매트릭스 구조해석

상위 문서 : 구조역학

종류

• 응력법(Force Method) : 힘을 미지수로 하여 적합조건식 구성, 부재력 해석. 활용도 낮음.
  • 연성법(Flexibility method)
  • 적합법(Compatibility method)
• 변위법(Displacement method) : 변위를 미지수로 하여 평형방정식 구성, 부재력 해석. 일반적으로 응력법보다 미지수 개수가 많지만, 컴퓨터로 풀기 용이하기 때문에 더 중요하다.
  • 강성도법(Stiffness method)

응력법

${\mathrm{\Delta }}_1+{\mathrm{\Delta }}_{11}{R}_1+{\mathrm{\Delta }}_{12}{R}_2+{\mathrm{\Delta }}_{13}{R}_3=0$

${\mathrm{\Delta }}_2+{\mathrm{\Delta }}_{21}{R}_1+{\mathrm{\Delta }}_{22}{R}_2+{\mathrm{\Delta }}_{23}{R}_3=0$

${\mathrm{\Delta }}_3+{\mathrm{\Delta }}_{31}{R}_1+{\mathrm{\Delta }}_{32}{R}_2+{\mathrm{\Delta }}_{33}{R}_3=0$

적합방정식

$\left\{\begin{array}{c}{\mathrm{\Delta }}_1\\ {\mathrm{\Delta }}_2\\ {\mathrm{\Delta }}_3\end{array}\right\}+\left[\begin{array}{ccc}{\mathrm{\Delta }}_{11}& {\mathrm{\Delta }}_{12}& {\mathrm{\Delta }}_{13}\\ {\mathrm{\Delta }}_{21}& {\mathrm{\Delta }}_{22}& {\mathrm{\Delta }}_{23}\\ {\mathrm{\Delta }}_{31}& {\mathrm{\Delta }}_{32}& {\mathrm{\Delta }}_{33}\end{array}\right]\left\{\begin{array}{c}{R}_1\\ R_2\\ R_3\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\\ 0\end{array}\right\}$

{Δ} + [F]{R} = {0}

{Δ} : 외력에 의한 변위벡터

[F] : 연성계수(Flexibility coefficient) 행렬

$\therefore \{R\}=-[F{]}^{-1}\{\mathrm{\Delta }\}$

변위법

강성도법

Stiffness method. 용수철을 생각했을 때 p의 힘을 주면 Δ의 변위가 생긴다고 하자. 이때 둘의 관계를 다음으로 나타낸다.

$p=k\mathrm{\Delta }$

k : 단위변위를 발생시키기 위한 힘. 강성(stiffness)$=\frac{EA}{L}$

EA : 축강성(Axial rigidity)

축력을 받는 부재의 강성방정식

$\left\{\begin{array}{c}{p}_i\\ p_j\end{array}\right\}=\left[\begin{array}{cc}\frac{EA}{L}& -\frac{EA}{L}\\ -\frac{EA}{L}& \frac{EA}{L}\end{array}\right]\left\{\begin{array}{c}{\mathrm{\Delta }}_i\\ {\mathrm{\Delta }}_j\end{array}\right\}$

휨부재의 강성방정식

$\left\{\begin{array}{c}{q}_i\\ M_i\\ q_j\\ M_j\end{array}\right\}=\left[\begin{array}{cccc}\frac{12EI}{L^3}& \frac{6EI}{L^2}& -\frac{12EI}{L^3}& \frac{6EI}{L^2}\\ \frac{6EI}{L^2}& \frac{4EI}{L}& -\frac{6EI}{L^2}& \frac{2EI}{L}\\ -\frac{12EI}{L^3}& -\frac{6EI}{L^2}& \frac{12EI}{L^3}& -\frac{6EI}{L^2}\\ \frac{6EI}{L^2}& \frac{2EI}{L}& -\frac{6EI}{L^2}& \frac{4EI}{L}\end{array}\right]\left\{\begin{array}{c}{v}_i\\ {\theta }_i\\ v_j\\ {\theta }_j\end{array}\right\}$

