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This information is sufficient to generate the flexibility matrix. Using these results formulate the flexibility matrix f .
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Step 3 Taking each of the releases that were introduced or reactions that were removed separately, calculate the reactions, internal forces and displacements associated with their corresponding unit forces and unit bi-actions. Step 2 Calculate the reactions, internal forces and displacements structure under the action of the applied known loads.
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These are the unknown that must be solved for. Similarly for each reaction that has been removed a corresponding unknown force must be introduced. For each release that has been introduced a corresponding bi-action of unknown magnitude must be introduced. This is the essence of the flexibility method.įlexibility Method – General Procedure Step 1 Introduce releases and/or remove external reactions until the structure is statically determinate. Hence the magnitude of the propping force can be determined. However the deflection at the beam’s end is known, the net deflection is zero. The magnitude of the propping force cannot be determined directly. That presupposes that the magnitude of the propping force is known.
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More importantly the deflected shape will be equal to the deflected shape for the cantilever without the prop plus the deflected shape due to the propping force. In the same way the bending moment at any point on the beam will equal the sum of the moments at that point for the two cases. If superposition can be applied then once the magnitude of the propping force is known then the overall reactions can be calculated by adding the reactions for the case without the propping force to the reactions due to the propping force. If the structure is made from a linear-elastic material and the deflections are small then superposition applies. The result of applying a vertical force at the cantilever’s end can be predicted using the equations of statics. Consider the effect of this vertical reaction if it were to act in isolation. However, the beam is propped and qualitative analysis indicates that the prop applies a vertical reaction to the cantilever’s end. Without the prop, the structure is statically determinate and it is possible to calculate the value of the three remaining reactions, the bending moments and the deflected shape of the structure. Consider how the structure would behave if the prop was removed. The cantilever’s end is propped and therefore does not deflect. This implies that one additional independent equation is all that is necessary to analyse the beam. The beam has four unknown reactions and hence is indeterminate to degree one. Qualitative analysis indicates that the directions of the external reactions are as shown. The method is sometimes called the force method because the unknown variables that are solved for using the flexibility method are either member forces or reactions. Flexibility Method - Overview The flexibility method uses knowledge of the deformations of the structure to provide additional equations.