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STRAIN ENERGY CONCEPT STRENGTH OF MATERIAL
- 1. SUBTOPIC -06 • STRAIN ENERGY AND IMPACT LOADING 1. Strain Energy:- when a body strained within elastic limit, energy stored in it. this energy is called strain energy. Strain energy= work done on a body .: u = 𝝈 𝟐 𝟐𝑬 x v Where, U= strain energy 𝜎=stress V=volume of a bar The unit of strain energy is N.m.
- 2. Resilience • Total strain energy stored in body, within elastic limit is called resilience. Resilience= 𝒖 = 𝝈 𝟐 𝟐𝑬 x v
- 3. Proof Resilience • The maximum strain energy that can be stored in a body at elastic limit is called proof resilience. • ∴ 𝒑𝒓𝒐𝒐𝒇 𝒓𝒆𝒔𝒊𝒍𝒊𝒆𝒏𝒄𝒆 = 𝒖 𝒑 = 𝝈 𝑬 𝟐 𝟐𝑬 𝒙 𝑽
- 4. Modulus of resilience(um) • The maximum strain energy that can be stored in a body per unit volume at elastic limit is called modulus of resilience. • 𝑴𝒐𝒅𝒖𝒍𝒖𝒔 𝒐𝒇 𝒓𝒆𝒔𝒊𝒍𝒊𝒆𝒏𝒄𝒆 = 𝝈 𝑬 𝟐 𝟐𝑬 𝒙 𝑽 𝑽 = 𝝈 𝑬 𝟐 𝟐𝑬 • The unit of modulus of resilience is N.mm/mm3
- 5. Instantaneous stress • When a body is subjected to sudden load or Impact load the stress produced is called instantaneous stress.
- 6. Method of applying load Method of applying load Gradual loading Sudden loading Impact loading Shock loading
- 7. Gradual load • The load is increasing gradually from zero to P. Stress, 𝝈 = 𝑷 𝑨
- 8. Sudden load • When a body is subjected to total load P at a time, without small increments. It is called sudden load. Stress, 𝝈 = 𝟐𝑷 𝑨
- 9. Impact load • When a load fall on a body from some height, it is called impact load. 𝝈 = 𝑷 𝑨 𝟏 + 𝟏 + 𝟐𝑬𝑨𝒉 𝑷. 𝒍 Strain energy is, u=work done u=P(h + 𝛿𝑙) ∴ 𝝈 𝟐 𝟐𝑬 𝐱 𝐕 = 𝐏(𝐡 + 𝜹𝒍)
- 11. • 𝝈 = stress due to impact load • 𝑷 = Impact load • 𝒉 = height of fall of load • 𝜹𝒍 = deformation of a body
- 12. Derive equation of strain energy for gradual load • In case of gradual load system, load gradually increases from zero to ‘ P’. • ∴ 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑙𝑜𝑎𝑑 = 0+𝑃 2 = 𝑃 2 • Now, • Work done=average load X deformation • = 𝑃 2 X 𝛿𝑙 • = 𝑃 2 X 𝜀. 𝑙 𝜎 = 𝑃 𝐴 𝜀 = 𝛿𝑙 𝑙
- 13. • 𝜎.𝐴 2 𝑥 𝜀. 𝐿 • 1 2 𝑥 𝜎 𝑥 𝜀 𝑥 𝐴𝑙 • 1 2 𝑥 𝜎 𝑥 𝜎 𝐸 𝑥V • 1 2 𝑥 𝜎2 𝐸 𝑥 𝑉 • 𝑼 = 𝝈 𝟐 𝟐𝑬 X V