beam3s¶

Purpose:

Compute section forces in a three dimensional beam element.

Syntax:

es = beam3s(ex, ey, ez, eo, ep, ed)
es = beam3s(ex, ey, ez, eo, ep, ed, eq)
[es, edi] = beam3s(ex, ey, ez, eo, ep, ed, eq, n)
[es, edi, eci] = beam3s(ex, ey, ez, eo, ep, ed, eq, n)

Description:

beam3s computes the section forces and displacements in local directions along the beam element beam3e.

The input variables ex, ey, ez, eo, ep and eq are defined in beam3e.

The element displacements, stored in ed, are obtained by the function extract_ed. If a distributed load is 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\(= \begin{bmatrix} N(0) & V_{\bar{y}}(0) & V_{\bar{z}}(0) & T(0) & M_{\bar{y}}(0) & M_{\bar{z}}(0) \\ N(\bar{x}_{2}) & V_{\bar{y}}(\bar{x}_{2}) & V_{\bar{z}}(\bar{x}_{2}) & T(\bar{x}_{2}) & M_{\bar{y}}(\bar{x}_{2}) & M_{\bar{z}}(\bar{x}_{2}) \\ N(\bar{x}_{n-1}) & V_{\bar{y}}(\bar{x}_{n-1}) & V_{\bar{z}}(\bar{x}_{n-1}) & T(\bar{x}_{n-1}) & M_{\bar{y}}(\bar{x}_{n-1}) & M_{\bar{z}}(\bar{x}_{n-1}) \\ N(L) & V_{\bar{y}}(L) & V_{\bar{z}}(L) & T(\bar{x}_{n-1}) & M_{\bar{y}}(L) & M_{\bar{z}}(L) \end{bmatrix}\)

edi\(= \begin{bmatrix} u(0) & v(0) & w(0) & \varphi(0) \\ u(\bar{x}_{2}) & v(\bar{x}_{2}) & w(\bar{x}_{2}) & \varphi(\bar{x}_{2}) \\ \vdots & \vdots & \vdots & \vdots \\ u(\bar{x}_{n-1}) & v(\bar{x}_{n-1}) & w(\bar{x}_{n-1}) & \varphi(\bar{x}_{n-1})\\ u(L) & v(L) & w(L) & \varphi(L) \end{bmatrix}\) \(\quad\) 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

\[\begin{split}\mathbf{\bar{a}}^e= \begin{bmatrix} \bar{u}_1 \\ \bar{u}_2 \\ \bar{u}_3 \\ \bar{u}_4 \\ \bar{u}_5 \\ \bar{u}_6 \\ \bar{u}_7 \\ \bar{u}_8 \\ \bar{u}_9 \\ \bar{u}_{10} \\ \bar{u}_{11} \\ \bar{u}_{12} \end{bmatrix} = \mathbf{G} \mathbf{a}^e\end{split}\]

where \(\mathbf{G}\) is described in beam3e and the transpose of \(\mathbf{a}^e\) is stored in ed.

The displacements associated with bar action, beam action in the \(\bar{x}\bar{y}\)-plane, beam action in the \(\bar{x}\bar{z}\)-plane, and torsion are determined as

\[\begin{split}\mathbf{\bar{a}}^e_{\text{bar}}= \begin{bmatrix} \bar{u}_1 \\ \bar{u}_7 \end{bmatrix}; \quad \mathbf{\bar{a}}^e_{\text{beam},\bar{z}}= \begin{bmatrix} \bar{u}_2 \\ \bar{u}_6 \\ \bar{u}_8 \\ \bar{u}_{12} \end{bmatrix}; \quad \mathbf{\bar{a}}^e_{\text{beam},\bar{y}}= \begin{bmatrix} \bar{u}_3 \\ -\bar{u}_5 \\ \bar{u}_9 \\ -\bar{u}_{11} \end{bmatrix}; \quad \mathbf{\bar{a}}^e_{\text{torsion}}= \begin{bmatrix} \bar{u}_4 \\ \bar{u}_{10} \end{bmatrix}\end{split}\]

The displacement \(u(\bar{x})\) and the normal force \(N(\bar{x})\) are computed from

\[u(\bar{x}) = \mathbf{N}_{\text{bar}} \mathbf{\bar{a}}^e_{\text{bar}} + u_p(\bar{x})\]

\[N(\bar{x}) = D_{EA} \mathbf{B}_{\text{bar}} \mathbf{\bar{a}}^e + N_p(\bar{x})\]

where

\[\mathbf{N}_{\text{bar}} = \begin{bmatrix} 1 & \bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\text{bar}} = \begin{bmatrix} 1-\frac{\bar{x}}{L} & \frac{\bar{x}}{L} \end{bmatrix}\]

\[\mathbf{B}_{\text{bar}} = \begin{bmatrix} 0 & 1 \end{bmatrix} \mathbf{C}^{-1}_{\text{bar}} = \begin{bmatrix} -\frac{1}{L} & \frac{1}{L} \end{bmatrix}\]

\[u_p(\bar{x}) = -\frac{q_{\bar{x}}}{D_{EA}}\left(\frac{\bar{x}^2}{2}-\frac{L\bar{x}}{2}\right)\]

\[N_p(\bar{x}) = -q_{\bar{x}}\left(\bar{x}-\frac{L}{2}\right)\]

in which \(D_{EA}\), \(L\), and \(q_{\bar{x}}\) are defined in beam3e and

\[\begin{split}\mathbf{C}^{-1}_{\text{bar}} = \begin{bmatrix} 1 & 0 \\ -\frac{1}{L} & \frac{1}{L} \end{bmatrix}\end{split}\]

