Learning topic
Vereshchagin Rule
Graphical multiplication of diagrams using the Vereshchagin rule to evaluate displacement integrals in framed structures.
The Vereshchagin rule is a graphical procedure for evaluating the Mohr integral on segments of constant bending stiffness. It replaces direct integration with the area of one diagram and an ordinate of the other.
For a segment with constant $EI$, the Mohr integral $\int M\bar M/(EI)\,dx$ can be evaluated graphically when one of the two multiplied diagrams is linear over that segment.
- Construct the real-load bending-moment diagram $M$ and the unit-load diagram $\bar M$ for the required displacement.
- Divide the beam or frame into segments where $EI$ is constant and the diagrams have convenient geometric shapes.
- For each simple part of one diagram, determine its algebraic area $\Omega$ and the coordinate of its centroid.
- On the other diagram, which must be linear over that segment, determine the ordinate $\bar M_c$ beneath the centroid of the first diagram area.
- Calculate the contribution $\Delta=\Omega\bar M_c/(EI)$, retaining the signs of the diagrams.
- Sum the contributions from all segments. A positive result acts in the direction of the introduced unit load; a negative result acts in the opposite direction.
If neither diagram is linear on a segment, the simple area-times-centroid-ordinate rule is generally not applicable without additional subdivision or direct integration.
Geometric interpretation
If one moment function is linear over a segment, the integral of the product can be evaluated using the first moment of area of the other diagram. The ordinate of the linear diagram must be taken directly beneath the centroid of the selected diagram area, not simply at the midpoint of the member segment.
Signs
Areas and ordinates are algebraic. Diagrams having the same sign over a contribution produce a positive term; opposite signs produce a negative term.
Subdivision
Trapezoidal, broken, or sign-changing diagrams can be decomposed into rectangles, triangles, or other simple shapes with known areas and centroid locations. Their contributions are calculated separately and summed.
Limitation
The simple rule must not be applied mechanically when both diagrams are nonlinear over the same segment. Use further subdivision, choose the diagram that is linear where possible, or evaluate the integral directly.
About this topic
This topic explains graphical multiplication of internal-force diagrams using the Vereshchagin rule, its applicability conditions, and its use in structural displacement calculations.