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Unformatted text preview: ch03.qxd 12/20/02 7:20 AM Page 95 CHAPTER 3 TwoDimensional
Problems in Elasticity 3.1 INTRODUCTION As has been pointed out in Sec. 1.1, the approaches in widespread use for determining the influence of applied loads on elastic bodies are the mechanics of materials
or elementary theory (also known as technical theory) and the theory of elasticity.
Both must rely on the conditions of equilibrium and make use of a relationship between stress and strain that is usually considered to be associated with elastic materials. The essential difference between these methods lies in the extent to which the
strain is described and in the types of simplifications employed.
The mechanics of materials approach uses an assumed deformation mode or
strain distribution in the body as a whole and hence yields the average stress at a
section under a given loading. Moreover, it usually treats separately each simple
type of complex loading, for example, axial centric, bending, or torsion. Although of
practical importance, the formulas of the mechanics of materials are best suited for
relatively slender members and are derived on the basis of very restrictive conditions. On the other hand, the method of elasticity does not rely on a prescribed deformation mode and deals with the general equations to be satisfied by a body in
equilibrium under any external force system.
The theory of elasticity is preferred when critical design constraints such as
minimum weight, minimum cost, or high reliability dictate more exact treatment or
when prior experience is limited and intuition does not serve adequately to supply
the needed simplifications with any degree of assurance. If properly applied, the
theory of elasticity should yield solutions more closely approximating the actual
distribution of strain, stress, and displacement.
Thus, elasticity theory provides a check on the limitations of the mechanics of
materials solutions. We emphasize, however, that both techniques cited are approximations of nature, each of considerable value and each supplementing the other. 95 ch03.qxd 12/20/02 7:20 AM Page 96 The influences of material anisotropy, the extent to which boundary conditions depart from reality, and numerous other factors all contribute to error. 3.2 FUNDAMENTAL PRINCIPLES OF ANALYSIS To ascertain the distribution of stress, strain, and displacement within an elastic
body subject to a prescribed system of forces requires consideration of a number of
conditions relating to certain physical laws, material properties, and geometry.
These fundamental principles of analysis, also referred to as the three aspects of
solid mechanics problems, are summarized as follows:
1. Conditions of equilibrium. The equations of statics must be satisfied throughout
the body.
2. Stress–strain relations. Material properties (constitutive relations, for example,
Hooke’s law) must comply with the known behavior of the material involved.
3. Conditions of compatibility. The geometry of deformation and the distribution
of strain must be consistent with the preservation of body continuity. (The matter of compatibility is not always broached in mechanics of materials analysis.)
In addition, the stress, strain, and displacement fields must be such as to conform to
the conditions of loading imposed at the boundaries. This is known as satisfying the
boundary conditions for a particular problem. If the problem is dynamic, the equations of equilibrium become the more general conservation of momentum; conservation of energy may be a further requirement.
The conditions described, and stated mathematically in the previous chapters,
are used to derive the equations of elasticity. In the case of a threedimensional
problem in elasticity, it is required that the following 15 quantities be ascertained:
six stress components, six strain components, and three displacement components.
These components must satisfy 15 governing equations throughout the body in addition to the boundary conditions: three equations of equilibrium, six stress–strain
relations, and six straindisplacement relations. Note that the equations of compatibility are derived from the strain–displacement relations, which are already included in the preceding description. Thus, if the 15 expressions are satisfied, the
equations of compatibility will also be satisfied. Threedimensional problems in
elasticity are often very complex. It may not always be possible to use the direct
method of solution in treating the general equations and given boundary conditions. Only a useful indirect method of solution will be presented in Secs. 6.3
and 6.4.
In many engineering applications, ample justification may be found for simplifying assumptions with respect to the state of strain and stress. Of special importance, because of the resulting decrease in complexity, are those reducing a
threedimensional problem to one involving only two dimensions. In this regard, we
shall discuss throughout the text various plane strain and plane stress problems.
This chapter is subdivided into two parts. In Part A, derivations of the governing differential equations and various approaches for solution of twodimensional
problems in Cartesian and polar coordinates are considered. Part B treats stress
96 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 97 concentrations in members whose cross sections manifest pronounced changes and
cases of load application over small areas. Part A—Formulation and Methods of Solution
3.3 PLANE STRAIN PROBLEMS Consider a long prismatic member subject to lateral loading (for example, a cylinder under pressure), held between fixed, smooth, rigid planes (Fig. 3.1). Assume
the external force to be functions of the x and y coordinates only. As a consequence, we expect all cross sections to experience identical deformation, including
those sections near the ends. The frictionless nature of the end constraint permits
x, y deformation, but precludes z displacement; that is, w = 0 at z = ; L/2. Considerations of symmetry dictate that w must also be zero at midspan. Symmetry arguments can again be used to infer that w = 0 at ; L/4, and so on, until every
cross section is taken into account. For the case described, the strain depends on x
and y only:
x z = 0w
= 0,
0z = 0u
,
0x y xz = = 0v
,
0y 0w
0u
+
= 0,
0x
0z xy = 0u
0v
+
0y
0x
yz = 0w
0v
+
=0
0y
0z (3.1)
(3.2) The latter expressions depend on 0u/0z and 0v/0z vanishing, since w and its derivatives are zero. A state of plane strain has thus been described wherein each point
remains within its transverse plane, following application of the load. We next proceed to develop the equations governing the behavior of bodies under plane
strain.
Substitution of z = yz = xz = 0 into Eq. (2.30) provides the following
stress–strain relationships: FIGURE 3.1. Plane strain in a cylindrical
body. 3.3 Plane Strain Problems 97 ch03.qxd 12/20/02 7:20 AM Page 98 x = 2G x y = 2G y + xy =G = yz = 0, 1 x
x y2 +
+ y2 (3.3) xy and
xz 1 + z 1 = x + y2 =1 x + y2 (3.4) Because z is not contained in the other governing expressions for plane strain, it is
determined independently by applying Eq. (3.4). The strain–stress relations, Eqs.
(2.28), for this case become
= 1E 2 x = 1E 2 y xy = xy G ¢
¢ x  1 y y  1 x ≤
≤ (3.5) Inasmuch as these stress components are functions of x and y only, the first two
equations of (1.11) yield the following equations of equilibrium of plane strain:
0 xy
0x
+
+ Fx = 0
x
0
0y
0 xy
0y
+
+ Fy = 0
0y
0x (3.6) The third equation of (1.11) is satisfied if Fz = 0. In the case of plane strain, therefore, no body force in the axial direction can exist.
A similar restriction is imposed on the surface forces. That is, plane strain will
result in a prismatic body if the surface forces px and py are each functions of x and
y and pz = 0. On the lateral surface, n = 0 (Fig. 3.2). The boundary conditions,
from the first two equations of (1.41), are thus given by
px = xl + xym py = xyl + ym (3.7) Clearly, the last equation of (1.41) is also satisfied.
In the case of a plane strain problem, therefore, eight quantities, x, y, xy, x,
, xy, u, and v , must be determined so as to satisfy Eqs. (3.1), (3.3), and (3.6) and
y
FIGURE 3.2. Surface forces. 98 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 99 the boundary conditions (3.7). How eight governing equations, (3.1), (3.3), and
(3.6), may be reduced to three is now discussed.
Three expressions for twodimensional strain at a point [Eq. (3.1)] are functions of only two displacements, u and v , and therefore a compatibility relationship
exists among the strains [Eq. (2.8)]:
02 y
02 xy
02 x
+
=
0x 0y
0y 2
0x2 (3.8) This equation must be satisfied for the strain components to be related to the displacements as in Eqs. (3.1). The condition as expressed by Eq. (3.8) may be transformed into one involving components of stress by substituting the strain–stress
relations and employing the equations of equilibrium. Performing the operations
indicated, using Eqs. (3.5) and (3.8), we have
02
[11  2
0y2 x  y] + 02
[11  2
0x2 02
y  x] = 2 xy 0x 0y (a) Next, the first and second equations of (3.6) are differentiated with respect to x and
y, respectively, and added to yield
02
2 xy 0x 0y = ¢ 02 y
0Fy
02 x
0Fx
+
¢
+
≤
2
2≤
0x
0y
0x
0y Finally, substitution of this into Eq. (a) results in ¢ 02
02
+ 2 ≤1
2
0x
0y x + y2 = 1
1 ¢ 0Fy
0Fx
+
≤
0x
0y (3.9) This is the equation of compatibility in terms of stress.
We now have three expressions, Eqs. (3.6) and (3.9), in terms of three unknown
quantities: x, y, and xy. This set of equations, together with the boundary conditions (3.7), is used in the solution of plane strain problems. For a given situation,
after determining the stress, Eqs. (3.5) and (3.1) yield the strain and displacement,
respectively. In Sec. 3.5, Eqs. (3.6) and (3.9) will further be reduced to one equation
containing a single variable. 3.4 PLANE STRESS PROBLEMS In many problems of practical importance, the stress condition is one of plane
stress. The basic definition of this state of stress has already been given in Sec. 1.8.
In this section we shall present the governing equations for the solution of plane
stress problems.
To exemplify the case of plane stress, consider a thin plate, as in Fig. 3.3,
wherein the loading is uniformly distributed over the thickness, parallel to the
plane of the plate. This geometry contrasts with that of the long prism previously
discussed, which is in a state of plane strain. To arrive at tentative conclusions with
regard to the stress within the plate, consider the fact that z, xz, and yz are zero
3.4 Plane Stress Problems 99 ch03.qxd 12/20/02 7:20 AM Page 100 FIGURE 3.3. Thin plate under plane stress. on both faces of the plate. Because the plate is thin, the stress distribution may be
very closely approximated by assuming that the foregoing is likewise true throughout the plate.
We shall, as a condition of the problem, take the body force Fz = 0 and Fx and
Fy each to be functions of x and y only. As a consequence of the preceding, the
stress is specified by
x,
z = y, xy = xz yz (a) =0 The nonzero stress components remain constant over the thickness of the plate and
are functions of x and y only. This situation describes a state of plane stress. Equations (1.11) and (1.41), together with this combination of stress, again reduce to the
forms found in Sec. 3.3. Thus, Eqs. (3.6) and (3.7) describe the equations of equilibrium and the boundary conditions in this case, as in the case of plane strain.
Substitution of Eq. (a) into Eq. (2.28) yields the following stress–strain relations
for plane stress:
x = 1
1
E y = 1
1
E xy = x  y2 y  x2 z = (3.10) xy G and
xz = = 0, yz E 1 x + y2 (3.11a) Solving for x + y from the sum of the first two of Eqs. (3.10) and inserting the result into Eq. (3.11a), we obtain
z 100 = 1 1 x + Chapter 3 y2 (3.11b) TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 101 Equations (3.11) define the outofplane principal strain in terms of the inplane
stresses 1 x, y2 or strains 1 x, y2.
