The bolts used for the connections of this steel framework are subjected to stress. In

this chapter we will discuss how engineers design these connections and their

fasteners.

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Stress

CHAPTER OBJECTIVES

In this chapter we will review some of the important principles of statics and

show how they are used to determine the internal resultant loadings in a body.

Afterwards the concepts of normal and shear stress will be introduced, and

specific applications of the analysis and design of members subjected to an

axial load or direct shear will be discussed.

1.1 Introduction

Mechanics of materials is a branch of mechanics that studies the

relationships between the external loads applied to a deformable body

and the intensity of internal forces acting within the body. This subject

also involves computing the deformations of the body, and it provides a

study of the body’s stability when the body is subjected to external forces.

In the design of any structure or machine, it is first necessary to use the

principles of statics to determine the forces acting both on and within its

various members. The size of the members, their deflection, and their

stability depend not only on the internal loadings, but also on the type of

material from which the members are made. Consequently, an accurate

determination and fundamental understanding of material behavior will

be of vital importance for developing the necessary equations used in

mechanics of materials. Realize that many formulas and rules for design,

as defined in engineering codes and used in practice, are based on the

fundamentals of mechanics of materials, and for this reason an

understanding of the principles of this subject is very important.

3

1

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Historical Development.

The origin of mechanics of materials

dates back to the beginning of the seventeenth century, when Galileo

performed experiments to study the effects of loads on rods and beams

made of various materials. For a proper understanding, however, it was

necessary to establish accurate experimental descriptions of a material’s

mechanical properties.Methods for doing this were remarkably improved

at the beginning of the eighteenth century.At that time both experimental

and theoretical studies in this subject were undertaken primarily in

France by such notables as Saint-Venant, Poisson, Lamé, and Navier.

Becausetheireffortswerebasedonmaterial-bodyapplicationsof mechanics,

theycalledthis study“strengthof materials.”Currently,however,it is usually

referred to as “mechanics of deformable bodies” or simply “mechanics of

materials.”

Over the years,after many of the fundamental problems of mechanics

of materials had been solved,it became necessary to use advanced

mathematical and computer techniques to solve more complex problems.

As a result,this subject expandedintoother subjects of advancedmechan-

ics such as the theory of elasticity and the theory of plasticity.Research in

these fields is ongoing,not only to meet the demands for solving advanced

engineering problems,but to justify further use and the limitations upon

which the fundamental theory of mechanics of materials is based.

1.2 Equilibrium of a Deformable Body

Since statics plays an important role in both the development and

application of mechanics of materials, it is very important to have a good

grasp of its fundamentals. For this reason we will review some of the

main principles of statics that will be used throughout the text.

External Loads.

A body can be subjected to several different types

of external loads; however, any one of these can be classified as either a

surface force or a body force, Fig.1–1.

Surface Forces.

As the name implies,surface forces are caused by the

direct contact of one body with the surface of another.In all cases these

forces are distributed over the area of contact between the bodies.If this

area is small in comparison with the total surface area of the body,then

the surface force can be idealized as a single concentrated force,which is

applied to a point on the body.For example,the force of the ground on

the wheels of a bicycle can be considered as a concentrated force when

studying the loading on the bicycle.If the surface loading is applied along

a narrow area,the loading can be idealized as a linear distributed load,

Here the loading is measured as having an intensity of

along the area and is represented graphically by a series of arrows along

the line s.The resultant force of w(s) is equivalent to the area under

the distributed loading curve,and this resultant acts through the centroid

Cor geometric center of this area.The loading along the length of a beam

is a typical example of where this idealization is often applied.

F

R

force>lengthw1s2.

w(s)

Concentrated force

idealization

Linear distributed

load idealization

Surface

force

Body

force

s

C

G

F

R

W

Fig.1–1

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F

F

Type of connection Reaction

Cable

Roller

One unknown: F

One unknown: F

F

Smooth support One unknown: F

External pin

Internal pin

F

x

F

y

F

x

F

y

Two unknowns: F

x

, F

y

F

x

F

y

M

Fixed support Three unknowns: F

x

, F

y

, M

Two unknowns: F

x

, F

y

Type of connection Reaction

u

u

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Body Force.

A body force is developed when one body exerts a force

on another body without direct physical contact between the bodies.

Examples include the effects caused by the earth’s gravitation or its

electromagnetic field. Although body forces affect each of the particles

composing the body, these forces are normally represented by a single

concentrated force acting on the body. In the case of gravitation, this

force is called the weight of the body and acts through the body’s center

of gravity.

Support Reactions.

The surface forces that developat the supports

or points of contact between bodies are called reactions.For two-

dimensional problems,i.e.,bodies subjected to coplanar force systems,

the supports most commonly encountered are shown in Table 1–1.Note

carefully the symbol used to represent each support and the type of

reactions it exerts on its contacting member.In general,one can always

determine the type of support reaction by imagining the attached member

as being translated or rotated in a particular direction.If the support

prevents translationinagivendirection,thenaforce must be developedon

the member in that direction.Likewise,if rotation is prevented,a couple

moment must be exerted on the member.For example,a roller support can

only prevent translation in the contact direction,perpendicular or normal

to the surface.Hence,the roller exerts a normal force F on the member at

the point of contact.Since the member can freely rotate about the roller,a

couple moment cannot be developed on the member.

TABLE 1–1

Many machine elements are pin connected

in order to enable free rotation at their

connections. These supports exert a force

on a member, but no moment.

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Equations of Equilibrium.

Equilibrium of a body requires both

a balance of forces, to prevent the body from translating or having

accelerated motion along a straight or curved path, and a balance of

moments, to prevent the body from rotating. These conditions can be

expressed mathematically by the two vector equations

(1–1)

Here, represents the sum of all the forces acting on the body, and

is the sum of the moments of all the forces about any point O

either on or off the body. If an x,y,z coordinate system is established

with the origin at point O, the force and moment vectors can be resolved

into components along the coordinate axes and the above two equations

can be written in scalar form as six equations, namely,

(1–2)

Often in engineering practice the loading on a body can be

represented as a system of coplanar forces. If this is the case, and the

forces lie in the x–y plane, then the conditions for equilibrium of the

body can be specified by only three scalar equilibrium equations; that is,

(1–3)

In this case, if point O is the origin of coordinates, then moments will

always be directed along the z axis, which is perpendicular to the plane

that contains the forces.

Successful application of the equations of equilibrium requires

complete specification of all the known and unknown forces that act

on the body. The best way to account for these forces is to draw the

body’s free-body diagram. Obviously, if the free-body diagram is drawn

correctly, the effects of all the applied forces and couple moments can be

accounted for when the equations of equilibrium are written.

©M

O

= 0

©F

y

= 0

©F

x

= 0

©M

x

= 0 ©M

y

= 0 ©M

z

= 0

©F

x

= 0 ©F

y

= 0 ©F

z

= 0

©M

O

©F

©M

O

= 0

©F = 0

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section

F

4

F

2

(

a

)

F

1

F

3

*The body’s weight is not shown, since it is assumed to be quite small, and therefore

negligible compared with the other loads.

Internal Resultant Loadings.

One of the most important

applications of statics in the analysis of mechanics of materials problems

is to be able to determine the resultant force and moment acting within a

body, which are necessary to hold the body together when the body is

subjected to external loads. For example, consider the body shown in

Fig.1–2a, which is held in equilibrium by the four external forces.*

In order to obtain the internal loadings acting on a specific region within

the body, it is necessary to use the method of sections. This requires that

an imaginary section or “cut” be made through the region where the

internal loadings are to be determined. The two parts of the body are

then separated, and a free-body diagram of one of the parts is drawn,

Fig.1–2b. Here it can be seen that there is actually a distribution of

internal force acting on the “exposed” area of the section. These forces

represent the effects of the material of the top part of the body acting on

the adjacent material of the bottom part.

Although the exact distribution of internal loading may be unknown,

we can use the equations of equilibrium to relate the external forces on

the body to the distribution’s resultant force and moment, and

at any specific point O on the sectioned area, Fig.1–2c. When doing so,

note that acts through point O, although its computed value will not

depend on the location of this point. On the other hand, does

depend on this location, since the moment arms must extend from O to

the line of action of each external force on the free-body diagram. It will

be shown in later portions of the text that point Ois most often chosen at

the centroid of the sectioned area, and so we will always choose this

location for O, unless otherwise stated. Also, if a member is long and

slender, as in the case of a rod or beam, the section to be considered is

generally taken perpendicular to the longitudinal axis of the member.

This section is referred to as the cross section.

M

R

O

F

R

M

R

O

,F

R

F

1

F

2

(

b

)

Fig.1–2

F

R

F

1

F

2

O

M

R

O

(c)

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Three Dimensions.

Later in this text we will show how to relate

the resultant loadings, and to the distribution of force on the

sectioned area, and thereby develop equations that can be used for

analysis and design. To do this, however, the components of and

acting both normal or perpendicular to the sectioned area and within the

plane of the area, must be considered, Fig.1–2d. Four different types of

resultant loadings can then be defined as follows:

Normal force, N.

This force acts perpendicular to the area. It is

developed whenever the external loads tend to push or pull on the two

segments of the body.

Shear force, V.

