Force method: frame, SFD and BMD

Reading of the figure: left column AB (4 m, A fixed); beam B-C-D with BC = 5 m carrying 5 kN/m and a 2 m overhang CD with 20 kN at D; a middle column CE (4 m) hinged at E, below C; 20 kN horizontal at B (towards the right). $EI$ is constant for all members.

$r = 3 + 2 = 5$, so $D_s = 2$. Release the hinge at E and take $X_1$ = horizontal reaction ($\rightarrow$), $X_2$ = vertical reaction ($\uparrow$) at E as redundants. The primary structure is fixed at A.

 20 kN->  B ============ C ====== D   5 kN/m on BC
          |             |        v 20 kN
          | 4 m         | 4 m
          |             o E (hinge)
          A (fixed)
          |<--- 5 m --->|<- 2 ->|

Unit and load moments (+ tension on the inner/lower face)

$M_0$ is the moment in the primary structure under the real loads (20 kN at B, 5 kN/m on BC, 20 kN at D), with the column CE unstressed; $m_1$ and $m_2$ are the moments due to unit forces at E. Their values at the key sections are given in the table below.

$$\begin{aligned} f_{11} &= \frac{122.667}{EI} \\ f_{12} &= \frac{90.000}{EI} \\ f_{22} &= \frac{141.667}{EI} \\ \Delta_{10} &= \frac{-4050.0}{EI} \\ \Delta_{20} &= \frac{-6574.0}{EI} \end{aligned}$$ $$\begin{aligned} 122.667\,X_1 + 90.000\,X_2 &= 4050.0 \\ 90.000\,X_1 + 141.667\,X_2 &= 6574.0 \end{aligned}$$ $$X_1 = H_E = -1.93\ \text{kN},\qquad X_2 = V_E = 47.63\ \text{kN}$$

Final bending moments (kNm)

At the joint C the member moments balance: -47.72 = -40.00 + (-7.72).

Shear force (kN)

Reactions

• E (hinge): $H_E = 1.93$ kN (towards the left), $V_E = 47.63$ kN ($\uparrow$)
• A (fixed): $H_A = 18.07$ kN (towards the left), $V_A = 2.63$ kN (downward), $M_A = 44.35$ kNm

Check: $\Sigma V$: $-2.63 + 47.63 = 5\times5 + 20 = 45$ kN; $\Sigma H$: $20 - 18.07 - 1.93 = 0$.

In BC the moment falls from $+27.93$ kNm at B through zero at 2.86 m from B to -47.72 kNm at C.

Answer: BMD ordinates: $M_A = 44.35$ kNm, $M_B = 27.93$ kNm, $M_C = -47.72$ kNm (beam side), $M_D = 0$ and the BM in BC changes sign at 2.86 m from B; the SFD values are in the table above.