Theory of structures I module 1
- 1. THEORY OF STRUCTURES I MODULE 1
- 2. CONTENTS Analysis of bars of varying sections Thermal stresses Stress analysis Principal Planes & Principal Stresses Mohr’s Circle
- 3. ANALYSIS OF BARS OF VARYING SECTIONS A bar of different lengths and different diameters is shown in figure. Let this bar is subjected to an axial load P. Though each section is subjected to the same axial load P, yet the stresses, strains and change in lengths will be different. The total change in length will be obtained by adding changes in length of the individual section.
- 4. Let, P = Axial load acting on the bar L1 = length of section 1 A1 = Cross sectional area of section 1 L2 & A2 = length and cross sectional area of section 2 L3& A3 = length and cross sectional area of section 3 E = Young’s modulus of the bar
- 5. Then, Stress for the section 1, σ 1 = ( load / Area of the section 1 ) = Similarly, Stress for the section 2 and section 3 are given as, σ 2 = σ3 = Strain of section 1, Strain of section 2 and 3 are,
- 6. But strain in section 1 = Change in length of section 1 or Length of section 1 Where dL1 = Change in length of section 1 = e 1 L1 = Similarly, changes in length of section 2 and of section 3 are obtained as dL2 = Change in length of section 2 = e 2 L2 = dL3 = Change in length of section 3 = e 3 L3 =
- 7. Total change in the length of the bar = dL = dL1 + dL2 + dL3 = Above equation is used when E of different sections is same. If, E of different sections is different, then total change in length of the bar is given by,
- 9. THERMAL STRESSES Thermal stress are the stress induced in a body due to change in temperature. Let, L = Original length of the body T = Rise in temperature E = Young’s modulus α = Coefficient of linear expansion dL = Extension of rod If the rod is free to expand, then extension of the rod is given by, dL = α . T . L
- 11. Compressive strain = Decrease in length Original length = α . T . L L + α . T . L ≈ α . T . L L = α . T But, Stress = E Strain Stress = Strain x E = α . T . E Load on the rod = Stress x Area = α . T . E . A
- 12. ASSIGNMENT – SUBMISSION ( 13/07/2015) Types of external loads Different types of beams Different types of supports relationship between elastic constants Relationship between modulus of elasticity and modulus of rigidity Relationship between Young’s modulus and bulk modulus
- 13. STRESS ANALYSIS
- 14. PRINCIPAL PLANES & PRINCIPAL STRESSES The planes which have no shear stresses, are known as PRINCIPAL PLANES These planes carry only normal stresses The normal stresses acting on a principal plane, are known as PRINCIPAL STRESSES
- 15. METHODS FOR DETERMINING STRESSES ON OBLIQUE SECTION Analytical method Graphical method
- 16. ANALYTICAL METHOD FOR DETERMINING STRESSES ON OBLIQUE SECTION A member subjected to a direct stress in one plane The member is subjected to like direct stresses in two mutually perpendicular directions
- 17. A MEMBER SUBJECTED TO A DIRECT STRESS IN ONE PLANE Normal stress n = cos2 θ Tangential stress t = sin2θ 2
- 18. THE MEMBER IS SUBJECTED TO LIKE DIRECT STRESSES IN TWO MUTUALLY PERPENDICULAR DIRECTIONS Normal stress n = 1 + 2 + 1 - 2 cos2θ 2 2 Tangential stress t = 1 - 2 sin2θ 2
- 19. MOHR’S CIRCLE Mohr’s circle is a graphical method of finding normal, tangential and resultant stresses on an oblique plane
- 20. MOHR’S CIRCLE WHEN A BODY IS SUBJECTED TO TWO MUTUALLY PERPENDICULAR PRINCIPAL TENSILE STRESSES OF UNEQUAL INTENSITIES
- 21. MOHR’S CIRCLE WHEN A BODY IS SUBJECTED TO TWO MUTUALLY PERPENDICULAR STRESSES WHICH ARE UNEQUAL AND UNLIKE