Because z is not contained in the other governing expressions for plane stress,
it can be obtained independently from Eqs. (3.11); then z = 0w/0z may be applied
to yield w. That is, only u and v are considered as independent variables in the governing equations. In the case of plane stress, therefore, the basic strain–displacement
relations are again given by Eqs. (3.1). Exclusion from Eq. (2.3) of z = 0w/0z
makes the plane stress equations approximate, as is demonstrated in the section
that follows.
The governing equations of plane stress will now be reduced, as in the case of
plane strain, to three equations involving stress components only. Since Eqs. (3.1)
apply to plane strain and plane stress, the compatibility condition represented by
Eq. (3.8) applies in both cases. The latter expression may be written as follows, substituting strains from Eqs. (3.10) and employing Eqs. (3.6): ¢ 02
02
+ 2 ≤1
0x2
0y x + y2 =  11 + 2 ¢ 0Fy
0Fx
+
≤
0x
0y (3.12) This equation of compatibility, together with the equations of equilibrium, represents a useful form of the governing equations for problems of plane stress.
To summarize the twodimensional situations discussed, the equations of equilibrium [Eqs. (3.6)], together with those of compatibility [Eq. (3.9) for plane strain
and Eq. (3.12) for plane stress] and the boundary conditions [Eqs. (3.7)], provide a
system of equations sufficient for determination of the complete stress distribution.
It can be shown that a solution satisfying all these equations is, for a given problem,
unique [Ref. 3.1]. That is, it is the only solution to the problem.
In the absence of body forces or in the case of constant body forces, the compatibility equations for plane strain and plane stress are the same. In these cases,
the equations governing the distribution of stress do not contain the elastic constants. Given identical geometry and loading, a bar of steel and one of Lucite
should thus display identical stress distributions. This characteristic is important in
that any convenient isotropic material may be used to substitute for the actual material, as, for example, in photoelastic studies.
It is of interest to note that by comparing Eqs. (3.5) with Eqs. (3.10) we can
form Table 3.1, which facilitates the conversion of a plane stress solution into a
plane strain solution, and vice versa. For instance, conditions of plane stress and
plane strain prevail in a narrow beam and a very wide beam, respectively. Hence, in
TABLE 3.1
Solution Plane stress Plane strain Plane strain 3.4 To Convert to: Plane stress Plane Stress Problems E is Replaced by: E
1 2
1+2
E
11 + 22 is Replaced by: 11+ 101 ch03.qxd 12/20/02 7:20 AM Page 102 a result pertaining to a thin beam, EI would become EI/11  22 for the case of a
wide beam. The stiffness in the latter case is, for = 0.3, about 10% greater owing
to the prevention of sidewise displacement (Secs. 5.2 and 13.3). 3.5 AIRY’S STRESS FUNCTION The preceding sections have demonstrated that the solution of twodimensional
problems in elasticity requires integration of the differential equations of equilibrium [Eqs. (3.6)], together with the compatibility equation [Eq. (3.9) or (3.12)] and
the boundary conditions [Eqs. (3.7)]. In the event that the body forces Fx and Fy
are negligible, these equations reduce to
0 xy
0x
+
= 0,
0x
0y ¢ 0 02
02
+ 2 ≤1
0x2
0y 0 y + 0y
x + xy 0x
y2 =0 =0 (a)
(b) together with the boundary conditions (3.7). The equations of equilibrium are identically satisfied by the stress function, £ 1x, y2, introduced by G. B. Airy, related to
the stresses as follows:
x = 02 £
,
0y2 y = 02 £
,
0x2 xy = 02 £
0x 0y (3.13) Substitution of (3.13) into the compatibility equation, Eq. (b), yields
04 £
04 £
04 £
+2 2 2+
= §4 £ = 0
0x4
0x 0y
0y4 (3.14) What has been accomplished is the formulation of a twodimensional problem in
which body forces are absent, in such a way as to require the solution of a single
biharmonic equation, which must of course satisfy the boundary conditions.
It should be noted that in the case of plane stress we have z = xz = yz = 0
and x, y, and xy independent of z. As a consequence, xz = yz = 0, and x, y,
z, and xy are independent of z. In accordance with the foregoing, from Eq. (2.9), it
is seen that in addition to Eq. (3.14), the following compatibility equations also
hold:
02 z
= 0,
0x2 02 z
= 0,
0y2 02 z
=0
0x0y (c) Clearly, these additional conditions will not be satisfied in a case of plane stress by
a solution of Eq. (3.14) alone. Therefore, such a solution of a plane stress problem
has an approximate character. However, it can be shown that for thin plates the
error introduced is negligibly small.
102 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 103 It is also important to note that, if the ends of the cylinder shown in Fig. 3.1 are
free to expand, we may assume the longitudinal strain z to be a constant. Such a
state may be called that of generalized plane strain. Therefore, we now have
= 1E 2 x = 1E 2 y xy = xy G and
z ¢ x y  1 x x =1 1 y ¢  + y2 +E ≤≤z z (3.15)
z (3.16) Introducing Eqs. (3.15) into Eq. (3.8) and simplifying, we again obtain Eq. (3.14) as
the governing differential equation. Having determined x and y, the constant
value of z can be found from the condition that the resultant force in the z direction acting on the ends of the cylinder is zero. That is,
4
where
3.6 z z dx dy = 0 (d) is given by Eq. (3.16). SOLUTION OF ELASTICITY PROBLEMS Unfortunately, solving directly the equations of elasticity derived may be a formidable task, and it is often advisable to attempt a solution by the inverse or semiinverse method. The inverse method requires examination of the assumed solutions
with a view toward finding one that will satisfy the governing equations and boundary conditions. The semiinverse method requires the assumption of a partial solution formed by expressing stress, strain, displacement, or stress function in terms of
known or undetermined coefficients. The governing equations are thus rendered
more manageable.
It is important to note that the preceding assumptions, based on the mechanics
of a particular problem, are subject to later verification. This is in contrast with the
mechanics of materials approach, in which analytical verification does not occur.
The applications of inverse, semiinverse, and direct methods are found in examples
to follow and in Chapters 5, 6, and 8.
A number of problems may be solved by using a linear combination of
polynomials in x and y and undetermined coefficients of the stress function £ .
Clearly, an assumed polynomial form must satisfy the biharmonic equation and
must be of second degree or higher in order to yield a nonzero stress solution of
Eq. (3.13), as described in the following paragraphs. In general, finding the desirable polynomial form is laborious and requires a systematic approach [Refs. 3.2 and
3.3]. The Fourier series, indispensible in the analytical treatment of many problems
in the field of applied mechanics, is also often employed (Secs. 10.10 and 13.6). 3.6 Solution of Elasticity Problems 103 ch03.qxd 12/20/02 7:20 AM Page 104 Another way to overcome the difficulty involved in the solution of Eq. (3.14) is
to use the method of finite differences. Here the governing equation is replaced by
series of finite difference equations (Sec. 7.3), which relate the stress function at
stations that are removed from one another by finite distances. These equations, although not exact, frequently lead to solutions that are close to the exact solution.
The results obtained are, however, applicable only to specific numerical problems.
Polynomial Solutions
An elementary approach to obtaining solutions of the biharmonic equation uses
polynomial functions of various degree with their coefficients adjusted so that
§4 £ = 0 is satisfied. A brief discussion of this procedure follows.
A polynomial of the second degree,
£2 = a2 2
c2 2
x + b2xy +
y
2
2 (3.17) satisfies Eq. (3.14). The associated stresses are
x = c2, y = a2, xy =  b2 All three stress components are constant throughout the body. For a rectangular
plate (Fig. 3.4a), it is apparent that the foregoing may be adapted to represent
simple tension 1c2 Z 02, double tension 1c2 Z 0, a2 Z 02, or pure shear 1b2 Z 02.
A polynomial of the third degree
£3 = a3 3
b3
c3
d3
x + x2y + xy 2 + y3
6
2
2
6 (3.18) fulfills Eq. (3.14). It leads to stresses
x = c3x + d3y, y = a3x + b3y, xy =  b3x  c3y For a3 = b3 = c3 = 0, these expressions reduce to
x = d3y, y = xy =0 representing the case of pure bending of the rectangular plate (Fig. 3.4b). FIGURE 3.4. Stress fields of (a) Eq. (3.17) and (b) Eq. (3.18). 104 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 105 A polynomial of the fourth degree,
£4 = a4 4
b4
c4
d4
e4 4
x + x3y + x2y2 + xy3 +
y
12
6
2
6
12 (3.19) satisfies Eq. (3.14) if e4 =  12c4 + a42. The corresponding stresses are
x
y xy = c4x2 + d4 xy  12c4 + a42y2
= a4 x2 + b4xy + c4y2
= b4 2
d4
x  2c4 xy  y2
2
2 A polynomial of the fifth degree
£5 = f5 5
a5 5
b5 4
c5
d5
e5 4
x+
x y + x3y 2 + x2y3 +
xy +
y
20
12
6
6
12
20 (3.20) fulfills Eq. (3.14) provided that 13a 5 + 2c5 + e52x + 1b5 + 2d5 + 3f52y = 0 It follows that
e5 =  3a5  2c5, b5 =  2d5  3f5 The components of stress are then
x y xy = c5 3
x + d5x2y  13a5 + 2c52xy2 + f5y3
3 = a5x3  13f5 + 2d52x2y + c5xy 2 + d5 3
y
3 = 113f5 + 2d52x3  c5x2y  d5xy2 + 113d5 + 2c52y3
3
3 Problems of practical importance may be solved by combining functions (3.17)
through (3.20), as required. With experience, the analyst begins to understand the
types of stress distributions arising from a variety of polynomials.
EXAMPLE 3.1
A narrow cantilever of rectangular cross section is loaded by a concentrated force at its free end of such magnitude that the beam weight may
be neglected (Fig. 3.5a). Determine the stress distribution in the beam.
Solution The situation described may be regarded as a case of plane
stress provided that the beam thickness t is small relative to the beam
depth 2h.
The following boundary conditions are consistent with the coordinate system in Fig. 3.5a:
1 xy2y = ; h = 0, 1 y2y = ; h = 0 (a) These conditions simply express the fact that the top and bottom edges
of the beam are not loaded. In addition to Eq. (a) it is necessary, on the
3.6 Solution of Elasticity Problems 105 ch03.qxd 12/20/02 7:20 AM Page 106 FIGURE 3.5. Example 3.1. Endloaded cantilever beam. basis of zero external loading in the x direction at x = 0, that x = 0
along the vertical surface at x = 0. Finally, the applied load P must be
equal to the resultant of the shearing forces distributed across the free
end:
+h P= Lh xyt dy (b) The negative sign agrees with the convention for stress discussed in
Sec. 1.4.
