Force method: reactions at D, frame with 110 kN

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 110 kN horizontal load acts at B (towards C), A is fixed and D is hinged.

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

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

Moment diagrams in the primary structure (+ tension on the inner face)

• $M_0$: in the beam, $-375$ kNm at B (UDL resultant $150\times2.5$) falling parabolically to 0 at C; in the column AB it varies linearly from $-375$ at B to $-(375 + 110\times4) = -815$ kNm at A.
• $m_1$ (unit horizontal force at D): CD from 0 to 4, beam constant 4, column AB 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)

$$\begin{aligned} f_{11} &= \frac{82.667}{EI} \\ f_{12} &= \frac{65.000}{EI} \\ f_{22} &= \frac{120.833}{EI} \\ \Delta_{10} &= \frac{-5423.3}{EI} \\ \Delta_{20} &= \frac{-13071.9}{EI} \end{aligned}$$

Compatibility

$$\begin{aligned} 82.667\,X_1 + 65.000\,X_2 &= 5423.3 \\ 65.000\,X_1 + 120.833\,X_2 &= 13071.9 \end{aligned}$$ $$X_1 = H_D = -33.72\ \text{kN},\qquad X_2 = V_D = 126.32\ \text{kN}$$

Final bending moments (kNm)

Maximum sagging moment in BC: 131.07 kNm at 0.79 m from B.

Reactions

• D (hinge): $H_D = 33.72$ kN (towards the left), $V_D = 126.32$ kN ($\uparrow$)
• A (fixed): $H_A = 76.28$ kN (towards the left), $V_A = 23.68$ kN ($\uparrow$), $M_A = 183.40$ kNm (anticlockwise)

Check: $\Sigma H$: $110 - 76.28 - 33.72 = 0$; $\Sigma V$: $23.68 + 126.32 = 30\times5 = 150$ kN.

Answer: at the hinged support D, the horizontal reaction is $33.72$ kN (towards the left) and the vertical reaction is $126.32$ kN (upward).