# A cantilever beam of span l is fixed at one end, the other end resting freely on the middle of a simply supported crossbeam of the same span and section. If the cantilever beam is now loaded with a uniform load of w per unit length, find the reaction at the free end offered by the cross beam.

A cantilever beam of span l is fixed at one end, the other end resting freely on the middle of a simply supported crossbeam of the same span and section. If the cantilever beam is now loaded with a uniform load of w per unit length, find the reaction at the free end offered by the cross beam.

#### SOLUTION

Consider a cantilever AB fixed at A and propped at B and carrying a uniformly distributed load over its entire span as shown in Fig.

Let the reaction at B on the cantilever be R. Deflection at B for the

Net deflection of free end of the cantilever. Applying compatibility equation,

Deflection at B due to an applied external load ⇒ ${\delta _A} = \frac{{w{l^2}}}{{8El}}$

Deflection at B due to the reaction at B ⇒ ${\delta _C} = – \frac{{R{l^3}}}{{3El}}$

Deflection at the center of the beam ⇒ ${\delta _N} = \frac{{R{l^3}}}{{48El}}$

Computing net deflection at B

δA + δB = δN

l = Span of the cantilever AB,
w = Uniformly distributed load per unit length over the entire span
E = Modulus of elasticity
R = Reaction at the prop.

$\begin{array}{l} \frac{{w{l^2}}}{{8El}} – \frac{{R{l^3}}}{{3El}} = \frac{{R{l^3}}}{{48El}}\\ \\ \frac{{17R}}{{48}} = \frac{{w{L^3}}}{8}\\ \\ R = 6w/17l \end{array}$

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