Calculation Procedure:- Subject the structure to a load for purposes of analysis
Steel and timber piles may be considered to be connected to the concrete pier by friction-
less hinges, and bearing piles that extend a relatively short distance into compact soil may
be considered to be hinge-supported by the soil.
Since four unknown quantities are present, the structure is statically indeterminate. A
solution to this problem therefore requires an analysis of the deformation of the structure.
As the load is applied, the pier, assumed to be infinitely rigid, rotates to some new po-
sition. This displacement causes each pile to rotate about its base and to undergo an axial
strain. The contraction or elongation of each pile is directly proportional to the perpendi-
cular distance/? from the axis of rotation to the longitudinal axis of that pile. Let P denote
the load induced in the pile. Then P = AL AEIL. Since AL is proportional to p and AE is
constant for the group, this equation may be transformed to
kp
P= f (37)where A: is a constant of proportionality.
If the center of rotation is established, the pile loads may therefore be found by scaling
the p distances. Westergaard devised a simple graphical method of locating the center of
rotation. This method entails the construction of string polygons, described in the first
calculation procedure of this handbook.
In Fig. 190 select any convenient point a on the action line of the load. Consider the
structure to be subjected to a load Ha that causes the pier to rotate about a as a center. The
object is to locate the action line of this hypothetical load.
It is often desirable to visualize that a load is applied to a body at a point that in reality
lies outside the body. This condition becomes possible if the designer annexes to the body
an infinitely rigid arm containing the given point. Since this arm does not deform, the
stresses and strains in the body proper are not modified.
- Scale the perpendicular distance from a to the longitudinal axis
of each pile; divide this distance by the relative length of the pile
In accordance with Eq. 37, the quotient represents the relative magnitude of the load in-
duced in the pile by the load Ha. If rotation is assumed to be counterclockwise, piles A and
B are in compression and D is in tension. - Using a suitable scale, construct the force polygon
This polygon is shown in Fig. I9b. Construct this polygon by applying the results ob-
tained in step 2. This force polygon yields the direction of the action line of H 0. - In Fig. 19b, select a convenient pole O and draw rays to the
force polygon - Construct the string polygon shown in Fig. 19c
The action line of H 0 passes through the intersection point Q of rays ah and dh, and its di-
rection appears in Fig. 1 9b. Draw this line. - Select a second point on the action line of the load
Choose point b on the action line of the 150-kip (667.2-kN) load, and consider the struc-
ture to be subjected to a load Hb that causes the pier to rotate about b as center. - Locate the action line of Hb
Repeat the foregoing procedure to locate the action line of Hb in Fig. 19c. (The construc-
tion has been omitted for clarity.) Study of the diagram shows that the action lines of Ha
and Hb intersect at M.