Handbook of Civil Engineering Calculations

Calculation Procedure:

1. Identify the "repeating group" of piles
   The concrete footing (Fig. 350) binds the piles, causing the surface along the top of the
   piles to remain a plane as bending occurs. Therefore, the pile group may be regarded as a
   structural member subjected to axial load and bending, the cross section of the member
   being the aggregate of the cross sections of the piles.
   Indicate the "repeating group" as shown in Fig. 35Z>.


2. Determine the area of the pile group and the moment of inertia
   Calculate the area of the pile group, locate its centroidal axis, and find the moment of in-
   ertia. Since all the piles have the same area, set the area of a single pile equal to unity.
   Then ,4 = 3+ 3+2 = 8.
   Take moments with respect to row A. Thus &t = 3(0) + 3(3) + 2(6); Jt = 2.625 ft
   (66.675 mm). Then /= 3(2.625)^2 + 3(0.375)^2 + 2(3.375)^2 = 43.9.


3. Compute the axial load and bending moment on the pile group
   The axial load P = 20,000(12) = 240,000 Ib (1067.5 JdST); then M= 240,000(3.25 - 2.625)
   = 150,000 lb-ft (203.4 kN-m).


4. Determine the pile load in each row
   Find the pile load in each row resulting from the combined axial load and moment. Thus,
   PIA = 240,000/8 = 30,000 Ib (133.4 kN) per pile; then MII= 150,000/43.9 = 3420. Also,
   pa = 30,000 - 3420(2.625) = 21,020 Ib (93.50 kN) per pile; pb = 30,000 + 3420(0.375) =
   31,280 Ib (139.13 kN) per pile; pc = 30,000 + 3420(3.375) = 41,540 Ib (184.76 kN) per
   pile.


5. Verify the above results
   Compute the total pile reaction, the moment of the applied load, and the pile reaction with
   respect to row A. Thus, R = 3(21,020) + 3(31,280) + 2(41,540) = 239,980 Ib (1067.43
   kN); then Ma = 240,000(3.25) = 780,000 lb-ft (1057.68 kN-m), and Mr = 3(31,280)(3) +
   2(41,540)(6) = 780,000 lb-ft (1057.68 kN-m). Since Ma = Mn the computed results are
   verified.

Deflection of Beams

In this handbook the slope of the elastic curve at a given section of a beam is denoted by

B, and the deflection, in inches, by y. The slope is considered positive if the section rotates

in a clockwise direction under the bending loads. A downward deflection is considered

positive. In all instances, the beam is understood to be prismatic, if nothing is stated to the

contrarv.

DOUBLE-INTEGRATION METHOD OF

DETERMINING BEAM DEFLECTION

The simply supported beam in Fig. 36 is subjected to a counterclockwise moment N ap-

plied at the right-hand support. Determine the slope of the elastic curve at each support

and the maximum deflection of the beam.