CVG 2140 Solution
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Transcript of CVG 2140 Solution
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Typewritten TextINSTRUCTORSOLUTIONSMANUAL
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CONTENTS Philpot MoM 3rd
1. Stress 1.1 Introduction 1.2 Normal Stress Under Axial Loading 1.3 Direct Shear Stress 1.4 Bearing Stress 1.5 Stresses on Inclined Sections 1.6 Equality of Shear Stresses on Perpendicular Planes
2. Strain 2.1 Displacement, Deformation, and the Concept of Strain 2.2 Normal Strain 2.3 Shear Strain 2.4 Thermal Strain
3. Mechanical Properties of Materials 3.1 The Tension Test 3.2 The StressStrain Diagram 3.3 Hookes Law 3.4 Poissons Ratio
4. Design Concepts 4.1 Introduction 4.2 Types of Loads 4.3 Safety 4.4 Allowable Stress Design 4.5 Load and Resistance Factor Design
5. Axial Deformation 5.1 Introduction 5.2 Saint-Venants Principle 5.3 Deformations in Axially Loaded Bars 5.4 Deformations in a System of Axially Loaded Bars 5.5 Statically Indeterminate Axially Loaded Members 5.6 Thermal Effects on Axial Deformation 5.7 Stress Concentrations
6. Torsion 6.1 Introduction 6.2 Torsional Shear Strain 6.3 Torsional Shear Stress 6.4 Stresses on Oblique Planes 6.5 Torsional Deformations 6.6 Torsion Sign Conventions 6.7 Gears in Torsion Assemblies
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6.8 Power Transmission 6.9 Statically Indeterminate Torsion Members 6.10 Stress Concentrations in Circular Shafts Under Torsional Loadings 6.11 Torsion of Noncircular Sections 6.12 Torsion of Thin-Walled Tubes: Shear Flow
7. Equilibrium of Beams 7.1 Introduction 7.2 Shear and Moment in Beams 7.3 Graphical Method for Constructing Shear and Moment Diagrams 7.4 Discontinuity Functions to Represent Load, Shear, and Moment
8. Bending 8.1 Introduction 8.2 Flexural Strains 8.3 Normal Stresses in Beams 8.4 Analysis of Bending Stresses in Beams 8.5 Introductory Beam Design for Strength 8.6 Flexural Stresses in Beams of Two Materials 8.7 Bending Due to Eccentric Axial Load 8.8 Unsymmetric Bending 8.9 Stress Concentrations Under Flexural Loadings
9. Shear Stress in Beams 9.1 Introduction 9.2 Resultant Forces Produced by Bending Stresses 9.3 The Shear Stress Formula 9.4 The First Moment of Area Q 9.5 Shear Stresses in Beams of Rectangular Cross Section 9.6 Shear Stresses in Beams of Circular Cross Section 9.7 Shear Stresses in Webs of Flanged Beams 9.8 Shear Flow in Built-Up Members 9.9 Shear Stress and Shear Flow in Thin-Walled Members 9.10 Shear Centers of Thin-Walled Open Sections
10. Beam Deflections 10.1 Introduction 10.2 Moment-Curvature Relationship 10.3 The Differential Equation of the Elastic Curve 10.4 Deflections by Integration of a Moment Equation 10.5 Deflections by Integration of Shear-Force or Load Equations 10.6 Deflections Using Discontinuity Functions 10.7 Method of Superposition
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11. Statically Indeterminate Beams 11.1 Introduction 11.2 Types of Statically Indeterminate Beams 11.3 The Integration Method 11.4 Use of Discontinuity Functions for Statically Indeterminate Beams 11.5 The Superposition Method
12. Stress Transformations 12.1 Introduction 12.2 Stress at a General Point in an Arbitrarily Loaded Body 12.3 Equilibrium of the Stress Element 12.4 Plane Stress 12.5 Generating the Stress Element 12.6 Equilibrium Method for Plane Stress Transformations 12.7 General Equations of Plane Stress Transformation 12.8 Principal Stresses and Maximum Shear Stress 12.9 Presentation of Stress Transformation Results 12.10 Mohrs Circle for Plane Stress 12.11 General State of Stress at a Point
13. Strain Transformations 13.1 Introduction 13.2 Two-Dimensional or Plane Strain 13.3 Transformation Equations for Plane Strain 13.4 Principal Strains and Maximum Shearing Strain 13.5 Presentation of Strain Transformation Results 13.6 Mohrs Circle for Plane Strain 13.7 Strain Measurement and Strain Rosettes 13.8 Generalized Hookes Law for Isotropic Materials
14. Thin-Walled Pressure Vessels 14.1 Introduction 14.2 Spherical Pressure Vessels 14.3 Cylindrical Pressure Vessels 14.4 Strains in Pressure Vessels
15. Combined Loads 15.1 Introduction 15.2 Combined Axial and Torsional Loads 15.3 Principal Stresses in a Flexural Member 15.4 General Combined Loadings 15.5 Theories of Failure
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16. Columns 16.1 Introduction 16.2 Buckling of Pin-Ended Columns 16.3 The Effect of End Conditions on Column Buckling 16.4 The Secant Formula 16.5 Empirical Column FormulasCentric Loading 16.6 Eccentrically Loaded Columns
17. Energy Methods 17.1 Introduction 17.2 Work and Strain Energy 17.3 Elastic Strain Energy for Axial Deformation 17.4 Elastic Strain Energy for Torsional Deformation 17.5 Elastic Strain Energy for Flexural Deformation 17.6 Impact Loading 17.7 Work-Energy Method for Single Loads 17.8 Method of Virtual Work 17.9 Deflections of Trusses by the Virtual-Work Method 17.10 Deflections of Beams by the Virtual-Work Method 17.11 Castiglianos Second Theorem 17.12 Calculating Deflections of Trusses by Castiglianos Theorem 17.13 Calculating Deflections of Beams by Castiglianos Theorem
Appendix A: Geometric Properties of an Area A.1 Centroid of an Area A.2 Moment of Inertia for an Area A.3 Product of Inertia for an Area A.4 Principal Moments of Inertia A.5 Mohrs Circle for Principal Moments of Inertia
Appendix B: Geometric Properties of Structural Steel Shapes
Appendix C: Table of Beam Slopes and Deflections
Appendix D: Average Properties of Selected Materials
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P1.1 A stainless steel tube with an outside diameter of 60 mm and a wall thickness of 5 mm is used as
a compression member. If the axial normal stress in the member must be limited to 200 MPa,
determine the maximum load P that the member can support.
Solution
The cross-sectional area of the stainless steel tube is
2 2 2 2 2( ) [(60 mm) (50 mm) ] 863.938 mm4 4
A D d
The normal stress in the tube can be expressed as
P
A
The maximum normal stress in the tube must be limited to 200 MPa. Using 200 MPa as the allowable
normal stress, rearrange this expression to solve for the maximum load P
2 2max allow (200 N/mm )(863.938 mm ) 172,788 172.8 kN NP A Ans.
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P1.2 A 2024-T4 aluminum tube with an outside diameter of 2.50 in. will be used to support a 27-kip
load. If the axial normal stress in the member must be limited to 18 ksi, determine the wall thickness
required for the tube.
Solution
From the definition of normal stress, solve for the minimum area required to support a 27-kip load
without exceeding a stress of 18 ksi
2
min
27 kips1.500 in.
18 ksi
P PA
A
The cross-sectional area of the aluminum tube is given by
2 2( )4
A D d
Set this expression equal to the minimum area and solve for the maximum inside diameter d
2 2 2
2 2 2
2 2 2
max
[(2.50 in.) ] 1.500 in.4
4(2.50 in.) (1.500 in. )
4(2.50 in.) (1.500 in. )
2.08330 in.
d
d
d
d
The outside diameter D, the inside diameter d, and the wall thickness t are related by
2D d t Therefore, the minimum wall thickness required for the aluminum tube is
min
2.50 in. 2.08330 in.0.20835 in. 0.208 in.
2 2
D dt
Ans.
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P1.3 Two solid cylindrical rods (1) and (2)
are joined together at flange B and loaded, as
shown in Figure P1.3/4. If the normal stress
in each rod must be limited to 40 ksi,
determine the minimum diameter required
for each rod.
