Principal of Virtual Work

• Developed by John Bernoulli in 1717• Based on conservation of energy• Used to obtain displacement and slope

at various points on a deformable body

• Real loads cause no movement of the supports but they can strain the material beyond the elastic limit.

• A series of external loads, P1, P2 and P3, will cause internal loadings, u within the body.

• The external loads will be displaced, Δe, and the internal loadings will undergo displacement, δ.

• Conservation of energy states that: Ue = Ui

• Thus; Σ P Δe = Σ u δ• Suppose displacement at point A, Δ,

need to be determined.

Application of real loads

• Place an imaginary or external virtual force P’ which acts in the same direction as Δ and has a “unit” magnitude, i.e. P’ = 1.

• External virtual load, P’, will create an internal virtual load, u, and each of them will undergo real displacement, Δ and dL, respectively, when the real loads are applied.

• Thus; 1. Δ = Σ u.dL• This method is called Method of

Virtual Force.Application of virtual unit load

Internal Virtual Work

• Material behaviour is assumed linear-elastic and stress does not exceed proportional limit.• Equations of elastic strain energy can be used to formulate equations of internal virtual work.• Strain energy expressions are one-half the product of the stress resultant and its displacement because

the loading was applied gradually.• For internal virtual work expressions, the virtual load is applied “fully” before the real loads cause

displacement.• Lowercase symbols, n, υ, m, and t, are the internal virtual loadings.

Exercise 1

Solution

Exercise 2

Solution

Exercise 3

Solution

Exercise 4

Solution

Exercise 5

Solution

Solution (Cont..)