STRENGTH OF MATERIALS

1. STRENGTH OF MATERIALS UNIT - 1 STRESS, STRAIN AND DEFORMATION OF SOLIDS

2. INTRODUCTION STRENGTH OF MATERIALS The solid mechanics as a subject may be defined as a branch of applied mechanics that deals with behaviours of solid bodies subjected to various types of loadings. This is usually subdivided into further two streams. • Mechanics of rigid bodies or simply Mechanics • Mechanics of deformable solids. The mechanics of deformable solids which is branch of applied mechanics is known by several names i.e. strength of materials, mechanics of materials etc.

3. INTRODUCTION Mechanics of solids: The mechanics of deformable solids is more concerned with the internal forces and associated changes in the geometry of the components involved. Of particular importance are the properties of the materials used, the strength of which will determine whether the components fail by breaking in service, and the stiffness of which will determine whether the amount of deformation they suffer is acceptable. Therefore, the subject of mechanics of materials or strength of materials is central to the whole activity of engineering design. Usually the objectives in analysis here will be the determination of the stresses, strains, and deflections produced by loads. Theoretical analyses and experimental results have an equal roles in this field.

4. Stress: defined as the force intensity or force per unit area. The rectangular bar is assumed to be cut into two halves at section XX. The each portion of this rectangular bar is in equilibrium under the action of load P and the internal forces acting at the section XX has been shown in above figure. Stress, The basic units of stress in S.I units i.e. (International system) are N / m2 (or Pa) 1 Pa (Pascal) = 1 N / m2

5. TYPES OF STRESSES : Only two basic stresses exists 1) Normal stress 2) shear shear stress. Other stresses either are similar to these basic stresses or are a combination of these e.g. bending stress is a combination tensile, compressive and shear stresses. Torsional stress, as encountered in twisting of a shaft is a shearing stress. Tensile and Compressive stresses : The normal stresses can be either tensile or compressive whether the stresses acts out of the area or into the area.

6. STRESSES: If the pulling force acts on the area which is caused to tensile stress. If the pushing force acts on the area which is caused to compressive stress.

7. Shear stresses : Let us consider now the situation, where the cross – sectional area of a block of material is subject to a distribution of forces which are parallel, rather than normal, to the area concerned. Such forces are associated with a shearing of the material, and are referred to as shear forces. The resulting force interistes are known as shear stresses. The resulting force intensities are known as shear stresses, the mean shear stress being equal to Where P is the total force and A the area over which it acts.

8. Stress-strain curve • The elastic behavior of a material can be studied by plotting a curve between the stress along the x axis and the corresponding strain along the y axis. This curve is called stress-strain curve. • elastic limit • permanent set • yield point • creeping

9. STRENGTH • It is defined as the capacity of a material to with stand, once the load is applied. It is expressed as force per unit area of cross-section. • Depending upon the value of stress, the strengths of a metals can be elastic or plastic • Depending upon the nature of stress, the strength of a metal can be tensile, compressive, shear, bending and torsional. Elastic Strength • It is the value of strength corresponding to transition from elastic to plastic range, i.e., when material changes its behaviors from elastic range to plastic range.

10. Plastic Strength It is the value of strength of the material which corresponds to plastic range and rupture. It is also termed as ultimate strength. Tensile Strength Tensile strength is the ultimate strength in tension and corresponding to the maximum load. Tensile strength = The tensile stress is expressed in N/m2

11. Shear Strength The shear strength of a metal is the value of load applied tangentially to shear it off across the resisting section. Shear strength = Bending Strength Bending strength of a metal is the value of load which can break the metal by bending it across the resisting section. Bending stress =

12. Torsional Strength Torsional Strength of a metal is the value of load applied to break the metal by twisting across the resisting section. Torsional strength = This is expressed in N/m2. Brittleness It may be defined as the property of a metal by which it will fracture without any appreciable deformation.

13. ULTIMATE STRENGTH • The strength of a material is a measure of the stress that it can take when in use. • The ultimate strength is the measured stress at failure but this is not normally used for design because safety factors are required.

14. STRAIN • The measure of the deformation produced by the influence of stress. For tensile and compressive loads. • For shear loads the strain is defined as the angle  This is measured in radians.

15. SHEAR STRESS AND STRAIN Area resisting shear Shear displacement (x) Shear Force Shear strain is angle  L Shear force

16. UNITS OF STRESS AND STRAIN • The basic unit for Force and Load is the Newton (N) which is equivalent to kg m/s2. One kilogramme (kg) weight is equal to 9.81 N.  • In industry the units of stress are normally Newton per millimetre square (N/mm2). • The MKS unit for pressure is the Pascal. 1 Pascal = 1 N/mm2 • Pressure and Stress have the same units 1 Pa = 1 N/mm2 • Strain is dimensionless.

17. MODULUS OF ELASTICITY If the strain is “Elastic" Hooke's law may be used to define Young's modulus is also called the modulus of elasticity or stiffness and is a measure of how much strain occurs due to a given stress. strain is dimensionless, therefore Young's modulus has the units of stress or pressure

18. THERMAL STRESSES • A temperature change results in a change in length or thermal strain. There is no stress associated with the thermal strain unless the elongation is restrained by the supports. • The thermal deformation and the deformation from the redundant support must be compatible.

19. For a slender bar subjected to axial loading: • The elongation in the x-direction is accompanied by a contraction in the other directions. Assuming that the material is isotropic (no directional dependence), • Poisson’s ratio is defined as POISSON’S RATIO

20. POISSON’S RATIO • This is a measure of the amount by which a solid "spreads out sideways" under the action of a load from above. • Note that a material like timber which has a "grain direction" will have a number of different Poisson's ratios corresponding to loading and deformation in different directions.

21. STRAIN ENERGY • Strain energy is a fundamental concept in applied mechanics • Consider axially loaded structural members subjected to static loads • Consider a prismatic bar subjected to a static load P • During the loading process, the load P moves slowly though the distance δ and does a certain amount of work • To find the work done by load P, we need to know the manner in which the force varies. Therefore we need to use a load-displacement diagram

22. Load – Displacement Diagram ‘ The work done by the load is equal to the area below the load – displacement curve’ Strain energy: The energy absorbed by the bar during the loading process

23. Elastic and Inelastic Strain Energy • Recall loading – unloading of a prismatic bar • The strain energy that recovers during unloading is called the elastic strain energy (triangle BCD) • Area OABDO represents energy that is lost in the process of permanently deforming the bar. This energy is called inelastic strain energy

24. Linearly Elastic Behaviour • If the material is linearly elastic (i.e. follows Hooke’s law) • Load – displacement curve is a straight line and the strain energy stored in the bar is: we also know… • Therefore we can express the strain energy of a linearly elastic bar in either of the following forms: …and for linearly elastic springs (replacing EA/L by k)…

25. Non-uniform bars • The total strain energy U of a bar consisting of several segments is equal to the sum of the strain energies of the individual segments • The strain energy of a nonprismatic bar with continuously varying axial force can be calculated by using equation 1 for the differential element dx and then integrating for the whole length of the bar. (1)