The shear force lies in the plane of the area and is

developed when the external loads tend to cause the two segments of the

body to slide over one another.

Torsional moment or torque,T.

This effect is developed when the

external loads tend to twist one segment of the body with respect to

the other.

Bending moment,M.

The bending moment is causedby the external

loads that tend to bend the body about an axis lying within the plane of

the area.

In this text, note that graphical representation of a moment or torque

is shown in three dimensions as a vector with an associated curl. By the

right-hand rule, the thumb gives the arrowhead sense of the vector and

the fingers or curl indicate the tendency for rotation (twist or bending).

Using an x,y,z coordinate system, each of the above loadings can be

determined directly from the six equations of equilibrium applied to

either segment of the body.

M

R

O

,F

R

M

R

O

,F

R

(d)

O

F

1

F

2

N

T

M

V

Torsional

Moment

Bending

Moment

Shear

Force

M

R

O

F

R

Normal

Force

O

(c)

M

R

O

F

1

F

2

F

R

Fig.1–2 (cont.)

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Coplanar Loadings.

If the body is subjected to a coplanar system

of forces,Fig.1–3a,then only normal-force,shear-force,and bending-

moment components will exist at the section,Fig.1–3b.If we use the

x,y,z coordinate axes, with origin established at point O, as shown on the

left segment, then a direct solution for N can be obtained by applying

and V can be obtained directly from Finally, the

bending moment can be determined directly by summing moments

about point O(the z axis), in order to eliminate the moments

caused by the unknowns Nand V.

©M

O

= 0,

M

O

©F

y

= 0.©F

x

= 0,

section

F

4

F

3

F

2

F

1

(a)

Fig.1–3

O

V

M

O

N

x

y

Bending

Moment

Shear

Force

Normal

Force

(b)

F

2

F

1

In order to design the members of this

building frame, it is first necessary to find

the internal loadings at various points along

their length.

Important Points

•

Mechanics of materials is a study of the relationship between the

external loads on a body and the intensity of the internal loads

within the body.

•

External forces can be applied to a body as distributed or

concentrated surface loadings, or as body forces which act through-

out the volume of the body.

•

Linear distributed loadings produce a resultant force having a

magnitude equal to the area under the load diagram, and having a

location that passes through the centroid of this area.

•

A support produces a force in a particular direction on its

attached member if it prevents translation of the member in that

direction, and it produces a couple moment on the member if it

prevents rotation.

•

The equations of equilibrium and must be

satisfied in order to prevent a body from translating with accel-

erated motion and from rotating.

•

When applying the equations of equilibrium, it is important to

first draw the body’s free-body diagram in order to account for all

the terms in the equations.

•

The method of sections is used to determine the internal resultant

loadings acting on the surface of the sectioned body. In general,

these resultants consist of a normal force, shear force, torsional

moment, and bending moment.

©M = 0©F = 0

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The following examples illustrate this procedure numerically and also

provide a review of some of the important principles of statics.

Procedure for Analysis

The method of sections is used to determine the resultant internal

loadings at a point located on the section of a body. To obtain these

resultants, application of the method of sections requires the

following steps.

Support Reactions

•

First decide which segment of the body is to be considered. If

the segment has a support or connection to another body, then

before the body is sectioned, it will be necessary to determine the

reactions acting on the chosen segment of the body. Draw the free-

body diagram for the entire body and then apply the necessary

equations of equilibrium to obtain these reactions.

Free-Body Diagram

•

Keep all external distributed loadings, couple moments, torques,

and forces acting on the body in their exact locations, then pass an

imaginary section through the body at the point where the

resultant internal loadings are to be determined.

•

If the body represents a member of a structure or mechanical

device, the section is often taken perpendicular to the longitudinal

axis of the member.

•

Draw a free-body diagram of one of the “cut” segments and

indicate the unknown resultants N, V, M, and T at the section.

These resultants are normally placed at the point representing the

geometric center or centroid of the sectioned area.

•

If the member is subjected to a coplanar system of forces, only N,

V, and Mact at the centroid.

•

Establish the x,y,z coordinate axes with origin at the centroid

and show the resultant components acting along the axes.

Equations of Equilibrium

•

Moments should be summed at the section, about each of the

coordinate axes where the resultants act. Doing this eliminates

the unknown forces N and V and allows a direct solution for M

(and T).

•

If the solution of the equilibrium equations yields a negative

value for a resultant, the assumed directional sense of the

resultant is opposite to that shown on the free-body diagram.

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EXAMPLE 1.1

Determine the resultant internal loadings acting on the cross section

at C of the beam shown in Fig.1–4a.

(

a

)

A

B

C

3 m 6 m

270 N/m

Fig.1–4

180 N/m

540 N

2 m

4 m

V

C

M

C

N

C

(

b

)

BC

1.5 m

0.5 m

1 m

180 N/m

90 N/m

540 N

135 N

V

C

M

C

N

C

1215 N

3645 Nm

CA

(c)

SOLUTION

Support Reactions.This problem can be solved in the most direct

manner by considering segment CB of the beam, since then the

support reactions at Ado not have to be computed.

Free-Body Diagram.Passing an imaginary section perpendicular to

the

longitudinal axis of the beam yields the free-body diagram of

segment CBshown in Fig.1–4b.It is important to keep the distributed

loading exactly where it is on the segment until after the section is

made.Only then should this loading be replaced by a single resultant

force.Notice that the intensity of the distributed loading at C is

found by proportion,i.e.,from Fig. 1–4a,

The magnitude of the resultant of the distributed load is

equal to the area under the loading curve (triangle) and acts through

the centroid of this area.Thus,which

acts fromCas shown in Fig.1–4b.

Equations of Equilibrium.Applying the equations of equilibrium

we have

Ans.

Ans

.

Ans.

NOTE:The negative sign indicates that acts in the opposite

direction to that shown on the free-body diagram. T

ry solving this

problem using segment AC, by first obtaining the support reactions at

A, which are given in Fig.1–4c.

M

C

M

C

= -1080 N

#

m

-M

C

- 540 N 12 m2 = 0d+©M

C

= 0;

V

C

= 540 N

V

C

- 540 N = 0+

c

©F

y

= 0;

N

C

= 0

-N

C

= 0:

+

©F

x

= 0;

1>316 m2 = 2 m

F =

1

2

1180 N>m216 m2 = 540 N,

w = 180 N>m.

w>6 m = 1270 N>m2>9 m,

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EXAMPLE 1.2

Determine the resultant internal loadings acting on the cross section

at Cof the machine shaft shown in Fig.1–5a.The shaft is supported by

bearings at Aand B, which exert only vertical forces on the shaft.

SOLUTION

We will solve this problem using segment AC of the shaft.

Support Reactions.A free-body diagram of the entire shaft is

shown in Fig.

1–5b. Since segment AC is to be considered, only the

reaction at Ahas to be determined. Why?

The negative sign for indicates that acts in the opposite sense to

that shown on the free-body diagram.

Free-Body Diagram.Passing an imaginary section perpendicular

to

the axis of the shaft through C yields the free-body diagram of

segment AC shown in Fig.1–5c.

Equations of Equilibrium.

Ans

.

Ans.

Ans.

NOTE:The negative signs for and indicate they act in the

opposite directions on the free-body diagram. As an exercise, calculate

the reaction at B and try to obtain the same results using segment

CBDof the shaft.

M

C

V

C

M

C

= -5.69 N

#

m

M

C

+ 40 N10.025 m2 + 18.75 N10.250 m2 = 0d+©M

C

= 0;

V

C

= -58.8 N

-18.75 N - 40 N - V

C

= 0+

c

©F

y

= 0;

N

C

= 0:

+

©F

x

= 0;

A

y

A

y

A

y

= -18.75 N

-A

y

10.400 m2 + 120 N10.125 m2 - 225 N10.100 m2 = 0d+©M

B

= 0;

225 N

C

D

200 mm

100 mm 100 mm

50 mm50 mm

800 N/m

B

(

a

)

A

Fig.1–5

0.275 m

0.125 m

(800 N

/

m)(0.150 m) = 120 N

0.100 m

225 N

A

y

B

y

(b)

B

(c)

40 N

18.75 N

0.250 m

0.025 m

M

C

V

C

C

A

N

C

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EXAMPLE 1.3

The hoist in Fig.1–6a consists of the beam AB and attached pulleys,

the cable, and the motor. Determine the resultant internal loadings

acting on the cross section at Cif the motor is lifting the 500-lb load W

with constant velocity. Neglect the weight of the pulleys and beam.

SOLUTION

The most direct way to solve this problem is to section both the cable

and the beam at C and then consider the entire left segment.

Free-Body Diagram.See Fig.1–6b.

Equations of Equilibrium.

Ans

.

Ans.

Ans.

NOTE:As an exercise, try obtaining these same results by

considering just the beam segment AC, i.e., remove the pulley at A

from the beam and show the 500-lb force components of the pulley

acting on the beam segment AC.

