MEC411-MECHANICS OF MATERIALS CHAPTER 1 STRESS & STRAIN INTRODUCTION The study on strength of materials refer to various methods of calculating the stresses and strains in structural members (beams, columns, shafts etc.) Knowledge of stresses and deflections allows for the safe design of structures that are capable of supporting their intended loads. TYPE OF FORCES STRESS Stress (๐): Stress is the applied force acting perpendicular to the surface area or system of forces that tends to deform a body. Stress measures the intensity of the force per given area o Normal stress (๐) results from the normal force ๐ and/or bending moment ๐๐ต o Shear stress (๐) results from shear stress ๐ and/or torsional moment ๐๐ก TYPE OF STRESS NORMAL STRESS When a force is applied to an elastic body, the body deforms. The way in which the body deforms depends upon the type of force applied to it. Compression force makes the body shorter. A tensile force makes the body longer Tensile and compressive forces are called DIRECT FORCES Stress is the force per unit area upon which it acts. ….. Unit is Pascal (Pa) or ( Simbol – Sigma) Note: Most of engineering fields used kPa, MPa, GPa. SHEAR STRESS Shear stress – results when a force tends to make part of the body or one side of a plane slide past the other. When a pair of force cut a material When a material is punched When a beam has a transverse load Shear stress is the force per unit area acting tangent to the surface area or the cross sectional area of the material being cut. •Shear stress, symbol is called Tau The sign convention for shear force and stress is based on how it shears the materials as shown below. SINGLE SHEAR STRESS DOUBLE SHEAR STRESS EXAMPLE The joint is fastened using 2 bolts as shown in the Figure. Determine the required diameter of the bolts if allowable shear stress for the bolts is = 110 MPA. allow STRAIN Normal strain (ε) is the deformation of a body which involved elongation or contraction. It is a dimensionless quantity. SHEAR STRAIN Shear strain is a strain which involved a shear deformation and changes in shape (angle) i.e. body twist due to torsion or a distorted cuboid as shown in Figure below. Strain changes the angles of an object. SHEAR MODULUS/MODULUS OF RIGIDITY Shear modulus also known as Modulus of rigidity is the measure of the rigidity of the body, given by the ratio of shear stress to shear strain. STRESS & STRAIN DIAGRAM • It is a tool for understanding material behaviour under load. A stress strain diagram help engineers to select the right materials for specific loading conditions. • It is a graph that represents how a material behaves under an increasing load and used by engineers when selecting materials for specific designs. A stress-strain diagram generally contains three regions: •Elastic region: This portion is generally represented as a linear relationship between stress and strain. If the load is released the specimen will return to its original dimensions. •Plastic region: In this portion, the specimen begins to yield. The maximum strength of the specimen occurs in this zone. The specimen endures some permanent deformation that remains after the load is released. •Rupture: The point at which a specimen breaks into two parts. STRESS STRAIN DIAGRAM Elastic limit โUpon reaching this point, if load is removed, the specimen still return to original shape Yielding โ A slight increase in stress above the elastic limit will result in breakdown of the material and cause it to deform permanently. โThis behaviour is called yielding โ The point where the stress-strain diagram becomes almost horizontal is called the yield point, and the corresponding stress is known as the yield stress or yield strength. STRESS STRAIN DIAGRAM Ultimate Stress •The ultimate stress or ultimate strength, as it is often called, is the highest stress on the stress-strain curve. Necking • At the ultimate stress, the cross-sectional area begins to decrease • it is caused by slip planes formed within material • As a result, “necking” begins to form • Specimen breaks at FRACTURE STRESS MODULUS OF ELASTICITY (E) As seen in the Figure, the stress-strain diagram is a straight line from the origin to a point called the proportional limit. This plot is a manifestation of Hooke’s law Stress is proportional to strain; that is, σ= E ะ where E is material property known as the modulus of elasticity or Young’s modulus. The units of E are the same as the units of, Pa or psi. For steel, E =29×106 psi, or 200 GPa, approximately. Note that Hooke’s law does not apply to the entire diagram;its validity ends at the proportional limit. Beyond this point, stress is no longer proportional to strain. MODULUS OF ELASTICITY (E) •Elastic materials always spring back into shape when released. They also obey HOOKE’s LAW. •This is the law of spring which states that deformation is directly proportional to the force. F/x = stiffness = kN/m •The stiffness is different for different material and sizes of the material. We may eliminate the size by using stress and strain instead of force and deformation: •The stiffness is now in terms of stress and strain only and this constant is called the MODULUS of ELASTICITY (E) DUCTILE MATERIALS Ductility: - Measure of the material property to deform before failure. - Ductile materials can be pulled or drawn into pipes, wire, and other structural shapes - Examples of ductile material : low carbon steel aluminum copper brass STRESS & STRAIN DIAGRAM: DUCTILE MATERIAL BRITTLE MATERIALS Brittleness: - Measure of the material’s inability to deform before failure. - The opposite of ductility. - Example of brittle materials : glass high carbon steel ceramics STRESS & STRAIN: BRITTLE MATERIAL ELASTIC VS. PLASTIC DOUBLE SHEAR โConsider a pin joint with a support on both ends as shown. This is called CLEVIS and CLEVIS PIN โ By balance of force, the force in the two supports is F/2 each โThe area sheared is twice the cross section of the pin โSo it takes twice as much force to break the pin as for a case of single shear โDouble shear arrangements doubles the maximum force allowed in the pin
