Propped Cantilever Reaction at the Prop

Mediumindeterminatecompatibility

A horizontal prismatic beam is fixed at A and supported by a vertical roller at B, 4.0 m from A. A uniform load of 6.0 kN/m acts downward over the whole span. EI is constant and both supports stay at their original elevation. Using linear-elastic small-deflection theory, find the vertical reaction at B.

A12.0 kN upward
B9.00 kN upward
C15.0 kN upward
D24.0 kN upward

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Answer: B. 9.00 kN upward

Release the roller and enforce compatibility: free-cantilever tip deflection wL⁴/(8EI) must equal the upward prop deflection R_B L³/(3EI). So R_B = 3wL/8 = 3(6)(4)/8 = 9.00 kN. Check: R_A = 24 − 9 = 15 kN. Wrong: 12 kN is simply supported wL/2, ignoring the fixed end; 15 kN is the reaction at the fixed end A (5wL/8), not B; 24 kN is the total load, as if B carried everything.

\frac{w L ^{4}}{8 E I}

= \frac{R _{B} L ^{3}}{3 E I}

\Rightarrow R_{B}

= \frac{3 w L}{8}

= \frac{3 ( 6 ) ( 4 )}{8}

= 9

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Propped Cantilever Reaction at the Prop

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