Study of Strain Energy due to Shear, Bending and Torsion

Strain Energy-Definition and Related Formulas, Strain Energy due to Shear Loading, Strain Energy due to Bending, Strain Energy due to Torsion and Examples

1. STUDY OF STRAIN ENERGY DUE TO SHEAR, BENDING AND TORSION SHANTILAL SHAH ENGINEERING COLLEGE, BHAVNAGAR Affiliated to Gujarat Technological University, Gandhinagar CIVIL ENGINEERING DEPARTMENT SUB: STRUCTURAL ANALYSIS- 1 SUB CODE: 2140306 PREPARED BY: SINGHANIA JAY SANJAY (170433106034)(1131)(D2D) VASAVA ASHISHKUMAR RAMSINGBHAI ( 170433106035)(1132)( D2D) VYAS NAMAN PRAKASHKUMAR ( 170433106036)(1133)( D2D) ZAPADIYA YOGESHKUMAR RANCHODBHAI ( 170433106037)(1134)( D2D) GUIDED BY : PROF. K.A.MEHTA PROF. D.P. ADVANI GROUP NO. = 30

2. Strain Energy What is Strain Energy? When a body is subjected to gradual, sudden or impact load , the body deforms and work is done upon it. If the elastic limit is not exceed, this work is stored in the body. This work done or energy stored in the body is called strain energy . Energy is stored in the body during deformation process and this energy is called “Strain Energy”. Strain energy = Work done

3. Resilience : Total strain energy stored in a body is called resilience. Where σ = stress V = volume of the body Proof Resilience : Maximum strain energy which can be stored in a body is called proof resilience. Where σ E = stress at Elastic Limit

4. Modulus of Resilience : Maximum strain energy which can be stored in a body per unit volume, at elastic limit is called modulus of resilience.

5. Strain Energy Due to Shear Loading If t is the uniform shear stress produce in the material by external forces applied within elastic limit, the energy stored due to Shear Loading is given by, Where, τ = shear stress G = Modulus of Rigidity

6. Consider a square block ABCD of length l , faces BC and AD are subjected to shear stress τ , Let face AD is fixed. The section ABCD will deform to AB1C1D through the angle ∅. ∅ = Shear strain ∅ is very small ∴ tan ∅ = ∅ ….. Shear Strain

7. Force P on Face BC, P= τ * BC * l When P is applied gradually, Gradual Load is: In Case of Average Force The elastic energy stored due to shear loading is known as Shear Resilience.

8. Strain Energy Due To Bending (Flexure) Consider two transverse section ‘1-1’ and ‘2-2’ of a beam, distance ‘dx’ apart as shown in fig. Consider a small strip of area ‘ da ’ at distance ‘y’ from the neutral axis. B.M. in small portion ‘dx’ will be constant.

9. Strain energy stored in small strip of area da. Strain Energy stored in entire section of a beam.

10. Now, for Strain Energy in Entire Beam, integrate between limits 0 to l. …Strain Energy due to Bending.

11. Strain Energy due to Torsion We have seen that, when a member is subjected to a uniform shear stress 𝛕, the strain energy stored in the member is: Consider a small elemental ring of thickness ‘dr’, at radius ‘r’.

12. Strain energy due to torsion for uniform shear stress, is the ...Strain Energy for One Ring

13. Total strain energy for whole section, is obtained by integrating over a range from r = 0 to r = D/2 for a solid shaft.

14. … Strain Energy due to Torsion

15. Example 1: An axial pull of 50 kN is suddenly applied to a steel bar 2m long and 1000 mm 2 in cross section. If modulus of elasticity of steel is 200 kN /mm 2 . Find, ( i ) Maximum Instantaneous Stress (ii) Maximum Instantaneous Extension (iii) Strain Energy (iv) Modulus of Resilience. Here, P = 50 kN (Sudden load) A = 1000 mm 2 l = 2m = 2000 mm E = 200 kN/mm 2 = 200× 10 3 N/mm 2

16. Maximum Instantaneous Stress: Maximum Instantaneous Extension:

17. Strain Energy (u): Modulus of Resilience (u m ):

18. Thank You