FIGURE P1.3/4
Solution
Cut a FBD through rod (1). The FBD should include the free end of the rod at A.
As a matter of course, we will assume that the internal force in rod (1) is tension
(even though it obviously will be in compression). From equilibrium,
1
1
15 kips 0
15 kips 15 kips (C)
yF F
F
Next, cut a FBD through rod (2) that includes the free end of the rod at A. Again,
we will assume that the internal force in rod (2) is tension. Equilibrium of this
FBD reveals the internal force in rod (2):
2
2
30 kips 30 kips 15 kips 0
75 kips 75 kips (C)
yF F
F
Notice that rods (1) and (2) are in compression. In this situation, we are
concerned only with the stress magnitude; therefore, we will use the force
magnitudes to determine the minimum required cross-sectional areas. If
the normal stress in rod (1) must be limited to 40 ksi, then the minimum
cross-sectional area that can be used for rod (1) is
211,min
15 kips0.375 in.
40 ksi
FA
The minimum rod diameter is therefore
2 21,min 1 10.375 in. 0.6909 0.691 9 i4
inn. .A d d
Ans.
Similarly, the normal stress in rod (2) must be limited to 40 ksi, which requires a minimum area of
222,min75 kips
1.875 in.40 ksi
FA
The minimum diameter for rod (2) is therefore
2 22,min 2 21.875 in. 1.54509 1.545 in.7 in.4
A d d
Ans.
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P1.4 Two solid cylindrical rods (1) and (2) are
joined together at flange B and loaded, as shown in
Figure P1.3/4. The diameter of rod (1) is 1.75 in.
and the diameter of rod (2) is 2.50 in. Determine the
normal stresses in rods (1) and (2).
FIGURE P1.3/4
Solution
Cut a FBD through rod (1). The FBD should include the free end of the rod at A. We
will assume that the internal force in rod (1) is tension (even though it obviously will
be in compression). From equilibrium,
1
1
15 kips 0
15 kips 15 kips (C)
yF F
F
Next, cut a FBD through rod (2) that includes the free end of the rod at A. Again, we
will assume that the internal force in rod (2) is tension. Equilibrium of this FBD
reveals the internal force in rod (2):
2
2
30 kips 30 kips 15 kips 0
75 kips 75 kips (C)
yF F
F
From the given diameter of rod (1), the cross-sectional area of rod (1) is
2 21 (1.75 in.) 2.4053 in.
4A
and thus, the normal stress in rod (1) is
11 21
15 kips6.23627 ksi
2.4053 in6.24 ksi )
.(C
F
A
Ans.
From the given diameter of rod (2), the cross-sectional area of rod (2) is
2 22 (2.50 in.) 4.9087 in.
4A
Accordingly, the normal stress in rod (2) is
22 22
75 kips15.2789 ksi
2.4053 in.15.28 ksi (C)
F
A
Ans.
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P1.5 Axial loads are applied with rigid bearing plates to the
solid cylindrical rods shown in Figure P1.5/6. The diameter
of aluminum rod (1) is 2.00 in., the diameter of brass rod (2)
is 1.50 in., and the diameter of steel rod (3) is 3.00 in.
Determine the axial normal stress in each of the three rods.
FIGURE P1.5/6
Solution
Cut a FBD through rod (1). The FBD should include the free end A. We will assume that the internal
force in rod (1) is tension (even though it obviously will be in compression). From equilibrium,
1 18 kips 4 kips 4 kips 0 16 kips 16 kips (C)yF F F
FBD through rod (1)
FBD through rod (2)
FBD through rod (3)
Next, cut a FBD through rod (2) that includes the free end A. Again, we will assume that the internal
force in rod (2) is tension. Equilibrium of this FBD reveals the internal force in rod (2):
2 28 kips 4 kips 4 kips 15 kips 15 kips 0 14 kips 14 kips (T)yF F F
Similarly, cut a FBD through rod (3) that includes the free end A. From this FBD, the internal force in
rod (3) is:
3
3
8 kips 4 kips 4 kips 15 kips 15 kips 20 kips 20 kips 0
26 kips 26 kips (C)
yF F
F
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From the given diameter of rod (1), the cross-sectional area of rod (1) is
2 21 (2.00 in.) 3.1416 in.
4A
and thus, the normal stress in aluminum rod (1) is
11 21
16 kips5.0930 ksi
3.1416 in5.09 ksi (C)
.
F
A
Ans.
From the given diameter of rod (2), the cross-sectional area of rod (2) is
2 22 (1.50 in.) 1.7671 in.
4A
Accordingly, the normal stress in brass rod (2) is
22 22
14 kips7.9224 ksi
1.7671 in.7.92 ksi (T)
F
A Ans.
Finally, the cross-sectional area of rod (3) is
2 23 (3.00 in.) 7.0686 in.
4A
and the normal stress in the steel rod is
33 23
26 kips3.6782 ksi
7.0686 in3.68 ksi (C)
.
F
A
Ans.
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P1.6 Axial loads are applied with rigid bearing plates to the
solid cylindrical rods shown in Figure P1.5/6. The normal
stress in aluminum rod (1) must be limited to 18 ksi, the
normal stress in brass rod (2) must be limited to 25 ksi, and
the normal stress in steel rod (3) must be limited to 15 ksi.
Determine the minimum diameter required for each of the
three rods.
FIGURE P1.5/6
Solution
The internal forces in the three rods must be determined. Begin with a FBD cut through rod (1) that
includes the free end A. We will assume that the internal force in rod (1) is tension (even though it
obviously will be in compression). From equilibrium,
1 18 kips 4 kips 4 kips 0 16 kips 16 kips (C)yF F F
FBD through rod (1)
FBD through rod (2)
FBD through rod (3)
Next, cut a FBD through rod (2) that includes the free end A. Again, we will assume that the internal
force in rod (2) is tension. Equilibrium of this FBD reveals the internal force in rod (2):
2 28 kips 4 kips 4 kips 15 kips 15 kips 0 14 kips 14 kips (T)yF F F
Similarly, cut a FBD through rod (3) that includes the free end A. From this FBD, the internal force in
rod (3) is:
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3
3
8 kips 4 kips 4 kips 15 kips 15 kips 20 kips 20 kips 0
26 kips 26 kips (C)
yF F
F
Notice that two of the three rods are in compression. In these situations, we are concerned only with the
stress magnitude; therefore, we will use the force magnitudes to determine the minimum required cross-
sectional areas, and in turn, the minimum rod diameters. The normal stress in aluminum rod (1) must be
limited to 18 ksi; therefore, the minimum cross-sectional area required for rod (1) is
21
1,min
1
16 kips0.8889 in.
18 ksi
FA
The minimum rod diameter is therefore
2 21,min 1 10.8889 in. 1.0638 in 1.064 in..
4A d d
Ans.
The normal stress in brass rod (2) must be limited to 25 ksi, which requires a minimum area of
22
2,min
2
14 kips0.5600 in.
25 ksi
FA
which requires a minimum diameter for rod (2) of
2 22,min 2 20.5600 in. 0.8444 in 0.844 in..
4A d d
Ans.
The normal stress in steel rod (3) must be limited to 15 ksi. The minimum cross-sectional area required
for this rod is:
23
3,min
3
26 kips1.7333 in.
15 ksi
FA
which requires a minimum diameter for rod (3) of
2 23,min 3 31.7333 in. 1.4856 in 1.486 in..4
A d d
Ans.
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P1.7 Two solid cylindrical rods support a load of
P = 50 kN, as shown in Figure P1.7/8. If the
normal stress in each rod must be limited to 130
MPa, determine the minimum diameter required
for each rod.
FIGURE P1.7/8
Solution
Consider a FBD of joint B. Determine the angle between rod (1) and the horizontal axis:
4.0 m
tan 1.600 57.99462.5 m
and the angle between rod (2) and the horizontal axis:
2.3 m
tan 0.7188 35.70673.2 m
Write equilibrium equations for the sum of forces in the
horizontal and vertical directions. Note: Rods (1) and (2)
are two-force members.
