Flexural Analysis in Dynamic Pinned Head Pile Testing

Abstract

A Laplace transform is used to solve the problem of the steady state and transient response of a pinned head pile embedded into a viscoelastic Winkler soil medium. The pile is modeled as an Euler–Bernoulli beam while the soil medium is modeled using a Winkler subgrade approach. Two analytical solutions are developed to specifically address both steady state and transient loads encountered during dynamic pile testing. After choosing a proper contour integration in the complex plane, inverse integration is evaluated. The steady state solutions are associated to the residues of the integration around the poles while the transient solutions are associated to the integration paths along the contour integration. The derived solutions are applied to a case history for which results of dynamic pile tests are available. Dynamic pile flexion is generated by delivering eccentric impact using a dynamic loading test module. Validity of the proposed solution is discussed basing on geotechnical campaign and recorded pile head bending moment and rotation rate.

Appendices

Appendix 1: Expressions of Integrals I₂, I₃ and I₄

The point $S_{1} = - \frac{k}{c} < 0$ is a branch point. Thus, the integrant around the branch point is calculated by replacing s by $s = - \frac{k}{c} + re^{i\theta }$:

$$I_{2} = \mathop {\lim }\limits_{r \mapsto 0} \frac{{M_{0} }}{\pi }\int\limits_{ - \pi }^{\pi } {\frac{{\beta^{2} \sin (\beta z)e^{ - \beta z} }}{{ - \left( {\frac{k}{c} + re^{i\theta } + iw} \right)c}}\theta d\theta } = 0$$

where

$$\beta = \sqrt{\frac{rc}{4EI}} e^{{i\frac{\theta }{4}}}$$

For the integration paths p and q, s is replaced with $s = - \frac{k}{c} + qe^{i\pi }$ and $s = - \frac{k}{c} + qe^{ - i\pi }$ respectively:

$$I_{3} = \frac{{M_{0} }}{\pi i}\int\limits_{0}^{ + \infty } {\frac{{\beta^{2} \sin (\beta z)e^{ - \beta z} }}{{qc\left( {\frac{k}{c} + q + iw} \right)}}} e^{{\left( { - \frac{k}{c} + qe^{i\pi } } \right)t}} dq$$

where

$$\beta = \sqrt{\frac{qc}{4EI}} e^{{i\frac{\pi }{4}}}$$

and:

$$I_{4} = \frac{{M_{0} }}{\pi i}\int\limits_{0}^{ + \infty } {\frac{{\beta^{2} \sin (\beta z)e^{ - \beta z} }}{{qc\left( {\frac{k}{c} + q + iw} \right)}}} e^{{\left( { - \frac{k}{c} + qe^{ - i\pi } } \right)t}} dq$$

where

$$\beta = \sqrt{\frac{qc}{4EI}} e^{{ - i\frac{\pi }{4}}}$$

Appendix 2: Expressions of Integrals I₅, I₆, I₇, I₈, I₉, I₁₀ and I₁₁

The residue of the pole is:

$$I_{5} = 2M_{0} \frac{{\sqrt{k/(4EI)} \sin (\root{4} \of {k/(4EI)}z)e^{{ - \root{4} \of {{k/\left( {4EI} \right)s}}}} }}{k}$$

The integration around branch points S ₂ and S ₃ is evaluated by considering the circle of radius r and by replacing s in Eq. (16) by re ^iθ and −k/c + re ^iθrespectively:

$$I_{6} = \mathop {\lim }\limits_{r \mapsto 0} \frac{{M_{0} }}{\pi }\int\limits_{ - \pi }^{0} {\frac{{\beta^{2} \sin (\beta z)e^{ - (\beta )z} }}{{( - k/c + re^{i\theta } )c}}e^{{\left( { - \frac{k}{c} + re^{i\theta } } \right)t}} \theta d\theta } = 0$$

$$I_{7} = \mathop {\lim }\limits_{r \mapsto 0} \frac{{M_{0} }}{\pi }\int\limits_{0}^{\pi } {\frac{{\beta^{2} \sin (\beta z)e^{ - \beta z} }}{{( - k/c + re^{i\theta } )c}}e^{{\left( { - \frac{k}{c} + re^{i\theta } } \right)t}} \theta d\theta } = 0$$

