The integral is always >0, since Cijkl is positive definite. Therefore I(+6n)=I(u)+a,a≥0,a=0分6u=0 and I()≥II(),vt [I(o)=Il(u) as sought Castigliano's First theorem M Given a body in equilibrium under the action of N concentrated forces FI. The potential energy of the external forces is given by Fur where the ur are the values of the displacement field at the point of applica- tion of the forces F. Imagine that somehow we can express the strain energy as a function of the ul, i.e U=U(a 3� The integral is always ≥ 0, since Cijkl is positive definite. Therefore: Π(u + δu) = Π(u) + a, a ≥ 0, a = 0 ⇔ δu = 0 and Π(v) ≥ Π(u), ∀v Π(v) = Π(u) ⇔ v = u as sought. Castigliano’s First theorem Given a body in equilibrium under the action of N concentrated forces FI . The potential energy of the external forces is given by: V = − N I=1 FIuI where the uI are the values of the displacement field at the point of applica￾tion of the forces FI . Imagine that somehow we can express the strain energy as a function of the uI , i.e.: U = U(u1, u2, . . . , uN ) = U(uI ) 3