M

C

= -2000 lb

#

ft

500 lb 14.5 ft2 - 500 lb 10.5 ft2 + M

C

= 0d+©M

C

= 0;

-500 lb - V

C

= 0

V

C

= -500 lb+

c

©F

y

= 0;

500 lb + N

C

= 0

N

C

= -500 lb:

+

©F

x

= 0;

4 ft

2 ft6 ft

0.5 ft

0.5 ft

A

C

D

B

W

(

a

)

(b)

4.5 ft

C

0.5 ft

500 lb

500 lb

V

C

N

C

M

C

A

Fig.1–6

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EXAMPLE 1.4

Determine the resultant internal loadings acting on the cross section

at G of the wooden beam shown in Fig.1–7a. Assume the joints at

A,B,C,D, and E are pin connected.

SOLUTION

Support Reactions.Here we will consider segment AG for the

analysis. A free-body diagram of the entire structure is shown in

Fig.

1–7b.Verify the computed reactions at Eand C.In particular,note

that BC is a two-force member since only two forces act on it. For this

reason the reaction at C must be horizontal as shown.

Since BA and BD are also two-force members, the free-body

diagram of joint B is shown in Fig.1–7c. Again, verify the magnitudes

of the computed forces and

Free-Body Diagram.Using the result for the left section AG

of the beam is shown in Fig.

1–7d.

Equations of Equilibrium.Applying the equations of equilibriumto

segment AG, we have

Ans.

Ans

.

Ans.M

G

= 6300 lb

#

ft

M

G

- 17750 lb2

A

3

5

B

12 ft2 + 1500 lb12 ft2 = 0d+©M

G

= 0;

V

G

= 3150 lb

-1500 lb + 7750 lb

A

3

5

B

- V

G

= 0+

c

©F

y

= 0;

7750 lb

A

4

5

B

+ N

G

= 0 N

G

= -6200 lb:

+

©F

x

= 0;

F

BA

,

F

BD

.F

BA

(

a

)

300 lb/ft

2 ft 2 ft 6 ft

1500 lb

A

B

G

D

C

3 ft

E

3 ft

6 ft

(6 ft) 4 ft

(6 ft)(300 lb/ft) 900 lb

1500 lb

E

y

2400 lb

E

x

6200 lb

F

BC

6200 lb

(

b

)

2

3

1

2

6200 lb

3

4

5

(c)

B

F

BA

7750 lb

F

BD

4650 lb

(d)

N

G

M

G

V

G

2 ft

3

4

5

7750 lb

1500 lb

A

G

Fig.1–7

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EXAMPLE 1.5

Determine the resultant internal loadings acting on the cross section

at Bof the pipe shown in Fig.1–8a. The pipe has a mass of and

is subjected to both a vertical force of 50 N and a couple moment of

at its end A. It is fixed to the wall at C.

SOLUTION

The problem can be solved by considering segment AB, which does

not involve the support reactions at C.

Free-Body Diagram.The x,y,z axes are established at B and the

free-body diagram of segment ABis shown in Fig.

1–8b. The resultant

force and moment components at the section are assumed to act in

the positive coordinate directions and to pass through the centroid of

the cross-sectional area at B. The weight of each segment of pipe is

calculated as follows:

These forces act through the center of gravity of each segment.

Equations of Equilibrium.Applying the six scalar equations of

equilibrium, we have*

Ans.

Ans

.

Ans.

Ans.

Ans.

Ans.

NOTE:What do the negative signs for and indicate?

Note that the normal force

whereas the shear

force is Also, the torsional

moment is and the bending moment is

M

B

= 2130.32

2

+ 102

= 30.3 N

#

m.

T

B

= 1M

B

2

y

= 77.8 N

#

m

V

B

= 2102

2

+ 184.32

2

= 84.3 N.

N

B

= 1F

B

2

y

= 0,

1M

B

2

y

1M

B

2

x

1M

B

2

z

= 0©1M

B

2

z

= 0;

1M

B

2

y

= -77.8 N

#

m

1M

B

2

y

+ 24.525 N 10.625 m2 + 50 N 11.25 m2 = 0©1M

B

2

y

= 0;

1M

B

2

x

= -30.3 N

#

m

- 9.81 N 10.25 m2 = 0

1M

B

2

x

+ 70 N

#

m - 50 N 10.5 m2 - 24.525 N 10.5 m2©1M

B

2

x

= 0;

1F

B

2

z

= 84.3 N

1F

B

2

z

- 9.81 N - 24.525 N - 50 N = 0©F

z

= 0;

1F

B

2

y

= 0©F

y

= 0;

1F

B

2

x

= 0©F

x

= 0;

W

AD

= 12 kg>m211.25 m219.81 N>kg2 = 24.525 N

W

BD

= 12 kg>m210.5 m219.81 N>kg2 = 9.81 N

70 N

#

m

2 kg>m

0.75 m

50 N

1.25 m

B

A

0.5 m

C

D

70 Nm

(a)

0.625 m

70 N∙m

(

b

)

y0.625 m

A

50 N

0.25 m

0.25 m

x

z

9.81 N

24.525 N

B

(F

B

)

z

(M

B

)

z

(M

B

)

y

(M

B

)

x

(F

B

)

y

(F

B

)

x

Fig.1–8

*The magnitude of each moment about an axis is equal to the magnitude of each

force times the perpendicular distance from the axis to the line of action of the force.

The direction of each moment is determined using the right-hand rule, with positive

moments (thumb) directed along the positive coordinate axes.

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 15

16

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PROBLEMS

1–1.Determine the resultant internal normal force

acting on the cross section through point A in each column.

In (a), segment BCweighs 180 lbft and segment CDweighs

250 lbft. In (b), the column has a mass of 200 kgm.

1–2.Determine the resultant internal torque acting on the

cross sections through points Cand Dof the shaft.The shaft

is fixed at B.

1–3.Determine the resultant internal torque acting on the

cross sections through points B and C.

*1–4.A force of 80 N is supported by the bracket as

shown. Determine the resultant internal loadings acting on

the section through point A.

3 ft

2 ft

2 ft

1 ft

B

A

C

500 lbf

t

350 lbft

600 lbft

Prob.1–3

3 kip

3 kip

5 kip

10 ft

4 ft

4 ft

8 in.8 in.

A

C

D

(a)

B

8 kN

3 m

1 m

6 kN6 kN

4.5 kN4.5 kN

200 mm200 mm

A

(b)

200 mm200 mm

Prob.1–1

200 mm

50 mm 50 mm

C

A

D

B

70 N m

150 mm300 mm

Prob.1–5

C

250 Nm

D

400 Nm

300 Nm

0.4 m

0.1 m

0.3 m

0.1 m

0.2 m

A

B

Prob.1–2

0.1 m

0.3 m

30

80 N

A

45

Prob.1–4

1–5.

Determine the resultant internal loadings acting on

the cross section through point Dof member AB.

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 16

P

ROBLEMS

17

1–6.The beam AB is pin supported at A and supported

by a cable BC. Determine the resultant internal loadings

acting on the cross section at point D.

1–7.Solve Prob. 1–6 for the resultant internal loadings

acting at point E.

1–9.The force acts on the gear tooth.

Determine the resultant internal loadings on the root of the

tooth, i.e., at the centroid point Aof section a–a.

F = 80 lb

1–10.The beam supports the distributed load shown.

Determine the resultant internal loadings on the cross

section through point C. Assume the reactions at the

supports Aand B are vertical.

1–11.The beam supports the distributed load shown.

Determine the resultant internal loadings on the cross

sections through points D and E. Assume the reactions at

the supports Aand B are vertical.

4 ft

B

C

6 ft

8 ft

3 ft

A

D

1200 lb

E

3 ft

Probs.1–6/7

*1–8.The boom DF of the jib crane and the column DE

have a uniform weight of 50 lbft. If the hoist and load

weigh 300 lb, determine the resultant internal loadings in

the crane on cross sections through points A, B, and C.

5 ft

7 ft

D F

C

B A

300 lb

2 ft 3 ft

E

8 ft

Prob.1–8

a

30

a

F 80 lb

0.23 in.

45

A

0.16 in.

Prob.1–9

6 ft

8 ft

C

A

B

300 lb/ft

4.5 ft

400 lb/ft

6 ft

4.5 ft

ED

Probs.1–10/11

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1–17.Determine the resultant internal loadings acting on

the cross section at point B.

M

4 ft

3 ft

4 ft

C B

1.5 ft

A

0.25 ft

4 ft

3 ft

D

Probs.1–15/16

*1–12.Determine the resultant internal loadings acting

on (a) section a–a and (b) section b–b. Each section is

located through the centroid, point C.

1–15.The 800-lb load is being hoisted at a constant speed

using the motor M, which has a weight of 90 lb. Determine

the resultant internal loadings acting on the cross section

through point B in the beam. The beam has a weight of

40 lbft and is fixed to the wall at A.

*1–16.Determine the resultant internal loadings acting

on the cross section through points C and Dof the beam in

Prob. 1–15.

45

8 ft

4 ft

45

A

C

B

b

a

a b

600 lb/ft

Prob.1–12

1–13.Determine the resultant internal normal and shear

forces in the member at (a) section a–a and (b) section b–b,

each of which passes through point A. Take The

650-Nloadis appliedalongthecentroidal axis of themember.

1–14.Determine the resultant internal normal and shear

forces in the member at section b–b, each as a function of

Plot these results for The 650-N load is

applied along the centroidal axis of the member.

0° … u … 90°.

u.

u = 60°.