where $\beta = \sqrt{\frac{rc}{4EI}} e^{{i\frac{\theta }{4}}}$

For the integration paths A B, s is replaced by s = qe ^−iπ however for the integration paths C and D, s is replaced by s = qe ^iπ respectively:

$$I_{8} = \frac{{M_{0} }}{\pi i}\int\limits_{k/c}^{ + \infty } {\frac{{\beta^{2} \sin (\beta z)e^{ - \beta z} }}{(k + sc)s}e^{{qe^{i\pi } t}} dq}$$

where

$$\beta = \root{4} \of {{\frac{ - (k - qc)}{4EI}}}e^{{ - i\frac{\pi }{4}}}$$

where

$$I_{9} = \frac{{M_{0} }}{\pi i}\int\limits_{0}^{k/c} {\frac{{\beta^{2} \sin (\beta z)e^{ - \beta z} }}{(k + sc)s}e^{{qe^{i\pi } t}} dq}$$

$$\beta = \root{4} \of {{\frac{k - qc}{4EI}}}$$

$$I_{10} = \frac{{M_{0} }}{\pi i}\int\limits_{0}^{k/c} {\frac{{\beta^{2} \sin (\beta z)e^{ - \beta z} }}{(k + sc)s}e^{{qe^{ - i\pi } t}} dq}$$

where

$$\beta = \root{4} \of {{\frac{k - qc}{4EI}}}e^{{i\frac{\pi }{2}}}$$

$$I_{11} = \frac{{M_{0} }}{\pi i}\int\limits_{k/c}^{ + \infty } {\frac{{\beta^{2} \sin (\beta z)e^{ - \beta z} }}{(k + sc)s}e^{{qe^{ - i\pi } t}} dq}$$

where

$$\beta = \sqrt{\frac{ - (k - qc)}{4EI}} e^{{i\frac{\pi }{4}}}$$

Appendix 3: Expressions of the Functions of F_i

$$F1(z) = \frac{{e^{(B - A)z} }}{2}\sin \left( {\left( {A + B} \right)z} \right) + \frac{{e^{ - (B + A)z} }}{2}\sin \left( {\left( {A - B} \right)z} \right)$$

$$F2(z) = - \frac{{e^{(B - A)z} }}{2}\cos ((A + B)z) + \frac{{e^{ - (B + A)z} }}{2}\cos ((A - B)z)$$

$$F3(z) = \frac{{e^{(B - A)z} }}{2}\left[ {\cos ((A + B)z - \varphi /4) - \sin \left( {\left( {A + B} \right)z - \varphi /4} \right) + \frac{{e^{ - (B + A)z} }}{2}\left[ {\cos \left( {\left( {A - B} \right)z + \varphi /4} \right) - \sin \left( {\left( {A - B} \right)z + \varphi /4} \right)} \right]} \right.$$

$$F4(z) = F3(z) = \frac{{e^{(B - A)z} }}{2}[\cos ((A + B)z - \varphi /4) + \sin ((A + B)z - \varphi /4) - \frac{{e^{ - (B + A)z} }}{2}[\cos ((A - B)z + \varphi /4) + \sin ((A - B)z + \varphi /4)]$$

$$F5(z) = \frac{{e^{(B - A)z} }}{2}\cos ((A + B)z - \varphi /2) + \frac{{e^{ - (B + A)z} }}{2}\cos ((A - B)z - \varphi /2)$$

$$F6(z) = - \frac{{e^{(B - A)z} }}{2}\sin ((A + B)z - \varphi /2) + \frac{{e^{ - (B + A)z} }}{2}\sin ((A - B)z + \varphi /2)$$

$$F7(z) = \frac{{e^{(B - A)z} }}{2}\sin ((A + B)z) + \frac{{e^{ - (B + A)z} }}{2}\sin ((A - B)z)$$

$$F8(z) = - \frac{{e^{(B - A)z} }}{2}\cos ((A + B)z) + \frac{{e^{ - (B + A)z} }}{2}\cos ((A - B)z)$$

where $A = \root{4} \of {r}(\cos (\phi /4)$ and $B = \root{4} \of {r}(\sin (\phi /4)$