beam1sΒΆ

Purpose:

Compute section forces in a one dimensional beam element.

_images/beam1s.svg
Syntax:



es = beam1s(ex, ep, ed)
es = beam1s(ex, ep, ed, eq)
[es, edi, eci] = beam1s(ex, ep, ed, eq, n)
Description:

beam1s computes the section forces and displacements in local directions along the beam element beam1e.

The input variables ex, ep and eq are defined in beam1e, and the element displacements, stored in ed, are obtained by the function extract_ed. If distributed loads are applied to the element, the variable eq must be included. The number of evaluation points for section forces and displacements are determined by n. If n is omitted, only the ends of the beam are evaluated.

The output variables

es\(= \begin{bmatrix} V(0) & M(0) \\ V(\bar{x}_{2}) & M(\bar{x}_{2}) \\ \vdots & \vdots \\ V(\bar{x}_{n-1}) & M(\bar{x}_{n-1})\\ V(L) & M(L) \end{bmatrix}\) \(\qquad\) edi\(= \begin{bmatrix} v(0) \\ v(\bar{x}_{2}) \\ \vdots \\ v(\bar{x}_{n-1})\\ v(L) \end{bmatrix}\) \(\qquad\) eci\(= \begin{bmatrix} 0 \\ \bar{x}_2 \\ \vdots \\ \bar{x}_{n-1} \\ L \end{bmatrix}\)

contain the section forces, the displacements, and the evaluation points on the local \(\bar{x}\)-axis. \(L\) is the length of the beam element.

Theory:

The nodal displacements in local coordinates are given by

\[\begin{split}\mathbf{\bar{a}}^e = \begin{bmatrix} \bar{u}_1 \\ \bar{u}_2 \\ \bar{u}_3 \\ \bar{u}_4 \end{bmatrix}\end{split}\]

where the transpose of \(\mathbf{a}^e\) is stored in ed.

The displacement \(v(\bar{x})\), the bending moment \(M(\bar{x})\) and the shear force \(V(\bar{x})\) are computed from

\[v(\bar{x}) = \mathbf{N} \mathbf{\bar{a}}^e + v_p(\bar{x})\]
\[M(\bar{x}) = D_{EI} \mathbf{B} \mathbf{\bar{a}}^e + M_p(\bar{x})\]
\[V(\bar{x}) = -D_{EI} \frac{d\mathbf{B}}{dx} \mathbf{\bar{a}}^e + V_p(\bar{x})\]

where

\[\mathbf{N} = \begin{bmatrix} 1 & \bar{x} & \bar{x}^2 & \bar{x}^3 \end{bmatrix} \mathbf{C}^{-1}\]
\[\mathbf{B} = \begin{bmatrix} 0 & 0 & 2 & 6\bar{x} \end{bmatrix} \mathbf{C}^{-1}\]
\[\frac{d\mathbf{B}}{dx} = \begin{bmatrix} 0 & 0 & 0 & 6 \end{bmatrix} \mathbf{C}^{-1}\]
\[v_p(\bar{x}) = \frac{q_{\bar{y}}}{D_{EI}} \left( \frac{\bar{x}^4}{24} - \frac{L \bar{x}^3}{12} + \frac{L^2 \bar{x}^2}{24} \right)\]
\[M_p(\bar{x}) = q_{\bar{y}} \left( \frac{\bar{x}^2}{2} - \frac{L \bar{x}}{2} + \frac{L^2}{12} \right)\]
\[V_p(\bar{x}) = -q_{\bar{y}} \left( \bar{x} - \frac{L}{2} \right)\]

in which \(D_{EI}\), \(L\), and \(q_{\bar{y}}\) are defined in beam1e and

\[\begin{split}\mathbf{C}^{-1} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ -\frac{3}{L^2} & -\frac{2}{L} & \frac{3}{L^2} & -\frac{1}{L} \\ \frac{2}{L^3} & \frac{1}{L^2} & -\frac{2}{L^3} & \frac{1}{L^2} \end{bmatrix}\end{split}\]