Compute the stiffness matrix and the self-weight equivalent nodal modal force amplitude vector for a 3-noded triangular

Question

Compute the stiffness matrix and the self-weight equivalent nodal modal force amplitude vector for a 3-noded triangular straight prism.

Accepted Answer

The faces of the prism are linear triangles (Figure 11.21). The shape functions are

[latex]N_{i}=\frac{1}{2A^{(e)}}[a_{i}+b_{i}x+c_{i}z][/latex]

where [latex]a_{i}[/latex], [latex]b_{i}[/latex] and [latex]c_{i}[/latex] are given by Eq.(4.32b) of [On4] with x, z for x, y. The element strain matrix is

[latex]B_{i}^{l}=\frac{1}{2A^{(e)}}\begin{bmatrix}b_{i} &0 &0 \ 0 &-\gamma N_{i} &0 \ 0 & 0 &c_{i} \ \gamma N_{i} &b_{i} &0 \ c_{i} &0 &b_{i} \ 0&c_{i} &\gamma N_{i} \end{bmatrix}con\gamma =\frac{l\pi }{b}[/latex]

The product [latex]B_{i}^{l^{T}}DB_{j}^{l}[/latex] contains terms such as [latex]N_{i}[/latex] and [latex]N_{i}N_{j}[/latex]. Analytical integration leads to

[latex][K_{ij}^{ll}]^{(e)}=\frac{b}{8A^{(e)}} imes [/latex]

[latex]\begin{bmatrix}(d_{11}b_{i}b_{j}+d_{55}c_{i}c_{j}+\frac{\gamma A^{(e)}}{3}(-d_{12}b_{i}+d_{44}b_{j})(d_{13}b_{i}c_{j}+d_{55}c_{i}b_{j})+d_{44}\alpha _{ij})\ \frac{\gamma A^{(e)}}{3}(-d_{21}b_{j}+c_{44}b_{i})(d_{44}b_{i}b_{j}+d_{66}c_{i}c_{j}+\frac{\gamma A^{(e)}}{3}(d_{66}c_{i}-d_{23}c_{j})+d_{22}\alpha _{ij})\ (d_{31}c_{i}b_{j}+d_{55}b_{i}c_{j}) \frac{\gamma A^{(e)}}{3}(d_{66}c_{j}-d_{32}c_{i})(d_{33}c_{i}c_{j}+d_{55}b_{i}b_{j}+d_{66}\alpha _{ij})\end{bmatrix}[/latex]

with [latex]\alpha _{ij}=\frac{\gamma ^{2}A^{(e)}}{6}[/latex] if i = j or [latex]\frac{\gamma ^{2}A^{(e)}}{12}[/latex] if [latex]i eq j[/latex]

The equivalent nodal modal force amplitude vector for self-weight is obtained from Eqs.(11.87)–(11.90). For [latex] ho =constant[/latex], [latex]g_{x}=g_{y}=0[/latex] and [latex]g_{z}=-g[/latex] ,

[latex]f_{i}^{l}=\frac{b}{2}b^{l}\iint_{A^{(e)}}^{}N_{i}^{T}dA=\frac{bA^{(e)}}{6}b^{l}=\frac{b ho g}{3l\pi }[1-(-1)^{l}]\begin{Bmatrix}0\ 0\ -1\end{Bmatrix}[/latex]

// (Eq.(4.32b)):[latex]\hat{\sigma }{}'=\iint_{A}^{}\begin{Bmatrix}E_{\varepsilon {x}'}\ G_{{y}'}(\frac{\partial {v}'_{c}}{\partial x}- heta _{x}')\ G_{{z}'}(\frac{\partial {w}'_{c}}{\partial z}- heta _{y}')\ {z}'E_{\varepsilon {x}'}\ -{y}'E_{\varepsilon {x}'}\ D_{t}\end{Bmatrix}[/latex]

(Eq.(11.87)):[latex][f_{i}^{l}]^{(e)}=C\iint_{A^{(e)}}^{} N_{i}^{T}b^{l}dA+C\oint_{S^{(e)}}^{}N_{i}^{T}t^{l}dA+p_{i}^{l}[/latex]

(Eq.(11.90)):[latex]b^{l}=\frac{2 ho }{l\pi }(1-(-1)^{l})[g^{x},g_{y},g_{z}]^{T}[/latex]//

Question : Compute the stiffness matrix and the self-weight equivalent ...