A

b

a

u

b

a

650 N

650 N

Probs.1–13/14

A

C

12 ft3 ft

60 lb/

ft

B

Prob.1–17

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 18

P

ROBLEMS

19

1–18.The beam supports the distributed load shown.

Determine the resultant internal loadings acting on the

cross section through point C. Assume the reactions at the

supports Aand B are vertical.

1–19.Determine the resultant internal loadings acting on

the cross section through point Din Prob. 1–18.

3 m

3 m

DC

A B

0.5 kN/m

1.5 kN/m

3 m

Probs.1–18/19

*1–20.The wishbone construction of the power pole

supports the three lines, each exerting a force of 800 lb on

the bracing struts. If the struts are pin connected at A, B,

and C, determine the resultant internal loadings at cross

sections through points D, E, and F.

4 ft

D

E

A

B

F

C

4 ft

6 ft

6 ft

800 lb

800 lb

800 lb

Prob.1–20

1–21.The drum lifter suspends the 500-lb drum. The linkage

is pin connected to the plate at A and B. The gripping action

on the drum chime is such that only horizontal and vertical

forces are exerted on the drum at G and H. Determine the

resultant internal loadings on the cross section through point I.

1–22.Determine the resultant internal loadings on the

cross sections through points K and J on the drum lifter in

Prob. 1–21.

8 in.

5 in.

3 in.

C

A B

E

D

F

I

K

60

60

2 in.

5 in.

J

HG

5 in.

500 lb

1–23.The pipe has a mass of 12 kgm. If it is fixed to the

wall at A, determine the resultant internal loadings acting on

the cross section at B. Neglect the weight of the wrench CD.

300 mm

200 mm

150 mm

60 N

60 N

400 mm

150 mm

B

A

x

y

z

C

D

Prob.1–23

Probs.1–21/22

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1–25.Determine the resultant internal loadings acting on

the cross section through point B of the signpost. The post

is fixed to the ground and a uniform pressure of acts

perpendicular to the face of the sign.

7 lb>ft

2

1–26.The shaft is supported at its ends by two bearings A

and B and is subjected to the forces applied to the pulleys

fixed to the shaft. Determine the resultant internal loadings

acting on the cross section through point D. The 400-N

forces act in the direction and the 200-N and 80-N forces

act in the direction. The journal bearings at A and B

exert only y and z components of force on the shaft.

+y

-z

*1–24.The main beam AB supports the load on the wing

of the airplane.The loads consist of the wheel reaction of

35,000 lb at C,the 1200-lb weight of fuel in the tank of the

wing,having a center of gravity at D,and the 400-lb weight of

the wing,having a center of gravity at E.If it is fixed to the

fuselage at A,determine the resultant internal loadings on

the beamat this point.Assume that the wing does not transfer

any of the loads to the fuselage,except through the beam.

6 ft

4 ft

2 ft

1.5 ft

D E

1 ft

A

B

C

35,000 lb

z

x

y

Prob.1–24

4 ft

z

y

6 ft

x

B

A

3 ft

2 ft

3 ft

7 lb/ft

2

Prob.1–25

B

C

200 mm

200 mm

300 mm

A

200 N

200 N

400 N

400 N

150 mm

400 mm

80 N

80 N

z

x

y

150 mm

D

Prob.1–27

B

C

200 mm

200 mm

300 mm

A

200 N

200 N

400 N

400 N

150 mm

400 mm

80 N

80 N

z

x

y

150 mm

D

Prob.1–26

1–27.

The shaft is supported at its ends by two bearings A

and B and is subjected to the forces applied to the pulleys

fixed to the shaft. Determine the resultant internal loadings

acting on the cross section through point C. The 400-N

forces act in the direction and the 200-N and 80-N forces

act in the direction. The journal bearings at A and B

exert only y and z components of force on the shaft.

+y

-z

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 20

A B

C

90

6 in.

P

ROBLEMS

21

1–29.The bolt shank is subjected to a tension of 80 lb.

Determine the resultant internal loadings acting on the

cross section at point C.

1–30.The pipe has a mass of . If it is fixed to the

wall at A, determine the resultant internal loadings acting

on the cross section through B.

12 kg>m

*1–28.Determine the resultant internal loadings acting

on the cross section of the frame at points F and G. The

contact at E is smooth.

1–31.The curved rod has a radius r and is fixed to the wall

at B. Determine the resultant internal loadings acting on

the cross section through A which is located at an angle

from the horizontal.

u

4 ft

1.5 ft

1.5 ft

3 ft

A

B

C

E

D

G

80 lb

5 ft

2 ft

2 ft

30

F

Prob.1–28

Prob.1–29

M

V

N

d

u

M dM

T dT

N dN

V dV

T

Prob.1–33

1 m

2 m

2 m

B

A

y

z

x

C

800 Nm

3

4

5

750 N

Prob.1–30

r

A

P

u

B

A

B

C

45

90

D

O

r

22.5

Prob.1–32

*1–32.

The curved rod AD of radius r has a weight per

length of If it lies in the horizontal plane, determine the

resultant internal loadings acting on the cross section

through point B. Hint: The distance from the centroid C of

segment AB to point Ois CO = 0.9745r.

w.

1–33.A differential element taken from a curved bar is

shown in the figure. Show that

and dT>du = M.dM>du = -T,

dV>du = -N,dN>du = V,

Prob.1–31

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1.3 Stress

It was stated in Section 1.2 that the force and moment acting at a

specified point on the sectioned area of a body, Fig.1–9, represents the

resultant effects of the actual distribution of force acting over the

sectioned area, Fig. 1–10a. Obtaining this distribution of internal loading

is of primary importance in mechanics of materials. To solve this problem

it is necessary to establish the concept of stress.

Consider the sectioned area to be subdivided into small areas, such as

shown dark shaded in Fig.1–10a. As we reduce to a smaller and

smaller size, we must make two assumptions regarding the properties of

the material. We will consider the material to be continuous, that is, to

consist of a continuumor uniform distribution of matter having no voids,

rather than being composed of a finite number of distinct atoms or

molecules. Furthermore, the material must be cohesive, meaning that all

portions of it are connected together, rather than having breaks, cracks,

or separations. A typical finite yet very small force acting on its

associated area is shown in Fig. 1–10a. This force, like all the others,

will have a unique direction, but for further discussion we will replace it

by its three components, namely, and which are taken

tangent, and normal to the area, respectively. As the area approaches

zero, so do the force and its components; however, the quotient of the

force and area will, in general, approach a finite limit. This quotient is

called stress, and as noted, it describes the intensity of the internal force on

a specific plane (area) passing through a point.

¢F

¢A

¢F

z

,¢F

y

,¢F

x

,

¢A,

¢F,

¢A¢A

F

1

F

2

F

1

F

A

F

F

z

z

y

x

F

x

F

y

z

(c)

x

y

(b)

zz

x

y

(a)

x

y

t

yz

s

y

t

yx

t

xz

s

x

t

xy

Fig.1–10

F

1

F

2

O

M

R

O

F

R

Fig.1–9

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 22

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1.3 S

TRESS

23

Normal Stress.

The intensity of force, or force per unit area, acting

normal to is defined as the normal stress, (sigma). Since is

normal to the area then

(1–4)

If the normal force or stress “pulls” on the area element as shown in

Fig. 1–10a, it is referred to as tensile stress, whereas if it “pushes” on

it is called compressive stress.

Shear Stress.

The intensity of force, or force per unit area, acting

tangent to is called the shear stress, (tau). Here we have shear

stress components,

Note that the subscript notation z in is used to reference the

direction of the outward normal line, which specifies the orientation of

the area Fig.1–11. Two subscripts are used for the shear-stress

components, and The z axis specifies the orientation of the area,

and x and y refer to the direction lines for the shear stresses.

General State of Stress.

If the body is further sectioned by

planes parallel to the x–z plane, Fig. 1–10b, and the y–z plane, Fig. 1–10c,

we can then “cut out” a cubic volume element of material that

represents the state of stress acting around the chosen point in the body,

Fig.1–12. This state of stress is then characterized by three components

acting on each face of the element. These stress components describe

the state of stress at the point only for the element orientated along the

x,y,z axes. Had the body been sectioned into a cube having some other

orientation, then the state of stress would be defined using a different set

of stress components.

Units.

In the International Standard or SI system, the magnitudes of

both normal and shear stress are specified in the basic units of newtons

per square meter This unit, called a pascal is

rather small, and in engineering work prefixes such as kilo-

symbolized by k, mega- symbolized by M, or giga-

symbolized by G, are used to represent larger, more realistic values of

stress.* Likewise, in the U.S. Customary or Foot-Pound-Second system

of units, engineers usually express stress in pounds per square inch (psi)

or kilopounds per square inch (ksi), where 1 kilopound 1kip2 = 1000 lb.

110

9

2,110

6

2,

110

3

2,

11 Pa = 1 N>m

2

21N>m

2

2.

t

zy

.t

zx

¢A,

s

z

t

zy

= lim

¢A:0

¢F

y

¢A

t

zx

= lim

¢A:0

¢F

x

¢A

t¢A

¢A

¢A

s

z

= lim

¢A:0

¢F

z

¢A

¢F

z

s¢A

*Sometimes stress is expressed in units of where However, in

the SI system, prefixes are not allowed in the denominator of a fraction and therefore it is

better to use the equivalent 1 N>mm

2

= 1 MN>m

2

= 1 MPa.

