Therefore, substituting for y„ and .vn
P, = 166.7 - 35.4 x 1.67 + 33.3 x 1.0 - 140.9kN P2 = 166.7 - 35.4 x 1.67 - 33.3 x 1.0 = 74.3 kN Pi 166.7 + 35.4 x 0.33 + 33.3 x 1.0 = 211.7 kN P4 = 166.7 + 35.4 x 0.33 - 33.3 x 1.0 = 145.1 kN Ps = 166.7 + 35.4 x 1.33 + 33.3 x 1.0 = 247.1 kN P6 = 166.7 +35.4 x 1.33 - 33.3 x 1.0 = 180.5 kN
Total =999.6« lOOOkN
When a pile group is unsymmetrieal about both co-ordinate axes it is necessary to consider the theory of bending about the principal axes which is dealt with in most textbooks on strength of materials. In this case the formulae for the pile loads are N
Note that eKK is the eccentricity about the XX axis, while eyy is the eccentricity about the YY axis, as in figure 10.18.
Piled foundations are sometimes required to resist horizontal forces in addition to the vertical loads. If the horizontal forces are small they can often be resisted by the passive pressure of the soil against vertical piles, otherwise if the forces are not small then raking piles must be provided as shown in figure 10.19(a).
To determine the load in each pile either a static method or an elastic method is available. The static method is simply a graphical analysis using Bow's notation as illustrated in figure 10.19(b). This method assumes that the piles arc pinned at their ends so that the induced loads are axial. The elastic method lakes into account the displacements and rotations of the piles which may be considered pinned or fixed at their ends, The pile foundation is analysed in a similar manner to a plane frame or space frame and available computer programs are commonly used.
Forces in raking piles
The pile cap must he rigid and capable of transferring the column loads to the piles. It should have sufficient thickness for anchorage of the column dowels and the pile reinforcement, and it must be checked for punching shear, diagonal shear, bending and bond. Piles are rarely positioned at the exact locations shown on the drawings, therefore this must be allowed for when designing and detailing the pile cap.
Two methods of design are common: design using beam theory or design using a truss analogy approach. In the former case the pile cap is treated as an inverted beam and is designed for the usual conditions of bending and shear. The truss analogy method is used to determine the reinforcement requirements where the span-to-depth ratio is less than 2 such that beam theory is not appropriate.
In the trass analogy the force from the supported column is assumed to be transmitted by a triangular trass action with concrete providing the compressive members of the truss and steel reinforcement providing the tensile tic force as shown in the two-pile cap in figure 10.20(a). The upper node of the trass is located at the centre of the loaded area and the lower nodes at the intersection of the tensile reinforcement with the centrelines of the piles. Where the piles are spaced at a distance greater than three times the pile diameter only the reinforcement within a distance of 1.5 times the pile diameter from the centre of (he pile should be considered as effective in providing the tensile resistance within the truss.
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