9

CHAPTER

Strength

of Materials

James R. Hutchinson

OUTLINE

AXIALLY LOADED MEMBERS 274

Modulus of Elasticity s Poisson’s Ratio

Deformations s Variable Load

THIN-WALLED CYLINDER

s

Thermal

280

GENERAL STATE OF STRESS

282

PLANE STRESS 283

Mohr’s Circle—Stress

STRAIN 286

Plane Strain

HOOKE’S LAW

288

TORSION 289

Circular Shafts s Hollow, Thin-Walled Shafts

BEAMS 292

Shear and Moment Diagrams

Stress s Deﬂection of Beams

Equation s Superposition

COMBINED STRESS

COLUMNS

s

s

Stresses in Beams s Shear

Fourth-Order Beam

307

309

SELECTED SYMBOLS AND ABBREVIATIONS

PROBLEMS

312

SOLUTIONS

311

319

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:

273

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274

Chapter 9

Strength of Materials

1. Equilibrium

2. Force-deformation relations

3. Compatibility

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,

σ=

P

A

(9.1)

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

ε=

∆L δ

=

L

L

where ∆L is denoted as d.

Figure 9.1 Axial member under force P

(9.2)

FundEng_Index.book Page 275 Wednesday, November 28, 2007 4:42 PM

Axially Loaded Members

275

Figure 9.2 Stress-strain curve for a typical

material

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

(9.3)

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...