Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros

Problem A reinforced concrete slab 8.0 ft square and 9.0 in. thick is lifted by four cables attached to the corners, as shown in the figure. The cables are attached to a hook at a point 5.0 ft above the top of the slab. Each cable has an effective cross-sectional area A = 0.12 in 2. Determine the tensile stress σ t in the cables due to the weight of the concrete slab. (See Table H-1, Appendix H, for the weight density of reinforced concrete.)

Problem Three different materials, designated A, B,and C, are tested in tension using test specimens having diameters of in. and gage lengths of 2.0 in. (see figure). At failure, the distances between the gage marks are found to be 2.13, 2.48, and 2.78 in., respectively. Also, at the failure cross sections the diameters are found to be0.484, 0.398, and in., respectively. Determine the percent elongation and percent reduction in area of each specimen, and then, using your own judgment, classify each material as brittle or ductile.

Problem A specimen of a methacrylate plastic is tested in tension at room temperature (see figure), producing the stress-strain data listed in the accompanying table. Plot the stress-strain curve and determine the proportional limit, modulus of elasticity (i.e., the slope of the initial part of the stress-strain curve), and yield stress at 0.2% offset. Is the material ductile or brittle?

Solution to Problem Material is brittle, because the strain after the proportional limit is exceeded is relatively small.

1.4: Elasticity What happens when the load is removed (i.e. the material is unloaded)? Tensile load is applied from O to A (fig 1.18a) and when load is removed the material follows the same curve back. This property is called elasticity If we load the same material from O to B (fig 1.18b) and then unloading occurs, the material follows the line BC. Line OC represents the residual or permanent strain. Line CD represents the elastic recovery of the material. During unloading the material is partially elastic

1.4: Plasticity “ Plasticity is the characteristic of a material which undergoes inelastic strains beyond the strain at the elastic limit ” When large deformations occur in a ductile material loaded in the plastic region, the material is undergoing plastic flow

1.4: Reloading of a material If the material is in the elastic range, it can be loaded, unloaded and loaded again without significantly changing the behaviour When loaded in the plastic range, the internal structure of the material is altered and the properties change If the material is reloaded (fig 1-19), CB is a linearly elastic region with the same slope as the slope of the tangent to the original loading curve at origin O By stretching steel or aluminium into the plastic range, the properties of the material are changed

1.4: Creep When loaded for periods of time, some materials develop additional strains and are said to creep Even though the load P remains constant after time t 0, the bar gradually lengthens Relaxation is a process at which, after time t 0, the stress in the wire gradually diminishes and eventually is reaching a constant value Creep is more important at high temperatures and has to be considered in the design of engines and furnaces

1.5: Hooke’s law Many structural materials such as metals, wood, plastics and ceramics behave both elastically and linearly when first loaded and their stress- strain curve begin with a straight line passing through origin (line OA) Linear elastic materials are useful for designing structures and machines when permanent deformations, due to yielding, must be avoided

1.5: Hooke’s law Robert Hooke ( ) The linear relationship between stress and strain for a bar in simple tension or compression is expressed by: σ = E ε σ is axial stress ε is axial strain E is modulus of elasticity Hooke’s law The above equation is a limited version of Hooke’s Law relating only the longitudinal stresses and strains that are developed during the uniaxial loading of a prismatic bar Robert Hooke was an English inventor, microscopist, physicist, surveyor, astronomer, biologist and artist, who played an important role in the scientific revolution, through both theoretical and experimental work.

1.5: Modulus of elasticity E is called modulus of elasticity or Young’s modulus and is a constant It is the slope of the stress – strain curve in the linearly elastic region Units of E are the same as the units of stress (i.e. psi for USCS and Pa for SI units) For stiff materials E is large (i.e. structural metals). E steel = GPa Plastics have lower E values than metals. E polyethylene = 0.7 – 1.4 GPa Appendix H, Table H-2 contains values of E for materials Thomas Young was an English polymath, contributing to the scientific understanding of vision, light, solid mechanics, energy, physiology, and Egyptology.

1.5: Poisson’s ratio When a prismatic bar is loaded in tension the axial elongation is accompanied by lateral contraction The lateral strain ε ’ at any point in a bar is proportional to the axial strain ε at the same point if the material is linearly elastic The ratio of the above two strains is known as Poisson’s ratio (ν) longitudinal extension lateral contraction ν = - (lateral strain / axial strain = - (ε ’ / ε )

1.5: Poisson’s ratio Simeon Denis Poisson ( ) Siméon-Denis Poisson was a French mathematician, geometer, and physicist. The minus sign in the equation is because the lateral strain is negative (width of the bar decreases) and the axial tensile strain is positive. Therefore, the Poisson’s ratio will have a positive value. When using the Poisson’s ratio equation we need to know that it applies only to a prismatic bar in uniaxial stress Poisson’s value of concrete = 0.1 – 0.2 Poisson’s value of rubber = 0.5 Appendix H, Table H-2 contains values of ν for various materials

1.5: Limitations Poisson’s ratio is constant in the linearly elastic range Material must be homogeneous (same composition at every point) Materials having the same properties in all directions are called isotropic If the properties differ in various directions the materials called anisotropic

Good luck with your homework Deadline: 28 January 2008