4. Energy method Castigliano's theorem Carlo Alberto Castigliano 1873 The partial derivative of the strain energy, considered as a function of the applied forces acting on a linearly elastic structure, with respect to one of these forces, is equal to the displacement in the direction of the force of its point of application ππ π1 , π2 , β― ππ = πΏπ πππ π = 1,2, β― π The forces and displacements can be generalized to moments and angular displacements (rotation angles). If moments are also applied to the structure then the angular displacement corresponding to an applied moment ππ is ππ π1 , π2 , β― ππ , π1 , π2 … = ππ πππ Example Consider a uniform thin circular bar with circular cross-section with the bending and torsional stiffness being EI and GJ, respectively. Determine the angle of twist of the cross-section at the tip where the load is applied. Example For a uniform cantilever with a distributed load p0 (force per unit length). The bending stiffness is EI. Find (i) Vertical deflection at the tip pf the beam, (ii) Angular displacement at the tip of the beam. Example A uniform linear elastic t beam is subjected to a distributed load q as shown. Find the displacement at D. The bending stiffness of the beam is EI. Ignore the axial deformation of the beam. Statically Indeterminate Structures • When equilibrium equations are insufficient in determining the internal forces and reaction forces, the structure becomes Statically Indeterminate. • In this case, displacement boundary conditions, which make the structure to be statically indeterminate, must be considered. For the cantilever shown, an extra support exists at the tip, resulting in zerodeflection at the tip. Solving problems of statically indeterminate structures • Replace selected displacement boundary conditions with unknown forces to reduce the problem to a statically determinate one. • Solve the 'new' problem, in terms of the applied and the unknown forces. • Use the displacement boundary condition to determine the unknown forces. • Determine the deformation. Example Consider the beam as shown. Determine the reaction force R. The beam is assumed to be linear elastic, uniform with a bending stiffness EI. Example A uniform linear elastic cantilever is supported by a linear spring at the tip. The spring has a constant k and initially unstretched. Find (i) the spring force, and (ii) the angular displacement at the tip of the beam.
