Solid 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. 4. Hooke’s Law Hooke's Law says that the stretch of a spring is directly proportional to the applied force. Engineers say "Stress is proportional to strain". This law is formulated in terms of the stress and strain and may be written as : where E is a material constant known as Young’s modulus. Example 1 Suppose that a 4-inch-diameter round bar is extended with a 50.000-lb axial load. The bar has an initial length of 5 feet and extends 0.006 inches. What is the Young’s modulus for the material from which the bar is made? . Solution We can obtain the Young’s modulus by using Hooke’s law, and Equation (1) and Equation (2). The stress in the bar is The strain is Thus, the Young’s modulus for this material is 5. Stress Concentration When an elastic body with a local geometrical irregularity is stressed, there usually is a localized variation in the stress state in the immediate neighborhood of the irregularity. The maximum stress levels at the irregularity may be several times larger than the nominal stress levels in the bulk of the body. This increase in stress caused by the irregularity in geometry is called a stress concentration. Stress Concentration Factor Where the stress concentration can not be avoided by a change in design, it is important to base the design on the local value of the stress rather than on an average value. The usual procedure in design is to obtain the local value of the stress by use of a stress concentration factor. σmax , maximum stress in the presence of a geometric irregularity or discontinuity, σnom, nominal stress which would exist at the point if the irregularity were not there. Typical Kt Curve 6. Fatigue Loads or deformations which will not cause fracture in a single application can result in fracture when applied repeatedly. Fracture may occur after a few cycles, or after millions of cycles. . This process of fracture under repeated loading is called fatigue. Fatigue is one of the three common causes of mechanical failure, the others being wear and corrosion. Consider a situation in which the stress at a point in a body varies with time. Experiments show that the alternating stress σa is the most important factor in determining the number of cycles of load a material can withstand before fracture, while the mean stress level σm is less important. Fatigue Curve Notes It is customary to designate the stress which can be withstood for some specified number of cycles as the fatigue strength of the material. . Fatigue cracks are most likely to form and grow from locations where holes or sharp corners cause stress concentrations. . In designing parts to withstand repeated stresses, it is important to avoid stress concentrations. Keyways, oil holes, and screw threads are potential sources of trouble and require special care in design in order to prevent fatigue failures. 7. Finite Element Method (FEM) The FEM is a numerical analysis technique for obtaining approximate solutions to engineering and design problems. Assume a problem with an infinite number of unknowns. The finite element discretization procedures reduce the problem to one of a finite number of unknowns: . dividing the solution region into discrete elements; expressing the unknown field variable in terms of assumed approximating functions within each element. Notes The discrete elements can be used to represent exceedingly complex geometric shapes, since these elements can be put together in various ways. . The approximating functions ( or interpolation functions) are defined in terms of the values of the field variables at specified points called nodes. Nodes usually lie on the element boundaries where adjacent elements are considered to be connected. Notes Clearly, the nature of the solution and the degree of approximation depend not only on the size and number of the element used, but also on the interpolation functions selected. . .In essence, a complex problem reduces to considering a series of greatly simplified problems. FEM Steps To summarize in general terms how the finite element method works, the following steps are listed. 1、 Discretize the continuum. The first step is to divide the continuum or solution region into elements. . 2、Select interpolation functions. The next step is to assign nodes to each element and then choose the type of interpolation function to represent the variation of the field variable over the element. 3. Find the element properties. Once the finite element model has been established, the matrix equations can be determined, thus expressing the properties of the individual elements. . 4. Assemble the element properties to obtain the system equations. Combine the matrix equations expressing the behavior of the elements and form the matrix equations expressing the behavior of the entire solution region or system. 5. Solve the system equations. The assembly process of the preceding step gives a set of simultaneous equations that must be solved to obtain the unknown nodal value of the field variable. . 6. Make additional computation if desired. Sometimes it is desirable to use the solution of the system equations to calculate other important parameters. Example 1 (ANSYS) Example 2（Crash Analysis for a Car） 8. Summary ? A solid material under applied load will deform. This deformation may be under tension, compression, shear, and torsion, depending on how the loads are applied. It is also possible that one or more of these loading cases may exist simultaneously. . ? In the realm of engineering design, it is important to know the behavior of a structure under applied loads. Unsafe designs are characterized by extensive deform-ations. In such cases, the engineer must redesign the component or structure. . ? The theory of solid mechanics is used to predict the amount of deformation under applied loads and hence whether or not a design is safe. .