Force equilibrium:P
Px= 0,P
Py= 0,Pz = 0 Moment equilibrium:PMx
= 0,PMy
= 0,PMz
= 0 Transverse shear in beams:V(x) =−
x
R
−∞
q(x)dx Bending moment in beams:M(x) =−
x
R
−∞
V(x)dx
Principal stresses in plane stress:
σ1, σ2=σx+σy
2 ±
r
τxy2 +(σx−σy)2
Mohr’s circle 4
Center:σx+σy
2 ,0 Radius:r=
r
σx−σy
2 2
+τxy2
Octahedral Stresses:
Normal:σoct=σ1+σ2+σ3
3 =σx+σy+σz
Shear: 3 τoct = 1
3 (σ1
−σ2)2+ (σ2−σ3)2+ (σ3−σ1)21/2
= 2 3
τ2
1/2+τ2/32 +τ1/32 1/2 Principal strains in plane strain:
1, 2=x+y
2 ±
s 1
2γxy
2
+x−y
2 2
Recommended Readings
Beer, F.P., Johnson, E.R., DeWolf, J., and Mazurek, D. (2011) Mechanics of Materials, 6th ed., McGraw-Hill.
Craig, R.R. (2011)Mechanics of Materials, 3rd ed., Wiley.
Hibbeler, R.C. (2010)Mechanics of Materials, 8th ed. Prentice- Hall, Upper Saddle River.
Popov, E.P. (1968)Introduction to Mechanics of Solids, Prentice- Hall.
Popov, E.P. (1999) Engineering Mechanics of Solids, 2nd ed., Prentice-Hall.
Riley, W.F., Sturges, L.D., and Morris, D.H. (2006)Mechanics of Materials, 6th ed., Wiley.
Shames, I.H., and Pitarresi, J.M. (2000) Introduction to Solid Mechanics, 3rd ed., Prentice-Hall.
Ugural, A.C. (2007)Mechanics of Materials, Wiley.
Questions
2.1 What is a concentrated load? What is a distributed load?
2.2 What kind of reaction occurs with a roller support?
What occurs with a pin?
2.3 Definestatic equilibrium.
2.4 What is a simply supported beam? What is a cantilever?
2.5 Why are singularity functions useful?
2.6 Under what conditions does a singularity functionnot equal zero?
2.7 Define the termsstressandstrain.
2.8 What is a tensor?
2.9 Definenormal stressandshear stress.
2.10 What is Mohr’s circle?
2.11 What is a principal stress?
2.12 What are the units for stress? What are the units for strain?
2.13 What are octahedral stresses?
2.14 What is elongation?
2.15 What is a rosette?
Qualitative Problems
2.16 Give three examples of (a) static loads; (b) sustained loads; (c) impact loads; and (d) cyclic loads.
2.17 Explain the sign convention for shear forces.
2.18 Explain the common sign conventions for bending mo- ments. Which is used in this book?
2.19 Without the use of equations, explain a methodology for producing shear and moment diagrams.
2.20 Give two examples of scalars, vectors, and tensors.
2.21 Explain the difference between plane stress and plain strain. Give an example of each.
2.22 Without the use of equations, qualitatively determine the bending moment diagram for a bookshelf.
2.23 Explain whyτxy=τyx.
2.24 Define and give two examples of (a) uniaxial stress state;
(b) biaxial stress state; and (c) triaxial stress state.
2.25 Sketch and describe the characteristics of a three- dimensional Mohr’s circle.
2.26 What are the similarities and differences between defor- mation and strain?
2.27 The text stated that0◦–45◦–90◦strain gage rosettes are common. Explain why.
2.28 Draw a free body diagram of a book on a table.
2.29 If the three principal stresses are determined to be 100 MPa,−50MPa and 75 MPa, which isσ2?
2.30 Derive Eq. (2.16).
Quantitative Problems
2.31 The stepped shaft A-B-C shown in Sketch ais loaded with the forcesP1and/orP2. Note thatP1gives a ten- sile stressσin B-C andσ/4 in A-B and thatP2 gives a bending stressσat B and 1.5σat A. What is the critical section
(a) If onlyP1is applied?
