Strain energy due to bending

    Consider a prismatic beam subjected to loads as shown in the Fig. 1.0. The loads are assumed to act on the beam in a plane containing the axis of symmetry of the cross section and the beam axis. It is assumed that the transverse cross sections (such as AB and CD), which are perpendicular to centroidal axis, remain plane and perpendicular to the centroidal axis of beam (as shown in Fig 1.0).

bending deformation

    Consider a small segment of beam of length ds subjected to bending moment as shown in the Fig. 1.0. Now one cross section rotates about another cross section by a small amount dθ. From the figure,

formula

where R is the radius of curvature of the bent beam and EI is the flexural rigidity of the beam. Now the work done by the moment M while rotating through angle dθ will be stored in the segment of beam as strain energy dU. Hence, 

formula1

Substituting for dθ in equation (2.0), we get,

formula

Now, the energy stored in the complete beam of span L may be obtained by integrating equation (3.0). Thus,

 formula

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