lecture 5 castigliono's theorem
TRANSCRIPT
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