Why is there no bending moment in the right column DEF in the real loading?

The roller gives no horizontal force, and the column carries only axial load

2080 Chaitra · 8 marks

Frame horizontal deflection at E

Short answer

Using the virtual work method, the horizontal deflection at E is 720/EI (kN·m³) to the right, obtained by integrating the product of real and virtual moments along all frame members.

Calculate the horizontal deflection at point E for the following frame. Assume flexural rigidity EI as constant throughout. [Figure: portal frame; left column A-B-C with A (hinge) at the bottom, B 2 m above A, C 2 m above B; top beam CD of length 4 m carrying 10 kN/m UDL; right column D-E-F with E 2 m below D and F (roller) 2 m below E; a 20 kN horizontal load acts at B pointing towards the frame.]

Given data

Portal frame (EI constant): A(0,0) hinge, B(0,2), C(0,4), D(4,4), E(4,2), F(4,0) roller. 20 kN horizontal at B (to the right); UDL 10 kN/m on CD (down). Required: horizontal deflection at E using the unit load method.

   C ============== D
   |   10 kN/m      |
   |                |
 20->B              E  <- unit load (virtual)
   |                |
   A (hinge)        F (roller)

Real loading: reactions

$\sum F_x = 0$ : $A_x = 20$ kN (left). $\sum M_A = 0$ : $F_y\times4 = 20\times2 + 40\times2 \Rightarrow F_y = 30$ kN; $A_y = 40 - 30 = 10$ kN.

Virtual load: unit horizontal load at E (to the right)

$A_x = 1$ (left). Moments about A: $F_y\times4 = 1\times2$ , so $F_y = 0.5$ (up) and $A_y = 0.5$ (down).

Bending moments ( $s$ from the first end of each member, in the order A-B-C-D-E-F)

Member	$s$ range (m)	$M$ (kNm)	$m$	$\int M m\,ds$
AB	0-2	$20s$	$s$	53.33
BC	0-2	40	$s+2$	240.0
CD	0-4	$40 + 10s - 5s^2$	$4 - s/2$	426.67
DE	0-2	0	$2-s$	0
EF	0-2	0	0	0

(The right column has no moment because the roller gives no horizontal force.)

Deflection

\Delta_{Eh} = \frac{1}{EI}\sum\int Mm\,ds = \frac{53.33 + 240 + 426.67}{EI} = \frac{720}{EI}

Answer: horizontal deflection of E $= \dfrac{720}{EI}$ kNm $^3$ /EI, towards the right (direction of the unit load).

✓ Answer checked by Claude Opus 5.5 (high effort)

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