2 1cos(35.7067 ) cos(57.9946 ) 0xF F F (a)
2 1sin(35.7067 ) sin(57.9946 ) 0yF F F P (b)
Unknown forces F1 and F2 can be found from the simultaneous solution of Eqs. (a) and (b). Using the
substitution method, Eq. (b) can be solved for F2 in terms of F1:
2 1cos(57.9946 )
cos(35.7067 )F F
(c)
Substituting Eq. (c) into Eq. (b) gives
1 1
1
1
cos(57.9946 )sin(35.7067 ) sin(57.9946 )
cos(35.6553 )
cos(57.9946 ) tan(35.7067 ) sin(57.9946 )
cos(57.9946 ) tan(35.7067 ) sin(57.9946 ) 1.2289
F F P
F P
P PF
For the given load of P = 50 kN, the internal force in rod (1) is therefore:
150 kN
40.6856 kN1.2289
F
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Backsubstituting this result into Eq. (c) gives force F2:
2 1
cos(57.9946 ) cos(57.9946 )(40.6856 kN) 26.5553 kN
cos(35.7067 ) cos(35.7067 )F F
The normal stress in rod (1) must be limited to 130 MPa; therefore, the minimum cross-sectional area
required for rod (1) is
21
1,min 2
1
(40.6856 kN)(1,000 N/kN)312.9664 mm
130 N/mm
FA
The minimum rod diameter is therefore
2 21,min 1 1312.9664 mm 19.9620 19.
496 mmmmA d d
Ans.
The minimum area required for rod (2) is
22
2,min 2
2
(26.5553 kN)(1,000 N/kN)204.2718 mm
130 N/mm
FA
which requires a minimum diameter for rod (2) of
2 22,min 2 2204.2718 mm 16.1272 16.
413 mmmmA d d
Ans.
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P1.8 Two solid cylindrical rods support a load
of P = 27 kN, as shown in Figure P1.7/8. Rod
(1) has a diameter of 16 mm and the diameter
of rod (2) is 12 mm. Determine the axial
normal stress in each rod.
FIGURE P1.7/8
Solution
Consider a FBD of joint B. Determine the angle between rod (1) and the horizontal axis:
4.0 m
tan 1.600 57.99462.5 m
and the angle between rod (2) and the horizontal axis:
2.3 m
tan 0.7188 35.70673.2 m
Write equilibrium equations for the sum of forces in the horizontal
and vertical directions. Note: Rods (1) and (2) are two-force
members.
2 1cos(35.7067 ) cos(57.9946 ) 0xF F F (a)
2 1sin(35.7067 ) sin(57.9946 ) 0yF F F P (b)
Unknown forces F1 and F2 can be found from the simultaneous solution of Eqs. (a) and (b). Using the
substitution method, Eq. (b) can be solved for F2 in terms of F1:
2 1cos(57.9946 )
cos(35.7067 )F F
(c)
Substituting Eq. (c) into Eq. (b) gives
1 1
1
1
cos(57.9946 )sin(35.7067 ) sin(57.9946 )
cos(35.6553 )
cos(57.9946 ) tan(35.7067 ) sin(57.9946 )
cos(57.9946 ) tan(35.7067 ) sin(57.9946 ) 1.2289
F F P
F P
P PF
For the given load of P = 27 kN, the internal force in rod (1) is therefore:
127 kN
21.9702 kN1.2289
F
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Backsubstituting this result into Eq. (c) gives force F2:
2 1
cos(57.9946 ) cos(57.9946 )(21.9702 kN) 14.3399 kN
cos(35.7067 ) cos(35.7067 )F F
The diameter of rod (1) is 16 mm; therefore, its cross-sectional area is:
2 21 (16 mm) 201.0619 mm
4A
and the normal stress in rod (1) is:
21
1 2
1
(21.9702 kN)(1,000 N/kN)109.2710 N/mm
201.0109.3 MPa (T)
619 mm
F
A Ans.
The diameter of rod (2) is 12 mm; therefore, its cross-sectional area is:
2 22 (12 mm) 113.0973 mm
4A
and the normal stress in rod (2) is:
22
2 2
2
(14.3399 kN)(1,000 N/kN)126.7924 N/mm
113.0126.8 MPa (T)
973 mm
F
A Ans.
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P1.9 A simple pin-connected truss is loaded
and supported as shown in Figure P1.9. All
members of the truss are aluminum pipes that
have an outside diameter of 4.00 in. and a wall
thickness of 0.226 in. Determine the normal
stress in each truss member.
FIGURE P1.9
Solution
Overall equilibrium:
Begin the solution by determining the
external reaction forces acting on the
truss at supports A and B. Write
equilibrium equations that include all
external forces. Note that only the
external forces (i.e., loads and
reaction forces) are considered at this
time. The internal forces acting in the
truss members will be considered
after the external reactions have been
computed. The free-body diagram
(FBD) of the entire truss is shown.
The following equilibrium equations
can be written for this structure:
2 kips
2 ki
0
ps
x
x
xF A
A
(6 ft) (5 kips)(14 ft) (2 kips)(7 ft)
14 kips
0
y
A yB
B
M
5 kips 0
9 kips
y y y
y
F A B
A
Method of joints:
Before beginning the process of determining the internal forces in the axial members, the geometry of
the truss will be used to determine the magnitude of the inclination angles of members AC and BC. Use
the definition of the tangent function to determine AC and BC:
7 fttan 0.50 26.565
14 ft
7 fttan 0.875 41.186
8 ft
AC AC
BC BC
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Joint A:
Begin the solution process by considering a FBD of joint A. Consider
only those forces acting directly on joint A. In this instance, two axial
members, AB and AC, are connected at joint A. Additionally, two
reaction forces, Ax and Ay, act at joint A. Tension forces will be
assumed in each truss member.
cos(26.565 ) 0x AC AB xF F F A (a)
sin(26.565 ) 0y AC yF F A (b)
Solve Eq. (b) for FAC:
9 kips
sin(26.565 ) sin(26.520.125 kip
65 )s
y
AC
AF
and then compute FAB using Eq. (a):
cos(26.565 )
(20.125 kips)cos(26.5 16.000 kips65 ) ( 2 kips)
AB AC xF F A
Joint B:
Next, consider a FBD of joint B. In this instance, the equilibrium
equations associated with joint B seem easier to solve than those that
would pertain to joint C. As before, tension forces will be assumed in
each truss member.
cos(41.186 ) 0x AB BCF F F (c)
sin(41.186 ) 0y BC yF F B (d)
Solve Eq. (d) for FBC:
14 kips
sin(41.186 ) sin(41.1821.260 kip
6s
)
y
BC
BF
Eq. (c) can be used as a check on our calculations:
cos(41.186 )
( 16.000 kips) ( 21.260 kips)cos(41.186 ) 0
x AB BCF F F
Checks!
Section properties:
For each of the three truss members:
2 2 24.00 in. 2(0.226 in.) 3.548 in. (4.00 in.) (3.548 in.) 2.67954 in.4
d A
Normal stress in each truss member:
2
16.000 kips5.971 ksi
2.679545.97 ksi (C)
in.
ABAB
AB
F
A
Ans.
2
20.125 kips7.510 ksi
2.679547.51 ksi (T)
in.
ACAC
AC
F
A Ans.
2
21.260 kips7.934 ksi
2.679547.93 ksi (C)
in.
BCBC
BC
F
A
Ans.
-
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P1.10 A simple pin-connected truss is loaded
and supported as shown in Figure P1.10. All
members of the truss are aluminum pipes that
have an outside diameter of 60 mm and a wall
thickness of 4 mm. Determine the normal
stress in each truss member.
FIGURE P1.10
Solution
Overall equilibrium:
Begin the solution by determining the
external reaction forces acting on the truss at
supports A and B. Write equilibrium
equations that include all external forces.
