Generate stiffness matrix of frame

Assumptions: A(0,0), B(0,3), C(4,3), D(4,2): column AB is 3 m high ($2EI$), beam BC is 4 m ($3EI$), column CD is 1 m high ($EI$, short column), A and D are fixed. Axial deformations are neglected. Coordinates (positive directions): 1 = rotation of B, 2 = rotation of C (both clockwise), 3 = horizontal sway of the beam (to the right). The chord rotations are $\psi_{AB}=\Delta/3$ and $\psi_{CD}=\Delta/1$.

Member constants

Generation of the stiffness matrix

• Unit rotation of B (coordinate 1): $k_{11}=\dfrac{4(2EI)}{3}+\dfrac{4(3EI)}{4}=5.6667EI$; $k_{21}=\dfrac{2(3EI)}{4}=1.5000EI$; $k_{31}=-\dfrac{6(2EI)}{3^2}=-1.3333EI$
• Unit rotation of C (coordinate 2): $k_{22}=\dfrac{4(3EI)}{4}+\dfrac{4EI}{1}=7.0000EI$; $k_{12}=1.5000EI$; $k_{32}=-\dfrac{6EI}{1^2}=-6.0000EI$
• Unit sway (coordinate 3): $k_{33}=\dfrac{12(2EI)}{3^3}+\dfrac{12EI}{1^3}=12.8889EI$; $k_{13}=-1.3333EI$; $k_{23}=-6.0000EI$

$$[K]=EI\begin{bmatrix}5.6667 & 1.5000 & -1.3333 \\ 1.5000 & 7.0000 & -6.0000 \\ -1.3333 & -6.0000 & 12.8889\end{bmatrix}$$

The matrix is symmetric ($k_{ij}=k_{ji}$). The short column CD (1 m) is laterally very stiff: it contributes $12EI$ to the sway stiffness $k_{33}$ and $-6EI$ to $k_{23}$, so most of the sway resistance, and most of the coupling between sway and the rotation of joint C, comes from CD. The equation of the frame is $\{P\}=[K]\{d\}$ with $\{d\}=\{\theta_B,\theta_C,\Delta\}^T$.