For purposes of illustration, three approaches will be employed to
determine the distribution of stress within the beam.
Method 1. Inasmuch as the bending moment varies linearly with x and
x at any section depends on y, it is reasonable to assume a general expression of the form
x = 02 £
= c1xy
0y2 (c) in which c1 represents a constant. Integrating twice with respect to y,
£ = 1c1xy3 + yf11x2 + f21x2
6 (d) where f11x2 and f21x2 are functions of x to be determined. Introducing
the £ thus obtained into Eq. (3.14), we have
y d4f1
dx4 + d4f2
dx4 =0 Since the second term is independent of y, a solution exists for all x and
y provided that d4f1/dx4 = 0 and d4f2/dx4 = 0, which, upon integrating,
leads to
f11x2 = c2x3 + c3x2 + c4x + c5
f21x2 = c6x3 + c7x2 + c8x + c9 where c2, c3, Á , are constants of integration. Substitution of f11x2 and
f21x2 into Eq. (d) gives
106 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 107 £ = 1 c1xy3 + 1c2x3 + c3x2 + c4x + c52y
6
+ c6x3 + c7x2 + c8x + c9 Expressions for
y xy = y and follow from Eq. (3.13): xy 02 £
= 61c2y + c62x + 21c3y + c72
0x2 (e) 02 £
==  1c1y2  3c2x2  2c3x  c4
2
0x 0y At this point, we are prepared to apply the boundary conditions. Substituting Eqs. (a) into (e), we obtain c2 = c3 = c6 = c7 = 0 and c4 =  1c1h2.
2
The final condition, Eq. (b), may now be written as
h  Lh
 h
xyt dy = Lh
 1
2
2 c1t1y  h22dy = P from which
c1 =  3P
P
=3
I
2th where I = 2th3 is the moment of inertia of the cross section about the
3
neutral axis. From Eqs. (c) and (e), together with the values of the constants, the stresses are found to be
x = Pxy
,
I y = 0, xy = P2
1h  y22
2I (3.21) The distribution of these stresses at sections away from the ends is
shown in Fig. 3.5b.
Method 2. Beginning with bending moments Mz = Px, we may assume a stress field similar to that for the case of pure bending:
x = ¢ Px
≤ y,
I xy = xy1x, y2, y = z = xz = yz =0 (f) Equation of compatibility (3.12) is satisfied by these stresses. On the
basis of Eqs. (f), the equations of equilibrium lead to
0 xy
0x
+
= 0,
0x
0y 0 xy 0x =0 (g) From the second expression, xy can depend only on y. The first equation of (g) together with Eqs. (f) yields
d xy dy 3.6 Solution of Elasticity Problems = Py
I 107 ch03.qxd 12/20/02 7:20 AM Page 108 from which
xy = Py2
+c
2I Here c is determined on the basis of 1 xy2y + ; h = 0: c =  Ph2/2I. The
resulting expression for xy satisfies Eq. (b) and is identical with the result previously obtained.
Method 3. The problem may be treated by superimposing the polynomials £ 2 and £ 4,
a2 = c2 = a4 = b4 = c4 = e4 = 0
Thus,
d4 3
xy
6 £ = £ 2 + £ 4 = b2xy +
The corresponding stress components are
x = d4xy, y = 0, xy =  b2  d4 2
y
2 It is seen that the foregoing satisfies the second condition of Eqs. (a).
The first of Eqs. (a) leads to d4 =  2b2/h2. We then obtain
xy =  b2 ¢ 1  y2
h2 ≤ which
when
substituted
into
condition
(b)
results
in
b2 =  3P/4ht = Ph2/2I. As before, xy is as given in Eqs. (3.21).
Observe that the stress distribution obtained is the same as that
found by employing the elementary theory. If the boundary forces result
in a stress distribution as indicated in Fig. 3.5b, the solution is exact.
Otherwise, the solution is not exact. In any case, recall, however, that
SaintVenant’s principle permits us to regard the result as quite accurate
for sections away from the ends.
Section 5.4 illustrates the determination of the displacement field
after derivation of the curvature–moment relation. 3.7 THERMAL STRESSES Consider the consequences of increasing or decreasing the uniform temperature of
an entirely unconstrained elastic body. The resultant expansion or contraction occurs in such a way as to cause a cubic element of the solid to remain cubic, while experiencing changes of length on each of its sides. Normal strains occur in each
direction unaccompanied by normal stresses. In addition, there are neither shear
strains nor shear stresses. If the body is heated in such a way as to produce a
nonuniform temperature field, or if the thermal expansions are prohibited from
108 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 109 taking place freely because of restrictions placed on the boundary even if the temperature is uniform, or if the material exhibits anisotropy in a uniform temperature
field, thermal stresses will occur. The effects of such stresses can be severe, especially since the most adverse thermal environments are often associated with design
requirements involving unusually stringent constraints as to weight and volume.
This is especially true in aerospace applications, but is of considerable importance,
too, in many everyday machine design applications.
Solution of thermal stress problems requires reformulation of the stress–strain
relationships accomplished by superposition of the strain attributable to stress and
that due to temperature. For a change in temperature T(x, y), the change of length,
L, of a small linear element of length L in an unconstrained body is L = LT.
Here , usually a positive number, is termed the coefficient of linear thermal expansion. The thermal strain t associated with the free expansion at a point is then
t T = (3.22) The total x and y strains, x and y, are obtained by adding to the thermal strains of
the type described, the strains due to stress resulting from external forces:
x = 1
1
E y = 1
1
E xy = x  y2 + T y  x2 + T (3.23a) xy G In terms of strain components, these expressions become
x = E
1 y = E
1 xy =G 2 2 1
1 x + y2  ET
1 y + x2  ET
1 (3.23b) xy Because free thermal expansion results in no angular distortion in an isotropic
material, the shearing strain is unaffected, as indicated. Equations (3.23) represent
modified strain–stress relations for plane stress. Similar expressions may be written
for the case of plane strain. The differential equations of equilibrium (3.6) are
based on purely mechanical considerations and are unchanged for thermoelasticity.
The same is true of the strain–displacement relations (2.3) and the compatibility
equation (3.8), which are geometrical in character. Thus, for given boundary conditions (expressed either as surface forces or displacements) and temperature distribution, thermoelasticity and ordinary elasticity differ only to the extent of the
strain–stress relationship.
By substituting the strains given by Eq. (3.23a) into the equation of compatibility (3.8), employing Eq. (3.6) as well, and neglecting body forces, a compatibility
equation is derived in terms of stress:
3.7 Thermal Stresses 109 ch03.qxd 12/20/02 7:20 AM Page 110 ¢ 02
02
+ 2 ≤1
0x2
0y + x y + ET2 = 0 (3.24) Introducing Eq. (3.13), we now have §4 £ + E §2 T = 0 (3.25) This expression is valid for plane strain or plane stress provided that the body
forces are negligible.
It has been implicit in treating the matter of thermoelasticity as a superposition
problem that the distribution of stress or strain plays a negligible role in influencing
the temperature field [Refs. 3.4 and 3.5]. This lack of coupling enables the temperature field to be determined independently of any consideration of stress or strain. If
the effect of the temperature distribution on material properties cannot be disregarded, the equations become coupled and analytical solutions are significantly
more complex, occupying an area of considerable interest and importance. Numerical solutions can, however, be obtained in a relatively simple manner through the
use of finite difference methods.
EXAMPLE 3.2
A rectangular beam of small thickness t, depth 2h, and length 2L is subjected to an arbitrary variation of temperature throughout its depth,
T = T1y2. Determine the distribution of stress and strain for the case in
which (a) the beam is entirely free of surface forces (Fig. 3.6a), and (b)
the beam is held by rigid walls that prevent the xdirected displacement
only (Fig. 3.6b).
Solution The beam geometry indicates a problem of plane stress. We
begin with the assumptions
x = x1y2, y = xy =0 (a) Direct substitution of Eqs. (a) into Eqs. (3.6) indicates that the equations
of equilibrium are satisfied. Equations (a) reduce the compatibility
equation (3.24) to the form
d2
1
dy2 x + ET2 = 0 (b) FIGURE 3.6. Example 3.2. Rectangular beam in plane thermal stress: (a) unsupported; (b) placed between two rigid walls. 110 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 111 from which
x =  ET + c1y + c2 (c) where c1 and c2 are constants of integration. The requirement that faces
y = ; h be free of surface forces is obviously fulfilled by Eq. (b).
a. The boundary conditions at the end faces are satisfied by determining the constants that assume zero resultant force and moment at
x = ; L:
h h
xt Lh
 dy = 0, Lh
 xyt dy = 0 (d) Substituting Eq. (c) into Eqs. (d), it is found that
h
h
c1 = 13/2h32 1h ETy dy and c2 = 11/2h2 1h ET dy. The normal
stress, upon substituting the values of the constants obtained, together with the moment of inertia I = 2h3t/3 and area A = 2ht, into
Eq. (c) is thus
x h
yt h
t
= E B T +
T dy +
Ty dy R
A Lh
I Lh
 (3.26) The corresponding strains are
x = x E T, + y = x E + T, xy =0 (e) The displacements can readily be determined from Eqs. (3.1).
From Eq. (3.26), observe that the temperature distribution for
T = constant results in zero stress, as expected. Of course, the strains
(e) and the displacements will, in this case, not be zero. It is also noted
that, when the temperature is symmetrical about the midsurface
1y = 02, that is, T1y2 = T1  y2, the final integral in Eq. (3.26) vanishes. For an antisymmetrical temperature distribution about the midsurface, T1y2 =  T1  y2, and the first integral in Eq. (3.26) is zero.
b. For the situation described, x = 0 for all y. With y = xy = 0 and
Eq. (c), Eqs. (3.23a) lead to c1 = c2 = 0, regardless of how T varies
with y. Thus,
x = E T (3.27) and
x = xy = 0, y = 11 + 2 T (f) Note that the axial stress obtained here can be large even for modest
temperature changes, as can be verified by substituting properties of
a given material.
3.7 Thermal Stresses 111 ch03.qxd 12/20/02 3.8 7:20 AM Page 112 BASIC RELATIONS IN POLAR COORDINATES Geometrical considerations related either to the loading or to the boundary of a
loaded system often make it preferable to employ polar coordinates, rather than
the Cartesian system used exclusively thus far. In general, polar coordinates are
used advantageously where a degree of axial symmetry exists. Examples include a
cylinder, a disk, a wedge, a curved beam, and a large thin plate containing a circular
hole.