1 mm = 10

-3

m.N>mm

2

,

x

y

z

T

zx

T

zy

s

z

Fig.1–11

x

y

z

z

zx

zy

yz

yx

xz

x

xy

y

s

s s

t

t

t

t

t

t

Fig.1–12

(1–5)

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1.4 Average Normal Stress

in an Axially Loaded Bar

Frequently structural or mechanical members are made long and

slender. Also, they are subjected to axial loads that are usually applied to

the ends of the member. Truss members, hangers, and bolts are typical

examples. In this section we will determine the average stress

distribution acting on the cross section of an axially loaded bar, such as

the one having the general form shown in Fig.1–13a. This section

defines the cross-sectional area of the bar, and since all such cross

sections are the same, the bar is referred to as being prismatic. If we

neglect the weight of the bar and section it as indicated, then, for

equilibrium of the bottom segment, Fig.1–13b, the internal resultant

force acting on the cross-sectional area must be equal in magnitude,

opposite in direction, and collinear to the external force acting at the

bottom of the bar.

Assumptions.

Before we determine the average stress distribution

acting over the bar’s cross-sectional area, it is necessary to make two

simplifying assumptions concerning the material description and the

specific application of the load.

1.It is necessary that the bar remains straight both before and after

the load is applied, and also, the cross section should remain flat or

plane during the deformation, that is, during the time the bar

changes its volume and shape. If this occurs, then horizontal and

vertical grid lines inscribed on the bar will deform uniformly when

the bar is subjected to the load, Fig.1–13c. Here we will not

consider regions of the bar near its ends, where application of the

external loads can cause localized distortions. Instead we will focus

only on the stress distribution within the bar’s midsection.

2.In order for the bar to undergo uniformdeformation,it is necessary

that P be applied along the centroidal axis of the cross section,and

the material must be homogeneous and isotropic.Homogeneous

material has the same physical and mechanical properties

throughout its volume,and isotropic material has these same

properties in all directions.Many engineering materials may be

approximated as being both homogeneous and isotropic as

assumed here.Steel,for example,contains thousands of randomly

oriented crystals in each cubic millimeter of its volume,and since

most problems involving this material have a physical size that is

much larger than a single crystal,the above assumption regarding

its material composition is quite realistic.It should be mentioned,

however,that steel can be made anisotropic by cold-rolling,(i.e.,

rolling or forging it at subcritical temperatures).Anisotropic

materials have different properties in different directions,and

although this is the case,if the anisotropy is oriented along the bar’s

P

P

(

a

)

P

P

External force

Cross-sectional

area

Internal force

(b)

(c)

P

P

Region of

uniform

deformation

of bar

Fig.1–13

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 24

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VERAGE

N

ORMAL

S

TRESS I N AN

A

XI ALLY

L

OADED

B

AR

25

axis,then the bar will also deform uniformly when subjected to an

axial load.For example,timber,due to its grains or fibers of wood,

is an engineering material that is homogeneous and anisotropic,

and due to the customary orientation of its fibers it is suited for the

following analysis.

Average Normal Stress Distribution.

Provided the bar is

subjected to a constant uniform deformation as noted, then this

deformation is the result of a constant normal stress Fig.1–13d. As a

result, each area on the cross section is subjected to a force

and the sum of these forces acting over the entire cross-

sectional area must be equivalent to the internal resultant force P at the

section. If we let and therefore then, recognizing

is constant, we have

(1–6)

Here

normal stress at any point on the cross-sectional area

resultant normal force, which is applied through the

centroid of the cross-sectional area. P is determined using the

method of sections and the equations of equilibrium

area of the bar

The internal load Pmust pass through the centroid of the cross-section

since the uniform stress distribution will produce zero moments about

any x and y axes passing through this point, Fig. 1–13d. When this occurs,

These equations are indeed satisfied, since by definition of the centroid,

and (See Appendix A.)

Equilibrium.

It should be apparent that only a normal stress exists

on any volume element of material located at each point on the cross

section of an axially loaded bar. If we consider vertical equilibrium of the

element, Fig.1–14, then applying the equation of force equilibrium,

s = s¿

s1¢A2 - s¿1¢A2 = 0©F

z

= 0;

1

x dA = 0.

1

y dA = 0

0 = -

L

A

x dF = -

L

A

xs dA = -s

L

A

x dA1M

R

2

y

= ©M

y

;

0 =

L

A

y dF =

L

A

ys dA = s

L

A

y dA1M

R

2

x

= ©M

x

;

A = cross-sectional

P = internal

s = average

s =

P

A

P = sA

L

dF =

L

A

s dA+

c

F

Rz

= ©F

z

;

s¢F:dF,¢A:dA

¢F = s ¢A,

¢A

s,

(d)

P

F sA

P

y

x

x

z

y

A

s

Fig.1–13 (cont.)

¿

A

s

s

Fig.1–14

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 25

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In other words, the two normal stress components on the element must

be equal in magnitude but opposite in direction. This is referred to as

uniaxial stress.

The previous analysis applies to members subjected to either tension

or compression, as shown in Fig.1–15. As a graphical interpretation, the

magnitude of the internal resultant force P is equivalent to the volume

under the stress diagram; that is,

Furthermore, as a consequence of the balance of moments, this resultant

passes through the centroid of this volume.

Although we have developed this analysis for prismatic bars, this

assumption can be relaxed somewhat to include bars that have a slight

taper. For example, it can be shown, using the more exact analysis of the

theory of elasticity, that for a tapered bar of rectangular cross section, for

which the angle between two adjacent sides is 15°, the average normal

stress, as calculated by is only 2.2% less than its value found

from the theory of elasticity.

Maximum Average Normal Stress.

In our analysis both the

internal force P and the cross-sectional area A were constant along the

longitudinal axis of the bar, and as a result the normal stress is

also constant throughout the bar’s length. Occasionally, however, the bar

may be subjected to several external loads along its axis, or a change in

its cross-sectional area may occur. As a result, the normal stress within

the bar could be different from one section to the next, and, if the

maximum average normal stress is to be determined, then it becomes

important to find the location where the ratio PAis a maximum. To do

this it is necessary to determine the internal force P at various sections

along the bar. Here it may be helpful to show this variation by drawing

an axial or normal force diagram. Specifically, this diagram is a plot of

the normal force P versus its position x along the bar’s length. As a sign

convention, P will be positive if it causes tension in the member, and

negative if it causes compression. Once the internal loading throughout

the bar is known, the maximum ratio of PAcan then be identified.

s = P>A

s = P>A,

P = sA 1volume = height * base2.

This steel tie rod is used to suspend a portion

of a staircase, and as a result it is subjected to

tensile stress.

P

P

P

P

Tension Compression

s

s

s

s

P

—

A

Fig.1–15

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Important Points

•

When a body that is subjected to an external load is sectioned,

there is a distribution of force acting over the sectioned area which

holds each segment of the body in equilibrium. The intensity of

this internal force at a point in the body is referred to as stress.

•

Stress is the limiting value of force per unit area, as the area

approaches zero. For this definition, the material at the point is

considered to be continuous and cohesive.

•

The magnitude of the stress components depends upon the type of

loading acting on the body, and the orientation of the element at

the point.

•

When a prismatic bar is made from homogeneous and isotropic

material, and is subjected to axial force acting through the

centroid of the cross-sectioned area, then the material within the

bar is subjected only to normal stress. This stress is assumed to be

uniform or averaged over the cross-sectional area.

Procedure for Analysis

The equation gives the average normal stress on the cross-

sectional area of a member when the section is subjected to an

internal resultant normal force P. For axially loaded members,

application of this equation requires the following steps.

Internal Loading

•

Section the member perpendicular to its longitudinal axis at the

point where the normal stress is to be determined and use the

necessary free-body diagram and equation of force equilibrium to

obtain the internal axial force P at the section.

Average Normal Stress

•

Determine the member’s cross-sectional area at the section and

compute the average normal stress

•

It is suggested that be shown acting on a small volume element

of the material located at a point on the section where stress is

calculated. To do this, first draw on the face of the element

coincident with the sectioned area A. Here acts in the same

direction as the internal force P since all the normal stresses on

the cross section act in this direction to develop this resultant. The

normal stress acting on the opposite face of the element can be

drawn in its appropriate direction.

s

s

s

s

s = P>A.

s = P>A

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EXAMPLE 1.6

The bar in Fig.1–16a has a constant width of 35 mm and a thickness of

10 mm. Determine the maximum average normal stress in the bar

when it is subjected to the loading shown.

(b)

9 kN

9 kN

12 kN

12 kN

P

AB

12 kN

P

BC

30 kN

P

CD

22 kN 22 kN

P(kN)

x

12

22

30

(c)

12 kN

22 k

N

9 kN

9 kN

4 kN

4 kN

35 mm

A D

B C

(a)

(

d

)

30 k

N

85.7 MPa

35 mm

10 mm

Fig.1–16

SOLUTION

Internal Loading.By inspection, the internal axial forces in regions

AB,BC, and CDare all constant yet have different magnitudes. Using

the method of sections, these loadings are determined in Fig. 1–16b;

and the normal force diagram which represents these results

graphically is shown in Fig. 1–16c. By inspection, the largest loading is

in region BC, where

Since the cross-sectional area of

the bar is constant, the largest average normal stress also occurs

within this region of the bar.

