- Test the accuracy of the construction
Select a third point c on the action line of the 150-kip (667.2-kN) load, and locate the ac-
tion line of the hypothetical load H 0 causing rotation about c. It is found that Hc also pass-
es through M. In summary, these hypothetical loads causing rotation about specific points
on the action line of the true load are all concurrent.
Thus, M is the center of rotation of the pier under the 150-kip (667.2-kN) load. This
conclusion stems from the following analysis: Load Ha applied at M causes zero deflec-
tion at a. Therefore, in accordance with Maxwell's theorem of reciprocal deflections, if
the true load is applied at a, it will cause zero deflection at M in the direction offfa. Simi-
larly, if the true load is applied at b, it will cause zero deflection at M in the direction of
Hb. Thus, Mremains stationary under the 150-kip (667.2-kN) load; that is, Mis the center
of rotation of the pier. - Scale the perpendicular distance from M to the longitudinal
axis of each pile
Divide this distance by the relative length of the pile. The quotient represents the relative
magnitude of the load induced in the pile by the 150-kip (667.2-kN) load. - Using a suitable scale, construct the force polygon by
applying the results from step 9
If the work is accurate, the resultant of these relative loads is parallel to the true load. - Scale the resultant; compute the factor needed to correct this
value to 150 k/ps (667.2 kN) - Multiply each relative pile load by this correction factor to
obtain the true load induced in the pile
LOAD DISTRIBUTION AMONG PILES WITH
FIXED BASES
Assume that the piles in Fig. I9a penetrate a considerable distance into a compact soil
and may therefore be regarded as restrained against rotation at a certain level. Outline a
procedure for determining the axial load and bending moment induced in each pile.
Calculation Procedure:- State the equation for the length of a dummy pile
Since the Westergaard construction presented in the previous calculation procedure ap-
plies solely to hinged piles, the group of piles now being considered is not directly
amenable to analysis by this method.
As shown in Fig. 2Oa, the pile AB functions in the dual capacity of a column and can-
tilever beam. In Fig. 2OZ?, let A' denote the position of A following application of the load.
If secondary effects are disregarded, the axial force P transmitted to this pile is a function
of Ay, and the transverse force S is a function of Ax.
Consider that the fixed support at B is replaced with a hinged support and a pile AC of
identical cross section is added perpendicular to AB, as shown in Fig. 2OZ?. If pile AC de-
forms an amount Ax under an axial force S, the forces transmitted by the pier at each point
of support are not affected by this modification of supports. The added pile is called a
dummy pile. Thus, the given pile group may be replaced with an equivalent group consist-
ing solely of hinged piles.