# Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA UNESCO EOLSS

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1 MECHANICS OF MATERIALS Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA Keywords: Solid mechanics, stress, strain, yield strength Contents 1. Introduction 2. Stress and Strain 2.1. Axial 2.2. Bending 2.3. Shear 2.4. Torsion 2.5. Biaxial States of Stress 3. Stress Transformations 3.1. Introduction to Yield Stress 3.2. Yield Criteria 4. Energy Methods 4.1. Strain Energy 4.2. Virtual Work Glossary Bibliography Biographical Sketch Summary Mechanics of Materials is a fundamental topic in the field of mechanical and civil/structural engineering. The primary goal of Mechanics of Materials is to understand how external loads applied to a structure cause the structure deform and develop internal stresses. This knowledge and understanding can be used to determine the safe allowable loading for an existing structure or to determine the dimensions and material properties required for a new structure to safely resist the expected loading. A solid understanding of Mechanics of Materials topics is considered a pre-requisite for further study in the areas of structural engineering and mechanical engineering design. This article provides a foundational treatment of topics typically found in a college/university level course in Mechanics of Materials. 1. Introduction The overall goal of structural design is to ensure that each element of a structure can safely resist any loads that are applied. The study of Mechanics of Materials is concerned with developing an understanding of how a structural element will respond under specified loading conditions. Some specific objectives of this type of analysis include:

3 Figure 1. General concept of normal stress and shear stress for one-dimensional loading For a general 3-D body subjected to generic loading, the state of stress at each point within the body can be summarized using a 3-D stress block. Each face of the 3-D block will experience one normal stress component and two shear stress components. The three unique normal stress components, σ x, σ y, and σ z, are denoted with subscripts corresponding to the x, y, or z axes that project normally from each face. For the block to remain in translational equilibrium, the normal stress component acting on the positive x -face must be equal in magnitude but opposite in direction to the stress component acting on the negative x -face. The same principal applies to the y and z faces. Each component of shear stress, τ, is denoted with two subscripts. The first subscript corresponds to the face on which the shear stress is acting, and the second subscript corresponds to the directional sense of the stress component. Positive sign conventions for these stress components are shown in Figure 2. Again, for the block to remain in translational equilibrium, the shear stress components on opposite faces must be equal in magnitude and opposite in direction. To ensure rotational equilibrium for the stress block, the following relationships must also be true: τ xy = τ yx ; τ xz = τ ; and zx τ yz = τ zy. As a result, the six-sided stress block, with three stress components acting on each face (a total of 18 stress components), can be accurately described with only six values: σ x, σ y, σ z, τ xy, τ xz and τ yz. Figure 2. 3-D state of stress for general body

4 2.1. Axial Axial Stress. Axially loaded structural elements often occur in the form of anchor bolts, hanging rods, cables, struts, or truss members. The internal axial force experienced by a structural element is typically determined using statics or more advanced structural analysis techniques. These internal axial forces develop normal stresses as shown in Figure 3. The general relationship between the internal axial force, P, and the normal stress, σ, that develops over the cross-sectional area of the member, A, is given by: P= σ da (1) A If the plane of interest is a sufficient distance from any external applied loads, the stress distribution is considered uniform and the average normal stress can be calculated as follows: P σ avg = (2) A The most common sign convention for axial stress is that tensile stress is considered positive and compressive stress is considered negative. This is the sign convention that will be used throughout the remainder of this document. It should be noted, however, that in reinforced concrete design the sign convention is often reversed so that compressive stress is considered positive and tensile stress is considered negative. Figure 3. Normal stress in axially loaded members. For locations near a point where a concentrated axial load is applied, the normal stress distribution is non-uniform. Non-uniform normal stress distributions can also result from axial loads if the resultant axial force does not pass through the centroid of the element s cross-section (i.e. an internal bending moment is present). A rule of thumb is that the normal stress distribution approaches uniformity a distance equal to the largest

6 ε t ν = (4) ε The relationship between axial stress, σ, and axial strain, ε, also depends on the material under consideration, and determining this relationship is one of the most common objectives of materials testing. For the case of common metals, such as steel and aluminum, there is a linear relationship between stress and strain as long as the stress in the material does not exceed the yield stress, σ y. The linear relationship between stress and strain is commonly expressed as Hooke s Law and is given by: ( for y ) σ = E ε σ σ (5) where E is the modulus of elasticity or Young s Modulus of the material under consideration. For the general case of 3-D loading, axial stress that develops in one direction due to external loading will tend to cause strain in all three orthogonal directions (one axial strain corresponding to the direction of the stress and two transverse strains that result from Poisson s ratio). The general relationship between strain and stress for 3-D loading is given as: σ ν σ x y ν σ z ε x = E E E ν σ σ x y ν σ z ε y = + (6) E E E ν σ ν σ x y σ z ε z = + E E E Deformations in Axially Loaded Structural Elements. The relationships described above for axial stress and strain can be combined to determine the total deformation experienced by a structural element subjected to a generic axial loading. Consider the A x, known end structural element shown in Figure 5 with variable cross-section, ( ) reactions, P A and P B, and variable applied axial load, P ( x) ( ). The internal axial force, P x, can be determined at any location between A and B using statics. A differential element, dx, will experience a portion of the total deformation, dδ AB. The total deformation between A and B is given by: δ AB L = dδ (7) 0 AB Recalling the fundamental definition of strain, dδ AB and dx are related by: dδ dx AB ε ( ) = or δab ε ( ) x d = x dx (8)

7 The strain experienced by dx, ε ( x), is related to stress, σ ( x) (5)):, by Hooke s Law (Eq. σ ( x) σ ( x) = ε ( x) E or ε ( x) = (9) E Finally, the stress can be related to the internal axial force, P( x ), and the crosssectional area, A( x ): Px ( ) σ ( x) = (10) Ax ( ) Substituting these expressions back into Eq. (7) results in the following integral expression for the total deformation between A and B: L Px ( ) δ AB = dx (11) Ax ( ) E 0 For the case of a prismatic member with constant cross-section, A, total length, L, and uniform internal axial force, P, Eq. (11) can be simplified to the following expression for the change in length: δ AB P L = AE (12) Figure 5. Determining axial deformations for structural element with varying crosssection and generic axial loading.

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