The displacement \(v(\bar{x})\), the bending moment \(M_{\bar{z}}(\bar{x})\) and the shear force \(V_{\bar{y}}(\bar{x})\) are computed from

\[v(\bar{x}) = \mathbf{N}_{\text{beam}} \mathbf{\bar{a}}^e_{\text{beam},\bar{z}} + v_p(\bar{x})\]

\[M_{\bar{z}}(\bar{x}) = D_{EI_{\bar{z}}} \mathbf{B}_{\text{beam}} \mathbf{\bar{a}}^e_{\text{beam},\bar{z}} + M_{\bar{z},p}(\bar{x})\]

\[V_{\bar{y}}(\bar{x}) = -D_{EI_{\bar{z}}} \frac{d\mathbf{B}_{\text{beam}}}{dx} \mathbf{\bar{a}}^e_{\text{beam},\bar{z}} + V_{\bar{y},p}(\bar{x})\]

where

\[\mathbf{N}_{\text{beam}} = \begin{bmatrix} 1 & \bar{x} & \bar{x}^2 & \bar{x}^3 \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]

\[\mathbf{B}_{\text{beam}} = \begin{bmatrix} 0 & 0 & 2 & 6\bar{x} \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]

\[\frac{d\mathbf{B}_{\text{beam}}}{dx} = \begin{bmatrix} 0 & 0 & 0 & 6 \end{bmatrix} \mathbf{C}^{-1}_{\text{beam}}\]

\[v_p(\bar{x}) = \frac{q_{\bar{y}}}{D_{EI_{\bar{z}}}}\left(\frac{\bar{x}^4}{24}-\frac{L \bar{x}^3}{12}+\frac{L^2 \bar{x}^2}{24}\right)\]

\[M_{\bar{z},p}(\bar{x}) = q_{\bar{y}}\left(\frac{\bar{x}^2}{2}-\frac{L \bar{x}}{2}+\frac{L^2}{12}\right)\]

\[V_{\bar{y},p}(\bar{x}) = -q_{\bar{y}}\left(\bar{x}-\frac{L}{2}\right)\]

in which \(D_{EI_{\bar{z}}}\), \(L\), and \(q_{\bar{y}}\) are defined in beam3e and

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

The displacement \(w(\bar{x})\), the bending moment \(M_{\bar{y}}(\bar{x})\) and the shear force \(V_{\bar{z}}(\bar{x})\) are computed from

\[w(\bar{x}) = \mathbf{N}_{\text{beam}} \mathbf{\bar{a}}^e_{\text{beam},\bar{y}} + w_p(\bar{x})\]

\[M_{\bar{y}}(\bar{x}) = -D_{EI_{\bar{y}}} \mathbf{B}_{\text{beam}} \mathbf{\bar{a}}^e_{\text{beam},\bar{y}} + M_{\bar{y},p}(\bar{x})\]

\[V_{\bar{z}}(\bar{x}) = -D_{EI_{\bar{y}}} \frac{d\mathbf{B}_{\text{beam}}}{dx} \mathbf{\bar{a}}^e_{\text{beam},\bar{y}} + V_{\bar{z},p}(\bar{x})\]

where

\[w_p(\bar{x}) = \frac{q_{\bar{z}}}{D_{EI_{\bar{y}}}}\left(\frac{\bar{x}^4}{24}-\frac{L \bar{x}^3}{12}+\frac{L^2 \bar{x}^2}{24}\right)\]

\[M_{\bar{y},p}(\bar{x}) = -q_{\bar{z}}\left(\frac{\bar{x}^2}{2}-\frac{L \bar{x}}{2}+\frac{L^2}{12}\right)\]

\[V_{\bar{z},p}(\bar{x}) = -q_{\bar{z}}\left(\bar{x}-\frac{L}{2}\right)\]

in which \(D_{EI_{\bar{y}}}\), \(L\), and \(q_{\bar{z}}\) are defined in beam3e and \(\mathbf{N}_{\text{beam}}\), \(\mathbf{B}_{\text{beam}}\), and \(\frac{d\mathbf{B}_{\text{beam}}}{dx}\) are given above.

The displacement \(\varphi(\bar{x})\) and the torque \(T(\bar{x})\) are computed from

\[\varphi(\bar{x}) = \mathbf{N}_{\text{torsion}} \mathbf{\bar{a}}^e_{\text{torsion}} + \varphi_p(\bar{x})\]

\[T(\bar{x}) = D_{GK} \mathbf{B}_{\text{torsion}} \mathbf{\bar{a}}^e + T_p(\bar{x})\]

where

\[\mathbf{N}_{\text{torsion}} = \mathbf{N}_{\text{bar}}\]

\[\mathbf{B}_{\text{torsion}} = \mathbf{B}_{\text{bar}}\]

\[\varphi_p(\bar{x}) = -\frac{q_{\omega}}{D_{GK}}\left(\frac{\bar{x}^2}{2}-\frac{L\bar{x}}{2}\right)\]

\[T_p(\bar{x}) = -q_{\omega}\left(\bar{x}-\frac{L}{2}\right)\]

in which \(D_{GK}\), \(L\), and \(q_{\omega}\) are defined in beam3e.