CASTIGLIANOS THEOREM:

In 1879, Alberto Castigliano, an Italian railroad engineer, published a book, in which he outlined

a method for determining the deflection or slope at a point in a structure, be it a truss, beam or

frame. This method, which is referred to as Castiglianos second theorem or method of least

work, applies only to structures having constant temperature, unyielding supports and linear

elastic material response. If the displacement of a point is to be determined, the theorem states

that it is equal to the first derivative of the strain energy in the structure with respect to force

acting at the point and in the direction of displacement. In a similar manner, the slope at a point

in a structure is equal to the couple moment acting at the point and in the direction of rotation.

To derive Castiglianos theorem, consider a body of any arbitrary shape which is subjected to a

series of n forces P1, P2,, Pn. The displacement i in the direction of Pi is equal to the first

derivative of the strain energy with respect to Pi.

ii

i

U

P

Trusses:

The strain energy for a member of a truss is given by 2

2i

N LU

AE . Substituting this into above

equation and omitting the subscript i, we have,

2

2

N L

AE

P

It is generally easier to perform the differentiation prior to summation. In the general case L, A

& E are constant for a given member, and therefore we may write,

N LN

P AE

Where, = External joint displacement; P = External force applied to the truss; N = Internal

force in a member caused by both the loads P and force in members; L = Length of members.