Beam Deflection (Statically Indeterminate)

*Example: Beam AB shown below is of length L and flexural rigidity EI. It is clamped at A

and supported by a roller at B. External loading consists of a concentrated moment, M_o, at B.

Find the support reactions at each end.

Strategy: Use equilibrium, geometry, and load-deflection simultaneously to analyze this

problem. Start with equilibrium. Clearly the beam is indeterminate since the support at B

is not needed for support. So replace the support at B with the force of constraint, F_B. Then

the FBD (free body diagram) is as shown below.

→ Σ F_x = 0 A_x = 0 (result)

↑ Σ Fy = 0 Ay + F_B = 0 So Ay = − F_B (1)

CCW Σ M_A = 0 − M_A + Mo - (F_B)L = 0 and M_A = Mo - (F_B)L (2)

So from equilibrium we have two equations, (1) and (2), with three unknowns

A_y , M_A, and F_B. The analysis must be supplemented with more information coming

from the geometry of deformation and the load-deflection response of the beam.

Strategy (continued): So next consider the geometry of deformation. From the given

constraints the deflection of the beam at end B is zero. Thus continue with analysis of the

load-deflection response of the beam to the externally applied moment, M_o, and the

force of constraint, F_B.

Click here to continue this example.

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