Stress Components

8 Key excerpts on "Stress Components"

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  eBook - ePub

  Elasticity

  Tensor, Dyadic, and Engineering Approaches

  • Pei Chi Chou, Nicholas J. Pagano, Nicholas J. Pagano(Authors)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  rθ , as well as a sign to indicate the direction of a particular component. However, when the plane under consideration is understood, we may eliminate one of the indices. The term stress is customarily used quite loosely to represent stress vector, stress component, or stress tensor. The precise meaning of the term is usually clear from the context in which it is used.

  Fig. 1.2 Force (a Vector) and Stress (Not a Vector)

  There are two convenient ways which are used to designate the components of a stress vector; the x, y, and z components, and the normal and tangential (shearing) components. The x, y, and z components of an external stress, i.e., the force per unit area acting on the boundary of a body, will be designated as (Fig. 1.3 ). The subscript gives the direction of the component, and the superscript μ defines the plane, i.e., the outward normal of the plane is in the μ direction. For internal stresses on imaginary cutting planes, the x, y, and z components are designated as p x

  , py , p

  z . Since these are most frequently utilized as an intermediate step in derivations, and the plane is understood, we use only a subscript.

  Fig. 1.3 External Stress Vector and Its Components

  Fig. 1.4 Normal and Shearing Stress Components

  Normal and shearing components of both external and internal stresses are more significant in some problems. We let σ represent the normal stress, i.e., the component of stress perpendicular to the plane on which it acts. The shear stress is denoted by the symbol τ. The latter is the component of stress which lie in the plane. Thus, referring to Fig. 1.4 , where p = dF/dA, we have

  p 2 = σ 2 + τ 2

  Let us next consider a three-dimensional state of stress acting on an element as shown in Fig. 1.5 . Body forces are omitted in the figure, since we are merely interested in establishing a nomenclature for stress at this time. In order to develop the terminology necessary to specify a stress component, we must define the plane on which the component acts. Unless otherwise specified, a plane is designated by the direction of its outward normal. In order to determine the direction of the outward normal, we must first establish a free body. The outward normal of any surface, then, points away from the interior of the free body. For example, the right-hand face of the element in Fig. 1.5 is called the x plane, since its outward normal points in the positive x direction. The left-hand face is termed the negative x

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  eBook - ePub

  Advanced Solid Mechanics

  Simplified Theory

  • Farzad Hejazi, Tan Kar Chun(Authors)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  Fig. 2.3 ).

  N o r m a l s t r e s s , σ =

  F A

  (2.1)

  where F is the normal force; A is the area of sectional plane.

  Figure 2.2     Girder subjected to normal force.

  (2.2)

  where

  V is the shear force;

  A

  v

  is the area of the shear plane.

  Figure 2.3     Girder subjected to shear force.

  2.2  Components of stress

  Solid mechanics studies the stresses and strains at any point on a body, usually illustrated using an infinitesimal cube enveloping the point. Since solids are three dimensional, we can define three mutually orthogonal axes by setting that point as centre. The stress and strain at this point can be categorised based on their direction, with each of them acting along a certain axis. Without other axes to be compared, the defined axes can be assumed not to have any inclination. This will be our reference axes, or global axes, as indicated in Fig. 2.4 .

  Figure 2.4     Stress Components at a point.

  Suppose a force is exerted along the x-axis. The developed stress acting along the x-axis is distributed over the plane yz, which is the plane normal to the x-axis. The resultant stress is normal stress along the x-axis, σ

  x

  . Meanwhile, the stress acting along the y-axis is distributed over the face of the plane yz, which often results in a change in the plane’s shape. The resulting stress is shear stress distributed along the y-axis due to the force acting parallelly on the normal plane of the x-axis, τ

  xy

  . Similarly, τ

  xz

  is known as the shear stress distributed along the z-axis due to the force acting parallelly on the normal plane of the x-axis. The notation for strain components can be interpreted in the same way.