Note that only the external forces (i.e., loads
and reaction forces) are considered at this
time. The internal forces acting in the truss
members will be considered after the external
reactions have been computed. The free-
body diagram (FBD) of the entire truss is
shown. The following equilibrium equations
can be written for this structure:
12 k
12
N 0
kNx
x xF A
A
(1 m) (15 kN)(4.3 m) 0
64.5 kNy
A yB
B
M
15 kN
49.5 kN
0
y
y y yF
A
A B
Method of joints:
Before beginning the process of determining the internal forces in the axial members, the geometry of
the truss will be used to determine the magnitude of the inclination angles of members AB and BC. Use
the definition of the tangent function to determine AB and BC:
1.5 mtan 1.50 56.310
1.0 m
1.5 mtan 0.454545 24.444
3.3 m
AB AB
BC BC
-
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Joint A:
Begin the solution process by considering a FBD of joint A. Consider
only those forces acting directly on joint A. In this instance, two axial
members, AB and AC, are connected at joint A. Additionally, two
reaction forces, Ax and Ay, act at joint A. Tension forces will be assumed
in each truss member.
cos(56.310 ) 0x AC AB xF F F A (a)
sin(56.310 ) 0y y ABF A F (b)
Solve Eq. (b) for FAB:
49.5 kN
sin(56.310 ) sin(56.310 )59.492 kN
y
AB
AF
and then compute FAC using Eq. (a):
cos(56.310 )
( 59.492 kN)cos(56.3 45.000 10 ) ( 12 kN) kN
AC AB xF F A
Joint C:
Next, consider a FBD of joint C. In this instance, the equilibrium
equations associated with joint C seem easier to solve than those that
would pertain to joint B. As before, tension forces will be assumed in
each truss member.
cos(24.444 ) 12 kN 0x AC BCF F F (c)
sin(24.444 ) 15 kN 0y BCF F (d)
Solve Eq. (d) for FBC:
15 kN
sin(24.444 )36.249 kNBCF
Eq. (c) can be used as a check on our calculations:
cos(24.444 ) 12 kN 0
(45.000 kN) ( 36.249 kN)cos(24.444 ) 12 kN 0
x AC BCF F F
Checks!
Section properties:
For each of the three truss members:
2 2 260 mm 2(4 mm) 52 mm (60 mm) (52 mm) 703.7168 mm4
d A
Normal stress in each truss member:
2
( 59.492 kN)(1,000 N/kN)84.539 MPa
7084.5 MPa (C)
3.7168 mm
ABAB
AB
F
A
Ans.
2
(45.000 kN)(1,000 N/kN)63.946 MPa
7063.9 MPa
3.7168)
mm(TACAC
AC
F
A Ans.
2
( 36.249 kN)(1,000 N/kN)51.511 MPa
7051.5 MPa (C)
3.7168 mm
BCBC
BC
F
A
Ans.
-
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P1.11 A simple pin-connected truss is loaded
and supported as shown in Figure P1.11. All
members of the truss are aluminum pipes that
have an outside diameter of 42 mm and a wall
thickness of 3.5 mm. Determine the normal
stress in each truss member.
FIGURE P1.11
Solution
Overall equilibrium:
Begin the solution by determining the external
reaction forces acting on the truss at supports A
and B. Write equilibrium equations that include all
external forces. Note that only the external forces
(i.e., loads and reaction forces) are considered at
this time. The internal forces acting in the truss
members will be considered after the external
reactions have been computed. The free-body
diagram (FBD) of the entire truss is shown. The
following equilibrium equations can be written for
this structure:
30 kN
30 kN
0y
y
yF A
A
(30 kN)(4.5 m) (15 kN)(1.6 m) (5.6 m)
19.821 kN
0
x
A x
B
M B
15 kN 0
15 kN 15 kN ( 19.821 kN 34.821 ) kN
x x
x x
x
x
F A B
A AB
Method of joints:
Before beginning the process of determining the internal forces in the axial members, the geometry of
the truss will be used to determine the magnitude of the inclination angles of members AC and BC. Use
the definition of the tangent function to determine AC and BC:
1.6 mtan 0.355556 19.573
4.5 m
4 mtan 0.888889 41.634
4.5 m
AC AC
BC BC
-
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Joint A:
Begin the solution process by considering a FBD of joint A. Consider
only those forces acting directly on joint A. In this instance, two axial
members, AB and AC, are connected at joint A. Additionally, two
reaction forces, Ax and Ay, act at joint A. Tension forces will be
assumed in each truss member.
cos(19.573 ) 0x x ACF A F (a)
sin(19.573 ) 0y y AC ABF A F F (b)
Solve Eq. (a) for FAC:
34.821 kN
cos(19.573 ) cos(19.573 )36.957 kNxAC
AF
and then compute FAB using Eq. (b):
sin(19.573 )
(30.000 kN) (36.957 kN)sin(19. 17.619 573 ) kN
AB y ACF A F
Joint B:
Next, consider a FBD of joint B. In this instance, the equilibrium
equations associated with joint B seem easier to solve than those that
would pertain to joint C. As before, tension forces will be assumed in
each truss member.
cos(41.634 ) 0x x BCF B F (c)
sin(41.634 ) 0y BC ABF F F (d)
Solve Eq. (c) for FBC:
( 19.821 kN)
cos(41.634 ) cos(41.634 )26.520 kNxBC
BF
Eq. (d) can be used as a check on our calculations:
sin(41.634 )
( 26.520 kN)sin(41.634 ) (17.619 kN) 0
y BC ABF F F
Checks!
Section properties:
For each of the three truss members:
2 2 242 mm 2(3.5 mm) 35 mm (42 mm) (35 mm) 423.3296 mm4
d A
Normal stress in each truss member:
2
(17.619 kN)(1,000 N/kN)41.620 MPa
4241.6 MPa
3.3296)
mm(TABAB
AB
F
A Ans.
2
(36.957 kN)(1,000 N/kN)87.301 MPa
4287.3 MPa
3.3296)
mm(TACAC
AC
F
A Ans.
2
( 26.520 kN)(1,000 N/kN)62.647 MPa
4262.6 MPa (C)
3.3296 mm
BCBC
BC
F
A
Ans.
-
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P1.12 The rigid beam BC shown in Figure P1.12 is
supported by rods (1) and (2) that have cross-sectional
areas of 175 mm2 and 300 mm
2, respectively. For a
uniformly distributed load of w = 15 kN/m, determine
the normal stress in each rod. Assume L = 3 m and a =
1.8 m.
FIGURE P1.12
Solution
Equilibrium: Calculate the internal forces in members (1) and (2).
1
2
1
2
1.8 m(3 m) (15 kN/m)(1.8 m) 0
2
1.8 m(3 m) (15 kN/m)(1.8 m) 3 m 0
8.100 kN
18.900
2
kN
C
B
M F
M
F
F
F
Stresses:
21
1 2
1
(8.100 kN)(1,000 N/kN)46.286 N/mm
175 m46.3 MPa
m
F
A Ans.
22
2 2
2
(18.900 kN)(1,000 N/kN)63.000 N/mm
300 m63.0 MPa
m
F
A Ans.
-
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P1.13 Bar (1) in Figure P1.15 has a cross-
sectional area of 0.75 in.2. If the stress in bar
(1) must be limited to 30 ksi, determine the
maximum load P that may be supported by
the structure.
FIGURE P1.13
Solution
Given that the cross-sectional area of bar (1) is 0.75 in.2 and its normal stress must be limited to 30 ksi,
the maximum force that may be carried by bar (1) is
21,max 1 1 (30 ksi)(0.75 in. ) 22.5 kipsF A
Consider a FBD of ABC. From the moment equilibrium
equation about joint A, the relationship between the force in
bar (1) and the load P is:
1
1
(6 ft) (10 ft) 0
6 ft
10 ft
AM F P
P F
Substitute the maximum force F1,max = 22.5 kips into this relationship to obtain the maximum load that
may be applied to the structure:
16 ft 6 ft
(22.5 kips)10 ft 10 ft
13.50 kipsP F Ans.
-
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P1.14 The rectangular bar shown in Figure
P1.14 is subjected to a uniformly distributed
axial loading of w = 13 kN/m and a
concentrated force of P = 9 kN at B.
Determine the magnitude of the maximum
normal stress in the bar and its location x.
Assume a = 0.5 m, b = 0.7 m, c = 15 mm, and
d = 40 mm.
FIGURE P1.14
Solution
Equilibrium: Draw a FBD for the interval between A and B where
0 x a . Write the following equilibrium equation:
(13 kN/m)(1.2 m ) (9 kN) 0
(13 kN/m)(1.2 m ) (9 kN)
xF x F
F x
The largest force in this interval occurs at x = 0 where F = 6.6
kN.
