Consistent deformation: frame, BMD, SFD, normal thrust

Reading of the figure (as assumed): portal frame of height 4 m and span 4 m. The left column AB ($2EI$) is fixed at A and carries a horizontal UDL of 10 kN/m over its lower 3 m (towards the right); the beam BC ($EI$, 2 m + 2 m) carries 15 kN downward at mid-span F; the right column CD ($2EI$) is hinged at D.

$r = 3 + 2 = 5$, so $D_s = 2$. By the consistent deformation method, release the hinge at D and take the reactions $X_1$ (horizontal, $\rightarrow$) and $X_2$ (vertical, $\uparrow$) at D as redundants. The displacements of D in the primary structure must be cancelled by the redundants, that is, the total displacement at D along each redundant is zero.

  B ----F---- C
  |   15 kN   |
 10->         | 4 m
 kN/m         |
  |           o D
  A (fixed)

Consistent deformation equations

$$\begin{aligned} \Delta_{10} + f_{11}X_1 + f_{12}X_2 &= 0 \\ \Delta_{20} + f_{21}X_1 + f_{22}X_2 &= 0 \end{aligned}$$

where $\Delta_{i0} = \int \dfrac{M_0\,m_i}{EI}dx$ (displacements of D in the primary structure) and $f_{ij} = \int\dfrac{m_im_j}{EI}dx$.

$$\begin{aligned} f_{11} &= \frac{85.333}{EI} \\ f_{12} &= \frac{48.000}{EI} \\ f_{22} &= \frac{53.333}{EI} \\ \Delta_{10} &= \frac{-256.9}{EI} \\ \Delta_{20} &= \frac{-430.0}{EI} \end{aligned}$$ $$\begin{aligned} 85.333\,X_1 + 48.000\,X_2 &= 256.9 \\ 48.000\,X_1 + 53.333\,X_2 &= 430.0 \end{aligned}$$ $$X_1 = H_D = -3.09\ \text{kN},\qquad X_2 = V_D = 10.84\ \text{kN}$$

Final bending moments $M = M_0 + X_1m_1 + X_2m_2$ (kNm, + tension on the inside)

Shear force (kN)

(Column shears are horizontal forces; positive means towards the right on the left face.)

Normal thrust (kN, compression negative)

Reactions

• D (hinge): $H_D = 3.09$ kN (towards the left), $V_D = 10.84$ kN ($\uparrow$)
• A (fixed): $H_A = 26.91$ kN (towards the left), $V_A = 4.16$ kN ($\uparrow$), $M_A = 31.63$ kNm (anticlockwise)

Check: $\Sigma H$: $30 - 26.91 - 3.09 = 0$ (UDL resultant $= 10\times3 = 30$ kN); $\Sigma V$: $4.16 + 10.84 = 15$ kN.

Answer: $H_D = 3.09$ kN, $V_D = 10.84$ kN; $H_A = 26.91$ kN, $V_A = 4.16$ kN, $M_A = 31.63$ kNm. The BMD, SFD and thrust values are tabulated above.