CTC / MTC 222 Strength of Materials Chapter 1 Basic Concepts

Strength of Materials A natural follow-up to the study of statics Statics – the study of forces acting on rigid bodies at rest Strength of materials – the study of the relationships between external forces acting on elastic bodies and the internal stresses and strains caused by these forces

Strength of Materials In statics, the bodies analyzed are assumed to be rigid Deformations or deflections of the bodies are neglected In strength of materials, the bodies analyzed are considered deformable Deformations and deflections of the bodies are important considerations

Course Objectives To provide students with the necessary tools and knowledge to analyze forces, stresses, strains, and deformations in mechanical and structural components. To help students understand how the properties of materials relate the applied loads to the corresponding strains and deformations.

Course Objectives To provide students with a portion of the knowledge necessary to design (or analyze) a product, machine or structure that is safe and stable under the loads exerted on it. Three “modes of failure” must be considered Failure by fracture Excessive deflection or deformation Instability or buckling Principles of strength of materials are required to ensure the component is safe with regard to strength, rigidity and stability.

Chapter Objectives Use units from the SI metric unit system and the U.S. Customary unit system correctly Define mass and weight and use these terms correctly Define stress, direct normal stress, direct shear stress and bearing stress Define single shear and double shear Define strain, normal strain and shearing strain Define Poisson’s ratio, modulus of elasticity in tension and modulus of elasticity in shear

Basic Unit Systems U.S. Customary Units Length – foot (ft) or inch (in) Mass – slug (lb-s 2 / ft) Force – pound (lb), or kip Time - seconds (s) SI Metric Units Length – meter (m) or millimeter (mm) Mass – kilogram (kg) Force – newton (N) (kg–m / s 2 ) Time - seconds (s)

Unit Conversions Convert all lengths in an equation to the same unit, all masses to the same unit, all forces to the same unit, all measures of time to the same unit, etc. Avoid “mixed units” To convert from one unit to another, multiply by a factor whose denominator in the first unit is equal to the numerator in the second unit. To convert 10 meters to feet, multiply by feet/1 meter 10 m x (3.281 ft / 1 m) = ft To convert 10 feet to meters, multiply by 1 meter/3.281 feet 10 ft x (1 m / ft) = m Conversion factors listed in Appendix A-26

Mass, Force and Weight Mass – the amount of substance in a body Absolute, not dependent on location A scalar quantity; has magnitude only Force – a push or pull exerted on a body by an external source A vector quantity; has magnitude and direction Weight – the force exerted on a body by gravity Relative, dependent on location

Mass, Force and Weight Mass, force, and weight are related by Newton’s 2 nd Law Force = mass x acceleration F = ma, or m = F / a When the force is the force of gravity, this can be expressed as: W = mg, or m = W / g g is the acceleration due to gravity g = 9.81 m / s 2 in SI units g = 32.2 ft / s 2 in US units Units of mass – kg in SI Units, slugs (#- s 2 /ft)in US units Units of weight – N (kg-m/s 2 ) in SI Units, lbs in US units

Density and Specific Weight Density – the amount of mass per unit volume Units – slugs / ft 3, kg / m 3 Specific Weight – the amount of weight per unit volume Units – lbs / ft 3, N / m 3

Stress Stress – the internal resistance to an external force offered by a unit area of the material from which a member is made, or, more simply, force per unit area Stress = force / area = F / A Units in US Customary system Pounds / in 2 (psi), kips / in 2 (ksi) Units in SI system Newtons / m 2, Newtons / mm 2, Kilonewtons / m 2 Also expressed in pascals (Pa), or more commonly, megapascals (MPa) 1 Pa = 1 N / m 2 1MPa = 1x 10 6 N / m 2 = 1x 10 3 KN / m 2 = 1 N / mm 2

Direct Normal Stress Normal Stress – a stress which acts perpendicular (or normal) to the cross section of the member Direct Normal Stress – a normal stress which is also uniform across the resisting area Units in US Customary system Pounds / in 2 (psi), kips / in 2 (ksi) Units in SI system Newtons / m 2, Newtons / mm 2, Kilonewtons / m 2 Also expressed in pascals (Pa), or more commonly, mega pascals (MPa) 1 Pa = 1 N / m 2 1 MPa = 1x 10 6 N / m 2 = 1x 10 3 KN / m 2 = 1 N / mm 2

Direct Normal Stress  Compressive Stress Tends to crush the material Shortens the member Tensile Stress Tends to pull the material apart Elongates the member σ = Applied Force/Cross-sectional Area = F/A Area A is perpendicular to the line of action of the force

Direct Shear Stress,  Shear – a cutting action Direct Shear Stress Shear force is resisted uniformly by the area of the part in shear Shear stress is uniform across the area  = Applied Force/Shear Area = F/A s Single shear – applied shear force is resisted by a single cross-section of the member Double shear – applied shear force is resisted by two cross-sections of the member

Bearing Stress, σ b Bearing Stress - developed when one body rests on another and transfers a load normal to it, tending to crush the supporting material σ b = Applied Load/Bearing Area = F/A b Area A b is the area over which the load is transferred For flat surfaces in contact, A b is the area of the smaller of the two surfaces For a pin in a close fitting hole, A b is the projected area, A b = Diameter of pin x material thickness

Contact Stress, σ c A type of bearing stress developed when a load is transmitted between two surfaces over a very small area. Examples: A cylindrical roller on a flat plate A sphere on a flat plate Area of bearing is theoretically zero. Due to elasticity of materials actual bearing area is very small Detailed analysis of these stresses, sometimes called Hertz stresses is beyond the scope of this course.

Stress Elements Stress element – a free body diagram of an infinitesimal portion of a member 3D element is a cube 2D element is a rectangle If faces are unit areas, forces represent stresses (force per unit area) Overall body is in equilibrium, therefore stress element is in equilibrium Tensile and compressive stresses are perpendicular to face Shear stresses are parallel to face

Strain  ε An applied load causes a load-carrying member to deform This deformation can be measured. It can also be calculated. Strain – unit deformation, calculated by dividing the total deformation by the original length Strain = ε = total deformation / original length Units Could be considered dimensionless Preferably reported as in/in, or mm/mm

Shearing Strain  Shearing action on parallel faces of stress element tend to deform it angularly The angle  measured in radians is the shearing strain

Poisson’s Ratio, μ An applied force causes a load-carrying member to deform in the direction of the force At the same time, there is also deformation in the direction perpendicular to the force For example, a tensile force will cause the member to elongate in the direction of the load (axial strain) At the same time, there is also deformation in the direction perpendicular to the force (lateral strain) Poisson’s Ratio is the ratio of the lateral strain to the axial strain Poisson’s Ratio = μ = Lateral strain/Axial strain = -ε L / ε a Values of Poisson’s ratio range from 0.1 to 0.5 Values for metallic materials range from 0.25 to 0.35

Modulus of Elasticity  E Modulus of Elasticity A measure of the stiffness of a material The ratio of the normal stress on an element to the corresponding strain Modulus of elasticity = normal stress / normal strain, or E = σ  A material with a higher value of E will deform less than one with a lower value of E The term “modulus of elasticity” without a modifier is assumed to refer to tensile modulus

Modulus of Elasticity  in Shear, G Modulus of Elasticity in Shear The ratio of the shearing stress on an element to the corresponding strain G = shearing stress / shearing strain =  /  Also called Modulus of Rigidity G is related to the tensile modulus by Poisson’s ratio G = E / [2 (1 + μ)]