In the interval between B and C where a x a b , and write the following equilibrium equation:
(13 kN/m)(1.2 m ) 0
(13 kN/m)(1.2 m )
xF x F
F x
The largest force in this interval occurs at x = a where F = 9.1
kN.
Maximum Normal Stress:
max(9.1 kN)(1,000 N/kN)
(15 mm15.17 MPa
)(40 mm)at 0.5 mx Ans.
-
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P1.15 The solid 1.25-in.-diameter rod shown in
Figure P1.15 is subjected to a uniform axial
distributed loading along its length of w = 750 lb/ft.
Two concentrated loads also act on the rod: P =
2,000 lb and Q = 1,000 lb. Assume a = 16 in. and b
= 32 in. Determine the normal stress in the rod at
the following locations:
(a) x = 10 in.
(b) x = 30 in.
FIGURE P1.15
Solution
(a) x = 10 in.
Equilibrium: Draw a FBD for the interval between A and B
where 0 x a , and write the following equilibrium equation:
(750 lb/ft)(1 ft/12 in.)(48 in. )
(2,000 lb) (1,000 lb) 0
(62.5 lb/in.)(48 in. ) 3,000 lb
xF x
F
F x
At x = 10 in., F = 5,375 lb.
Stress: The normal stress at this location can be calculated as follows.
2 2
2
(1.25 in.) 1.227185 in.4
5,375 lb4,379 4,380 p.944 psi
1si
.227185 in.
A
Ans.
(b) x = 30 in.
Equilibrium: Draw a FBD for the interval between B and C
where a x a b , and write the following equilibrium equation:
(750 lb/ft)(1 ft/12 in.)(48 in. )
(1,000 lb) 0
(62.5 lb/in.)(48 in. ) 1,000 lb
xF x
F
F x
At x = 30 in., F = 2,125 lb.
Stress: The normal stress at this location can be calculated as follows.
2
1,730 ps2,125 lb
1,731.606 psi1.227185 i
in.
Ans.
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P1.16 Two 6 in. wide wooden boards are
to be joined by splice plates that will be
fully glued on the contact surfaces. The
glue to be used can safely provide a shear
strength of 120 psi. Determine the smallest
allowable length L that can be used for the
splice plates for an applied load of P =
10,000 lb. Note that a gap of 0.5 in. is
required between boards (1) and (2).
FIGURE P1.16
Solution
Consider a FBD of board (2). The glue on the splice plates provides resistance to the 10,000 lb applied
load on both the top and bottom surfaces of board (2). Denoting the shear resistance on a glue surface as
V, equilibrium in the horizontal direction requires
0
10,000 lb5,000 lb
2
xF P V V
V
In other words, each glue surface must be large enough so that 5,000 lb of shear resistance can be
provided to board (2). Since the glue has a shear strength of 120 psi, the area of each glue surface on
board (2) must be at least
2min5,000 lb
41.6667 in.120 psi
A
The boards are 6-in. wide; therefore, glue must be spread along board (2) for a length of at least
2
glue joint
41.6667 in.6.9444 in.
6 in.L
Although weve discussed only board (2), the same rationale applies to board (1). For both boards (1) and (2), the glue must be applied along a length of at least 6.9444 in. on both the top and bottom of the
boards in order to resist the 10,000 lb applied load.
The glue applied to boards (1) and (2) must be matched by glue applied to the splice plates. Therefore,
the splice plates must be at least 6.9444 in. + 6.9444 in. = 13.8889 in. long. However, we are told that a
0.5-in. gap is required between boards (1) and (2); therefore, the splice plates must be 0.5-in. longer.
Altogether, the length of the splice plates must be at least
min 6.9444 in. 6.9444 in. 0.5 in 14.39 in..L Ans.
-
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P1.17 For the clevis connection shown in Figure
P1.17, determine the maximum applied load P that
can be supported by the 10-mm-diameter pin if the
average shear stress in the pin must not exceed 95
MPa.
FIGURE P1.17
Solution
Consider a FBD of the bar that is connected by the clevis,
including a portion of the pin. If the shear force acting on each
exposed surface of the pin is denoted by V, then the shear force
on each pin surface is related to the load P by:
0 2xF P V V P V
The area of the pin surface exposed by the FBD is simply the cross-sectional area of the pin:
2 2 2pin pin (10 mm) 78.539816 mm
4 4A d
If the average shear stress in the pin must be limited to 95 MPa, the maximum shear force V on a single
cross-sectional surface must be limited to
2 2bolt (95 N/mm )(78.539816 mm ) 7,461.283 NV A
Therefore, the maximum load P that may be applied to the connection is
2 2(7,461.283 N) 14,922.565 14.92 kNNP V Ans.
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P1.18 For the connection shown in Figure P1.18,
determine the average shear stress produced in the 3/8-
in. diameter bolts if the applied load is P = 2,500 lb.
FIGURE P1.18
Solution
There are four bolts, and it is assumed that each bolt supports an equal portion of the external load P.
Therefore, the shear force carried by each bolt is
2,500 lb
625 lb4 bolts
V
The bolts in this connection act in single shear. The cross-sectional area of a single bolt is
2 2 2 2bolt bolt (3 / 8 in.) (0.375 in.) 0.110447 in.
4 4 4A d
Therefore, the average shear stress in each bolt is
2
bolt
625 lb5,658.8427 psi
0.110447 in.5,660 psi
V
A Ans.
-
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P1.19 The five-bolt connection shown in Figure P1.19 must
support an applied load of P = 265 kN. If the average shear stress
in the bolts must be limited to 120 MPa, determine the minimum
bolt diameter that may be used for this connection.
FIGURE P1.19
Solution
There are five bolts, and it is assumed that each bolt supports an equal portion of the external load P.
Therefore, the shear force carried by each bolt is
265 kN
53 kN 53,000 N5 bolts
V
Since the average shear stress must be limited to 120 MPa, each bolt must provide a shear area of at
least:
22
53,000 N441.6667 mm
120 N/mmVA
Each bolt in this connection acts in double shear; therefore, two cross-sectional bolt surfaces are
available to transmit shear stress in each bolt.
2
2
bolt
441.6667 mm220.8333 mm per surface
2 surfaces per bolt 2 surfaces
VAA
The minimum bolt diameter must be
2 2bolt bolt 16.77 m220.8333 mm 16.7682 mm m
4d d
Ans.
-
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P1.20 A coupling is used to connect a 2 in. diameter
plastic pipe (1) to a 1.5 in. diameter pipe (2), as
shown in Figure P1.20. If the average shear stress in
the adhesive must be limited to 400 psi, determine
the minimum lengths L1 and L2 required for the joint
if the applied load P is 5,000 lb.
FIGURE P1.24
Solution
To resist a shear force of 5,000 lb, the area of adhesive required on each pipe is
2
adhesive
5,000 lb12.5 in.
400 psiV
VA
Consider the coupling on pipe (1). The adhesive is applied to the circumference of the pipe, and the
circumference C1 of pipe (1) is
1 1 (2.0 in.) 6.2832 in.C D
The minimum length L1 is therefore
2
1
1
12.5 in.1.9894 in.
6.2832 i1.989 i
n n.
.
VALC
Ans.
Consider the coupling on pipe (2). The circumference C2 of pipe (2) is
2 2 (1.5 in.) 4.7124 in.C D
The minimum length L2 is therefore
2
2
2
12.5 in.2.6526 in.
4.7124 2.65 in.
in.
VALC
Ans.
-
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1.21 A hydraulic punch press is used to
punch a slot in a 0.50-in. thick plate, as
illustrated in Fig. P1.21. If the plate shears
at a stress of 30 ksi, determine the
minimum force P required to punch the
slot.
FIGURE P1.21
Solution
The shear stress associated with removal of the slug exists on its perimeter. The perimeter of the slug is
given by
perimeter 2(3.00 in.) + (0.75 in.) 8.35619 in.
Thus, the area subjected to shear stress is
2perimeter plate thickness (8.35619 in.)(0.50 in.) 4.17810 in.VA
Given that the plate shears at = 30 ksi, the force required to remove the slug is therefore
2min (30 ksi)(4.17810 in. ) 125.343 kips 125.3 kipsVP A Ans.
