Page 71 - Handbook of Civil Engineering Calculations, Second Edition

1.54 STRUCTURAL STEEL ENGINEERING AND DESIGN

Calculation Procedure:

1. Identify the “repeating group” of piles
The concrete footing (Fig. 35a) 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. 35b.
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 A 3 3 2 8.
Take moments with respect to row A. Thus 8x 3(0) 3(3) 2(6); x 2.625 ft
2
2
2
(66.675 mm). Then I 3(2.625) 3(0.375) 2(3.375) 43.9.
3. Compute the axial load and bending moment on the pile group
The axial load P 20,000(12) 240,000 lb (1067.5 kN); 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,
P/A 240,000/8 30,000 lb (133.4 kN) per pile; then M/I 150,000/43.9 3420.
Also, p a 30,000 3420(2.625) 21,020 lb (93.50 kN) per pile; p b 30,000
3420(0.375) 31,280 lb (139.13 kN) per pile; p c 30,000 3420(3.375) 41,540 lb
(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 lb (1067.43
kN); then M a 240,000(3.25) 780,000 lb·ft (1057.68 kN·m), and M r 3(31,280)(3)
2(41,540)(6) 780,000 lb·ft (1057.68 kN·m). Since M a M r , 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
, 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
contrary.

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.