Force method: frame reactions and BMD

Reading of the figure: portal frame ABCD, columns AB and CD are 4 m high ($EI$), the beam BC is 5 m long ($2EI$) with 30 kN/m, a 100 kN horizontal load acts at B (towards C), A is fixed and D is hinged.

$r = 3 + 2 = 5$, so $D_s = 2$. Release the hinge at D and take $X_1$ = horizontal reaction ($\rightarrow$) and $X_2$ = vertical reaction ($\uparrow$) at D.

 100 kN
  ->  B ============ C   30 kN/m
      |              |
      | 4 m          | 4 m
      |              |
     ///A            o D (hinge)
      |<---- 5 m --->|

Primary structure (cantilever fixed at A)

• $M_0$ (+ tension on the inner face): at B, $-30\times5\times2.5 = -375$ kNm (the 100 kN load acts at the joint B); at A, $-(375 + 100\times4) = -775$ kNm; the beam falls parabolically from $-375$ at B to 0 at C; column CD has no moment.
• $m_1$ (unit horizontal force at D): CD from 0 at D to 4 at C, beam constant 4, column from 4 at B down to 0 at A.
• $m_2$ (unit vertical force at D): beam from 5 at B to 0 at C, column AB constant 5.

Flexibility coefficients (in $1/EI$ units)

$$\begin{aligned} f_{11} &= \frac{82.667}{EI} \\ f_{12} &= \frac{65.000}{EI} \\ f_{22} &= \frac{120.833}{EI} \\ \Delta_{10} &= \frac{-5316.7}{EI} \\ \Delta_{20} &= \frac{-12671.9}{EI} \end{aligned}$$

Compatibility ($\Delta_{Dx} = \Delta_{Dy} = 0$)

$$\begin{aligned} 82.667\,X_1 + 65.000\,X_2 &= 5316.7 \\ 65.000\,X_1 + 120.833\,X_2 &= 12671.9 \end{aligned}$$ $$X_1 = H_D = -31.44\ \text{kN},\qquad X_2 = V_D = 121.79\ \text{kN}$$

Final bending moments (kNm)

The maximum sagging moment in the beam is 121.42 kNm at 0.94 m from B.

Reactions

• At D (hinge): $H_D = 31.44$ kN (towards the left), $V_D = 121.79$ kN ($\uparrow$)
• At A (fixed): $H_A = 68.56$ kN (towards the left), $V_A = 28.21$ kN ($\uparrow$), $M_A = 166.07$ kNm (anticlockwise)

Check: $\Sigma H$: $100 - 68.56 - 31.44 = 0$; $\Sigma V$: $28.21 + 121.79 = 30\times5 = 150$ kN.

Answer: at the hinged support D, the horizontal reaction is $31.44$ kN (towards the left) and the vertical reaction is $121.79$ kN (upward). End moments: $M_A = 166.07$ kNm, $M_B = 108.15$ kNm, $M_C = -125.78$ kNm (signs as in the table).