-
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P1.22 The handle shown in Figure P1.22 is
attached to a 40-mm-diameter shaft with a
square shear key. The forces applied to the
lever are P = 1,300 N. If the average shear
stress in the key must not exceed 150 MPa,
determine the minimum dimension a that must
be used if the key is 25 mm long. The overall
length of the handle is L = 0.70 m. FIGURE P1.22
Solution
To determine the shear force V that must be resisted by the shear key, sum moments about the center of
the shaft (which will be denoted O):
700 mm 700 mm 40 mm(1,300 N) (1,300 N) 0
2 2 2
45,500 N
OM V
V
Since the average shear stress in the key must not exceed 150 MPa, the shear area required is
22
45,500 N303.3333 mm
150 N/mmV
VA
The shear area in the key is given by the product of its length L (i.e., 25 mm) and its width a. Therefore,
the minimum key width a is
2303.3333 mm
12.1333 mm25 m
12.1 m
3 mmVA
aL
Ans.
-
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P1.23 An axial load P is supported by the short steel
column shown in Figure P1.23. The column has a cross-
sectional area of 14,500 mm2. If the average normal stress
in the steel column must not exceed 75 MPa, determine the
minimum required dimension a so that the bearing stress
between the base plate and the concrete slab does not
exceed 8 MPa. Assume b = 420 mm.
FIGURE P1.23
Solution
Since the normal stress in the steel column must not exceed 75 MPa, the maximum column load is
2 2max (75 N/mm )(14,500 mm ) 1,087,500 NP A
The maximum column load must be distributed over a large enough area so that the bearing stress
between the base plate and the concrete slab does not exceed 8 MPa; therefore, the minimum plate area
is
2
min 2
1,087,500 N135,937.5 mm
8 N/mmb
PA
The area of the plate is a b. Since b = 420, the minimum length of a must be
2
min
2
135,937.5 mm
135,937.5 mm
420 mm324 mm
A a b
a
Ans.
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P1.24 The two wooden boards shown in Figure P1.24 are
connected by a 0.5-in.-diameter bolt. Washers are installed
under the head of the bolt and under the nut. The washer
dimensions are D = 2 in. and d = 5/8 in. The nut is tightened
to cause a tensile stress of 9,000 psi in the bolt. Determine
the bearing stress between the washer and the wood.
FIGURE P1.24
Solution
The tensile stress in the bolt is 9,000 psi; therefore, the tension force that acts in the bolt is
2 2bolt bolt bolt (9,000 psi) (0.5 in.) (9,000 psi)(0.196350 in. ) 1,767.146 lb
4F A
The contact area between the washer and the wood is
2 2 2washer (2 in.) (0.625 in.) 2.834796 in.
4A
Thus, the bearing stress between the washer and the wood is
2
1,767.146 lb
2.834796 in.623 psib Ans.
-
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P1.25 For the beam shown in Figure P1.25, the
allowable bearing stress for the material under
the supports at A and B is b = 800 psi. Assume w = 2,100 lb/ft, P = 4,600 lb, a = 20 ft,
and b = 8 ft. Determine the size of square
bearing plates required to support the loading
shown. Dimension the plates to the nearest
in.
FIGURE P1.25
Solution
Equilibrium: Using the FBD shown,
calculate the beam reaction forces.
20 ft(20 ft) (2,100 lb/ft)(20 ft) (4,600 lb)(28 ft) 0
2
27,440 lb
A y
y
M B
B
20 ft(20 ft) (2,100 lb/ft)(20 ft) (4,600 lb)(8 ft) 0
2
19,160 lb
B y
y
M A
A
Bearing plate at A: The area of the bearing plate required for support A is
219,160 lb
23.950 in.800 psi
AA
Since the plate is to be square, its dimensions must be
2 use 5 in. 23.95 5 in. bearing plate at0 in. 4.894 in . Awidth Ans.
Bearing plate at B: The area of the bearing plate required for support B is
227,440 lb
34.300 in.800 psi
BA
Since the plate is to be square, its dimensions must be
2 use 6 in. 34.30 6 in. bearing plate at0 in. 5.857 in . Bwidth Ans.
-
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P1.26 The d = 15-mm-diameter solid rod shown in
Figure P1.26 passes through a D = 20-mm-diameter
hole in the support plate. When a load P is applied
to the rod, the rod head rests on the support plate.
The support plate has a thickness of b = 12 mm.
The rod head has a diameter of a = 30 mm and the
head has a thickness of t = 10 mm. If the normal
stress produced in the rod by load P is 225 MPa,
determine:
(a) the bearing stress acting between the support
plate and the rod head.
(b) the average shear stress produced in the rod
head.
(c) the punching shear stress produced in the
support plate by the rod head.
FIGURE P1.26
Solution
The cross-sectional area of the rod is:
2 2
rod (15 mm) 176.715 mm4
A
The tensile stress in the rod is 225 MPa; therefore, the tension force in the rod is
2 2rod rod rod (225 N/mm )(176.715 mm ) 39,760.782 NF A
(a) The contact area between the support plate and the rod head is
2 2 2contact (30 mm) (20 mm) 392.699 mm
4A
Thus, the bearing stress between the support plate and the rod head is
2
39,760.782 N
392.699 mm101.3 MPab Ans.
(b) In the rod head, the area subjected to shear stress is equal to the perimeter of the rod times the
thickness of the head.
2(15 mm)(10 mm) 471.239 mmVA
and therefore, the average shear stress in the rod head is
2
39,760.782 N
471.84.4 MP
ma
239 m Ans.
(c) In the support plate, the area subjected to shear stress is equal to the product of the rod head
perimeter and the thickness of the plate.
2(30 mm)(12 mm) 1,130.973 mmVA
and therefore, the average punching shear stress in the support plate is
2
39,760.782 N
1,13035.2 MPa
.973 mm Ans.
-
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P1.27 The rectangular bar is connected to the support bracket with
a circular pin, as shown in Figure P1.27. The bar width is w = 1.75
in. and the bar thickness is 0.375 in. For an applied load of P =
5,600 lb, determine the average bearing stress produced in the bar
by the 0.625-in.-diameter pin.
FIGURE P1.27
Solution
The average bearing stress produced in the bar by the pin is based on the projected area of the pin. The
projected area is equal to the pin diameter times the bar thickness. Therefore, the average bearing stress
in the bar is
5,600 lb
23,893.33 psi(0.625 in.)(0.375 i
23,n.
900)
psib Ans.
-
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P1.28 The steel pipe column shown in Figure P1.28 has an
outside diameter of 8.625 in. and a wall thickness of 0.25 in.
The timber beam is 10.75 in wide, and the upper plate has the
same width. The load imposed on the column by the timber
beam is 80 kips. Determine
(a) The average bearing stress at the surfaces between the pipe
column and the upper and lower steel bearing plates.
(b) The length L of the rectangular upper bearing plate if its
width is 10.75 in. and the average bearing stress between the
steel plate and the wood beam is not to exceed 500 psi.
(c) The dimension a of the square lower bearing plate if the average bearing stress between the lower bearing plate and
the concrete slab is not to exceed 900 psi.
Figure P1.28
Solution
(a) The area of contact between the pipe column and one of the bearing plates is simply the cross-
sectional area of the pipe. To calculate the pipe area, we must first calculate the pipe inside diameter d:
2 2 8.625 in. 2(0.25 in.) 8.125 in.D d t d D t
The pipe cross-sectional area is
2 2 2 2 2pipe (8.625 in.) (8.125 in.) 6.5777 in.
4 4A D d
Therefore, the bearing stress between the pipe and one of the bearing plates is
2
80 kips12.1623 ksi
6.5777 in.12.16 ksib
b
P
A Ans.
(b) The bearing stress between the timber beam and the upper bearing plate must not exceed 500 psi
(i.e., 0.5 ksi). To support a load of 80 kips, the contact area must be at least
280 kips 160 in.
0.5 ksib
b
PA
If the width of the timber beam is 10.75 in., then the length L of the upper bearing plate must be
2160 in.
14.8837 in.beam width 10.75
14.88 in
.in.
bAL Ans.
