REVIEW AND SUMMARY

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Stress and Strain—Axial Loading We noted in Sec. 2.5 that the initial portion of the stress-strain dia- gram is a straight line. This means that for small deformations, the stress is directly proportional to the strain:

s5 EP (2.4)

This relation is known as Hooke’s law and the coefficient E as the modulus of elasticity of the material. The largest stress for which Eq.

(2.4) applies is the proportional limit of the material.

Materials considered up to this point were isotropic, i.e., their properties were independent of direction. In Sec. 2.5 we also con- sidered a class of anisotropic materials, i.e., materials whose proper- ties depend upon direction. They were fiber-reinforced composite materials, made of fibers of a strong, stiff material embedded in lay- ers of a weaker, softer material (Fig. 2.71). We saw that different moduli of elasticity had to be used, depending upon the direction of loading.

If the strains caused in a test specimen by the application of a given load disappear when the load is removed, the material is said to behave elastically, and the largest stress for which this occurs is called the elastic limit of the material [Sec. 2.6]. If the elastic limit is exceeded, the stress and strain decrease in a linear fashion when the load is removed and the strain does not return to zero (Fig. 2.72), indicating that a permanent set or plastic deformation of the material has taken place.

In Sec. 2.7, we discussed the phenomenon of fatigue, which causes the failure of structural or machine components after a very large number of repeated loadings, even though the stresses remain in the elastic range. A standard fatigue test consists in determining the number n of successive loading-and-unloading cycles required to cause the failure of a specimen for any given maximum stress level s, and plotting the resulting s-n curve. The value of s for which failure does not occur, even for an indefinitely large number of cycles, is known as the endurance limit of the material used in the test.

Section 2.8 was devoted to the determination of the elastic defor- mations of various types of machine and structural components under various conditions of axial loading. We saw that if a rod of length L and uniform cross section of area A is subjected at its end to a centric axial load P (Fig. 2.73), the corresponding defor- mation is

d5 PL

AE (2.7)

If the rod is loaded at several points or consists of several parts of various cross sections and possibly of different materials, the defor- mation d of the rod must be expressed as the sum of the deforma- tions of its component parts [Example 2.01]:

d5 a_i PiLi

AiEi (2.8)

Elastic limit. Plastic deformation

Fatigue. Endurance limit

Elastic deformation under axial loading Hooke’s law Modulus of elasticity

Layer of material

Fibers y

z x

Fig. 2.71

C

A D

Rupture B

Fig. 2.72

L

C A C

B B

P Fig. 2.73

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Section 2.9 was devoted to the solution of statically indeterminate problems, i.e., problems in which the reactions and the internal forces cannot be determined from statics alone. The equilibrium equations derived from the free-body diagram of the member under consideration were complemented by relations involving deforma- tions and obtained from the geometry of the problem. The forces in the rod and in the tube of Fig. 2.74, for instance, were determined by observing, on one hand, that their sum is equal to P, and on the other, that they cause equal deformations in the rod and in the tube [Example 2.02]. Similarly, the reactions at the supports of the bar of Fig. 2.75 could not be obtained from the free-body diagram of the bar alone [Example 2.03]; but they could be determined by express- ing that the total elongation of the bar must be equal to zero.

In Sec. 2.10, we considered problems involving temperature changes.

We first observed that if the temperature of an unrestrained rod AB of length L is increased by DT, its elongation is

d_T 5a1¢T2L (2.21) where a is the coefficient of thermal expansion of the material. We noted that the corresponding strain, called thermal strain, is

P_T5a¢T (2.22)

and that no stress is associated with this strain. However, if the rod AB is restrained by fixed supports (Fig. 2.76), stresses develop in the

Statically indeterminate problems

Problems with temperature changes

P Tube (A₂, E₂)

Rod (A₁, E₁)

End plate L

Fig. 2.74

P L₁

L₂

R_A

R_B

(a) (b)

L A

B

A

B

C C

P

Fig. 2.75

rod as the temperature increases, because of the reactions at the supports. To determine the magnitude P of the reactions, we detached the rod from its support at B (Fig. 2.77) and considered separately the deformation d_T of the rod as it expands freely because of the temperature change, and the deformation d_P caused by the force P required to bring it back to its original length, so that it may be reat- tached to the support at B. Writing that the total deformation d 5 d_T 1 d_P is equal to zero, we obtained an equation that could be solved for P. While the final strain in rod AB is clearly zero, this will generally not be the case for rods and bars consisting of elements of different cross sections or materials, since the deformations of the various elements will usually not be zero [Example 2.06].

Fig. 2.76 L

A B

Fig. 2.77 L

(b)

(c)

L A

A B

B

P

(a) T

A B

P

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Stress and Strain—Axial Loading

When an axial load P is applied to a homogeneous, slender bar (Fig. 2.78), it causes a strain, not only along the axis of the bar but in any transverse direction as well [Sec. 2.11]. This strain is referred to as the lateral strain, and the ratio of the lateral strain over the axial strain is called Poisson’s ratio and is denoted by n (Greek letter nu). We wrote

n5 2lateral strain

axial strain (2.25)

Recalling that the axial strain in the bar is P_x5s_xyE, we expressed as follows the condition of strain under an axial loading in the x direction:

P_x 5 s_x

E

  

^{Py5 Pz5 2ns}_E^{x (2.27)}

This result was extended in Sec. 2.12 to the case of a multiaxial loading causing the state of stress shown in Fig. 2.79. The resulting strain condition was described by the following relations, referred to as the generalized Hooke’s law for a multiaxial loading.

