribution graphs plotted in Fig. 2.20 are normalised by the maximum shear stress, either in the 1-direction or in 3-direction, for each case so that the relative magnitudes of the horizontal and the vertical shear stresses can be compared. Fig. 2.20(a) and Fig. 2.20(b) show a similar shear stress distribution, implying that a rigid cap with the cap rigidity increased by increasing the Young’s modulus results in a similar torsional moment distribution at the cross-section 2-2 as a flexible cap. Nevertheless, a rigid cap with the cap rigidity increased by thickening its depth shows a shear stress distribution, as shown in Fig. 2.20(c), similar to that in Fig. 2.18(b) rather than Fig. 2.17. This concludes that the shear stresses at the cross-section 2-2 for thickened rigid pile cap are “bending-induced” instead of “torsion-induced” in nature.

In a routine reinforced concrete design of a rigid cap, sufficient stirrups are provided to resist the bending-induced shear stress. Hence, there is no need to provide additional reinforcement to resist the “torsion” induced shear stress, which does not theoretically exist according to the assumption of a rigid cap.
 
2.6    SUMMARY
The rigid cap assumption and the flexible cap assumption for the pile cap design are usually referred to as the conventional rigid method and the approximate flexible method for the mat foundation design. However, the behaviours and the force transfer mechanism for a pile cap and a mat foundation are different. This is because the piles underneath the pile cap share most of the subjected loads whereas the rest of the loading is supported by the underlying soil, which is not the case for a mat foundation. Therefore, to compare the difference between the rigid cap assumption and the flexible cap assumption, the difference between the two methods for the mat foundation design should not be used and reanalysis is required.
The structural behaviours of a pile group with a rigid cap and that with a flexible cap are compared in Section 2.3. When subjected to vertical loading without bending moment, the rigid cap shares the load among all piles identically whereas the flexible cap results in a large out-of-plane deflection and uneven distribution of pile loads, due to the allowing of bending and twisting of the flexible pile cap. When subjected to vertical loading with a bending moment, the rigid pile cap remains a planar deformed shape and distributes efficiently the load among all piles, avoiding existence of some exceptionally large pile loads. The flexible pile cap, however, results in a nonlinear deformed shape, leading to negative pile loads, exceptionally large pile loads in the loaded area and the possibility of pile failure. Based on the difference in the structural behaviours, both the advantages and the drawbacks of the rigid cap assumption are discussed in Section 2.4,and it is concluded that the rigid cap assumption is preferentially used in pile foundation design. 高层建筑钢筋混凝土承台设计英文文献和翻译(7):http://www.751com.cn/fanyi/lunwen_1015.html