30079_10a.pdf

CHAPTER 10

STRESS ANALYSIS

Franklin E. Fisher

Mechanical Engineering Department

Loyola Marymount University

Los Angeles, California

and

Senior Staff Engineer

Hughes Aircraft Company (Retired)

10.1 STRESSES, STRAINS, STRESS

INTENSITY

10.8

COLUMNS

229

191

10.8.1 Definitions

229

10.1.1 Fundamental Definitions

191

10.8.2 Theory

230

10.1.2 Work and Resilience

197

10.8.3 Wooden Columns

232

10.8.4 Steel Columns

232

10.2 DISCONTINUITIES, STRESS

CONCENTRATION

10.9 CYLINDERS, SPHERES, AND

PLATES

199

235

10.3 COMBINED STRESSES

199

10.9.1 Thin Cylinders and

Spheres under Internal

Pressure

10.4 CREEP

203

235

10.9.2 Thick Cylinders and

Spheres

10.5 FATIGUE

205

235

10.5.1 Modes of Failure

206

10.9.3 Plates

237

10.6 BEAMS

207

10.9.4 Trunnion

237

10.6.1 Theory of Flexure

207

10.9.5 Socket Action

237

10.6.2 Design of Beams

212

10.10 CONTACT STRESSES

242

10.6.3 Continuous Beams

217

10.6.4 Curved Beams

220

10.11 ROTATING ELEMENTS

244

10.6.5 Impact Stresses in Bars and

Beams

10.11.1 Shafts

244

220

10.11.2 Disks

244

10.6.6 Steady and Impulsive

Vibratory Stresses

10.11.3 Blades

244

224

10.12 DESIGNSOLUTION

SOURCES AND

GUIDELINES

10.7 SHAFTS, BENDING, AND

TORSION

224

244

10.7.1 Definitions

224

10.12.1 Computers

244

10.7.2 Determination of Torsional

Stresses in Shafts

10.12.2 Testing

245

225

10.7.3 Bending and Torsional

Stresses

229

10.1 STRESSES, STRAINS, STRESS INTENSITY

10.1.1 Fundamental Definitions

Static Stresses

TOTAL STRESS on a section mn through a loaded body is the resultant force S exerted by one part

of the body on the other part in order to maintain in equilibrium the external loads acting on the

Revised from Chapter 8, Kent's Mechanical Engineer's Handbook, 12th ed., by John M. Lessells

and G. S. Cherniak.

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

191

part. Thus, in Figs. 10.1, 10.2, and 10.3 the total stress on section mn due to the external load P

is S. The units in which it is expressed are those of load, that is, pounds, tons, etc.

UNIT STRESS more commonly called stress cr, is the total stress per unit of area at section mn. In

general it varies from point to point over the section. Its value at any point of a section is the total

stress on an elementary part of the area, including the point divided by the elementary total stress

on an elementary part of the area, including the point divided by the elementary area. If in Figs

10.1, 10,2, and 10.3 the loaded bodies are one unit thick and four units wide, then when the total

stress S is uniformly distributed over the area, a = PIA = P/4. Unit stresses are expressed in

pounds per square inch, tons per square foot, etc.

TENSILE STRESS OR TENSION is the internal total stress S exerted by the material fibers to resist the

action of an external force P (Fig. 10.1), tending to separate the material into two parts along the

line mn. For equilibrium conditions to exist, the tensile stress at any cross section will be equal

and opposite in direction to the external force P. If the internal total stress S is distributed uniformly

over the area, the stress can be considered as unit tensile stress a = SIA.

COMPRESSIVE STRESS OR COMPRESSION is the internal total stress S exerted by the fibers to resist

the action of an external force P (Fig. 10.2) tending to decrease the length of the material. For

equilibrium conditions to exist, the compressive stress at any cross section will be equal and

opposite in direction to the external force P. If the internal total stress S is distributed uniformly

over the area, the unit compressive stress a = SIA.

SHEAR STRESS is the internal total stress S exerted by the material fibers along the plane mn (Fig.

10.3) 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 P. If the internal total stress S is uniformly distributed

over the area, the unit shear stress r = SIA.

NORMAL STRESS is the component of the resultant stress that acts normal to the area considered

(Fig. 10.4).

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 either 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. 10.4, the total strain is equal to 8. 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

8 in an original gage length /, the unit strain e = 8/1. 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.

Fig. 10.1 Tensile stress.

Fig. 10.2 Compressive

Fig. 10.3 Shear stress.

stress.

Fig. 10.4 Normal and shear stress components of resultant stress on section

mn and strain due to tension.

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

ELASTICITY is that property of a material that enables it to deform or undergo strain and return to

its original shape upon the removal of the load.

HOOKE'S LAW states that within certain limits (not to exceed the proportional limit) the elongation

of a bar produced by an external force is proportional to the tensile stress developed. Hooke's law

gives the simplest relation between stress and strain.

PLASTICITY is that state of matter where permanent deformations or strains may occur without

fracture. A material is plastic if the smallest load increment produces a permanent deformation. A

perfectly plastic material is nonelastic and has no ultimate strength in the ordinary meaning of that

term. Lead is a plastic material. A prism tested in compression will deform permanently under a k

small load and will continue to deform as the load is increased, until it flattens to a thin sheet.

Wrought iron and steel are plastic when stressed beyond the elastic limit in compression. When

stressed beyond the elastic limit in tension, they are partly elastic and partly plastic, the degree of

plasticity increasing as the ultimate strength is approached.

