Flexibility matrix: reactions at E and A, BMD

Reading of the figure: A (0, 0) is fixed; column AB is 10 m high ($2EI$) with a 20 kN horizontal load (towards the right) at 7 m above A; beam BD is 10 m long ($2EI$) with 75 kN/m over its whole length; column DE is 5 m long ($EI$), so E is 5 m above the level of A, and E is hinged.

$D_s = 3 + 2 - 3 = 2$. Release the hinge at E: $X_1$ = horizontal reaction ($\rightarrow$), $X_2$ = vertical reaction ($\uparrow$). The primary structure is a cantilever fixed at A.

 B ================= D    75 kN/m
 |                   |
 |  20 kN ->         | 5 m
 |                   E (hinge)
 | 10 m
 A (fixed)

Primary-structure moments (+ tension on the inner face)

• $M_0$: at B, $-75\times10\times5 = -3750$ kNm; at the 20 kN level, $-3750$; at A, $-(3750 + 20\times7) = -3890$ kNm; the beam falls parabolically from $-3750$ at B to 0 at D.
• $m_1$ (unit horizontal force at E): DE from 0 at E to 5 at D; beam constant 5; column from 5 at B to $-5$ at A.
• $m_2$ (unit vertical force at E): beam from 10 at B to 0 at D; column constant 10.

Flexibility matrix $[F]$ and load vector $\{\Delta_0\}$ (in $1/EI$ units)

$$\begin{aligned} f_{11} &= \frac{208.333}{EI} \\ f_{12} &= \frac{125.000}{EI} \\ f_{22} &= \frac{666.667}{EI} \\ \Delta_{10} &= \frac{-30596.7}{EI} \\ \Delta_{20} &= \frac{-236825.0}{EI} \end{aligned}$$

Compatibility: $[F]\{X\} = -\{\Delta_0\}$

$$\begin{aligned} 208.333\,X_1 + 125.000\,X_2 &= 30596.7 \\ 125.000\,X_1 + 666.667\,X_2 &= 236825.0 \end{aligned}$$ $$X_1 = H_E = -74.68\ \text{kN},\qquad X_2 = V_E = 369.24\ \text{kN}$$

Final bending moments $M = M_0 + X_1m_1 + X_2m_2$ (kNm)

Maximum sagging moment in the beam: 535.52 kNm at 5.08 m from B.

Reactions

• E (hinge): $H_E = 74.68$ kN (towards the left), $V_E = 369.24$ kN ($\uparrow$)
• A (fixed): $H_A = 54.68$ kN (towards the right), $V_A = 380.76$ kN ($\uparrow$), $M_A = 175.80$ kNm (clockwise)

Check: $\Sigma H$: $20 + 54.68 - 74.68 = 0$; $\Sigma V$: $380.76 + 369.24 = 75\times10 = 750$ kN.

Answer: at E, $H_E = 74.68$ kN (towards the left) and $V_E = 369.24$ kN (upward); at A, $H_A = 54.68$ kN (towards the right), $V_A = 380.76$ kN (upward) and $M_A = 175.80$ kNm (clockwise).