1. Structural analysis 2
Enrollment no. 1-23

2. ENERGY PRINCIPLES Contents: Castigliano’s theorems
Maxwell’s reciprocal theorem
Principle of virtual work
Application of energy theorems for computing deflections in beams

3. Castigliano’s theorems
Castigliano’s first theorem:
For linearly elastic structure, the Castigliano’s first theorem may be defined 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.

4. Castigliano’s First theorem derivation
Consider an elastic beam AB subjected to loads W1 and W2, acting at points 1 and 2 respectively

5. Castigliano’s First theorem derivationIf

6. Castigliano’s First theorem derivation
Then
Similarly,
Considering the work done = Ui

7. Castigliano’s First theorem derivation
Now applying W2 at Point 2 first and then applying W1 at Point 1,
Similarly,
Strain energy, Ui

8. Castigliano’s First theorem derivation
Considering equation (III) and (IV), and equating them, it can be shown that
This is called Betti – Maxwell’s reciprocal theorem
Deflection at point 2 due to a unit load at point 1 is equal to the deflection at point 1 due to a unit load at point 2. 

9. Castigliano’s First theorem derivation
From Eqn. (III),
From Eqn. (IV),
This is Castigliano’s first theorem.

10. Castigliano’s second theorem
Similarly the energy Ui can be express in terms of spring stiffnesses k11, k12 (or k21), & k22 and deflections δ1 and δ2; then it can be shown that
This is Castigliano’s second theorem.
When rotations are to be determined,

11. Maxwell’s reciprocal theorem
In any beam (or) truss, the deflection at any point C due to load W at any point B is the same as the deflection at B due to the same load W applied at C. A B C D W

12. Problems
W kN/m
‘L’ m

13. Problems
A simply supported beam is loaded as shown in Figure. Determine the strain energy stored due to bending and deflection at centre of the beam.
W kN
L/2 m A B C
X m

14. Solution:
W kN
L/2 m A B C
X m

15. Problems
A simply supported beam is loaded as shown in Figure. Determine the strain energy stored due to bending E= 210 GN/m2.
6 cm
9 cm
15 kN
3 m
1.5 m A B C
X m

16. Consider the section X X at a distance X from A, 15 kN C B A X m 3 m
Consider the section X X at a distance X from A,

18. Question paper problems

19. A.U. Question paper problems
2. A simply supported beam of span 8 m carries two concentrated loads of 32 kN and 48 kN at 3m and 6 m from left support. Calculate the deflection at the centre by strain energy principle (Nov/Dec 2007).

20. Problems
Find the slope at the free end of a cantilever beam subjected to u.d.l. of w kN/m for the whole span using energy principle.
W kN/m
‘L’ m

21. Problems W kN/m ‘L’ m A B X Since there is no external
W kN/m
‘L’ m A B X
Since there is no external
moment at B, assume a
moment M at B. M
Moment at any section XX from free end is given by,

22. Problems
Substitute M=0 in the above equation, we get L

23. Problems
Find the slope at the centre of a cantilever beam subjected to u.d.l. of w kN/m for the whole span using energy principle.
W kN/m
‘L’ m

24. Problems Y
W kN/m
‘L’ m A B X C
Since there is no external moment at C, assume a
moment M at C. M X
Moment at any section XX between B and C from free end is given by,
Moment at any section YY between C and A from free end is given by,

25. Problems C
W kN/m
‘L’ m A B M X Y

26. Problems
Substitute M=0
L/2 L

27. Question paper problems
A simply supported beam of span 3 m is carrying a point load of 20 kN at 1m from left support in addition to a u.d.l. of 10 kN/m spread over the right half span. Using castigliano’s theorem determine the deflection under the point load. Take EI is constant throughout. (May/June 2012)

28. Question paper problems
20 kN
1.5 m
1 m A B C D
3 m
W kN
1.5 m
1 m A B C D
0.67W+3.75
0.33W+11.25
10 kN/m z X Y

29. ENERGY PRINCIPLES References:
Ferdinand P. Beer, E. Russell Johnston Jr., John T. Dewolf, “Mechanics of Materials” McGraw-Hill, New York, 2006.
Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004.
Rattan.S.S., "Strength of Materials", Tata McGraw Hill Education Pvt. Ltd., New Delhi, 2011.
Rajput R.K., "Strength of Materials (Mechanics of Solids)", S.Chand & company Ltd., New Delhi, 2010.
Ramamrutham S., “Theory of structures” Dhanpat Rai & Sons, New Delhi 1990.