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Maxwell-Betti Law of real work is a basic theorem in the structural analysis. Using this theorem, it will be established that the flexibility coefficients in compatibility equations, formulated to solve indeterminate structures by the flexibility method, form a symmetric matrix and this will reduce the number of deflection computations. The Maxwell-Betti law also has applications in the construction of influence lines diagrams for statically indeterminate structures. The Maxwell-Betti law, which applies to any stable elastic structure (a beam, truss, or frame, for example) on unyielding supports and at constant temperature, states: |
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The deflection of point A in direction 1 due to unit load at point B in direction 2 is equal in the magnitude to the deflection of point B in direction 2 produced by a unit load applied at A in direction 1. |
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The Figure 4.31 explains the Maxwell-Betti Law of reciprocal displacements in which, the displacement
 is equal to the displacement  .
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In order to prove the reciprocal theorem, consider the simple beams shown in Figure 4.32.
Let a vertical force  at point B produces a vertical deflection  at point A and  at point B as shown in Figure 4.32(a). Similarly, in Figure 4.32(b) the application of a vertical force  at point A produces a vertical deflections  and  at points A and B , respectively. Let us evaluate the total work done by the two forces  and  when they are applied in different order to the zero to their final value. |
