Frame horizontal deflection at free end

Given data

Frame fixed at the base A (0,0). Left column AB: 4 m high, $2EI$, with 50 kN horizontal (to the right) at 2 m above the base. Beam BC: 6 m, $1.5EI$, UDL 20 kN/m (down). Right column CD: 3 m, $EI$, hanging down from C to the free end D, where a 30 kN horizontal load acts towards the left. Required: horizontal deflection at the free end D by virtual work.

    B =========== C        UDL 20 kN/m on BC
    |             |
 50->| (2 m)      | 3 m
    |             D <- 30 kN (free end)
    A (fixed)

Virtual load

Apply a unit horizontal load at D, to the left (the same direction as the 30 kN). Distances $s$ are measured from D along DC, from C along CB, and from B along BA. Since the structure is a cantilever fixed at A, no reactions are needed to find $M$ and $m$ (moments are taken on the free-end side).

The column is split at the 50 kN load, where the expression for $M$ changes.

Integrals

$$\Delta_{Dh} = \frac{3633.33}{EI}$$

Answer: horizontal deflection of the free end $= 3633.3/EI$ (kNm$^3$/EI), towards the left (the direction of the unit load and the 30 kN force). Assumed: the 50 kN load acts to the right and the 30 kN load to the left.