매트릭스-직접강도법

원리

• $F=T^{T}Q$
• $Q=ku$
• $u=Tv$
• $F=(T^{T}kT)v=Kv$

암기사항

Plane Trusses

• $k=\frac{EA}{L}\left[\begin{array}{cccc}1& 0& -1& 0\\ 0& 0& 0& 0\\ -1& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$

• $T=\left[\begin{array}{cccc}\cos \theta & \sin \theta & 0& 0\\ -\sin \theta & \cos \theta & 0& 0\\ 0& 0& \cos \theta & \sin \theta \\ 0& 0& -\sin \theta & \cos \theta \end{array}\right]$

• $\cos \theta =\frac{X_e-X_b}{L}$

Beams

• $k=\frac{EI}{L^3}\left[\begin{array}{cccc}12& 6L& -12& 6L\\ 6L& 4L^2& -6L& 2L^2\\ -12& -6L^2& 12& -6L\\ 6L& 2L^2& -6L& 4L^2\end{array}\right]$

Plane Frames

• $k=\frac{EI}{L^3}\left[\begin{array}{cccccc}\frac{A{L}^2}{I}& 0& 0& -\frac{A{L}^2}{I}& 0& 0\\ 0& 12& 6L& 0& -12& 6L\\ 0& 6L& 4L^2& 0& -6L& 2L^2\\ -\frac{A{L}^2}{I}& 0& 0& \frac{A{L}^2}{I}& 0& 0\\ 0& -12& -6L& 0& 12& -6L\\ 0& 6L& 2L^2& 0& -6L& 4L^2\end{array}\right]$

• $T=\left[\begin{array}{cccccc}\cos \theta & \sin \theta & 0& 0& 0& 0\\ -\sin \theta & \cos \theta & 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& \cos \theta & \sin \theta & 0\\ 0& 0& 0& -\sin \theta & \cos \theta & 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$

Member Releases

• MT=1 : $k=\frac{EI}{L^3}\left[\begin{array}{cccccc}\frac{A{L}^2}{I}& 0& 0& -\frac{A{L}^2}{I}& 0& 0\\ 0& 3& 0& 0& -3& 3L\\ 0& 0& 0& 0& 0& 0\\ -\frac{A{L}^2}{I}& 0& 0& \frac{A{L}^2}{I}& 0& 0\\ 0& -3& 0& 0& 3& -3L\\ 0& 3L& 0& 0& -3L& 3L^2\end{array}\right]$

• MT=2 : $k=\frac{EI}{L^3}\left[\begin{array}{cccccc}\frac{A{L}^2}{I}& 0& 0& -\frac{A{L}^2}{I}& 0& 0\\ 0& 3& 3L& 0& -3& 0\\ 0& 3L& 3L^2& 0& -3L& 0\\ -\frac{A{L}^2}{I}& 0& 0& \frac{A{L}^2}{I}& 0& 0\\ 0& -3& -3L& 0& 3& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$

• MT=3 : $k=\frac{EI}{L}\left[\begin{array}{cccccc}1& 0& 0& -1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ -1& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$

• Beam Member의 경우 1행&열, 4행&열 제거

Support Displacement

Temperature Change

Fabrication Errors