2. Benchmark solution#

2.1. Calculation method used for the reference solution#

The problem is handled by a one-degree-of-freedom model. The column is considered to be an undamped and non-heavy slender beam of stiffness \(k=3E{I}_{Z}/{l}^{\mathrm{3 }}=\mathrm{3,942}{.10}^{\mathrm{7 }}N/m\). The superstructure located at the top of the column is modelled by a point mass \(m=\mathrm{43,8}{10}^{\mathrm{3 }}\mathrm{kg}\).

The two load cases lead to the calculation of the response of a system to a degree of freedom subjected to acceleration \(\gamma (t)\) of any form:

\(\ddot{{x}_{r}}+{\omega }^{2}{x}_{r}=-\gamma (t)\) with \(\omega =\sqrt{\frac{k}{m}}=\sqrt{\frac{3E{I}_{z}}{m{l}^{3}}}\) the natural frequency of the system and \({x}_{r}\) the relative displacement of point \(B\) with respect to point \(A\). The solution is obtained by integrating the Duhamel integral [bib3]:

\({x}_{r}(t)=-\frac{m}{\omega }\underset{0}{\overset{t}{\int }}\gamma (t)\mathrm{sin}\omega (t-\tau )d\tau\)

2.2. Benchmark results#

Displacement relative to point \(B\).

For a triangular imposed acceleration, we can calculate the Duhamel integral analytically [bib3]:

\(t<{t}_{0}\): \({x}_{r}=-\frac{{P}_{0}}{{\omega }^{2}{t}_{0}}(t-\frac{\mathrm{sin}\omega t}{\omega })\)

\({t}_{0}<t<2{t}_{0}\): \({x}_{r}=-\frac{{P}_{0}}{{\omega }^{2}{t}_{0}}(2{t}_{0}-t-\frac{2\mathrm{sin}\omega (t-{t}_{0})}{\omega }-\frac{\mathrm{sin}\omega t}{\omega })\)

\({t}_{0}<t<2{t}_{0}\): \({x}_{r}=-\frac{{P}_{0}}{{\omega }^{3}{t}_{0}}(2\mathrm{sin}\omega (t-{t}_{0})-\mathrm{sin}\omega (t-2{t}_{0})-\mathrm{sin}\omega t)\)

2.3. Uncertainty about the solution#

None if we calculate the Duhamel integral analytically [bib3]. In order of the precision of the numerical integration method used to calculate the Duhamel integral ([bib1], [bib2]): Simpson method with 40 points per period.

2.4. Bibliographical references#

  1. R.W. Clough and J. Penzien: Dynamics of structures New York, Mac Graw-Hill, 1975, p.102-105

  2. Guide VPCS AFNOR Technical- 1990

  3. J.S. Przemieniecki: Theory of matrix structural analysis New York, MacGraw-Hill, 1968, p.351-357