lecture 5 castigliono's theorem

Unit 1- Stress and Strain

  Lecture -1 - Introduction, state of plane stress

  Lecture -2 - Principle Stresses and Strains

  Lecture -3 - Mohr's Stress Circle and Theory of Failure

  Lecture -4- 3-D stress and strain, Equilibrium equations and impact loading

  Lecture -5 – Castigliono's Theorem

Topics Covered

Castigliono’s First Theorem

  Let P1, P2 ,...., Pn be the forces acting at x1 , x2 ,......, xn from the left end on a simply supported beam of span L .Let u1 , u2 ,..., un be the displacements at the loading P1, P2 ,...., Pn respectively as shown in figure.

Castigliono’s First Theorem

  Now, assume that the material obeys Hooke’s law and invoking the principle of superposition, the work done by the external forces is given by

  Work done by external forces is stored in structure as strain energy. €

W =12P1u1 +

12P2u2 + ....+ 1

2Pnun

€

U =12P1u1 +

12P2u2 + ....+ 1

2Pnun

Castigliono’s First Theorem

  u1 (deflection at point of application of P1) can be expressed as

  In general

  = flexibility coeff at i due to unit force applied at j.

  Work done by external forces is stored in structure as strain energy.

€

u1 = a11P1 + a12P2 + ....+ a1nPn

€

U =12P1 a11P1 + a12P2 + ..[ ] +

12P2 a21P1 + a22P2 + ..[ ] + ....+ 1

2Pn an1P1 + an2P2 + ..[ ]

€

u1 = ai1P1 + ai2P2 + ....+ ainPn

€

aij

Castigliono’s First Theorem

  In general

  Differentiating the strain energy with force P1

  This is nothing but displacement at the loading point

€

a ji = aij

€

U =12a11P1

2 + a22P22 + ..+ annPn

2[ ] + a12P1P2 + a13P1P3 + ..+ a1nP1Pn[ ]

€

∂U∂P1

= a11P1 + a12P2 + ..+ a1nPn[ ]

€

∂U∂Pn

= un

Castigliono’s First Theorem

  Castigliano’s first theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action.

€

∂U∂Pn

= un

Castigliono’s Second Theorem

  Castigliano’s second theorem may be stated as the first partial derivative of the strain energy of the structure with respect to any particular displacement gives the force.

€

∂U∂un

= Pn