Solid Mechanics and Mechanical Engineering Objectives After learning this chapter, you should be able to do the following "o Differentiate between the different types of basic loading conditions Understand the basic approach of the Finite Element Method( FEM) 1 Introduction During the analysis of an engineering design, a mechanical engineer is often faced with predicting the deformation of a In some cases, the inverse problem is solved. That is, the maximum amount of desired de load that will produce the deformation is desired Solid Mechanics: Structural Mechanics, Mechanics of Materials, Elastic Mechanics. Plastic Mechanics 2. Stress and strain Normal Stress, often symbolized by the Greek letter sigma, is defined as the force perpendicular to the cross sectional area divided by the cross sectional area.(axial stress) Axial Strain, a non-dimensional parameter, is defined as the ratio of the deformation in length to the original length. Strain is often represented by the greek symbol epsilon( Application I (Tension Compression Suppose the force is perpendicular to the longitudinal axis. The stress will be a Shear Stress, defined as force parallel to an area divided by the area. Just as an axial stress results in an axial strain, so does shear stress produce a Shear Strain(n) Application 2----Shearing Force Let's consider a shaft, to which an external torgue is applied(such as in power transmission). The shaft is said to be in torsion. The effect of torsion is to create an angular displacement of one end of the shaft with respect to the other. For a shaft of circular cross section, the relationship between the shear stress and torque is where is the polar moment of inertia. Application 3---Transmission Shaft Notes In general, more than one type of stress may be active in a solid body, due to combined loading conditions. (tension compression, shear, torsion, etc. )When faced with an engineering problem, an engineer must recognize if more than state of stress exists. Because stresses are vector quantities, care must be taken when adding the terms together Application 4----Transmission system of machine tools Notes The simple loading cases considered in this chapter form the basics of the study of strength of materials The Finite Element Method is often used to solve problems involving complicated geometries or loading conditions for structural analysis 3 Poisson Effect When a tensile load is applied to a uniform beam the increase in the len gth of the beam is accompanied by a decrease the lateral dimension of the beam The decrease or the increase in the lateral dimension is due to a lateral strain, which is proportional to the strain along the axial direction The ratio of the lateral strain to the axial strain is related to the poisson ratio. named after the mathematician who calculated the ratio by molecular theory The minus sign in Equation is needed in order to keep track of the sign in the strain. For example, because tension corresponds to a decrease in the lateral direction, the lateral strain is negativeSolid Mechanics and Mechanical Engineering Objectives After learning this chapter, you should be able to do the following : ❖ Differentiate between the different types of basic loading conditions. . ❖ Understand the basic approach of the Finite Element Method(FEM). 1. Introduction During the analysis of an engineering design, a mechanical engineer is often faced with predicting the deformation of a body. . In some cases, the inverse problem is solved. That is, the maximum amount of desired deformation is known and the load that will produce the deformation is desired. Solid Mechanics : Structural Mechanics、Mechanics of Materials、Elastic Mechanics、Plastic Mechanics 2. Stress and Strain . Normal Stress, often symbolized by the Greek letter sigma, is defined as the force perpendicular to the cross sectional area divided by the cross sectional area. (axial stress) . . Axial Strain, a non-dimensional parameter, is defined as the ratio of the deformation in length to the original length. Strain is often represented by the Greek symbol epsilon( ). Application 1——（Tension & Compression） Suppose the force is perpendicular to the longitudinal axis. The stress will be a Shear Stress, defined as force parallel to an area divided by the area..Just as an axial stress results in an axial strain, so does shear stress produce a Shear Strain (γ). Application 2——Shearing Force Let’s consider a shaft, to which an external torque is applied (such as in power transmission). The shaft is said to be in torsion. The effect of torsion is to create an angular displacement of one end of the shaft with respect to the other. For a shaft of circular cross section, the relationship between the shear stress and torque is where J is the polar moment of inertia. Application 3——Transmission Shaft Notes In general, more than one type of stress may be active in a solid body, due to combined loading conditions.(tension, compression, shear, torsion, etc.) When faced with an engineering problem, an engineer must recognize if more than state of stress exists.. Because stresses are vector quantities, care must be taken when adding the terms together. Application 4——Transmission system of machine tools Notes The simple loading cases considered in this chapter form the basics of the study of strength of materials. .. The Finite Element Method is often used to solve problems involving complicated geometries or loading conditions for structural analysis. 3. Poisson Effect When a tensile load is applied to a uniform beam, the increase in the length of the beam is accompanied by a decrease in the lateral dimension of the beam. . The decrease or the increase in the lateral dimension is due to a lateral strain, which is proportional to the strain along the axial direction. The ratio of the lateral strain to the axial strain is related to the Poisson ratio, named after the mathematician who calculated the ratio by molecular theory. The minus sign in Equation is needed in order to keep track of the sign in the strain. For example, because tension corresponds to a decrease in the lateral direction, the lateral strain is negative