(b) If onlyP2is applied?
(c) If bothP1andP2are applied?
l 5l
B
A C
P1 P2
Sketcha, used in Problems 2.31 and 2.32.
2.32 The stepped shaft in Sketchahas loadsP1andP2. Find the load classification ifP1’s variation is sinusoidal and P2is the load from a weight
(a) If onlyP1is applied (b) If onlyP2is applied
(c) If bothP1andP2are applied
2.33 A bar hangs freely from a frictionless hinge. A horizon- tal forceP is applied at the bottom of the bar until it inclines45◦ from the vertical direction. Calculate the horizontal and vertical components of the force on the hinge if the acceleration due to gravity isg, the bar has a constant cross section along its length, and the total mass isma.Ans.Rx= 12mag,Ry=mag.
2.34 Sketchbshows the forces acting on a rectangle. Is the rectangle in equilibrium?Ans.No.
30 N 7 cm
5 cm
20 N
10 N 20 N 18 N
18 N 20 N
A B
C D
Sketchb, used in Problem 2.34
2.35 Sketch cshows the forces acting on a triangle. Is the triangle in equilibrium?Ans.Yes.
17.81 N
65.37 N
39.76 N
40 N
30 N 5 cm
6.5 cm
2.5 cm 30°
10°
Sketchc, used in Problem 2.35
Quantitative Problems 47 2.36 Given the components shown in Sketchesdande, draw
the free-body diagram of each component and calculate the forces.
y B
P = 2 kN
A
0 x
0.9 m
1
2 3
60°
Sketchd, used in Problem 2.36 y
1
2 0.15-m radius
x 30° 60°
Sketche, used in Problem 2.36
2.37 Sketch f shows a cube with side lengthsa and eight forces acting at the corners. Is the cube in equilibrium?
Ans.Yes.
P P
z
x
y P
P a
P P
P P
Sketchf, used in Problem 2.37
2.38 A 5-m-long beam is loaded as shown in Sketchg. The beam cross section is constant along its length. Draw the shear and moment diagrams and locate the critical section.Ans.|Vmax|= 6.8kN,|Mmax|= 6.4kN-m.
2 m 2 m
P = 10 kN P = 0.2 kN
x y
1 m
5 m
Sketchg, used in Problem 2.38
2.39 Sketchhshows a 0.06-m-diameter steel shaft supported by self-aligning bearings at A and B (which can provide radial but not bending loads on the shaft). Two gears attached to the shaft cause applied forces as shown. The shaft weight can be neglected. Determine the forces at A and B and the maximum bending moment. Draw shear and moment diagrams.Ans.|Mmax|=296 Nm.
0.2 m 0.3 m
2 kN
A B
1.0 kN 0.14 m
Sketchh, used in Problem 2.39
2.40 A beam is loaded as shown in Sketchi. Determine the reactions and draw the shear and moment diagrams for P = 500N.Ans. Ay =By = 1000N,|Mmax|= 2000 Nm.
P
x y
4 m 5 m 5 m 4 m
P 2P
Sketchi, used in Problem 2.40
2.41 Sketchjshows a simply supported beam loaded with a forceP at a position one-third of the length from one of the supports. Determine the largest shear force and bending moment in the beam. Also, draw the shear and moment diagrams.Ans.|Mmax|=29P l.
P
x y
l/3
l
Sketchj, used in Problem 2.41
2.42 Sketchkshows a simply supported beam loaded by two equally large forces P at a distancel/4from its ends.
Determine the largest shear force and bending moment in the beam, and find the critical location with respect to bending. Also, draw the shear and moment diagrams.
Ans.|Mmax|= 14P l.
P P
l/4 l/2 l/4
x y
Sketchk, used in Problem 2.42
2.43 The beam shown in Sketch lis loaded by the forceP.
Draw the shear and moment diagrams for the beam, in- dicating maximum values.Ans.|M|max=P l.