Average Normal Stress.Applying Eq. 1–6, we have

Ans.

NOTE:The stress distribution acting on an arbitrary cross section of

the

bar withinregionBCis showninFig.1–16d.Graphically the volume

(or “block”) represented by this distribution of stress is equivalent to

the load of 30 kN;that is,30 kN = 185.7 MPa2135 mm2110 mm2.

s

BC

=

P

BC

A

=

30110

3

2N

10.035 m210.010 m2

= 85.7 MPa

P

BC

= 30 kN.

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EXAMPLE 1.7

The 80-kg lamp is supported by two rods AB and BC as shown in

Fig.1–17a. If AB has a diameter of 10 mm and BC has a diameter of

8 mm, determine the average normal stress in each rod.

SOLUTION

Inter

nal Loading.We must first determine the axial force in each

rod. A free-body diagram of the lamp is shown in Fig.

1–17b. Applying

the equations of force equilibrium yields

By Newton’s third law of action, equal but opposite reaction, these

forces subject the rods to tension throughout their length.

Average Normal Stress.Applying Eq. 1–6, we have

Ans.

Ans

.

NOTE:The average normal stress distribution acting over a cross

section of rod AB is shown in Fig.

1–17c, and at a point on this cross

section, an element of material is stressed as shown in Fig. 1–17d.

s

BA

=

F

BA

A

BA

=

632.4 N

p10.005 m2

2

= 8.05 MPa

s

BC

=

F

BC

A

BC

=

395.2 N

p10.004 m2

2

= 7.86 MPa

F

BA

= 632.4 NF

BC

= 395.2 N,

F

BC

A

3

5

B

+ F

BA

sin 60° - 784.8 N = 0+

c

©F

y

= 0;

F

BC

A

4

5

B

- F

BA

cos 60° = 0:

+

©F

x

= 0;

A

60

B

C

3

4

5

(a)

Fig.1–17

(

b

)

60

F

BA

F

BC

y

x

80(9.81) 784.8 N

B

3

4

5

632.4 N

8.05 MPa

8.05 MPa

(c)(d)

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EXAMPLE 1.8

The casting shown in Fig.1–18a is made of steel having a specific

weight of Determine the average compressive stress

acting at points Aand B.

g

st

= 490 lb>ft

3

.

SOLUTION

Internal Loading.A free-body diagram of the top segment of the

casting where the section passes through points A and B is shown

in

Fig.1–18b. The weight of this segment is determined from

Thus the internal axial force P at the section is

Average Compressive Stress.The cross-sectional area at the

section is

and so the average compressive stress

becomes

Ans.

NOTE:The stress shown on the volume element of material in

Fig.

1–18c is representative of the conditions at either point A or B.

Notice that this stress acts upward on the bottomor shaded face of the

element since this face forms part of the bottomsurface area of the cut

section,and on this surface,the resultant internal force P is pushing

upward.

= 9.36 psi

= 1347.5 lb>ft

2

= 1347.5 lb>ft

2

11 ft

2

>144 in

2

2

s =

P

A

=

2381 lb

p10.75 ft2

2

A = p10.75 ft2

2

,

P = 2381 lb

P - 1490 lb>ft

3

212.75 ft2 p10.75 ft2

2

= 0

P - W

st

= 0+

c

©F

z

= 0;

W

st

= g

st

V

st

.

0.75 ft

0.75 ft

2.75 ft

y

z

x

(a)

A

B0.75 ft

0.4 ft

Fig.1–18

2.75 ft

(b)

A

P

(c)

9.36 ps

i

B

W

st

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EXAMPLE 1.9

Member ACshown in Fig.1–19a is subjected to a vertical force of 3 kN.

Determine the position x of this force so that the average compressive

stress at the smooth support C is equal to the average tensile stress in

the tie rod AB.The rod has a cross-sectional area of and the

contact area at Cis 650 mm

2

.

400 mm

2

SOLUTION

Internal Loading.The forces at A and C can be related by

considering the free-body diagram for member AC, Fig.

1–19b. There

are three unknowns, namely, and x. To solve this problem

we will work in units of newtons and millimeters.

(1)

(2)

Average Normal Stress.A necessary third equation can be written

that requires the tensile stress in the bar AB and the compressive

stress at C to be equivalent, i.e.,

Substituting this into Eq. 1, solving for

then solving for we

obtain

The position of the applied load is determined from Eq. 2,

Ans.

NOTE:as required.0 6 x 6 200 mm,

x = 124 mm

F

C

= 1857 N

F

AB

= 1143 N

F

C

,F

AB

,

F

C

= 1.625F

AB

s =

F

AB

400 mm

2

=

F

C

650 mm

2

-3000 N1x2 + F

C

1200 mm2 = 0d+©M

A

= 0;

F

AB

+ F

C

- 3000 N = 0+

c

©F

y

= 0;

F

C

,F

AB

,

x

A

B

C

200 mm

(a)

3 kN

Fig.1–19

(b)

x

3 kN

A

200 mm

F

AB

F

C

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1.5 Average Shear Stress

Shear stress has been defined in Section 1.3 as the stress component that

acts in the plane of the sectioned area. In order to show how this stress

can develop, we will consider the effect of applying a force F to the bar in

Fig.1–20a. If the supports are considered rigid, and F is large enough, it

will cause the material of the bar to deform and fail along the planes

identified by AB and CD. A free-body diagram of the unsupported

center segment of the bar, Fig. 1–20b, indicates that the shear force

must be applied at each section to hold the segment in

equilibrium.Theaverage shear stress distributedover eachsectionedarea

that develops this shear force is defined by

(1–7)

Here

average shear stress at the section, which is assumed to be the

same at each point located on the section

internal resultant shear force at the section determined from

the equations of equilibrium

area at the section

The distribution of average shear stress is shown acting over the

sections in Fig. 1–20c. Notice that is in the same direction as V, since

the shear stress must create associated forces all of which contribute to

the internal resultant force Vat the section.

The loading case discussed in Fig. 1–20 is an example of simple or

direct shear, since the shear is caused by the direct action of the applied

load F. This type of shear often occurs in various types of simple

connections that use bolts, pins, welding material, etc. In all these cases,

however, application of Eq. 1–7 is only approximate. A more precise

investigation of the shear-stress distribution over the critical section

often reveals that much larger shear stresses occur in the material than

those predicted by this equation. Although this may be the case,

application of Eq. 1–7 is generally acceptable for many problems in

engineering design and analysis. For example, engineering codes allow

its use when considering design sizes for fasteners such as bolts and for

obtaining the bonding strength of joints subjected to shear loadings. In

this regard, two types of shear frequently occur in practice, which

deserve separate treatment.

t

avg

A =

V =

t

avg

=

t

avg

=

V

A

V = F>2

F

(a)

B

D

A

C

(b)

F

V

V

(c)

F

t

avg

Fig.1–20

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Single Shear.

The steel and wood joints shown in Figs.1–21a and

1–21c, respectively, are examples of single-shear connections and are

often referred to as lap joints. Here we will assume that the members are

thin and that the nut in Fig. 1–21a is not tightened to any great extent so

friction between the members can be neglected. Passing a section

between the members yields the free-body diagrams shown in

Figs.1–21b and 1–21d. Since the members are thin, we can neglect the

moment created by the force F. Hence for equilibrium, the cross-

sectional area of the bolt in Fig. 1–21b and the bonding surface between

the members in Fig. 1–21d are subjected only to a single shear force

This force is used in Eq. 1–7 to determine the average shear

stress acting on the colored section of Fig. 1–21d.

Double Shear.

When the joint is constructed as shown in Fig.1–22a

or 1–22c, two shear surfaces must be considered. These types of

connections are often called double lap joints. If we pass a section

between each of the members, the free-body diagrams of the center

member are shown in Figs. 1–22b and 1–22d. Here we have a condition

of double shear. Consequently, acts on each sectioned area and

this shear must be considered when applying t

avg

= V>A.

V = F>2

V = F.

F

F

(

a

)

F

(

b

)

V F

V F

F

(

c

)

F

(

d

)

F

Fig.1–21

The pin on this tractor is subjected to double

shear.

F

(

d

)

F

(

c

)

F

(

b

)

V

V

F

(

a

)

F

2

F

2

F

2

F

2

V

V

F

2

F

2

F

2

F

2

Fig.1–22

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Equilibrium.

Consider a volume element of material taken at a

point located on the surface of any sectioned area on which the average

shear stress acts, Fig. 1–23a. If we consider force equilibrium in the y

direction, then

force

stress area

And in a similar manner, force equilibrium in the z direction yields

Finally, taking moments about the x axis,

moment

force arm

stress area

so that

In other words, force and moment equilibrium requires the shear stress

acting on the top face of the element, to be accompanied by shear stress

acting on three other faces, Fig. 1–23b. Here all four shear stresses must

have equal magnitude and be directed either toward or away from each

other at opposite edges of the element. This is referred to as the

complementary property of shear, and under the conditions shown in

Fig.1–23, the material is subjected to pure shear.

