Demonstration From Betti's Theorem
Influence lines are based on Betti's theorem. From there, consider two external force systems, and, each one associated with a displacement field whose displacements measured in the force's point of application are represented by and .
Consider that the system represents actual forces applied to the structure, which are in equilibrium. Consider that the system is formed by a single force, . The displacement field associated with this forced is defined by releasing the structural restraints acting on the point where is applied and imposing a relative unit displacementwhich is cinematically admissible in the negative direction, represented as . From Betti's theorem, we obtain the following result:
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