Method of Virtual Work (Unit Load Method) for slope and deflection
Unit load method also referred to as method of virtual work was developed by John Bernoulli in 1717. It is used to determine the slope and deflection at a specified point of a beam, frame, or truss.
In this method a virtual unit load is applied at the point of deflection (δ) and in the direction of deflection, whereas a unit virtual moment is applied at the point of desired slope.
By applying the principle of work and energy we get
external virtual work = internal virtual work
External virtual work =1kN.δ
Therefore, δ = Internal virtual work
Based on internal virtual work for different types of loading the deflection formula is given as follows;
In case of flexural members (beam, frame) the total internal virtual work is obtained by integration.
Where δ = deflection at point C
Mx = bending moment at x due to actual loading
mx = bending moment at x due to virtual unit load applied at point of deflection, C.
L = beam span
E = Modulus of elasticity of beam material
I = moment of inertia of beam section.
Note: In case of calculation of slope at a point we should apply a unit moment at the specified point of slope and write the equation for mx
See our solved Problem 7-3 for deflection of beam by unit load method.
In case of truss the total internal virtual work is obtained by taking the algebraic sum
δC = deflection of joint C
N=axial force in a member due to real loading
n = axial force in the member due to virtual unit load applied at the joint C
L= length of member
A= area of cross-section of the member
E = modulus of elasticity of the member
Note: compressive forces are taken as negative whereas tensile forces are taken as positive.
See our solved Problem 7-5 for deflection of Truss by unit load method
For other problems please visit our Problem Solver
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