P
l l
x y
Sketchl, used in Problem 2.43
2.44 Sketchmshows a simply supported beam with a con- stant load per unit length, wo, imposed over its entire length. Determine the shear force and bending moment as functions ofx. Draw a graph of these functions. Also, find the critical section with the largest bending mo- ment.Ans.Mmax=18wol2.
y
wo
x
l
Sketchm, used in Problem 2.44
2.45 Sketchnshows a simply supported beam loaded with a ramp function over its entire length, the largest value beingP/l. Determine the shear force and the bending moment and the critical section with the largest bending moment. Also, draw the shear and moment diagrams.
Ans.|Mmax|= 9√32 P l.
y
x
l
P_l
Sketchn, used in Problem 2.44
2.46 The simply supported beam shown in SketchohasP1
= 5 kN, P2 = 10 kN,wo = 5 kN/m, andl = 12 m. Use singularity functions to determine the shear force and bending moment as functions ofx. Also, draw the shear force and bending moment diagrams. Ans. |Mmax| = 52.5 kN-m.
x
wosin(πx/l) y
Sketcho, used in Problem 2.46
2.47 Sketchpshows a sinusoidal distributed force applied to a beam. Determine the reactions and largest shear force and bending moment for each section of the beam.Ans.
Reactions =lwπo,|Mmax|= l22πwo.
x
wosin(πx/l) y
Sketchp, used in Problem 2.47
2.48 Find the lengthcthat gives the smallest maximum bend- ing moment for the load distribution shown in Sketchq.
Ans.x= 0.207l.
l wo
c c
Sketchq, used in Problem 2.48
2.49 Draw the shear and moment diagrams and give the re- action forces for the load distribution shown in Sketchr.
Ans.R=woa,Mmax=12woa2.
wo wo
a 2 a a
Sketchr, used in Problem 2.49
2.50 Use singularity functions for the force system shown in Sketch s to determine the load intensity, the shear force, and the bending moment in the beam. From a force analysis determine the reaction forces R1 and R2. Also, draw the shear and moment diagrams. Ans.
R1=−71N,R2=221 N,|M|max=29.2 N-m.
y
100 mm 100 mm x 150 N
250 N R2
R1
A B C
0
250 N 150 mm 100 mm D
Sketchs, used in Problem 2.50
Quantitative Problems 49 2.51 Use singularity functions for the force system shown in
Sketchtto determine the load intensity, the shear force, and the bending moment. Draw the shear and moment diagrams. Also, from a force analysis determine the re- action forcesR1andR2.Ans.R1=361 N,R2 =189 N, Mmax=48.2 N-m.
y
75 mm 100 mm x
25 mm 150 N 250 N
150 N
A BC
0 250 mm D
R2 R1
Sketcht, used in Problem 2.51
2.52 Draw a free-body diagram of the forces acting on the simply supported beam shown in Sketchu, withwo= 6 kN/m andl=10 m. Use singularity functions to draw the shear force and bending moment diagrams. Ans.
Mmax=63.89 kN-m.
y
x
wo wo
l/3 l/3 l/3
Sketchu, used in Problem 2.52 2.53 Repeat Example 2.8 using singularity functions.
2.54 Sketchvshows a simply supported beam withwo = 6 kN/m andl= 10m. Draw a free-body diagram of the forces acting along the beam as well as the coordinates used. Use singularity functions to determine the shear force and the bending moment.Ans.Mmax= 25kN-m.
wo y
x wo l/2
l/2
Sketchv, used in Problem 2.54
2.55 An additional concentrated force with an intensity of 20 kN is applied downward at the center of the simply sup- ported beam shown in Sketchv. Draw a free-body dia- gram of the forces acting on the beam. Assumel = 10 m andwo = 5kN/m. Use singularity functions to de- termine the shear force and bending moments and draw the diagrams.Ans.Mmax= 75kN-m.