The polar coordinate system 1r, 2 and the Cartesian system (x, y) are related
by the following expressions (Fig. 3.7a):
x = r cos , r2 = x2 + y2 y = r sin , = tan1 (a) y
x These equations yield
x
0r
=
= cos ,
r
0x
y
sin
0
= 2=,
r
r
0x y
0r
=
= sin
r
0y
(b) x
cos
0
= 2=
r
r
0y Any derivatives with respect to x and y in the Cartesian system may be transformed
into derivatives with respect to r and by applying the chain rule:
0r 0
00
0
=
+
= cos
0x
0x 0r
0x 0 sin
0
r
0r 0
0 0r 0
00
0
=
+
= sin
0y
0y 0r
0y 0 cos
0
+
r
0r 0
0 (c) Relations governing properties at a point not containing any derivatives are not affected by the curvilinear nature of the coordinates, as is observed next. FIGURE 3.7. (a) Polar coordinates; (b) stress element in polar coordinates. 112 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 113 Equations of Equilibrium
Consider the state of stress on an infinitesimal element abcd of unit thickness described by polar coordinates (Fig. 3.7b). The r and directed body forces are denoted by Fr and F . Equilibrium of radial forces requires that  ¢ r + dr sin 0r
dr ≤ 1r + dr2d 0r d
+¢
2 r + rrd ¢ 0r
d
d ≤ dr cos
2
0 + r 0
0 d ≤ dr sin dr cos d
2 d
+ Frrdr d = 0
2 Inasmuch as d is small, sin1d /22 may be replaced by d /2 and cos1d /22 by 1. Additional simplication is achieved by dropping terms containing higherorder infinitesimals. A similar analysis may be performed for the tangential direction. When
both equilibrium equations are divided by r dr d , the results are
0r
1 0r
+
+
r0
0r
10
r0 r r + Fr = 0 2r
0r
+F =0
+
+
r
0r (3.28) In the absence of body forces, Eqs. (3.28) are satisfied by a stress function
£ 1r, 2 for which the stress components in the radial and tangential directions are
given by
r r 1 0£
1 02 £
+2
r 0r
r 02
2
0£
=
0r2
= = 1 0£
1 02 £
0 1 0£
= ¢
≤
2
r 0r 0
r0
0r r 0 (3.29) Strain–Displacement Relations Consider now the deformation of the infinitesimal element abcd, denoting the r
and displacements by u and v , respectively. The general deformation experienced
by an element may be regarded as composed of (1) a change in length of the sides,
as in Figs. 3.8a and b, and (2) rotation of the sides, as in Figs. 3.8c and d.
In the analysis that follows, the small angle approximation sin L is employed, and arcs ab and cd are regarded as straight lines. Referring to Fig. 3.8a, it is
observed that a u displacement of side ab results in both radial and tangential
strain. The radial strain r, the deformation per unit length of side ad, is associated
only with the u displacement:
r 3.8 Basic Relations in Polar Coordinates = 0u
0r (3.30a) 113 ch03.qxd 12/20/02 7:20 AM Page 114 FIGURE 3.8. Deformation and displacement of an element in polar coordinates. The tangential strain owing to u, the deformation per unit length of ab, is
1 2u = 1r + u2 d  r d
u
=
r
rd (d) Clearly, a v displacement of element abcd (Fig. 3.8b) also produces a tangential
strain,
1 2v = 10v/0 2 d
1 0v
=
r0
rd (e) since the increase in length of ab is 10v/0 2d . The resultant tangential strain, combining Eqs. (d) and (e), is
= 1 0v
u
+
r0
r (3.30b) Figure 3.8c shows the angle of rotation eb ¿ f of side a ¿ b ¿ due to a u displacement. The associated strain is
1 r 10u/0 2 d
1 0u
=
r0
rd 2u = (f) The rotation of side bc associated with a v displacement alone is shown in Fig. 3.8d.
Since an initial rotation of b – through an angle v /r has occurred, the relative rotation gb – h of side bc is
1 r 2v = v
0v
r
0r (g) The sum of Eqs. (f) and (g) provides the total shearing strain
114 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 115 1 0u
v
0v
+
r0
r
0r = r (3.30c) The strain–displacement relationships in polar coordinates are thus given by Eqs.
(3.30). Hooke’s Law
To write Hooke’s law in polar coordinates, we need only replace subscripts x by r
and y by in the appropriate Cartesian equations. In the case of plane stress, from
Eqs. (3.10) we have r = 1
1
E = r 1
1
E = 1
G  2  r r2 (3.31) r For plane strain, Eqs. (3.5) lead to
1+
[11  2
E
1+
[11  2
E = r = = r 1
G   r r] (3.32) r Transformation Equations
Replacement of the subscripts x ¿ by r and y ¿ by
r =
=1
= r cos2 x x y 2 sin +
x2 sin + y y sin2 +2 cos + 2 cos xy 2 xy
y =
=1 r = r cos2
r 2 sin + sin2 2 sin cos
+ 2 cos 2
+
+2 xy
r, r
r
r sin cos 1cos2 xy We can also express x, xy, and y in terms of
placing with  in Eqs. (1.13). Thus,
x in Eqs. (1.13) results in
 sin2 2 sin cos
r , and (Problem 3.26) by re sin cos 1cos2  sin2 2 Basic Relations in Polar Coordinates (3.34) sin cos Similar transformation equations may also be written for the strains
3.8 (3.33) r, r , and .
115 ch03.qxd 12/20/02 7:20 AM Page 116 Compatibility Equation
It can be shown that Eqs. (3.30) result in the following form of the equation of compatibility:
02
1 02
+2
r0
0r2 r
2 + 20
10r
1 02 r
1 0r
=
+2
r 0r
r 0r
r 0r 0
r0 (3.35) To arrive at a compatibility equation expressed in terms of the stress function £ , it
is necessary to evaluate the partial derivatives 02 £ /0x2 and 02 £ /0y2 in terms of r and
by means of the chain rule together with Eqs. (a). These derivatives lead to the
Laplacian operator:
§2 £ = 1 0£
1 02 £
02 £
02 £
02 £
+
=
+
+2
2
2
2
r 0r
r 02
0x
0y
0r (3.36) The equation of compatibility in alternative form is thus
§4 £ = ¢ 10
1 02
02
+
+ 2 2 ≤ 1 §2 £ 2 = 0
r 0r
r0
0r2 (3.37) For the axisymmetrical, zero body force case, the compatibility equation is, from
Eq. (3.9) [referring to (3.36)],
§21 r + 2= d21 r + dr 2 2 + 1 d1 r +
r
dr 2 =0 (3.38) The remaining relationships appropriate to twodimensional elasticity are found in
a manner similar to that outlined in the foregoing discussion.
EXAMPLE 3.3
A large thin plate is subjected to uniform tensile stress o at its ends, as
shown in Fig. 3.9. Determine the field of stress existing within the plate.
Solution For purposes of this analysis, it will prove convenient to locate the origin of coordinate axes at the center of the plate as shown.
The state of stress in the plate is expressed by
x = o, y = xy =0 2 The stress function, £ = oy /2, satisfies the biharmonic equation, Eq.
(3.14). The geometry suggests polar form. The stress function £ may be
transformed by substituting y = r sin , with the following result:
FIGURE 3.9. Example 3.3. A plate in uniaxial tension. 116 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 117 £= 1
4 or 11  cos 2 2 2 (h) The stresses in the plate now follow from Eqs. (h) and (3.29):
r =
= r 1
2
1
2 = o11 o11
1
2o + cos 2 2
 cos 2 2 (3.39) sin 2 Clearly, substitution of y = xy = 0 could have led directly to the foregoing result, using the transformation expressions of stress, Eqs. (3.33). Part B—Stress Concentrations
3.9 STRESSES DUE TO CONCENTRATED LOADS Let us now consider a concentrated force P or F acting at the vertex of a very large
or semiinfinite wedge (Fig. 3.10). The load distribution along the thickness (z direction) is uniform. The thickness of the wedge is taken as unity, so P or F is the load
per unit thickness. In such situations, it is convenient to use polar coordinates and
the semiinverse method.
In actuality, the concentrated load is assumed to be a theoretical line load and
will be spread over an area of small finite width. Plastic deformation may occur locally. Thus, the solutions that follow are not valid in the immediate vicinity of the
application of load. FIGURE 3.10. Wedge of unit thickness subjected to a concentrated
load per unit thickness: (a) knife edge or pivot; (b)
wedge cantilever. 3.9 Stresses Due to Concentrated Loads 117 ch03.qxd 12/20/02 7:20 AM Page 118 Compression of a Wedge (Fig. 3.10a).
Assume the stress function
£ = cPr sin (a) where c is a constant. It can be verified that Eq. (a) satisfies Eq. (3.37) and compatibility is ensured. For equilibrium, the stresses from Eqs. (3.29) are
r = 2cP cos
,
r = 0, r =0 (b) The force resultant acting on a cylindrical surface of small radius, shown by the
dashed lines in Fig. 3.10a, must balance P. The boundary conditions are therefore
expressed by
=
2 L
0 = 0, r 1 r (c) =; cos 2rd =  P (d) Conditions (c) are fulfilled by the last two of Eqs. (b). Substituting the first of Eqs.
(b) into condition (d) results in
4cP L
0 cos2 Integrating and solving for c: c =  1/12
knife edge is therefore
r r1 = P cos
1
2 + d = P
+ sin 2 2. The stress distribution in the sin 2 2 , = 0, r =0 (3.40) This solution is due to J. H. Michell [Ref. 3.6].
The distribution of the normal stresses x over any cross section m  n perpendicular to the axis of symmetry of the wedge is not uniform (Fig. 3.10a). Applying
Eq. (3.34) and substituting r = L/cos in Eq. (3.40), we have
x = r cos2 = L1 P cos4
+ 1
2 sin 2 2 (3.41) The foregoing shows that the stresses increase as L decreases. Observe also that the
normal stress is maximum at the center of the cross section 1 = 02 and a minimum at = . The difference between the maximum and minimum stress, ¢ x, is,
from Eq. (3.41),
¢ x = P11  cos4 2 L1 + 1
2 sin 2 2 (e) = 10°, ¢ x =  0.172P/L is about 6% of the average normal
For instance, if
stress calculated from the elementary formula 1 x2elem =  P/A =  P/2L tan = 118 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 119  2.836P/L. For larger angles, the difference is greater; the error in the mechanics
of materials solution increases (Prob. 3.31). It may be demonstrated that the stress
distribution over the cross section approaches uniformity as the taper of the wedge
diminishes. Analogous conclusions may also be drawn for a conical bar. Note that
Eqs. (3.40) can be applied as well for the uniaxial tension of tapered members by
assigning r a positive value.