STRESS-STRAIN RELATIONSHIP gives the relation between unit stress and unit strain when plotted

on a stress-strain diagram in which the ordinate represents unit stress and the abscissa represents

unit strain. Figure 10.5 shows a typical tension stress-strain curve for medium steel. The form of

the curve obtained will vary according to the material, and the curve for compression will be

different from the one for tension. For some materials like cast iron, concrete, and timber, no part

of the curve is a straight line.

Fig. 10.5 Stress-strain relationship showing determination of apparent elastic limit.

PROPORTIONAL LIMIT is that unit stress at which unit strain begins to increase at a faster rate than

unit stress. It can also be thought of as the greatest stress that a material can stand without deviating

from Hooke's law. It is determined by noting on a stress-strain diagram the unit stress at which

the curve departs from a straight line.

ELASTIC LIMIT is the least stress that will cause permanent strain, that is, the maximum unit stress

to which a material may be subjected and still be able to return to its original form upon removal

of the stress.

JOHNSON'S APPARENT ELASTIC LIMIT. In view of the difficulty of determining precisely for some

materials the proportional limit, J. B. Johnson proposed as the "apparent elastic limit" the point

on the stress-strain diagram at which the rate of strain is 50% greater than at the original. It is

determined by drawing OA (Fig. 10.5) with a slope with respect to the vertical axis 50% greater

than the straight-line part of the curve; the unit stress at which the line O'A' which is parallel to

OA is tangent to the curve (point B, Fig. 10.5) is the apparent elastic limit.

YIELD POINT is the lowest stress at which strain increases without increase in stress. Only a few

materials exhibit a true yield point. For other materials the term is sometimes used as synonymous

with yield strength.

YIELD STRENGTH is the unit stress at which a material exhibits a specified permanent deformation

or state. It is a measure of the useful limit of materials, particularly of those whose stress-strain

curve in the region of yield is smooth and gradually curved.

ULTIMATE STRENGTH is the highest unit stress a material can sustain in tension, compression, or

shear before rupturing.

RUPTURE STRENGTH OR BREAKING STRENGTH is the unit stress at which a material breaks or

ruptures. It is observed in tests on steel to be slightly less than the ultimate strength because of a

large reduction in area before rupture.

MODULUS OF ELASTICITY (Young's modulus) in tension and compression is the rate of change of

unit stress with respect to unit strain for the condition of uniaxial stress within the proportional

limit. For most materials the modulus of elasticity is the same for tension and compression.

MODULUS OF RIGIDITY (modulus of elasticity in shear) is the rate of change of unit shear stress

with respect to unit shear strain for the condition of pure shear within the proportional limit. For

metals it is equal to approximately 0.4 of the modulus of elasticity.

TRUE STRESS is defined as a ratio of applied axial load to the corresponding cross-sectional area.

The units of true stress may be expressed in pounds per square inch, pounds per square foot, etc.,

a = A

where cr is the true stress, pounds per square inch, P is the axial load, pounds, and A is the smallest

value of cross-sectional area existing under the applied load P, square inches.

TRUE STRAIN is defined as a function of the original diameter to the instantaneous diameter of the

test specimen:

d Q

q = 2 log e — in./in.

where q = true strain, inches per inch, d 0 = original diameter of test specimen, inches, and d =

instantaneous diameter of test specimen, inches.

TRUE STRESS-STRAIN RELATIONSHIP is obtained when the values of true stress and the correspond-

ing true strain are plotted against each other in the resulting curve (Fig. 10.6). The slope of the

nearly straight line leading up to fracture is known as the coefficient of strain hardening. It as well

as the true tensile strength appear to be related to the other mechanical properties.

DUCTILITY is the ability of a material to sustain large permanent deformations in tension, such as

drawing into a wire.

MALLEABILITY is the ability of a material to sustain large permanent deformations in compression,

such as beating or rolling into thin sheets.

BRITTLENESS is that property of a material that permits it to be only slightly deformed without

rupture. Brittleness is relative, no material being perfectly brittle, that is, capable of no deformation

before rupture. Many materials are brittle to a greater or less degree, glass being one of the most

brittle of materials. Brittle materials have relatively short stress-strain curves. Of the common

structural materials, cast iron, brick, and stone are brittle in comparison with steel.

TOUGHNESS is the ability of the material to withstand high unit stress together with great unit strain,

without complete fracture. The area OAGH, or OJK, under the curve of the stress-strain diagram

Fig. 10.6 True stress-strain relationship.

(Fig. 10.7), is a measure of the toughness of the material. The distinction between ductility and

toughness is that ductility deals only with the ability to deform, whereas toughness considers both

the ability to deform and the stress developed during deformation.

STIFFNESS is the ability to resist deformation under stress. The modulus of elasticity is the criterion

of the stiffness of a material.

HARDNESS is the ability to resist very small indentations, abrasion, and plastic deformation. There

is no single measure of hardness, as it is not a single property but a combination of several

properties.

CREEP or flow of metals is a phase of plastic or inelastic action. Some solids, as asphalt or paraffin,

flow appreciably at room temperatures under extremely small stresses; zinc, plastics, fiber-

reinforced plastics, lead, and tin show signs of creep at room temperature under moderate stresses.

At sufficiently high temperatures, practically all metals creep under stresses that vary with tem-

perature, the higher the temperature the lower being the stress at which creep takes place. The

deformation due to creep continues to increase indefinitely and becomes of extreme importance in

members subjected to high temperatures, as parts in turbines, boilers, super-heaters, etc.

Fig. 10.7 Toughness comparison.

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