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Chapter: Civil : Principles of Solid Mechanics : Two Dimensional Solutions for Straight and Circular Beams

Pure bending of a Beam of Circular Arc

Consider a circular beam of constant rectangular cross-section bent by end moments in the plane of curvature.

Pure bending of a Beam of Circular Arc

 

Consider a circular beam of constant rectangular cross-section bent by end moments in the plane of curvature (Figure 6.5). Since the moment diagram is con-stant, stresses are only a function of r and a stress function of type I is called for:












Equations (6.24) and (6.25) can now be evaluated for the displacements since all the constants A, B, C, D, F, H, and K are known.

 

The rotation and the displacements for a particular beam can be computed from the Winkler and elasticity solutions for comparison. Results for a deep beam (Figure 6.6) are presented in Table 6.2. The Winkler or elementary val-ues of the displacements at the free end, θ = π/2, are computed by integration


Comparing the two solutions, we see that again, agreement is excellent. At the neutral axis, ur is the same. The elasticity values are computed assuming v = 1/4 and therefore K = - 5.585 M/aE. If, instead v=0, then K = - 5.78 M/aE and the agreement in vθ would essentially be exact. The discrepancy in vθ at the inner and outer fibers, which is still small, is due primarily to the difference in the rotation that accumulates when integrated over the arc length and is then multiplied by the distance from the neutral axis in computing the longitudinal displacement.

 

Although the radial displacement at the centerline depends slightly on Pois-son’s ratio due to the K term, the small radial change in thickness:


Thus, considering both stresses and displacements, the close agreement of values computed from the simple Winkler analysis indicates that the added refinement from elasticity theory, while very satisfying, is of little practical significance.


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Civil : Principles of Solid Mechanics : Two Dimensional Solutions for Straight and Circular Beams : Pure bending of a Beam of Circular Arc |


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