Bending of a Wedge (Fig. 3.10b)
We now employ £ = cFr 1 sin 1, with
force. The equilibrium condition is
1 /22 +
1
L /22  1 12r d r cos 1 measured from the line of action of the = 2cF 1 from which, after integration, c =  1/12
90°  , we have
F cos 1 /22 + cos2 1
L /22  1 d 1 = F  sin 2 2. Thus, by replacing F sin 1 with It is seen that if 1 is larger than /2 the radial stress is positive, that is, tension
exists. Because sin = y/r, cos = x/r, and r = 2x2 + y2, the normal and shearing stresses at a point over any cross section m  n, using Eqs. (3.34) and (3.42),
may be expressed as
r = r1  1
2 1 sin 2 2 x = = r1 =F y = r sin2 sin 2 2 , = 0, r = 0 (3.42) r1  1 sin 2 2
2
x2y sin 2 1x2 + y222
F sin3
=r1  1 sin 2 2
2
y3
F 1
2 (3.43) sin 2 1x2 + y222
F sin2 cos
= r sin cos = r1  1 sin 2 2
2
xy2
F
= 1 sin 2 1x2 + y222
2
=  xy 1
2 F sin cos2 2
r cos =  1
2 Using Eqs. (3.43), it can be shown that (Prob. 3.33) across a transverse section
x = L of the wedge: x is a maximum for
= ; 30°, y is a maximum for
= ; 60°, and xy is a maximum for = ; 45°.
To compare the results given by Eqs. (3.43) with the results given by the elementary formulas for stress, consider the series 3.9 Stresses Due to Concentrated Loads 119 ch03.qxd 12/20/02 7:20 AM Page 120 sin 2 =2  12 23
12 25
+
3!
5! It follows that, for small angle , we can disregard all but the first two terms of this
series to obtain
2 = sin 2 + 12 23
6 (f) By introducing the moment of inertia of the cross
I = 2y3 = 2x3 # tan3 , and Eq. (f), we find from Eqs. (3.43) that
3
3
x Fxy tan
=B¢
I ≤ cos
3 4 R, xy Fy2 tan
=B¢
I section ≤ cos4 R m  n, 3 (g) For small values of , the factor in the bracket is approximately equal to unity. The
expression for x then coincides with that given by the flexure formula,  My/I, of
the mechanics of materials. In the elementary theory, the lateral stress y given by
the second of Eqs. (3.43) is ignored. The maximum shearing stress xy obtained
from Eq. (g) is twice as great as the shearing stress calculated from VQ/Ib of the elementary theory and occurs at the extreme fibers (at points m and n) rather than
the neutral axis of the rectangular cross section.
In the case of loading in both compression and bending, superposition of the
effects of P and F results in the following expression for combined stress in a pivot
or in a wedge–cantilever:
r = r1 P cos
1
2 + sin 2 2  F cos r1  1
2 1 sin 2 2 , = 0, r =0 (3.44) The foregoing provides the local stresses at the support of a beam of narrow rectangular cross section.
Concentrated Load on a Straight Boundary (Fig. 3.11a)
By setting = /2 in Eq. (3.40), the result
r = 2P cos
,
r = 0, r =0 (3.45) is an expression for radial stress in a very large plate (semiinfinite solid) under
normal load at its horizontal surface. For a circle of any diameter d with center on
the x axis and tangent to the y axis, as shown in Fig. 3.11b, we have, for point A of
the circle, d # cos = r. Equation (3.45) then becomes
r = 2P
d (3.46) We thus observe that, except for the point of load application, the stress is the same
at all points on the circle.
The stress components in Cartesian coordinates may be obtained readily by
following a procedure similar to that described previously for a wedge:
120 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 121 FIGURE 3.11. (a) Concentrated load on a straight boundary of a large plate; (b) a circle
of constant radial stress. x = 2P
cos4
x y = 2P
sin2
x xy = 2P
sin cos3
x x3
1x2 + y222
xy 2
2P
=2
1x + y222
x2y
2P
=1x2 + y222 2P
=cos2 (3.47) The state of stress is shown on a properly oriented element in Fig. 3.11a.
3.10 STRESS DISTRIBUTION NEAR CONCENTRATED
LOAD ACTING ON A BEAM The elastic flexure formula for beams gives satisfactory results only at some distance away from the point of load application. Near this point, however, there is a
significant perturbation in stress distribution, which is very important. In the case of
a beam of narrow rectangular cross section, these irregularities can be studied by
using the equations developed in Sec. 3.9.
Consider the case of a simply supported beam of depth h, length L, and width
b, loaded at the midspan (Fig. 3.12a). The origin of coordinates is taken to be the FIGURE 3.12. Beam subjected to a concentrated load P at the midspan.
3.10 Stress Distribution Near Concentrated Load Acting on a Beam 121 ch03.qxd 12/20/02 7:20 AM Page 122 center of the beam, with x the axial axis as shown in the figure. Both force P and
the supporting reactions are applied along lines across the width of the beam. The
bending stress distribution, using the flexure formula, is expressed by
œ
x = My
6P L
=
¢  x≤y
I
bh3 2 where I = bh3/12 is the moment of inertia of the cross section. The stress at the
loaded section is obtained by substituting x = 0 into the preceding equation:
œ
x 3PL
y
bh3 = (a) To obtain the total stress along section AB, we apply the superposition of the
bending stress distribution and stresses created by the line load, given by Eq. (3.45)
for a semiinfinite plate. Observe that the radial pressure distribution created by a
line load over quadrant ab of cylindrical surface abc at point A (Fig. 3.12b) produces a horizontal force
/2 L
0 1 r sin 2r d = /2 2P L
0 sin cos d = P (b) and a vertical force
/2 L
 /2 1 /2 r cos 2r d = L
 2P cos2 d = P (c) /2 applied at A (Fig. 3.12c). In the case of a beam (Fig. 3.12a), the latter force is balanced by the supporting reactions that give rise to the bending stresses [Eq. (a)].
On the other hand, the horizontal forces create tensile stresses at the midsection of
the beam of
P
bh (d) Ph y
6P
=y
2I
bh2 (e) –
x = as well as bending stresses of
–
x = Here Ph/2 is the bending moment of forces P/ about the point 0.
Combining the stresses of Eqs. (d) and (e) with the bending stress given by Eq.
(a), we obtain the axial normal stress distribution over beam cross section AB:
x = 3P
2h
L≤y +
3¢
bh P
bh 3PL
4h
3PL
P
1 0.637
≤=
2¢
2
3L
bh
2bh
2bh (3.48) At point B(0, h/2), the tensile stress is
1 x2B =
122 Chapter 3 (3.49) TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 123 The second term represents a correction to the simple beam formula owing to the
presence of the line load. It is observed that for short beams this stress is of considerable magnitude. The axial normal stresses at other points in the midsection are
determined in a like manner.
The foregoing procedure leads to the poorest accuracy for point B, the point of
maximum tensile stress. A better approximation [see Ref. 3.7] of this stress is given
by
1 x2B = 3PL
P
 0.508
bh
2bh2 (3.50) Another more detailed study demonstrates that the local stresses decrease very
rapidly with increase of the distance (x) from the point of load application. At a
distance equal to the depth of the beam, they are usually negligible. Furthermore,
along the loaded section, the normal stress x does not obey a linear law.
In the preceding discussion, the disturbance caused by the reactions at the ends
of the beam, which are also applied as line loads, are not taken into account. To determine the radial stress distribution at the supports of the beam of narrow rectangular cross section, Eq. (3.44) can be utilized. Clearly, for the beam under
consideration, we use F = 0 and replace P by P/2 in this expression. 3.11 STRESS CONCENTRATION FACTORS The discussion of Sec. 3.9 shows that, for situations in which the cross section of a
loadcarrying member varies gradually, reasonably accurate results can be expected
if we apply equations derived on the basis of constant section. On the other hand,
where abrupt changes in the cross section exist, the mechanics of materials approach cannot predict the high values of stress that actually exist. The condition referred to occurs in such frequently encountered configurations as holes, notches,
and fillets. While the stresses in these regions can in some cases (for example, Table
3.2) be analyzed by applying the theory of elasticity, it is more usual to rely on experimental techniques and, in particular, photoelastic methods. The finite element
method (Chapter 7) is very efficient for this purpose.
It is to be noted that irregularities in stress distribution associated with abrupt
changes in cross section are of practical importance in the design of machine elements subject to variable external forces and stress reversal. Under the action of
stress reversal, progressive cracks (Sec. 4.4) are likely to start at certain points at
which the stress is far above the average value. The majority of fractures in machine
elements in service can be attributed to such progressive cracks.
It is usual to specify the high local stresses owing to geometrical irregularities
in terms of a stress concentration factor, k. That is,
k= maximum stress
nominal stress (3.51) Clearly, the nominal stress is the stress that would exist in the section in question in
the absence of the geometric feature causing the stress concentration. The technical
3.11 Stress Concentration Factors 123 ch03.qxd 12/20/02 7:20 AM Page 124 FIGURE 3.13. Example 3.4. Circular hole in a plate subjected to
uniaxial tension: (a) tangential stress distribution for
= ; /2; (b) tangential stress distribution along periphery of the hole. literature contains an abundance of specialized information on stress concentration
factors in the form of graphs, tables, and formulas.*
EXAMPLE 3.4
A large, thin plate containing a small circular hole of radius a is subjected to simple tension (Fig. 3.13a). Determine the field of stress and
compare with those of Example 3.3.
Solution The boundary conditions appropriate to the circumference
of the hole are
r = r = 0, r=a (a) For large distances away from the origin, we set r, , and r equal to
the values found for a solid plate in Example 3.3. Thus, from Eq. (3.39),
for r = q ,
r =
= 1
2 o 11
1
2 o11 + cos 2 2  cos 2 2, r = 1
2o sin 2 (b) For this case, we assume a stress function analogous to Eq. (h) of Example 3.3,
£ = f11r2 + f21r2 cos 2 (c) in which f1 and f2 are yet to be determined. Substituting Eq. (c) into the
biharmonic equation (3.37) and noting the validity of the resulting expression for all , we have *See, for example, Refs. 3.8 through 3.11. 124 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 125 ¢ ¢ d2f1
d2
1d
1 df1
+
≤¢ 2 +
≤=0
2
r dr
r dr
dr
dr (d) 4f2
d2f2
d2
1d
4
1 df2
+
 2≤ ¢ 2 +
 2≤ =0
2
r dr
r dr
dr
r
dr
r (e) The solutions of Eqs. (d) and (e) are (Prob. 3.35) f1 = c1r2 ln r + c2r2 + c3 ln r + c4 f2 = c5r2 + c6r4 + (f) c7
+ c8
r2 (g) where the c’s are the constants of integration. The stress function is then
obtained by introducing Eqs. (f) and (g) into (c). By substituting £ into
Eq. (3.29), the stresses are found to be
r = c111 + 2 ln r2 + 2c2 +
= c113 + 2 ln r2 + 2c2  r = ¢ 2c5 + 6c6r2  c3
6c7
4c8
 ¢ 2c5 + 4 + 2 ≤ cos 2
2
r
r
r c3
6c7
+ ¢ 2c5 + 12c6r2 + 4 ≤ cos 2
r2
r (h) 6c7
2c8
 2 ≤ sin 2
4
r
r The absence of c4 indicates that it has no influence on the solution.
