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Wednesday, April 13, 2016

Formulation of Diagram Combination Equations Based on Vereschagin’s Rule

The famous Maxwell-Mohr’s integral forms the backbone of force method of structural analysis, and computing of displacements in beams and frames. It is more ideal for hand calculation purposes than the direct stiffness method which may involve large matrices and are very not convenient for manual calculation. The Mohr’s integral may be solved by direct multiplication and integration of bending moment equations (one linearly and the other of any arbitrary form), or by using the bending moment diagram multiplication/graphical method which is based on Vereschagin’s rule.

 In the paper downloadable in this post, a lot of formulars were derived for combining different shapes of bending moment diagrams for use in the graphical method than can be found in many structural engineering textbooks.

Diagram multiplication method presents most effective way for computation of any displacement (linear, angular, mutual, etc.) of bending structures, particularly for framed structures. The advantage of this method is that the integration procedure according to Maxwell–Mohr integral is replaced by elementary algebraic procedure on two bending moment diagrams in the actual and unit states. This method was developed by Russian engineer Vereschagin in 1925 and is often referred as the Vereschagin's rule, in which the area of the bending moment diagram in the actual state multiplies the ordinate that its centroid makes with the unit state diagram in order to obtain the deformation.

Example on the Application of Vereschagin's Rule on the Analysis of Indeterminate Frames

On the Deformation of Statically Indeterminate Frames Using Force Method

In the post, the following diagrams were combined as examples;


The procedure for the combination of the two shapes shown above is presented in the screenshot shown below;


In this case, we have to split the shapes at the point of contraflexure, so we have two shapes with areas A1 and A2 combining with the triangle below them. So as usual we have;

To download the paper the full paper on equation formulation where series of shapes encountered in analysis have been combined, click HERE

To know how Vereschagin's rule is applied in analysis, click HERE

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