P_x5 1s_x E 2 nsy

E 2 ns_z E P_y5 2ns_x

E 1 sy

E 2 ns_z

E (2.28)

P_z5 2ns_x E 2 nsy

E 1 s_z E

If an element of material is subjected to the stresses s_x, s_y, s_z, it will deform and a certain change of volume will result [Sec. 2.13].

The change in volume per unit volume is referred to as the dilatation of the material and is denoted by e. We showed that

e5122n

E 1s_x1s_y1s_z2 (2.31) When a material is subjected to a hydrostatic pressure p, we have

e5 2p

k (2.34)

where k is known as the bulk modulus of the material:

k5 E

31122n2 (2.33)

Lateral strain. Poisson’s ratio

Multiaxial loading

z y

P x A

Fig. 2.78

x

y

y

x _z

z

Fig. 2.79

Dilatation

Bulk modulus

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zy

yz ^yx

zx

z ^x

y

z

y

x xy

xz

Q

Fig. 2.80

1 1

z

y

x yx

xy 2 xy

2 xy

Fig. 2.81

As we saw in Chap. 1, the state of stress in a material under the most general loading condition involves shearing stresses, as well as nor- mal stresses (Fig. 2.80). The shearing stresses tend to deform a cubic element of material into an oblique parallelepiped [Sec. 2.14]. Con- sidering, for instance, the stresses t_xy and t_yx shown in Fig. 2.81 (which, we recall, are equal in magnitude), we noted that they cause the angles formed by the faces on which they act to either increase or decrease by a small angle g_xy; this angle, expressed in radians, defines the shearing strain corresponding to the x and y directions.

Defining in a similar way the shearing strains g_yz and g_zx, we wrote the relations

t_xy5 Gg_xy t_yz5 Gg_yz t_zx5 Gg_zx (2.36, 37) which are valid for any homogeneous isotropic material within its proportional limit in shear. The constant G is called the modulus of rigidity of the material and the relations obtained express Hooke’s law for shearing stress and strain. Together with Eqs. (2.28), they form a group of equations representing the generalized Hooke’s law for a homogeneous isotropic material under the most general stress condition.

We observed in Sec. 2.15 that while an axial load exerted on a slender bar produces only normal strains—both axial and transverse—

on an element of material oriented along the axis of the bar, it will produce both normal and shearing strains on an element rotated through 458 (Fig. 2.82). We also noted that the three constants E, n, and G are not independent; they satisfy the relation.

E

2G 511n (2.43)

which may be used to determine any of the three constants in terms of the other two.

Stress-strain relationships for fiber-reinforced composite materials were discussed in an optional section (Sec. 2.16). Equations similar to Eqs. (2.28) and (2.36, 37) were derived for these materials, but we noted that direction-dependent moduli of elasticity, Poisson’s ratios, and moduli of rigidity had to be used.

Shearing strain. Modulus of rigidity

y

1 x

1

1x

1x

(a)

P

(b)

2

2 ' '

P P'

P'

Fig. 2.82

Fiber-reinforced composite materials

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Stress and Strain—Axial Loading In Sec. 2.17, we discussed Saint-Venant’s principle, which states that except in the immediate vicinity of the points of application of the loads, the distribution of stresses in a given member is inde- pendent of the actual mode of application of the loads. This prin- ciple makes it possible to assume a uniform distribution of stresses in a member subjected to concentrated axial loads, except close to the points of application of the loads, where stress concentrations will occur.

Stress concentrations will also occur in structural members near a discontinuity, such as a hole or a sudden change in cross section [Sec.

2.18]. The ratio of the maximum value of the stress occurring near the discontinuity over the average stress computed in the critical section is referred to as the stress-concentration factor of the discon- tinuity and is denoted by K:

K5 s_max

s_ave (2.48)

Values of K for circular holes and fillets in flat bars were given in Fig. 2.64 on p. 108.

In Sec. 2.19, we discussed the plastic deformations which occur in structural members made of a ductile material when the stresses in some part of the member exceed the yield strength of the material.

Our analysis was carried out for an idealized elastoplastic material characterized by the stress-strain diagram shown in Fig. 2.83 [Exam- ples 2.13, 2.14, and 2.15]. Finally, in Sec. 2.20, we observed that when an indeterminate structure undergoes plastic deformations, the stresses do not, in general, return to zero after the load has been removed. The stresses remaining in the various parts of the structure are called residual stresses and may be determined by adding the maximum stresses reached during the loading phase and the reverse stresses corresponding to the unloading phase [Example 2.16].

Saint-Venant’s principle

Stress concentrations

D ⑀

A

C Rupture

␴ Y

␴

Y

Fig. 2.83

Plastic deformotions

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