According to the boundary conditions (b), c1 = c6 = 0 in Eq. (h),
because as r : q the stresses must assume finite values. Then, according to the conditions (a), the equations (h) yield
2c2 + c3
= 0,
a2 2c5 + 6c7
4c8
+ 2 = 0,
4
a
a 2c5  6c7
2c8
 2 =0
4
a
a Also, from Eqs. (b) and (h) we have
o =  4c5, o = 4c2 Solving the preceding five expressions, we obtain c2 = o/4,
c3 =  a2 o/2, c5 =  o/4, c7 =  a4 o/4, and c8 = a2 o/2. The determination of the stress distribution in a large plate containing a small circular hole is completed by substituting these constants into Eq. (h): r 3.11 1
2o = r = 1
2o = B ¢1 B ¢1 + 1
2o ¢1  Stress Concentration Factors a2
3a4
4a2
≤ + ¢ 1 + 4  2 ≤ cos 2 R
r2
r
r
a2
3a4
≤  ¢ 1 + 4 ≤ cos 2 R
r2
r
3a4
2a2
+ 2 ≤ sin 2
r4
r (3.52a)
(3.52b)
(3.52c) 125 ch03.qxd 12/20/02 7:20 AM Page 126 FIGURE 3.14. Example 3.4. Graph of tangential
and radial stresses for
= /2
versus the distance from the center
of the plate shown in Fig. 3.13a. The tangential stress distribution along the edge of the hole, r = a,
is shown in Fig. 3.13b using Eq. (3.52b). We observe from the figure
that
1 2max = 3 o,
1 2min =  o, = ; /2
= 0, =; The latter indicates that there exists a small area experiencing compressive
stress. On the other hand, from Eq. (3.39) for = ; /2, 1 2max = o.
The stress concentration factor, defined as the ratio of the maximum stress
at the hole to the nominal stress o is therefore k = 3 o/ o = 3.
To depict the variation of r1r, /22 and 1r, /22 over the distance
from the origin, dimensionless stresses are plotted against the dimensionless radius in Fig. 3.14. The shearing stress r 1r, /22 = 0. At a distance of twice the diameter of the hole, that is, r = 4a, we obtain
L 1.037 o and r L 0.088 o. Similarly, at a distance r = 9a, we have
L 1.006 o and r L 0.018 o, as is observed in the figure. Thus, simple
tension prevails at a distance of approximately nine radii; the hole has a
local effect on the distribution of stress. This is a verification of SaintVenant’s principle.
The results expressed by Eqs. (3.52) are applied, together with the method of
superposition, to the case of biaxial loading. Distributions of maximum stress
1r, /22, obtained in this way (Prob. 3.36), are given in Fig. 3.15. Such conditions
of stress concentration occur in a thinwalled spherical pressure vessel with a small
circular hole (Fig. 3.15a) and in the torsion of a thinwalled circular tube with a
small circular hole (Fig. 3.15b).
126 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 127 FIGURE 3.15. Tangential stress distribution for = ; /2 in the
plate with the circular hole subject to biaxial
stresses: (a) uniform tension; (b) pure shear.
FIGURE 3.16. Elliptical hole in a plate under uniaxial tension. It is noted that a similar stress concentration is caused by a small elliptical hole
in a thin, large plate (Fig. 3.16). It can be shown that the maximum tensile stress at
the ends of the major axis of the hole is given by
max = o ¢1 + 2 ≤
b
a (3.53) Clearly, the stress increases with the ratio b/a. In the limit as a : 0, the ellipse becomes a crack of length 2b, and a very high stress concentration is produced; material will yield plastically around the ends of the crack or the crack will propagate. To
prevent such spreading, holes may be drilled at the ends of the crack to effectively
increase the radii to correspond to a smaller b/a. Thus, a high stress concentration is
replaced by a relatively smaller one.
3.12 NEUBER’S DIAGRAM Several geometries of practical importance, given in Table 3.2, were the subject of
stress concentration determination by Neuber on the basis of mathematical analysis, as in the preceding example. Neuber’s diagram (a nomograph), which is used
with the table for determining the stress concentration factor k for the configurations shown, is plotted in Fig. 3.17. In applying Neuber’s diagram, the first step is
the calculation of the values of 2h/a and 2b/a.
Given a value of 2b/a, we proceed vertically upward to cut the appropriate curve
designated by the number found in column 5 of the table, then horizontally to the left
to the ordinate axis. This point is then connected by a straight line to a point on the left
abscissa representing 2h/a, according to either scale e or f as indicated in column 4 of
3.12 Neuber’s Diagram 127 ch03.qxd 12/20/02 7:20 AM Page 128 TABLE 3.2
Type of change of section A Type of
loading P
2bt
3M
2b2t Tension
Bending B P
bt
6M
b 2t Tension
Bending C Nominal
stress Scale for
2h/a Curve
for k f 1 f 2 f 3 f 4 P
2bt f 5 Bending
D Tension 3Mh
2t1c3  h32 e 5 f 6 f 7 e 8 e 9 Tension
Bending
Direct shear
Torsional shear P
b2
4M
b3
1.23V
b2
2T
b3 the table. The value of k is read off on the circular scale at a point located on a normal
from the origin. [The values of (theoretical) stress concentration factors obtained from
Neuber’s nomograph agree satisfactorily with those found by the photoelastic method.]
Consider, for example, the case of a member with a single notch (Fig. B in the
table), and assume that it is subjected to axial tension P only. For given
a = 7.925 mm, h = 44.450 mm, and b = 266.700 mm, 2h/a = 2.37 and
2b/a = 5.80. Table 3.2 indicates that scale f and curve 3 are applicable. Then, as
just described, the stress concentration factor is found to be k = 3.25. The path followed is denoted by the broken lines in the diagram. The nominal stress P/bt, multiplied by k, yields maximum theoretical stress, found at the root of the notch.
EXAMPLE 3.5
A circular shaft with a circumferential circular groove (notch) is subjected to axial force P, bending moment M, and torque T (Fig. D of
Table 3.2). Determine the maximum principal stress.
128 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 129 FIGURE 3.17. Neuber’s nomograph. ¢ ≤+
C2 Solution For the loading described, the principal stresses occur at a
point at the root of the notch which, from Eq. (1.16), are given by
2 = 1,2 x 2 ; x 2
xy , 3 =0 (a) where x and xy represent the normal and shear stresses in the reduced
cross section of the shaft, respectively. We have
x = ka My
P
+ kb
,
A
I xy = kt xy = kt Tr
J or
x = ka P
4M
+ kb 3 ,
2
b
b 2T
b3 (b) Here ka, kb, and kt denote the stress concentration factors for axial
force, bending moment, and torque, respectively. These factors are determined from curves 6, 7, and 9 in Fig. 3.17. Thus, given a set of shaft dimensions and the loading, formulas (a) and (b) lead to the value of the
maximum principal stress 1.
In addition, note that a shear force V may also act on the shaft (as in
Fig. D of Table 3.2). For slender members, however, this shear contributes very little to the deflection (Sec. 5.4) and to the maximum stress.
3.13 CONTACT STRESSES Application of a load over a small area of contact results in unusually high stresses.
Situations of this nature are found on a microscopic scale whenever force is transmitted through bodies in contact. There are important practical cases when the
3.13 Contact Stresses 129 ch03.qxd 12/20/02 7:20 AM Page 130 geometry of the contacting bodies results in large stresses, disregarding the stresses
associated with the asperities found on any nominally smooth surface. The Hertz
problem relates to the stresses owing to the contact of a sphere on a plane, a sphere
on a sphere, a cylinder on a cylinder, and the like. The practical implications with respect to ball and roller bearings, locomotive wheels, valve tappets, and numerous
machine components are apparent.
Consider, in this regard, the contact without deformation of two bodies having
spherical surfaces of radii r1 and r2, in the vicinity of contact. If now a collinear pair
of forces P acts to press the bodies together, as in Fig. 3.18, deformation will occur,
and the point of contact O will be replaced by a small area of contact. A common
tangent plane and common normal axis are denoted Ox and Oy, respectively. The
first steps taken toward the solution of this problem are the determination of the
size and shape of the contact area as well as the distribution of normal pressure acting on the area. The stresses and deformations resulting from the interfacial pressure are then evaluated.
The following assumptions are generally made in the solution of the contact
problem:
1. The contacting bodies are isotropic and elastic.
2. The contact areas are essentially flat and small relative to the radii of curvature
of the undeformed bodies in the vicinity of the interface.
3. The contacting bodies are perfectly smooth, and therefore only normal pressures
need be taken into account.
The foregoing set of assumptions enables an elastic analysis to be conducted.
Without going into the derivations, we shall, in the following paragraphs, introduce
some of the results.* It is important to note that, in all instances, the contact pressure varies from zero at the side of the contact area to a maximum value c at its
center.
FIGURE 3.18. Spherical surfaces of two bodies compressed
by forces P. *A summary and complete list of references dealing with contact stress problems are
given by Refs. 3.12 through 3.16. 130 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 131 Two Spherical Surfaces in Contact
Because of forces P (Fig. 3.18), the contact pressure is distributed over a small circle
of radius a given by
a = 0.88 B P1E1 + E22r1r2
R
E1E21r1 + r22 1/3 (3.54) where E1 and E2 ( r1 and r2 ) are the respective moduli of elasticity (radii) of the
spheres. The force P causing the contact pressure acts in the direction of the normal
axis, perpendicular to the tangent plane passing through the contact area. The
maximum contact pressure is found to be
c = 1.5 P
a2 (3.55) This is the maximum principal stress owing to the fact that, at the center of the
contact area, material is compressed not only in the normal direction but also in
the lateral directions. The relationship between the force of contact P, and the relative displacement of the centers of the two elastic spheres, owing to local deformation, is
1
121
1
+
= 0.77 B P ¢
≤ ¢ + ≤R
r1
r2
E1
E2 1/3 2 (3.56) In the special case of a sphere of radius r contacting a body of the same material but having a flat surface (Fig. 3.19a), substitution of r1 = r, r2 = q , and
E1 = E2 = E into Eqs. (3.54) through (3.56) leads to
2Pr
a = 0.88 ¢
≤,
E
1/3 PE 2
= 0.62 ¢ 2 ≤ ,
4r
1/3 c P2
= 1.54 ¢ 2 ≤
2E r 1/3 (3.57) For the case of a sphere in a spherical seat of the same material (Fig. 3.19b) substituting r2 =  r2 and E1 = E2 = E in Eqs. (3.54) through (3.56), we obtain
a = 0.88 B
= 1.54 B 2Pr1r2
R,
E1r2  r12
1/3 P21r2  r12
2E 2r1r2 R 1/3 c = 0.62 B PE 2 ¢ r2  r1 2
≤R
2r1r2 1/3 (3.58) FIGURE 3.19. Contact load: (a) in sphere on
a plane; (b) in ball in a spherical seat. 3.13 Contact Stresses 131 ch03.qxd 12/20/02 7:20 AM Page 132 FIGURE 3.20. Contact load: (a) in two cylindrical rollers; (b) in
cylinder on a plane. Two Parallel Cylindrical Rollers
Here the contact area is a narrow rectangle of width 2b and length L (Fig. 3.20a).