휨-축력 부재의 강성방정식

$\left\{\begin{array}{c}{p}_i\\ q_i\\ M_i\\ p_j\\ q_j\\ M_j\end{array}\right\}=\left[\begin{array}{cccccc}\frac{EA}{L}& 0& 0& -\frac{EA}{L}& 0& 0\\ 0& \frac{12EI}{L^3}& \frac{6EI}{L^2}& 0& -\frac{12EI}{L^3}& \frac{6EI}{L^2}\\ 0& \frac{6EI}{L^2}& \frac{4EI}{L}& 0& -\frac{6EI}{L^2}& \frac{2EI}{L}\\ -\frac{EA}{L}& 0& 0& \frac{EA}{L}& 0& 0\\ 0& -\frac{12EI}{L^3}& -\frac{6EI}{L^2}& 0& \frac{12EI}{L^3}& -\frac{6EI}{L^2}\\ 0& \frac{6EI}{L^2}& \frac{2EI}{L}& 0& -\frac{6EI}{L^2}& \frac{4EI}{L}\end{array}\right]\left\{\begin{array}{c}{u}_i\\ v_i\\ {\theta }_i\\ u_j\\ v_j\\ {\theta }_j\end{array}\right\}$

경사재 부재력 좌표변환

$\left\{\begin{array}{c}P\\ Q\end{array}\right\}=\left[\begin{array}{cc}\cos \beta & -\sin \beta \\ \sin \beta & \cos \beta \end{array}\right]\left\{\begin{array}{c}p\\ q\end{array}\right\}$

경사재의 강성행렬

$\left[\begin{array}{c}{P}_i\\ Q_i\\ P_j\\ Q_j\end{array}\right]=\frac{EA}{L}\left[\begin{array}{cccc}{c}^2& sc& -c^2& -sc\\ sc& s^2& -sc& -s^2\\ -c^2& -sc& c^2& sc\\ -sc& -s^2& sc& s^2\end{array}\right]\left[\begin{array}{c}{U}_i\\ V_i\\ U_j\\ V_j\end{array}\right]\begin{array}{c}s=\sin \beta \\ c=\cos \beta \end{array}$ (for a truss element at angle β)

휨과 축력을 받는 경사재의 강성행렬

$\left\{\begin{array}{c}{P}_i\\ Q_i\\ M_i\\ P_j\\ Q_j\\ M_j\end{array}\right\}=\left[\begin{array}{c}{c}^2\frac{AE}{L}+s^2\frac{12EI}{L^3}\\ sc(\frac{AE}{L}-\frac{12EI}{L^3})& s^2\frac{AE}{L}+c^2\frac{12EI}{L^3}& & Symmetric& \\ -s\frac{6EI}{L^2}& c\frac{6EI}{L^2}& \frac{4EI}{L}\\ -c^2\frac{AE}{L}-s^2\frac{12EI}{L^3}& -sc(\frac{AE}{L}-\frac{12EI}{L^3})& s\frac{6EI}{L^2}& c^2\frac{AE}{L}+s^2\frac{12EI}{L^3}\\ -sc(\frac{AE}{L}-\frac{12EI}{L^3})& -s^2\frac{AE}{L}-c^2\frac{12EI}{L^3}& -c\frac{6EI}{L^2}& sc(\frac{AE}{L}-\frac{12EI}{L^3})& s^2\frac{AE}{L}+c^2\frac{12EI}{L^3}\\ -s\frac{6EI}{L^2}& c\frac{6EI}{L^2}& \frac{2EI}{L}& s\frac{6EI}{L^2}& -c\frac{6EI}{L^2}& \frac{4EI}{L}\end{array}\right]\left\{\begin{array}{c}{U}_i\\ V_i\\ {\theta }_i\\ U_j\\ V_j\\ {\theta }_j\end{array}\right\}$