Although we have considered here a case of simple shear as caused by

the direct action of a load, in later chapters we will show that shear stress

can also arise indirectly due to the action of other types of loading.

t

zy

= t

œ

zy

= t

yz

= t

œ

yz

= t

t

zy

= t

yz

-t

zy

1¢x ¢y2 ¢z + t

yz

1¢x ¢z2 ¢y = 0©M

x

= 0;

t

yz

= t

œ

yz

.

t

zy

= t

œ

zy

t

zy

1¢x ¢y2 - t

œ

zy

¢x ¢y = 0©F

y

= 0;

Pure shear

(a) (b)

Section plane

x

y

z

y

z

x

t¿

zy

t

zy

t¿

yz

t

yz

t

t

t

t

Fig.1–23

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Procedure for Analysis

The equation is used to compute only the average shear

stress in the material. Application requires the following steps.

Internal Shear

•

Section the member at the point where the average shear stress is

to be determined.

•

Draw the necessary free-body diagram, and calculate the internal

shear force V acting at the section that is necessary to hold the

part in equilibrium.

Average Shear Stress

•

Determine the sectioned area A, and compute the average shear

stress

•

It is suggested that be shown on a small volume element of

material located at a point on the section where it is determined.

To do this, first draw on the face of the element, coincident

with the sectioned area A. This shear stress acts in the same

direction as V. The shear stresses acting on the three adjacent

planes can then be drawn in their appropriate directions

following the scheme shown in Fig. 1–23.

t

avg

t

avg

t

avg

= V>A.

t

avg

= V>A

Important Points

•

If two parts which are thin or small are joined together, the

applied loads can cause shearing of the material with negligible

bending. If this is the case, it is generally suitable for engineering

analysis to assume that an average shear stress acts over the cross-

sectional area.

•

Oftentimes fasteners, such as nails and bolts, are subjected to

shear loads. The magnitude of a shear force on the fastener is

greatest along a plane which passes through the surfaces being

joined. A carefully drawn free-body diagram of a segment of the

fastener will enable one to obtain the magnitude and direction of

this force.

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EXAMPLE 1.10

The bar shown in Fig.1–24a has a square cross section for which the

depth and thickness are 40 mm. If an axial force of 800 N is applied

along the centroidal axis of the bar’s cross-sectional area, determine

the average normal stress and average shear stress acting on the

material along (a) section plane a–a and (b) section plane b–b.

SOLUTION

Part (a)

Inter

nal Loading.The bar is sectioned, Fig.1–24b, and the internal

resultant loading consists only of an axial force for which

Average Stress.The average normal stress is determined from

Eq. 1–6.

Ans.

No shear stress exists on the section, since the shear force at the

section is zero.

Ans.

NOTE:The distribution of average normal stress over the cross

section is shown in Fig. 1–24c.

t

avg

= 0

s =

P

A

=

800 N

10.04 m210.04 m2

= 500 kPa

P = 800 N.

a

a

b

b

800 N

20 mm

60

(a)

20 mm

(b)

800 N

P 800 N

(c)

500 kPa

500 kPa

Fig.1–24

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Part (b)

Inter

nal Loading.If the bar is sectioned along b–b, the free-body

diagram of the left segment is shown in Fig.

1–24d. Here both a

normal force (N) and shear force (V) act on the sectioned area. Using

x, y axes, we require

or, more directly, using axes,

Solving either set of equations,

Average Stresses.In this case the sectioned area has a thickness

and depth of 40 mm and

respectively,

Fig. 1–24a. Thus the average normal stress is

Ans.

and the average shear stress is

Ans.

NOTE:The stress distribution is shown in Fig.1–24e.

t

avg

=

V

A

=

400 N

10.04 m210.04619 m2

= 217 kPa

s =

N

A

=

692.8 N

10.04 m210.04619 m2

= 375 kPa

40 mm>sin 60° = 46.19 mm,

V = 400 N

N = 692.8 N

V - 800 N sin 30° = 0+Q©F

y¿

= 0;

N - 800 N cos 30° = 0+R©F

x¿

= 0;

y¿x¿,

V sin 60° - N cos 60° = 0+

c

©F

y

= 0;

-800 N + N sin 60° + V cos 60° = 0:

+

©F

x

= 0;

V

800 N

60

(

d

)

30

y

y¿

x¿

x

30

60

N

800

N

(e)

375 kPa

217 kPa

375 kPa

Fig.1–24 (cont.)

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EXAMPLE 1.11

The wooden strut shown in Fig.1–25a is suspended from a 10-mm-

diameter steel rod, which is fastened to the wall. If the strut supports

a vertical load of 5 kN, compute the average shear stress in the rod at

the wall and along the two shaded planes of the strut, one of which is

indicated as abcd.

SOLUTION

Inter

nal Shear.As shown on the free-body diagram in Fig.1–25b,

the rod resists a shear force of 5 kN where it is fastened to the wall. A

free-body diagram of the sectioned segment of the strut that is in

contact with the rod is shown in Fig.

1–25c. Here the shear force

acting along each shaded plane is 2.5 kN.

Average Shear Stress.For the rod,

Ans.

For the strut,

Ans.

NOTE:The average-shear-stress distribution on the sectioned rod

and strut segment is shown in Figs.

1–25d and 1–25e, respectively.

Also shown with these figures is a typical volume element of the

material taken at a point located on the surface of each section. Note

carefully how the shear stress must act on each shaded face of these

elements and then on the adjacent faces of the elements.

t

avg

=

V

A

=

2500 N

10.04 m210.02 m2

= 3.12 MPa

t

avg

=

V

A

=

5000 N

p10.005 m2

2

= 63.7 MPa

c

5 kN

(a)

20 mm

40 mm

b

a

d

(

b

)

5 kN

V 5 k

N

force of

strut on rod

a

d

c

5 kN

V 2.5 kN

V 2.5 kN

force of

rod on strut

(c)

b

(d)

5 kN

63.7 MPa

Fig.1–25

(

e

)

5 kN

3.12 MPa

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The inclined member in Fig.1–26a is subjected to a compressive force

of 600 lb. Determine the average compressive stress along the smooth

areas of contact defined by AB and BC, and the average shear stress

along the horizontal plane defined by EDB.

(b)

3

4

5

600 lb

F

AB

F

BC

(

c

)

V

360 l

b

(d)

3

45

600 lb

160 psi

240 psi

(e)

360 lb

80 psi

(a)

1 in.

3

4

5

600 lb

1.5 in.

3 in.

2 in.

A

C

B

D

E

Fig.1–26

SOLUTION

Internal Loadings.The free-body diagram of the inclined member

is shown in Fig.

1–26b. The compressive forces acting on the areas of

contact are

Also, from the free-body diagram of the top segment of the bottom

member, Fig.1–26c, the shear force acting on the sectioned horizontal

plane EDB is

Average Stress.The average compressive stresses along the

horizontal and vertical planes of the inclined member are

Ans.

Ans

.

These stress distributions are shown in Fig.1–26d.

The average shear stress acting on the horizontal plane defined by

EDB is

Ans.

This stress is shown distributed over the sectioned area in Fig.1–26e.

t

avg

=

360 lb

13 in.211.5 in.2

= 80 psi

s

BC

=

480 lb

12 in.211.5 in.2

= 160 psi

s

AB

=

360 lb

11 in.211.5 in.2

= 240 psi

V = 360 lb:

+

©F

x

= 0;

F

BC

- 600 lb

A

4

5

B

= 0

F

BC

= 480 lb+

c

©F

y

= 0;

F

AB

- 600 lb

A

3

5

B

= 0

F

AB

= 360 lb:

+

©F

x

= 0;

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1–35.The anchor shackle supports a cable force of 600 lb.

If the pin has a diameter of 0.25 in., determine the average

shear stress in the pin.

1–37.The thrust bearing is subjected to the loads shown.

Determine the average normal stress developed on cross

sections through points B, C, and D. Sketch the results on a

differential volume element located at each section.

PROBLEMS

1–34.The column is subjected to an axial force of 8 kN,

which is applied through the centroid of the cross-sectional

area. Determine the average normal stress acting at section

a–a. Show this distribution of stress acting over the area’s

cross section.

*1–36.While running the foot of a 150-lb man is

momentarily subjected to a force which is 5 times his

weight. Determine the average normal stress developed in

the tibia T of his leg at the mid section a–a. The cross

section can be assumed circular, having an outer diameter

of 1.75 in. and an inner diameter of 1 in. Assume the fibula

F does not support a load.

0.25 in.

600 lb

Prob.1–35

750 lb

a

T

F

a

Prob.1–36

8 kN

a

a

75 mm

10 mm

10 mm

10 mm

75 mm

70 mm

70 mm

Prob.1–34

500 N

200 N

65 mm

140 mm

100 mm

B

D

C

150 N

150 N

Prob.1–37

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41

1–38.The small block has a thickness of 5 mm. If the

stress distribution at the support developed by the load

varies as shown, determine the force F applied to the block,

and the distance d to where it is applied.

1–42.The 50-lb lamp is supported by three steel rods

connected by a ring at A.Determine which rod is subjected

to the greater average normal stress and compute its value.

Take The diameter of eachrodis giveninthe figure.