(c) The bearing stress between the concrete slab and the lower bearing plate must not exceed 900 psi
(i.e., 0.9 ksi). To support the 80-kip pipe load, the contact area must be at least
280 kips 88.8889 in.
0.9 ksib
b
PA
Since the lower bearing plate is square, its dimension a must be
288.8889 in. 9.4 9.43 in281 n. .ibA a a a Ans.
-
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P1.29 A clevis-type pipe hanger supports an 8-in-
diameter pipe as shown in Figure P1.29. The hanger
rod has a diameter of 1/2 in. The bolt connecting the
top yoke and the bottom strap has a diameter of 5/8 in.
The bottom strap is 3/16-in.-thick by 1.75-in.-wide by
36-in.-long. The weight of the pipe is 2,000 lb.
Determine the following:
(a) the normal stress in the hanger rod
(b) the shear stress in the bolt
(c) the bearing stress in the bottom strap
FIGURE P1.29
Solution
(a) The normal stress in the hanger rod is
2 2
rod
rod 2
(0.5 in.) 0.196350 in.4
2,000 lb10,185.917 psi
0.196350 in.10,190 psi
A
Ans.
(b) The cross-sectional area of the bolt is:
2 2
bolt (0.625 in.) 0.306796 in.4
A
The bolt acts in double shear; therefore, its average shear stress is
bolt 22,000 lb
3,259.493 psi2(0.306796 in.
3,260 i)
ps Ans.
(c) The bearing stress in the bottom strap is based on the projected area of the bolt in contact with the
strap. Also, keep in mind that there are two ends of the strap that contact the bolt. The bearing stress is
thus
2,000 lb
8,533.334 psi2(0.625 in.)(3/16
8,530 psiin.)
b Ans.
-
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P1.30 Rigid bar ABC shown in Figure P1.30 is
supported by a pin at bracket A and by tie rod (1).
Tie rod (1) has a diameter of 5 mm, and it is
supported by double-shear pin connections at B and
D. The pin at bracket A is a single-shear
connection. All pins are 7 mm in diameter.
Assume a = 600 mm, b = 300 mm, h = 450 mm, P
= 900 N, and = 55. Determine the following: (a) the normal stress in rod (1)
(b) the shear stress in pin B
(c) the shear stress in pin A
FIGURE P1.30
Solution
Equilibrium: Using the FBD shown, calculate
the reaction forces that act on rigid bar ABC.
1
1
sin(36.87 )(600 mm)
(900 N)sin (55 )(900 mm) 0
1,843.092 N
AM F
F
(1,843.092 N)cos(36.87 ) (900 N)cos(55 ) 0
958.255 N
x x
x
F A
A
(1,843.092 N)sin (36.87 ) (900 N)sin (55 ) 0
368.618 N
y y
y
F A
A
The resultant force at A is
2 2(958.255 N) ( 368.618 N) 1,026.709 NA
(a) Normal stress in rod (1).
2 2
rod
rod 2
(5 mm) 19.635 mm4
1,843.092 N
19.635 mm93.9 MPa
A
Ans.
(b) Shear stress in pin B. The cross-sectional area of a 7-mm-diameter pin is:
2 2
pin (7 mm) 38.485 mm4
A
Pin B is a double shear connection; therefore, its average shear stress is
pin 21,843.092 N
2(38.485 m23.9 M a
)P
mB Ans.
(c) Shear stress in pin A.
Pin A is a single shear connection; therefore, its average shear stress is
pin 21,026.709 N
38.485 m26.7 MPa
mA Ans.
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P1.31 The bell crank shown in Figure P1.31 is in
equilibrium for the forces acting in rods (1) and (2).
The bell crank is supported by a 10-mm-diameter
pin at B that acts in single shear. The thickness of
the bell crank is 5 mm. Assume a = 65 mm, b =
150 mm, F1 = 1,100 N, and = 50. Determine the following:
(a) the shear stress in pin B
(b) the bearing stress in the bell crank at B
FIGURE P1.31
Solution
Equilibrium: Using the FBD shown, calculate the
reaction forces that act on the bell crank.
2
2
(1,100 N)sin(50 )(65 mm)
(150 mm) 0
365.148 N
BM
F
F
(1,100 N)cos(50 )
365.148 N 0
341.919 N
x x
x
F B
B
(1,100 N)sin(50 ) 0
842.649 N
y y
y
F B
B
The resultant force at B is
2 2(341.919 N) ( 842.649 N) 909.376 NB
(a) Shear stress in pin B. The cross-sectional area of the 10-mm-diameter pin is:
2 2
pin (10 mm) 78.540 mm4
A
Pin B is a single shear connection; therefore, its average shear stress is
pin 2909.376 N
78.540 mm11.58 MPaB Ans.
(b) Bearing stress in the bell crank at B. The average bearing stress produced in the bell crank by the
pin is based on the projected area of the pin. The projected area is equal to the pin diameter times the
bell crank thickness. Therefore, the average bearing stress in the bell crank is
909.376 N
(10 mm)(5 mm)18.19 MPab Ans.
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P1.32 The beam shown in Figure P1.32 is
supported by a pin at C and by a short link
AB. If w = 30 kN/m, determine the average
shear stress in the pins at A and C. Each
pin has a diameter of 25 mm. Assume L =
1.8 m and = 35.
FIGURE P1.32
Solution
Equilibrium: Using the FBD shown,
calculate the reaction forces that act on the
beam.
1
1
1.8 msin(35 )(1.8 m) (30 kN/m)(1.8 m) 0
2
47.0731 kN
CM F
F
(47.0731 kN)cos(35 ) 0
38.5600 kN
x x
x
F C
C
1.8 m(1.8 m) (30 kN/m)(1.8 m) 0
2
27.0000 kN
B y
y
M C
C
The resultant force at C is
2 2(38.5600 kN) (27.0000 kN) 47.0731 kNC
Shear stress in pin A. The cross-sectional area of a 25-mm-diameter pin is:
2 2
pin (25 mm) 490.8739 mm4
A
Pin A is a single shear connection; therefore, its average shear stress is
pin 247,073.1 N
490.8739 m95.9 MPa
mA Ans.
Shear stress in pin C.
Pin C is a double shear connection; therefore, its average shear stress is
pin 247,073.1 N
2(490.8739 m47.9 M a
)P
mC Ans.
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P1.33 The bell-crank mechanism shown
in Figure P1.33 is in equilibrium for an
applied load of P = 7 kN applied at A.
Assume a = 200 mm, b = 150 mm, and = 65. Determine the minimum diameter d
required for pin B for each of the following
conditions:
(a) The average shear stress in the pin may
not exceed 40 MPa.
(b) The bearing stress in the bell crank
may not exceed 100 MPa.
(c) The bearing stress in the support
bracket may not exceed 165 MPa.
FIGURE P1.33
Solution
Equilibrium: Using the FBD shown, calculate
the reaction forces that act on the bell crank.
2
2
(7,000 N)sin(65 )(200 mm)
(150 mm) 0
8,458.873 N
BM
F
F
(7,000 N)cos(65 )
8,458.873 N 0
11,417.201 N
x x
x
F B
B
(7,000 N)sin(65 ) 0
6,344.155 N
y y
y
F B
B
The resultant force at B is
2 2( 11,417.201 N) (6,344.155 N) 13,061.423 NB
(a) The average shear stress in the pin may not exceed 40 MPa. The shear area required for the pin
at B is
22
13,061.423 N326.536 mm
40 N/mmVA
Since the pin at B is supported in a double shear connection, the required cross-sectional area for the pin
is
2pin 163.268 mm2
VAA
and therefore, the pin must have a diameter of
24 (163.268 m 14.m 4) 2 mmd
Ans.
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(b) The bearing stress in the bell crank may not exceed 100 MPa. The projected area of pin B on the
bell crank must equal or exceed
22
13,061.423 N130.614 mm
100 N/mmbA
The bell crank thickness is 8 mm; therefore, the projected area of the pin is Ab = (8 mm)d. Calculate the
required pin diameter d:
2130.614 mm
816.
3
mm3 mmd Ans.