  Table 2.1

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  eBook - ePub

  Mechanics of Materials For Dummies

  • James H. Allen(Author)
  • 2011(Publication Date)
  • For Dummies

    (Publisher)

  Part II Analyzing Stress In this part . . .

  S

  tresses are one of the fundamentals of mechanics of materials because they relate internal forces to specific section properties. I start this section by describing the basic categories of stress and then show you how to determine the maximum and minimum values (known as principal stresses) and their orientation angles through several basic transformation techniques. I conclude the part by demonstrating how to calculate stresses for different load types, including axial forces, bending moments, shear forces, and torsional moments.

  Passage contains an image

  Chapter 6 Remain Calm, It’s Only Stress! In This Chapter

  Defining the basics of stress

  Computing average stresses

  Grasping stress’s units

  Working with stresses at a point

  Understanding assumptions about plane stress problems

  I

  n the “old days,” engineers used a trial-and-error approach or relied on previous experience. But you don’t really want to build a bridge and load it until it fails just to find out how much it could carry before you broke it. That proposition sounds expensive (not to mention potentially dangerous)! So how do you actually determine whether a particular object made from a particular material can carry a particular load? A more scientific approach involves calculating the actual intensity of the force (known as stress) on an object and then comparing this stress to the intensity of a force that a material is capable of withstanding before it fails. With that information, you can begin predicting the most-critically stressed location in a particular object, which is a fundamental design skill for engineers.

  In this chapter, I introduce the basic concept of stress because the intensity of a force affects different types of objects in different ways. I start the explanation with the simplest of stress calculations: average stresses. You can then use these average stresses to develop a general relationship about the state of stress at a particular point. Finally, I introduce the concept of plane stress, which is a significant assumption in the formulation of many of the equations and design relationships in Part III.

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  eBook - ePub

  Mechanical Engineers' Handbook, Volume 1

  Materials and Engineering Mechanics

  • Myer Kutz, Myer Kutz(Authors)
  • 2015(Publication Date)
  • Wiley

    (Publisher)

  .
• SHEAR STRESS is the internal total stress exerted by the material fibers along the plane mn (Fig. 3 ) to resist the action of the external forces, tending to slide the adjacent parts in opposite directions. For equilibrium conditions to exist, the shear stress at any cross section will be equal and opposite in direction to the external force . If the internal total stress is uniformly distributed over the area, the unit shear stress .
• NORMAL STRESS is the component of the resultant stress that acts normal to the area considered (Fig. 4 ).
• AXIAL STRESS is a special case of normal stress and may be either tensile or compressive. It is the stress existing in a straight homogeneous bar when the resultant of the applied loads coincides with the axis of the bar.
• SIMPLE STRESS exists when tension, compression, or shear is considered to operate singly on a body.
• TOTAL STRAIN on a loaded body is the total elongation produced by the influence of an external load. Thus, in Fig. 4 , the total strain is equal to . It is expressed in units of length, that is, inches, feet, etc.
• UNIT STRAIN, or deformation per unit length, is the total amount of deformation divided by the original length of the body before the load causing the strain was applied. Thus, if the total elongation is in an original gauge length , the unit strain . Unit strains are expressed in inches per inch and feet per foot.
• TENSILE STRAIN is the strain produced in a specimen by tensile stresses, which in turn are caused by external forces.
• COMPRESSIVE STRAIN is the strain produced in a bar by compressive stresses, which in turn are caused by external forces.
• SHEAR STRAIN is a strain produced in a bar by the external shearing forces.
• POISSON'S RATIO is the ratio of lateral unit strain to longitudinal unit strain under the conditions of uniform and uniaxial longitudinal stress within the proportional limit. It serves as a measure of lateral stiffness. Average values of Poisson's ratio for the usual materials of construction are:

  Material	Steel	Wrought iron	Cast iron	Brass	Concrete
  Poisson's ratio	0.300	0.280	0.270	0.340	0.100