1–43.Solve Prob. 1–42 for

*1–44.The 50-lb lamp is supported by three steel rods

connected by a ring at A. Determine the angle of orientation

of AC such that the average normal stress in rod AC is

twice the average normal stress in rod AD. What is the

magnitude of stress in each rod? The diameter of each rod is

given in the figure.

u

u = 45°.

u = 30°.

1–39.The lever is held to the fixed shaft using a tapered

pin AB, which has a mean diameter of 6 mm. If a couple is

applied to the lever, determine the average shear stress in

the pin between the pin and lever.

1–41.The cinder block has the dimensions shown. If

it is subjected to a centrally applied force of

determine the average normal stress in the material. Show

the result acting on a differential volume element of the

material.

P = 800 lb,

60 mm

120 mm

40 MPa

60 MPa

F

d

180 mm

Prob.1–38

20 N 20 N

250 mm

250 mm

12 mm

A

B

Prob.1–39

0.25 in.

A

D C

B

0.35 in.

0.3 in.

u

45

Probs.1–42/43/44

1 in.

1 in.

4 in.

2 in.

2 in.

1 in.

1 in.

2 in.

3 in.

3 in.

P

Probs.1–40/41

*1–40.

The cinder block has the dimensions shown. If the

material fails when the average normal stress reaches 120 psi,

determine the largest centrally applied vertical load P it can

support.

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 41

42

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1–45.The shaft is subjected to the axial force of 30 kN. If

the shaft passes through the 53-mm diameter hole in the

fixed support A, determine the bearing stress acting on the

collar C. Also, what is the average shear stress acting along

the inside surface of the collar where it is fixed connected to

the 52-mm diameter shaft?

1–49.The open square butt joint is used to transmit a

force of 50 kip from one plate to the other. Determine the

average normal and average shear stress components that

this loading creates on the face of the weld, section AB.

1–46.The two steel members are joined together using a

60° scarf weld. Determine the average normal and average

shear stress resisted in the plane of the weld.

*1–48.The board is subjected to a tensile force of 85 lb.

Determine the average normal and average shear stress

developed in the wood fibers that are oriented along

section a–a at 15° with the axis of the board.

30 mm

25 mm

60

8 kN

8 kN

Prob.1–46

15

3 in.

a

1 in.

85 lb85 lb

a

Prob.1–48

52

0.5 in.

Prob.1–50

10 mm

30 kN

52 mm

60 mm

40 mm

53 mm

A

C

Prob.1–45

C

A B

20

30 mm

40 mm

775 N

Prob.1–47

30

30

50 kip

50 kip

2 in.

6 in.

A

B

Prob.1–49

1–47.

The J hanger is used to support the pipe such that

the force on the vertical bolt is 775 N. Determine the

average normal stress developed in the bolt BC if the bolt

has a diameter of 8 mm. Assume Ais a pin.

1–50.The specimen failed in a tension test at an angle of

52° when the axial load was 19.80 kip. If the diameter of the

specimen is 0.5 in., determine the average normal and

average shear stress acting on the area of the inclined

failure plane. Also, what is the average normal stress acting

on the cross section when failure occurs?

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 42

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43

1–51.A tension specimen having a cross-sectional area A

is subjected to an axial force P. Determine the maximum

average shear stress in the specimen and indicate the

orientation of a section on which it occurs.u

1–55.The row of staples AB contained in the stapler is

glued together so that the maximum shear stress the glue

can withstand is Determine the minimum

force F that must be placed on the plunger in order to shear

off a staple from its row and allow it to exit undeformed

through the groove at C. The outer dimensions of the staple

are shown in the figure. It has a thickness of 0.05 in. Assume

all the other parts are rigid and neglect friction.

t

max

= 12 psi.

*1–56.Rods ABandBChave diameters of 4 mmand6 mm,

respectively.If the load of 8 kN is applied to the ring at B,

determine the average normal stress in each rod if

1–57.Rods ABand BChave diameters of 4 mm and 6 mm,

respectively. If the vertical load of 8 kN is applied to the ring

at B, determine the angle of rod BC so that the average

normal stress in each rod is equivalent. What is this stress?

u

u = 60°.

*1–52.The joint is subjected to the axial member force of

5 kN. Determine the average normal stress acting on

sections AB and BC. Assume the member is smooth and is

50-mm thick.

1–54.The two members used in the construction of an

aircraft fuselage are joined together using a 30° fish-mouth

weld.Determine the average normal andaverage shear stress

on the plane of each weld.Assume each inclined plane

supports a horizontal force of 400 lb.

P

u

P

A

Prob.1–51

5 kN

40 mm

A

C

B

50 mm

60

45

Prob.1–52

800 lb 800 lb

30

1 in.

1 in.

1.5 in.

30

Prob.1–54

1–53.

The yoke is subjected to the force and couple

moment. Determine the average shear stress in the bolt

acting on the cross sections through Aand B. The bolt has a

diameter of 0.25 in. Hint:The couple moment is resisted by

a set of couple forces developed in the shank of the bolt.

500 lb

80 lbft

2 in.

2.5 in.

A

B

60

Prob.1–53

F

0.5 in.

0.3 in.

C

A B

Prob.1–55

8 kN

u

C

BA

Probs.1–56/57

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1–58.The bars of the truss each have a cross-sectional

area of Determine the average normal stress in

each member due to the loading State whether

the stress is tensile or compressive.

1–59.The bars of the truss each have a cross-sectional

area of If the maximumaverage normal stress inany

bar is not toexceed20ksi,determinethemaximummagnitude

P of the loads that can be applied to the truss.

1.25 in

2

.

P = 8 kip.

1.25 in

2

.

1–63.The railcar docklight is supported by the

diameter pin at A. If the lamp weighs 4 lb, and the extension

arm AB has a weight of , determine the average

shear stress in the pin needed to support the lamp. Hint:

The shear force in the pin is caused by the couple moment

required for equilibrium at A.

0.5 lb>ft

1

8

-in.-

*1–64.The two-member frame is subjected to the

distributed loading shown. Determine the average normal

stress and average shear stress acting at sections a–a and

b–b. Member CB has a square cross section of 35 mm on

each side. Take w = 8 kN>m.

*1–60.The plug is used to close the end of the cylindrical

tube that is subjected to an internal pressure of

Determine the average shear stress which the glue exerts on

the sides of the tube needed to hold the cap in place.

p = 650 Pa.

1–62.Solve Prob. 1–61 for pin B. The pin is subjected to

double shear and has a diameter of 0.2 in.

3 ft

4 ft 4 ft

P

0.75 P

E D

A

B C

Probs.1–58/59

3 ft

1.25 in.

B

A

Prob.1–63

P

40 m

m

35 mm

25 mm

Prob.1–60

A

20 lb

20 lb

5 in.

1.5 in.2 in.1 in.

E

C

B

D

Probs.1–61/62

4 m

B

A

C

3 m

b

b

a

a

w

Prob.1–64

1–61.

The crimping tool is used to crimp the end of the

wire E. If a force of 20 lb is applied to the handles,

determine the average shear stress in the pin at A. The pin

is subjected to double shear and has a diameter of 0.2 in.

Only a vertical force is exerted on the wire.

HEBBMC01_0132209918.QXD 7/9/07 2:51 PM Page 44

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ROBLEMS

45

1–67.The beam is supported by a pin at Aand a short link

BC. If determine the average shear stress

developed in the pins at A, B, and C. All pins are in double

shear as shown, and each has a diameter of 18 mm.

*1–68.The beam is supported by a pin at A and a short

link BC. Determine the maximum magnitude P of the loads

the beam will support if the average shear stress in each pin

is not to exceed 80 MPa. All pins are in double shear as

shown, and each has a diameter of 18 mm.

P = 15 kN,

P

2

P

1

P

n

A

m

A

2

A

1

d

1

d

2

d

n

L

1

L

2

L

m

x

Prob.1–66

30

C

1 m

P 4P 4P 2P

1.5 m 1.5 m

AB

0.5m 0.5m

Probs.1–67/68

1–65.

Member A of the timber step joint for a truss is

subjected to a compressive force of 5 kN. Determine the

average normal stress acting in the hanger rod C which has

a diameter of 10 mm and in member B which has a

thickness of 30 mm.

C

10

mm

F

C

m

m

4

0

m

m

30

6

0

5

kN

F

B

E

A

A

A

B

D

F

Prob.1–65

A

C

B

1.5 ft2 ft

0.5 ft

200 lb

u

Prob.1–69

1–66.Consider the general problem of a bar made from

msegments, each having a constant cross-sectional area

and length If there are n loads on the bar as shown,

write a computer program that can be used to determine

the average normal stress at any specified location x. Show

an application of the program using the values

A

2

= 1 in

2

.P

2

= -300 lb,

d

2

= 6 ft,L

2

= 2 ft,A

1

= 3 in

2

,P

1

= 400 lb,d

1

= 2 ft,

L

1

= 4 ft,

L

m

.

A

m

1–69.The frame is subjected to the load of 200 lb.

Determine the average shear stress in the bolt at A as a

function of the bar angle Plot this function,

and indicate the values of for which this stress is a

minimum. The bolt has a diameter of 0.25 in. and is

subjected to single shear.

u

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