James R. Hutchinson
AXIALLY LOADED MEMBERS 274
Modulus of Elasticity s Poisson’s Ratio
Deformations s Variable Load
GENERAL STATE OF STRESS
PLANE STRESS 283
Circular Shafts s Hollow, Thin-Walled Shafts
Shear and Moment Diagrams
Stress s Deﬂection of Beams
Equation s Superposition
Stresses in Beams s Shear
SELECTED SYMBOLS AND ABBREVIATIONS
Mechanics of materials deals with the determination of the internal forces (stresses) and the deformation of solids such as metals, wood, concrete, plastics and composites. In mechanics of materials there are three main considerations in the solution of problems:
FundEng_Index.book Page 274 Wednesday, November 28, 2007 4:42 PM
Strength of Materials
2. Force-deformation relations
Equilibrium refers to the equilibrium of forces. The laws of statics must hold for the body and all parts of the body. Force-deformation relations refer to the relation of the applied forces to the deformation of the body. If certain forces are applied, then certain deformations will result. Compatibility refers to the compatibility of deformation. Upon loading, the parts of a body or structure must not come apart. These three principles will be emphasized throughout.
AXIALLY LOADED MEMBERS
If a force P is applied to a member as shown in Fig. 9.1(a), then a short distance away from the point of application the force becomes uniformly distributed over the area as shown in Fig. 9.1(b). The force per unit area is called the axis or normal stress and is given the symbol s. Thus,
The original length between two points A and B is L as shown in Fig. 9.1(c). Upon application of the load P, the length L grows by an amount ∆L. The ﬁnal length is L + ∆L as shown in Fig. 9.1(d). A quantity measuring the intensity of deformation and being independent of the original length L is the strain e, deﬁned as
where ∆L is denoted as d.
Figure 9.1 Axial member under force P
FundEng_Index.book Page 275 Wednesday, November 28, 2007 4:42 PM
Axially Loaded Members
Figure 9.2 Stress-strain curve for a typical
The relationship between stress and strain is determined experimentally. A typical plot of stress versus strain is shown in Fig. 9.2. On initial loading, the plot is a straight line until the material reaches yield at a stress of Y. If the stress remains less than yield then subsequent loading and reloading continues along that same straight line. If the material is allowed to go beyond yield, then during an increase in the load the curve goes from A to D. If unloading occurs at some point B, for example, then the material unloads along the line BC which has approximately the same slope as the original straight line from 0 to A. Reloading would occur along the line CB and then proceed along the line BD. It can be seen that if the material is allowed to go into the plastic region (A to D) it will have a permanent strain offset on unloading.
Modulus of Elasticity
The region of greatest concern is that below the yield point. The slope of the line between 0 and A is called the modulus of elasticity and is given the symbol E, so s = Ee
This is Hooke’s Law for axial loading; a more general form will be considered in a later section. The modulus of elasticity is a function of the material alone and not a function of the shape or size of the axial member. The relation of the applied force in a member to its axial deformation can be found by inserting the deﬁnitions of the...