The maximum contact pressure is given by
c b=B where = 2P
bL 4Pr1r2
1¢
L1r1 + r22
E1 2
1 (3.59) + 1E2 2
2 ≤R 1/2 (3.60) In this expression, Ei1 i2 and ri, with i = 1, 2, are the moduli of elasticity (Poisson’s
ratio) of the two rollers and the corresponding radii, respectively. If the cylinders
have the same elastic modulus E and Poisson’s ratio = 0.3, these expressions reduce to
c = 0.418 PE r1 + r2
,
r1r2
AL b = 1.52 r1r2
P
A EL r1 + r2 (3.61) Figure 3.20b shows the special case of contact between a circular cylinder of radius
r and a flat surface, both bodies of the same material. After rearranging the terms
and taking r1 = r and r2 = q in Eqs. (3.61), we have
c = 0.418 PE
,
A Lr b = 1.52 Pr
A EL (3.62) Two Curved Surfaces of Different Radii
Consider now two rigid bodies of equal elastic moduli E, compressed by force P
(Fig. 3.21). The load lies along the axis passing through the centers of the bodies
and through the point of contact and is perpendicular to the plane tangent to both
132 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:20 AM Page 133 FIGURE 3.21. Curved surfaces of different radii of two bodies compressed by forces P. bodies at the point of contact. The minimum and maximum radii of curvature of the
œ
œ
surface of the upper body are r1 and r1; those of the lower body are r2 and r2 at the
œ
œ
point of contact. Thus, 1/r1, 1/r1, 1/r2, and 1/r2 are the principal curvatures. The sign
convention of the curvature is such that it is positive if the corresponding center of
curvature is inside the body. If the center of the curvature is outside the body, the
œ
œ
curvature is negative. (For example, in Fig. 3.22a, r1, r1 are positive, while r2, r2 are
negative.)
Let be the angle between the normal planes in which radii r1 and r2 lie. Subsequent to loading, the area of contact will be an ellipse with semiaxes a and b
(Table C.1). The maximum contact pressure is
c = 1.5 3 Pm
,
An P
ab In this expression the semiaxes are given by
a = ca b = cb (3.63) 3 Pm
An (3.64) Here
m= 4
,
1
1
1
1
+ œ+
+œ
r1
r2
r1
r2 n= 4E
311  2 2 (3.65) The constants ca and cb are read in Table 3.3. The first column of the table lists values of , calculated from
cos 3.13 Contact Stresses = B
A (3.66) 133 ch03.qxd 12/20/02 7:21 AM Page 134 TABLE 3.3
ca cb 3.778
2.731
2.397
2.136
1.926
1.754
1.611
1.486
1.378
1.284
1.202
1.128
1.061
1.000 0.408
0.493
0.530
0.567
0.604
0.641
0.678
0.717
0.759
0.802
0.846
0.893
0.944
1.000 (degrees) 20
30
35
40
45
50
55
60
65
70
75
80
85
90
where
2
A= ,
m 1
1
1
1
1
B = B¢ ≤ +¢ ≤
r2
2 r1
r¿1
r¿2
2 + 2¢ 2 1
1
1
1
≤¢ ≤ cos 2 R
r1
r ¿ 1 r2
r¿2 1/2 (3.67) By applying Eq. (3.63), many problems of practical importance may be treated,
for example, contact stresses in ball bearings (Fig. 3.22a), contact stresses between a FIGURE 3.22. Contact load: (a) in a singlerow ball bearing; (b) in a cylindrical wheel and rail. 134 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:21 AM Page 135 cylindrical wheel and a rail (Fig. 3.22b), and contact stresses in cam and pushrod
mechanisms.
EXAMPLE 3.6
A railway car wheel rolls on a rail. Both rail and wheel are made of steel
= 0.3. The wheel has a radius of
for which E = 210 GPa and
r1 = 0.4 m, and the cross radius of the rail top surface is r2 = 0.3 m (Fig.
3.22b). Determine the size of the contact area and the maximum contact
pressure, given a compression load of P = 90 kN.
Solution For the situation described, 1/r ¿ 1 = 1/r ¿ 2 = 0, and, because
the axes of the members are mutually perpendicular, = /2. The first
of Eqs. (3.65) and Eqs. (3.67) reduce to
m= 4
,
1/r1 + 1/r2 A= 11
1
¢ + ≤,
r2
2 r1 B= ; 11
1
¢  ≤ (3.68)
r2
2 r1 The proper sign in B must be chosen so that its values are positive. Now
Eq. (3.66) has the form
cos =; 1/r1  1/r2
1/r1 + 1/r2 (3.69) Substituting the given numerical values into Eqs. (3.68), (3.69), and
the second of (3.65), we obtain
m=
cos 4
= 0.6857,
1/0.4 + 1/0.3 =; 1/0.4  1/0.3
= 0.1428
1/0.4 + 1/0.3 n= 41210 * 1092
= 3.07692 * 1011
310.912 or = 81.79° Corresponding to this value of , interpolating in Table 3.3, we have
ca = 1.1040, cb = 0.9113 The semiaxes of the elliptical contact are found by applying Eqs. (3.64):
a = 1.1040 B b = 0.9113 B 90,000 * 0.6857 1/3
R = 0.00646 m
3.07692 * 1011 90,000 * 0.6857 1/3
R = 0.00533 m
3.07692 * 1011 The maximum contact pressure, or maximum principal stress, is thus
c 3.13 Contact Stresses = 1.5 90,000
= 1248 MPa
10.00646 * 0.005332 135 ch03.qxd 12/20/02 7:21 AM Page 136 A hardened steel material is capable of resisting this or somewhat
higher stress levels for the body geometries and loading conditions described in this section.
PROBLEMS
Secs. 3.1 through 3.7
3.1. A stress distribution is given by
x = pyx3  2c1xy + c2y y = pxy 3  2px3y xy 3
22
2 px y = (a) + c1y2 + 1
4
2 px + c3 where the p and c’s are constants. (a) Verify that this field represents a solution for a thin plate of thickness t (Fig. P3.1); (b) obtain the corresponding stress function; (c) find the resultant normal and shearing boundary
forces ( Py and Vx ) along edges y = 0 and y = b of the plate.
3.2. If the stress field given by Eq. (a) of Prob. 3.1 acts in the thin plate shown
in Fig. P3.1 and p is a known constant, determine the c’s so that edges
x = ; a are free of shearing stress and no normal stress acts on edge
x = a.
3.3. In bending of a rectangular plate (Fig. P3.3), the state of stress is expressed
by
x = c1y + c2xy xy = c31b2  y22 (a) What conditions among the constants (the c’s) make the preceding expressions possible? Body forces may be neglected. (b) Draw a sketch
showing the boundary stresses on the plate.
3.4. Given the following stress field within a structural member,
x
y
z = a[y2 + b1x2  y22] xy 2 2 = ab1x + y 2
2 =  2abxy yz = a[x + b1y  x 2]
2 = xz =0 2 where a and b are constants. Determine whether this stress distribution
represents a solution for a plane strain problem. The body forces are
omitted.
y
b
x
b
a FIGURE P3.1.
136 FIGURE P3.3.
Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:21 AM Page 137 3.5. Determine whether the following stress functions satisfy the conditions of
compatibility for a twodimensional problem:
£ 1 = ax2 + bxy + cy 2 (a) £ 2 = ax3 + bx2y + cxy 2 + dy3 (b) Here a, b, c, and d are constants. Also obtain the stress fields that arise from
£ 1 and £ 2.
3.6. Figure P3.6 shows a long, thin steel plate of thickness t, width 2h, and
length 2a. The plate is subjected to loads that produce the uniform stresses
o at the ends. The edges at y = ; h are placed between the two rigid walls.
Show that, by using an inverse method, the displacements are expressed by
u= 1E 2
o x, v = 0, w= 11 + 2
E oz 3.7. Determine whether the following stress distribution is a valid solution for a
twodimensional problem:
x =  ax2y y 1
=  ay3
3 xy = axy 2 where a is a constant. Body forces may be neglected.
3.8. The strain distribution in a thin plate has the form B ax3
axy 2 axy2
R
ax2y in which a is a small constant. Show whether this strain field is a valid solution of an elasticity problem. Body forces may be disregarded.
3.9. The components of the displacement of a thin plate (Fig. P3.9) are given by
u =  c1y2 + x2 v = 2cxy Here c is a constant and v represents Poisson’s ratio. Determine the stresses
x, y, and xy. Draw a sketch showing the boundary stresses on the plate.
3.10. Consider a rectangular plate with sides a and b of thickness t (Fig. P3.10).
(a) Determine the stresses x, y, and xy for the stress function £ = px3y,
where p is a constant. (b) Draw a sketch showing the boundary stresses on
the plate. (c) Find the resultant normal and shearing boundary forces
( Px, Py, Vx, and V ) along all edges of the plate.
y FIGURE P3.6.
Problems 137 ch03.qxd 12/20/02 7:21 AM Page 138 y b
x
b
a a FIGURE P3.9. FIGURE P3.10. 3.11. Redo Prob. 3.10 for the case of a square plate of side dimensions a and
p
£= 2a2 1x2y2 + 1xy32
3 where p is a constant.
3.12. Resolve Prob. 3.10 a and b for the stress function of the form
p
£= b3 xy 213b  2y2 where p represents a constant.
3.13. A vertical force P per unit thickness is applied on the horizontal boundary
of a semiinfinite solid plate of unit thickness (Fig. 3.11a). Show that the
stress function £ =  1P/ 2y tan11y/x2 results in the following stress field
within the plate:
2P
x = x3
,
1x2 + y222 2P
y = xy2 ,
1x2 + y222 2P
xy = yx2 1x2 + y222 Also plot the resulting stress distribution for x and xy at a constant depth
L below the boundary.
