Moment distribution: frame, 150 kN, 60 kN/m

Assumptions (figure partly illegible): column AB (2I) is 10 m high, fixed at A; the girder is B-C-D with BC = 4 m (2I) carrying a 50 kN load at mid-span (2 m from B), CD = 5 m (I) carrying 60 kN/m, and D on a roller; column CE (I, 6 m) runs down from C to its fixed base E. A second 50 kN load acts vertically at C (it passes through the axis of column CE and produces no bending moment). The 150 kN load at B acts to the right. The roller at D allows horizontal movement, so the frame sways; the moment distribution is done for the non-sway case and the sway case.

Distribution factors ($k=EI/L$; the hinged-end member CD gets the factor 3/4)

• Joint B: relative stiffness $\Sigma k$ = 0.7000; BA: $k=EI/L$ = 0.2000, DF = 0.2857; BC: $k=EI/L$ = 0.5000, DF = 0.7143
• Joint C: relative stiffness $\Sigma k$ = 0.8167; CB: $k=EI/L$ = 0.5000, DF = 0.6122; CD: $k=EI/L$ = 0.1500 (far end hinged: 3/4 factor), DF = 0.1837; CE: $k=EI/L$ = 0.1667, DF = 0.2041

Case I: sway prevented by an imaginary support at B

Fixed-end moments: BC $\mp\dfrac{50\times4}{8}=\mp25$ kNm; CD $\mp\dfrac{60\times5^2}{12}=\mp125$ kNm (modified at C for the hinge at D: -187.50 kNm).

The horizontal reaction of the imaginary support (positive to the right) is $R_1=-157.61$ kN; this is the force that the imaginary support must supply to prevent sway and that the sway correction has to remove.

Case II: sway $\Delta$ with no loads

Give the girder a sway to the right such that the fixed-end moment in AB is $-100$ kNm at each end. Since $FEM\propto EI/L^2$ and both columns move equally, column CE (EI, 6 m) gets $-100\times\dfrac{(1/36)}{(2/100)}=-138.89$ kNm.

The horizontal reaction of the imaginary support is $R_2=57.62$ kN.

Combination

$R_1+kR_2=0\Rightarrow k=-\dfrac{-157.61}{57.62}=2.7353$

Checks: Joint B: $0.00$; Joint C: $0.00$; hinge D: $M_{DC}=0$.

Answer: $M_{AB}=-252.28$, $M_{BA}=-231.03$, $M_{BC}=231.03$, $M_{CB}=377.68$, $M_{CD}=-97.63$, $M_{CE}=-280.05$, $M_{EC}=-329.97$ kNm (clockwise end moments +).