The foundation can be analyzed using a 1 inch unit thickness which will have a load of
P = (6 kip/in^{2}) (1 in) (2 in) = 12 kip
This load will push down on the foundation from the top to the bottom.
Next, it will be easier to analyze the deflection if the foundation is split into two sections, 1 and 2. The deflection of the top section can be determined by
δ_{1} = PL_{1}/A_{1}E_{1}
= (-12,000) (10) / [(2)(4×10^{6})] = -0.0150 in
However, since the cross sectional area of the bottom section varies, the deflection needs to be integrated over its length using
The load P and stiffness E are constant, but the area, A changes over its length. The rate of change is linear, and can be modeled as
A(x) = 1 (4 - x/10) = (40 - x)/10
The variable of integration, x, is zero at the base where the area 4 in^{2} and goes to 20 in at the top of the bottom section where the area is 2 in^{2}. This gives,
= 0.030 (ln 20 - ln 40) = -0.02079 in
The total deflection is the sum of both sections,
δ_{total} = δ_{1} + δ_{2}
= -0.01500 - 0.02079 = -0.03577 in |