3.14. The thin cantilever shown in Fig. P3.14 is subjected to uniform shearing
stress o along its upper surface 1y = + h2 while surfaces y =  h and
x = L are free of stress. Determine whether the Airy stress function FIGURE P3.14. 138 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:21 AM Page 139 £= 1
4o ¢ xy  xy3
Ly 3
xy2
Ly 2
 2+
+ 2≤
h
h
h
h satisfies the required conditions for this problem.
3.15. Figure P3.15 shows a thin cantilever beam of unit thickness carrying a uniform load of intensity p per unit length. Assume that the stress function is
expressed by
£ = ax2 + bx2y + cy3 + dy 5 + ex2y3
in which a, Á , e are constants. Determine (a) the requirements on
a, Á , e so that £ is biharmonic; (b) the stresses x, y, and xy.
3.16. Consider a thin square plate with sides a. For a stress function
£ = 1p/a2211x2y2  1y42, determine the stress field and sketch it along the
2
6
boundaries of the plate. Here p represents a uniformly distributed loading
per unit length. Note that the origin of the x, y coordinate system is located
at the lowerleft corner of the plate.
3.17. Consider a thin cantilever loaded as shown in Fig. P3.17. Assume that the
bending stress is given by
Mzy
x = = I p2
xy
2I (P3.17) and z = xz = yz = 0. Determine the stress components y and xy as
functions of x and y.
3.18. Show that for the case of plane stress, in the absence of body forces, the
equations of equilibrium may be expressed in terms of displacements u and
v as follows:
1+
02u
02u
+ 2+
2
10x
0y
1+
02v
02v
+ 2+
2
10y
0x 0 0u
0v
¢ + ≤=0
0x 0x
0y
0 0v
0v
¢ + ≤=0
0y 0y
0x (P3.18) [Hint: Substitute Eqs. (3.10) together with (2.3) into (3.6).]
3.19. Determine whether the following compatible stress field is possible within
an elastic uniformly loaded cantilever beam (Fig. P3.17):
y
p
h
h x
L FIGURE P3.15. Problems 139 ch03.qxd 12/20/02 7:21 AM Page 140 FIGURE P3.17. x xy y p
p3
15x2 + 2h22y +
y
10I
3I
px 2
=1h  y22
2I
p
=  12h3  3h2y + y32
6I
= (P3.19) Here I = 2th3/3 and the body forces are omitted. Given p = 10 kN/m,
L = 2 m, h = 100 mm, t = 40 mm, = 0.3, and E = 200 GPa, calculate
the magnitude and direction of the maximum principal strain at point Q.
3.20. A prismatic bar is restrained in the x (axial) and y directions, but free to expand in z direction. Determine the stresses and strains in the bar for a temperature rise of T1 degrees.
3.21. Under free thermal expansion, the strain components within a given elastic
solid are x = y = z = T and xy = yz = xz = 0. Show that the temperature field associated with this condition is of the form
T = c1x + c2y + c3z + c4
in which the c’s are constants.
3.22. Redo Prob. 3.6 adding a temperature change T1, with all other conditions
remaining unchanged.
3.23. Determine the axial force Px and moment Mz that the walls in Fig. 3.6b
apply to the beam for T = a1y + a2, where a1 and a2 are constant.
3.24. A copper tube of 800mm2 crosssectional area is held at both ends as in Fig.
P3.24. If at 20°C no axial force Px exists in the tube, what will Px be when the
temperature rises to 120°C? Let E = 120 GPa and = 16.8 * 106per °C. FIGURE P3.24. 140 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 7:21 AM Page 141 Secs. 3.8 through 3.10
3.25. Show that the case of a concentrated load on a straight boundary (Fig.
3.11a) is represented by the stress function
P
£= r sin and derive Eqs. (3.45) from the result.
3.26. Verify that Eqs. (3.34) are determined from the equilibrium of forces acting on the elements shown in Fig. P3.26.
3.27. Demonstrate that the biharmonic equation §4 £ = 0 in polar coordinates
can be written as ¢ 10
1 02
1 0£
1 02 £
02
02 £
+
+ 2 2≤ ¢ 2 +
+2
≤=0
2
r 0r
r 0r
r0
r 02
0r
0r 3.28. Show that the compatibility equation in polar coordinates, for the axisymmetrical problem of thermal elasticity, is given by
1d
d£
¢r
≤+E T=0
r dr
dr (P3.28) 3.29. Assume that moment M acts in the plane and at the vertex of the
wedge–cantilever shown in Fig. P3.29. Given a stress function
£= M1sin 2  2 cos 2 2
21sin 2  2 cos 2 2 (P3.29a) determine (a) whether £ satisfies the condition of compatibility; (b) the
stress components r, , and r ; and (c) whether the expressions
2M sin 2
2M cos2
,
= 0,
=
r
r
r2
r2
represent the stress field in a semiinfinite plate (that is, for (P3.29b) = FIGURE P3.26. Problems = /2 ). FIGURE P3.29. 141 ch03.qxd 12/20/02 7:21 AM Page 142 FIGURE P3.30. 3.30. Referring to Fig. P3.30, verify the results given by Eqs. (b) and (c) of Sec. 3.10.
3.31. Consider the pivot of unit thickness subject to force P per unit thickness at
its vertex (Fig. 3.10a). Determine the maximum values of x and xy on a
plane a distance L from the apex through the use of r given by Eq. (3.40)
and the formulas of the elementary theory: (a) take
= 15°; (b) take
= 60°. Compare the results given by the two approaches.
3.32. Solve Prob. 3.31 for = 30°.
3.33. Redo Prob. 3.31 in its entirety for the wedge–cantilever shown in Fig. 3.10b.
3.34. A uniformly distributed load of intensity p is applied over a short distance
on the straight edge of a large plate (Fig. P3.34). Determine stresses x, y,
and xy in terms of p, 1, and 2, as required. [Hint: Let dP = pdy denote
the load acting on an infinitesimal length dy = rd /cos (from geometry)
and hence dP = prd /cos . Substitute this into Eqs. (3.47) and integrate
the resulting expressions.]
Secs. 3.11 through 3.13
3.35. Verify the result given by Eqs. (f) and (g) of Sec. 3.11 (a) by rewriting Eqs.
(d) and (e) in the following forms, respectively,
df1
1d
d 1d
¢r
≤Rr = 0
br B
r dr
dr r dr
dr d 1 d 3d 1 d 2
r
1r f22 R r ≤ = 0
¢
B
br
dr r3 dr
dr r3 dr (P3.35) FIGURE P3.34.
142 Chapter 3 TwoDimensional Problems in Elasticity ch03.qxd 12/20/02 3.36.
3.37. 3.38. 3.39. 3.40. 3.41. 3.42. 7:21 AM Page 143 and by integrating (P3.35) and (b) by expanding Eqs. (d) and (e), setting
t = ln r, and thereby transforming the resulting expressions into two ordinary differential equations with constant coefficients.
Verify the results given in Fig. 3.15 by employing Eq. (3.52b) and the
method of superposition.
For the flat bar of Fig. C of Table 3.2, let b = 17h, c = 18h, and a = h (circular hole). Referring to Neuber’s nomograph (Fig. 3.17), determine the
value of k for the bar loaded in tension.
A 20mmthick steel bar with a slot (25 mm radii at ends) is subjected to an
axial load P, as shown in Fig. P3.38. What is the maximum stress for
P = 180 kN?
For the flat bar in Fig. A of Table 3.2, let h = 3a and b = 15a. Referring to
Neuber’s nomograph (Fig. 3.17), find the value of k for the bar subjected to
(a) axial tensile load, and (b) bending.
A thinwalled circular cylindrical vessel of diameter d and wall thickness t
is subjected to internal pressure p (see Table 1.1). Given a small circular
hole in the vessel wall, show that the maximum tangential and axial
= 5pd/4t and a = pd/4t, respectively.
stresses at the hole are
The shaft shown in Fig. D of Table 3.2 has the following dimensions:
a = 6 mm, h = 12 mm, and b = 200 mm. The shaft is subjected simultaneously to a torque T = 4 kN # m, a bending moment M = 2 kN # m, and an
axial force P = 10 kN. Calculate at the root of the notch (a) the maximum
principal stress, (b) the maximum shear stress, and (c) the octahedral stresses.
Redo Prob. 3.41 for a = 4 mm, h = 6 mm, b = 120 mm, T = 3 kN # m,
M = 1.5 kN # m, and P = 0. 3.43. A 50mmdiameter ball is pressed into a spherical seat of diameter 75 mm
by a force of 500 N. The material is steel 1E = 200 GPa, = 0.32. Calculate (a) the radius of the contact area; (b) the maximum contact pressure;
and (c) the relative displacement of the centers of the ball and seat.
3.44. Calculate the maximum contact pressure c in Prob. 3.43 for the cases
when the 50mmdiameter ball is pressed against (a) a flat surface, and (b)
an identical ball.
3.45. Calculate the maximum pressure between a steel wheel of radius
r1 = 400 mm and a steel rail of crown radius of the head r2 = 250 mm (Fig.
3.22b) for P = 4 kN. Use E = 200 GPa and = 0.3.
3.46. A concentrated load of 2.5 kN at the center of a deep steel beam is applied
through a 10mmdiameter steel rod laid across the 100mm beam width.
50 mm
150 mm P
slot FIGURE P3.38.
Problems 143 ch03.qxd 12/20/02 7:21 AM Page 144 Compute the maximum contact pressure and the width of the contact between rod and beam surface. Use E = 200 GPa and = 0.3.
3.47. Two identical 400mmdiameter steel rollers of a rolling mill are pressed
together with a force of 2 MN/m. Using E = 200 GPa and = 0.25, compute the maximum contact pressure and width of contact.
3.48. Determine the size of the contact area and the maximum pressure between
two circular cylinders with mutually perpendicular axes. Denote by r1 and
r2 the radii of the cylinders. Use r1 = 500 mm, r2 = 200 mm, P = 5 kN,
E = 210 GPa, and = 0.25.
3.49. Solve Prob. 3.48 for the case of two cylinders of equal radii,
r1 = r2 = 200 mm.
3.50. Determine the maximum pressure at the contact point between the outer
race and a ball in the singlerow ball bearing assembly shown in Fig. 3.22a.
The ball diameter is 50 mm; the radius of the grooves, 30 mm; the diameter
of the outer race, 250 mm; and the highest compressive force on the ball,
P = 1.8 kN. Take E = 200 GPa and = 0.3.
3.51. Redo Prob. 3.50 for a ball diameter of 40 mm and a groove radius of
22 mm. Assume the remaining data to be unchanged. 144 Chapter 3 TwoDimensional Problems in Elasticity ...
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