# Advanced Mechanics of Materials and Applied Elasticity: Analysis of Stress

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## 1.6 Internal Force-Resultant and Stress Relations

Distributed forces within a load-carrying member can be represented by a statically equivalent system consisting of a force and a moment vector acting at any arbitrary point (usually the centroid) of a section. These internal force resultants, also called stress resultants, exposed by an imaginary cutting plane containing the point through the member, are usually resolved into components normal and tangent to the cut section (Fig. 1.4). The sense of moments follows the right-hand screw rule, often represented by double-headed vectors, as shown in the figure. Each component can be associated with one of four modes of force transmission:

1. The axial force P or N tends to lengthen or shorten the member.
2. The shear forces Vy and Vz tend to shear one part of the member relative to the adjacent part and are often designated by the letter V.
3. The torque or twisting moment T is responsible for twisting the member.
4. The bending moments My and Mz cause the member to bend and are often identified by the letter M. Figure 1.4 Positive forces and moments on a cut section of a body and components of the force on an infinitesimal area dA.

A member may be subject to any or all of the modes simultaneously. Note that the same sign convention is used for the force and moment components that is used for stress; a positive force (or moment) component acts on the positive face in the positive coordinate direction or on a negative face in the negative coordinate direction.

A typical infinitesimal area dA of the cut section shown in Fig. 1.4 is acted on by the components of an arbitrarily directed force dF, expressed using Eq. (1.5) as dFx = `s` x dA, dFy = `t` xy dA, and dFz = `t` xz dA. Clearly, the stress components on the cut section cause the internal force resultants on that section. Thus, the incremental forces are summed in the x, y, and z directions to give

Equation 1.9a In a like manner, the sums of the moments of the same forces about the x, y, and z axes lead to

Equation 1.9b where the integrations proceed over area A of the cut section. Equations (1.9) represent the relations between the internal force resultants and the stresses. In the next paragraph, we illustrate the fundamental concept of stress and observe how Eqs. (1.9) connect internal force resultants and the state of stress in a specific case.

Consider a homogeneous prismatic bar loaded by axial forces P at the ends (Fig. 1.5a). A prismatic bar is a straight member having constant cross-sectional area throughout its length. To obtain an expression for the normal stress, we make an imaginary cut (section a–a) through the member at right angles to its axis. A free-body diagram of the isolated part is shown in Fig. 1.5b, wherein the stress is substituted on the cut section as a replacement for the effect of the removed part. Equilibrium of axial forces requires that P = `s` x dA or P = A s x . The normal stress is therefore

Equation 1.10  Figure 1.5 (a) Prismatic bar in tension; (b) Stress distribution across cross section.

where A is the cross-sectional area of the bar. Because V y , V z , and T all are equal to zero, the second and third of Eqs. (1.9a) and the first of Eqs. (1.9b) are satisfied by t xy = t xz = 0. Also, My = Mz = 0 in Eqs. (1.9b) requires only that s x be symmetrically distributed about the y and z axes, as depicted in Fig. 1.5b. When the member is being extended as in the figure, the resulting stress is a uniaxial tensile stress; if the direction of forces were reversed, the bar would be in compression under uniaxial compressive stress. In the latter case, Eq. (1.10) is applicable only to chunky or short members owing to other effects that take place in longer members.*

Similarly, application of Eqs. (1.9) to torsion members, beams, plates, and shells is presented as the subject unfolds, following the derivation of stress–strain relations and examination of the geometric behavior of a particular member. Applying the method of mechanics of materials, we develop other elementary formulas for stress and deformation. These, also called the basic formulas of mechanics of materials, are often used and extended for application to more complex problems in advanced mechanics of materials and the theory of elasticity. For reference purposes to preliminary discussions, Table 1.1 lists some commonly encountered cases. Note that in thin-walled vessels (r/t 10) there is often no distinction made between the inner and outer radii because they are nearly equal. In mechanics of materials, r denotes the inner radius. However, the more accurate shell theory (Sec. 13.11) is based on the average radius, which we use throughout this text. Each equation presented in the table describes a state of stress associated with a single force, torque, moment component, or pressure at a section of a typical homogeneous and elastic structural member [Ref. 1.7]. When a member is acted on simultaneously by two or more load types, causing various internal force resultants on a section, it is assumed that each load produces the stress as if it were the only load acting on the member. The final or combined stress is then determined by superposition of the several states of stress, as discussed in Section 2.2.

#### Table 1.1. Commonly Used Elementary Formulas for Stressa

1. Prismatic Bars of Linearly Elastic Material where

 s x = normal axial stress I = moment of inertia about neutral axis (N.A.) t = shearing stress due to torque t xy = shearing stress due to vertical shear force J = polar moment of inertia of circular cross section P = axial force b = width of bar at which t xy is calculated T = torque V = vertical shear force r = radius M = bending moment about z axis Q = first moment about N.A. of the area beyond the point at which t xy is calculated A = cross-sectional area y, z = centroidal principal axes of the area
2. Thin-Walled Pressure Vessels where

 s q = tangential stress in cylinder wall p = internal pressure s a = axial stress in cylinder wall t = wall thickness s = membrane stress in sphere wall r = mean radius

The mechanics of materials theory is based on the simplifying assumptions related to the pattern of deformation so that the strain distributions for a cross section of the member can be determined. It is a basic assumption that plane sections before loading remain plane after loading. The assumption can be shown to be exact for axially loaded prismatic bars, for prismatic circular torsion members, and for prismatic beams subjected to pure bending. The assumption is approximate for other beam situations. However, it is emphasized that there is an extraordinarily large variety of cases in which applications of the basic formulas of mechanics of materials lead to useful results. In this text we hope to provide greater insight into the meaning and limitations of stress analysis by solving problems using both the elementary and exact methods of analysis.