to the displacement profile curves (Fig. 4-22). The fitted lines were in good agreement

with the displacement data (average R2 = 0.997). The polynomial displacement profile

equation of the beam was used to determine the curvature at specific points along the

length of the pile. The second derivative of the polynomial displacement profile equation

is the curvature equation.

 Displacement Location from North Support (in)
 0 30 60 90 120 150 180
 0 --I I I I I
 y =2.6E-13x6 1.3E-10x + 2.1E-08x 1.3E-06x3 + 3.6E-05x2 -4.0E-03x- 5.8E-02

 -0.1


 -0.2


 -0.3
 0 P= 43.7 kips
 Poly. (P = 43.7 kips)
 -0.4


Figure 4-22. Displacement curve and fitted polynomial line for control beam at 43.7 kips

 To model the behavior of the beams a program that generated the sectional

moment-curvature (M-PD) relationship was used (Consolazio, Fung and Ansley 2004).

 The program describes cross sectional geometry using piecewise linear segments

for both the exterior and interior boundaries of the section. Interior boundaries represent

voids in the section. Curved boundaries such as the boundaries of circular sections are

approximated by using multiple linear segments. The reinforcing bars are modeled as

cross-sectional areas located at specific locations.

 Some of the assumptions associated with the program are the following:

* Sign convention is positive for compression and negative for tension

* Plain sections remain plain which means that the strain distribution for the cross-
 section is linear