(c) The bearing stress in the support bracket may not exceed 165 MPa. The pin at B bears on two 6-
mm-thick support brackets. Thus, the minimum pin diameter required to satisfy the bearing stress limit
on the support bracket is
22
13,061.423 N79.160 mm
165 N/mmbA
279.160 mm
2(6 mm)6.60 mmd Ans.
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P1.34 A structural steel bar with a 25 mm 75 mm rectangular cross section is subjected to an axial
load of 150 kN. Determine the maximum normal and shear stresses in the bar.
Solution
The maximum normal stress in the steel bar is
max(150 kN)(1,000 N/kN)
(25 mm)(75 mm)80 MPa
F
A Ans.
The maximum shear stress is one-half of the maximum normal stress
maxmax
240 MPa
Ans.
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P1.35 A steel rod of circular cross section will be used to carry an axial load of 92 kips. The
maximum stresses in the rod must be limited to 30 ksi in tension and 12 ksi in shear. Determine the
required diameter for the rod.
Solution
Based on the allowable 30 ksi tension stress limit, the minimum cross-sectional area of the rod is
2
min
max
92 kips3.0667 in.
30 ksi
FA
For the 12-ksi shear stress limit, the minimum cross-sectional area of the rod must be
2
min
max
92 kips3.8333 in.
2 2(12 ksi)
FA
Therefore, the rod must have a cross-sectional area of at least 3.8333 in.2 in order to satisfy both the
normal and shear stress limits.
The minimum rod diameter D is therefore
2 2min min3.8333 in. 2.2092 in. 2.21 in.
4d d
Ans.
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P1.36 An axial load P is applied to the
rectangular bar shown in Figure P1.36. The
cross-sectional area of the bar is 400 mm2.
Determine the normal stress perpendicular to
plane AB and the shear stress parallel to
plane AB if the bar is subjected to an axial
load of P = 70 kN.
FIGURE P1.36
Solution
The angle for the inclined plane is 35. The normal force N perpendicular to plane AB is
found from
cos (40 kN)cos35 57.3406 kNN P
and the shear force V parallel to plane AB is
sin (70 kN)sin35 40.1504 kNV P
The cross-sectional area of the bar is 400 mm2, but the area along inclined plane AB is
2
2400 mm 488.3098 mmcos cos35
n
AA
The normal stress n perpendicular to plane AB is
2
(57.3406 kN)(1,000 N/kN)117.4268 MPa
488117.4 MP
.3098 mman
n
N
A Ans.
The shear stress nt parallel to plane AB is
2
(40.1504 kN)(1,000 N/kN)82.2231 MPa
482.2 MP
88.309 a
8 mmnt
n
V
A Ans.
-
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P1.37 An axial load P is applied to the 1.75 in.
by 0.75 in. rectangular bar shown in Figure
P1.37. Determine the normal stress
perpendicular to plane AB and the shear stress
parallel to plane AB if the bar is subjected to
an axial load of P = 18 kips.
FIGURE P1.37
Solution
The angle for the inclined plane is 60. The normal force N perpendicular to plane AB is
found from
cos (18 kips)cos60 9.0 kipsN P
and the shear force V parallel to plane AB is
sin (18 kips)sin60 15.5885 kipsV P
The cross-sectional area of the bar is (1.75 in.)(0.75 in.) = 1.3125 in.2, but the area along inclined plane
AB is
2
21.3125 in./ cos 2.6250 in.cos60
nA A
The normal stress n perpendicular to plane AB is
2
9.0 kips3.4286 ksi
2.6250 in3.43 ks
.in
n
N
A Ans.
The shear stress nt parallel to plane AB is
2
15.5885 kips5.9385 ksi
2.62505.94 ks
ii
n.nt
n
V
A Ans.
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P1.38 A compression load of P = 80 kips is applied to a 4 in. by 4
in. square post, as shown in Figure P1.38/39. Determine the normal
stress perpendicular to plane AB and the shear stress parallel to
plane AB.
FIGURE P1.38/39
Solution
The angle for the inclined plane is 55. The normal force N perpendicular to plane AB is found from
cos (80 kips)cos55 45.8861 kipsN P
and the shear force V parallel to plane AB is
sin (80 kips)sin55 65.5322 kipsV P
The cross-sectional area of the post is (4 in.)(4 in.) = 16 in.2, but the area
along inclined plane AB is
2
216 in./ cos 27.8951 in.cos55
nA A
The normal stress n perpendicular to plane AB is
2
45.8861 kips1.6449 ksi
27.8951 1.645 ksi
in.n
n
N
A Ans.
The shear stress nt parallel to plane AB is
2
65.5322 kips2.3492 ksi
27.89512.35 ks
ii
n.nt
n
V
A Ans.
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P1.39 Specifications for the 50 mm 50 mm square bar
shown in Figure P1.38/39 require that the normal and shear
stresses on plane AB not exceed 120 MPa and 90 MPa,
respectively. Determine the maximum load P that can be
applied without exceeding the specifications.
FIGURE P1.38/39
Solution
The general equations for normal and shear stresses on an inclined plane in terms of the angle are
(1 cos2 )2
n
P
A (a)
and
sin 22
nt
P
A (b)
The cross-sectional area of the square bar is A = (50 mm)2 = 2,500 mm
2, and the angle for plane AB is
55.
The normal stress on plane AB is limited to 120 MPa; therefore, the maximum load P that can be
supported by the square bar is found from Eq. (a):
2 22 2(2,500 mm )(120 N/mm )
911,882 N1 cos2 1 cos2(55 )
nAP
The shear stress on plane AB is limited to 90 MPa. From Eq. (b), the maximum load P based the shear
stress limit is
2 22 2(2,500 mm )(90 N/mm )
478,880 Nsin 2 sin 2(55 )
ntAP
Thus, the maximum load that can be supported by the bar is
max 479 kNP Ans.
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P1.40 Specifications for the 6 in. 6 in. square post shown in
Figure P1.40 require that the normal and shear stresses on
plane AB not exceed 800 psi and 400 psi, respectively.
Determine the maximum load P that can be applied without
exceeding the specifications.
FIGURE P1.40
Solution
The general equations for normal and shear stresses on an inclined plane in terms of the angle are
(1 cos2 )2
n
P
A (a)
and
sin 22
nt
P
A (b)
The cross-sectional area of the square post is A = (6 in.)2 = 36 in.
2, and the angle for plane AB is 40.
The normal stress on plane AB is limited to 800 psi; therefore, the maximum load P that can be
supported by the square post is found from Eq. (a):
22 2(36 in. )(800 psi)
49,078 lb1 cos2 1 cos2(40 )
nAP
The shear stress on plane AB is limited to 400 psi. From Eq. (b), the maximum load P based the shear
stress limit is
22 2(36 in. )(400 psi)
29,244 lbsin 2 sin 2(40 )
ntAP
Thus, the maximum load that can be supported by the post is
max 29,200 l 29.2 ipb k sP Ans.
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P1.41 A 90 mm wide bar will be used to carry an axial
tension load of 280 kN as shown in Figure P1.41. The
normal and shear stresses on plane AB must be limited
to 150 MPa and 100 MPa, respectively. Determine the
minimum thickness t required for the bar.
FIGURE P1.41
Solution
The general equations for normal and shear stresses on an inclined plane in terms of the angle are
(1 cos2 )2
n
P
A (a)
and
sin 22
nt
P
A (b)
The angle for plane AB is 50.
The normal stress on plane AB is limited to 150 MPa; therefore, the minimum cross-sectional area A
required to support P = 280 kN can be found from Eq. (a):
2
2
(280 kN)(1,000 N/kN)(1 cos2 ) (1 cos2(50 )) 771.2617 mm
2 2(150 N/mm )n
PA
The shear stress on plane AB is limited to 100 MPa; therefore, the minimum cross-sectional area A
required to support P = 280 kN can be found from Eq. (b):
2
2
(280 kN)(1,000 N/kN)sin 2 sin 2(50 ) 1,378.7309 mm
2 2(100 N/mm )nt
PA
To satisfy both the normal and shear stress requirements, the cross-sectional area must be at least Amin =
1,379.7309 mm2. Since the bar width is 90 mm, the minimum bar thickness t must be
2
min
1,378.7309 mm15.3192 mm
90 m15.3
m2 mmt Ans.
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