Stiffness matrix: frame with 40 kN/m, 60 kN

Assumptions (figure partly illegible): A(0,0), B(0,3), C(4,3), D(4,-1): the left column is 3 m high (2 m + 1 m) and the right column is 4 m high (2 m + 2 m), with the beam BC = 4 m horizontal at the top; EI is constant; the bases A and D are fixed. Loads: 40 kN/m on BC (downward); 60 kN horizontal (to the right) on AB at 2 m above A; 60 kN horizontal (to the left) at mid-height of CD. Coordinates: 1 = $\theta_B$, 2 = $\theta_C$ (clockwise), 3 = horizontal sway $\Delta$ of the beam (right +). Axial deformation is neglected.

Chord rotations

$\psi_{AB}=\Delta/3$ and $\psi_{CD}=\Delta/4$.

Stiffness matrix (column by column)

Member stiffnesses: AB ($L=3$): $\dfrac{4EI}{3}=1.3333EI$, $\dfrac{2EI}{3}=0.6667EI$, $\dfrac{6EI}{9}=0.6667EI$, $\dfrac{12EI}{27}=0.4444EI$; CD ($L=4$): $EI$, $0.5EI$, $0.375EI$, $0.1875EI$; BC ($L=4$): $EI$, $0.5EI$.

• $k_{11}=\dfrac{4EI}{3}+\dfrac{4EI}{4}=2.3333EI$, $k_{12}=\dfrac{2EI}{4}=0.5000EI$, $k_{13}=-\dfrac{6EI}{3^2}=-0.6667EI$
• $k_{22}=\dfrac{4EI}{4}+\dfrac{4EI}{4}=2.0000EI$, $k_{23}=-\dfrac{6EI}{4^2}=-0.3750EI$
• $k_{33}=\dfrac{12EI}{3^3}+\dfrac{12EI}{4^3}=0.6319EI$

$$[K]=EI\begin{bmatrix}2.3333 & 0.5000 & -0.6667 \\ 0.5000 & 2.0000 & -0.3750 \\ -0.6667 & -0.3750 & 0.6319\end{bmatrix}$$

Load vector (for completeness)

Fixed-end moments: BC $\mp\dfrac{40\times4^2}{12}=\mp53.33$; AB (60 kN at 2 m from A, $L=3$): $FEM_{AB}=-13.33$, $FEM_{BA}=26.67$; CD (60 kN to the left at 2 m from C): $FEM_{CD}=-30.00$, $FEM_{DC}=30.00$ kNm. The equivalent loads are

$$\{P\}=\begin{Bmatrix}26.667\\-23.333\\14.444\end{Bmatrix}$$

and $[K]\{d\}=\{P\}$ gives

$$EI\theta_{B}=26.3388,\quad EI\theta_{C}=-9.8519,\quad EI\Delta_{1}=44.7970$$

End moments (kNm, clockwise +)

Checks:

• Joint B: (31.92) + (-31.92) = 0.00 kNm (applied clockwise moment 0.00) ✓
• Joint C: (56.65) + (-56.65) = 0.00 kNm (applied clockwise moment 0.00) ✓

Answer: $[K]=EI\begin{bmatrix}2.333&0.500&-0.667\\0.500&2.000&-0.375\\-0.667&-0.375&0.6319\end{bmatrix}$ (coordinates $\theta_B,\theta_C,\Delta$).