2.56 Draw a free-body diagram of the beam shown in Sketch wand use singularity functions to determine the shear force and the bending moment diagrams. Determine the maximum moment.Ans.|M|max|= 16.19kN-m.
4 m 1.5 m 1.5 m
3 kN/m 6 kN
Sketchw, used in Problems 2.56 and 2.57 2.57 Use direct integration to determine the shear force
and bending moment diagrams for the beam shown in Sketch w. Determine the maximum moment. Ans.
|M|max|= 16.19kN-m.
2.58 Draw the shear and bending moment diagrams for the beam shown in Sketchx. Determine the magnitude and location of the maximum moment. Ans. |Mmax| = 2.5 kN-m.
M = 5 kN-m
700 mm 700 mm
Sketchx, used in Problem 2.58
2.59 Using singularity functions, draw the shear and mo- ment diagrams for the beam shown in Sketchy. Use P = 20kN andl= 4m.Ans.|Mmax|= 40kN-m.
P l/2
l/2
P
Sketchy, used in Problem 2.59
2.60 Determine the location and magnitude of maximum shear stress and bending moment for the beam shown in Sketchz. Use wo = 10kN/m andl = 5m. Ans.
|Vmax|= 25kN,|Mmax|= 95.75kN-m.
l/2 l/2
wo
Sketchz, used in Problem 2.60
2.61 Sketch the shear and bending moment diagrams for the beam shown in Sketch aa. Determine the maximum shear force and bending moment.Ans.|Vmax|= 18kN,
|Mmax|= 55kN-m.
2 kN/m 10 kN 4 kN
2 m 1.5 m 0.5 m
Sketchaa, used in Problem 2.61
2.62 A steel bar is loaded by a tensile forceP = 20 kN. The cross section of the bar is circular with a radius of 10 mm. What is the normal tensile stress in the bar? Ans.
σ= 63.66MPa.
2.63 A stainless-steel bar of square cross section is subjected to a tensile force ofP = 10kN. Calculate the required cross section to provide a tensile stress in the bar of 90 MPa.Ans.l= 10.54mm.
2.64 What is the maximum length,lmax, of a copper wire if its weight should not produce a stress higher than 70 MPa when it is hanging vertically? The density of copper is 8900 kg/m3, and the density of air is so small relative to that of copper that it may be neglected. The acceleration of gravity is9.81m/s2.Ans.lmax= 801m.
2.65 A machine with a mass of 5000 kg will be lifted by a steel rod with an ultimate tensile strength of 860 MPa. A safety factor of 4 is to be used. Determine the diameter needed for the steel rod.Ans.d= 17.04mm.
2.66 A string on a guitar is made of nylon and has a diameter of 0.5 mm. It is tightened with a forceP = 12N. What is the stress in the string?Ans.σ= 61.12MPa.
2.67 Determine the normal and shear stresses due to axial and shear forces at sections A and B in Sketchbb. The cross sectional area of the rod is 0.025 m2andθ = 30◦. Ans. At section AA,σ = 200kPa,τ = 346.4kPa. At section BB,τ=400 kPa.
1 m
1 m
P = 10 kN A
A
B B
θ
Sketchbb, used in Problem 2.67
2.68 Determine the normal and shear stresses in sections A and B of Sketchcc. The cross-sectional area of the rod is 0.00250 m2. Ignore bending and torsional effects. Ans.
In AA,σ=−3.464MPa,τ =−2.00MPa.
0.3 m 0.3 m 2 m
1 m
10 kN x’
z’
y’
30°
45°
B B
x z
A y A
Sketchcc, used in Problem 2.68
2.69 Sketch dd shows a distributed load on a semi-infinite plane. The stress in polar coordinates based on plane stress is
σr=−2wocosθ πr σθ=τrθ=τθr= 0
Determine the expressionsσx,σy, andτxyin terms ofr andθ.Ans.σx=-2wocos3θ
πr .
r σr σθ
θ
x y
P
Sketchdd, used in Problem 2.69
2.70 Sketcheeshows loading of a thin but infinitely wide and long plane. Determine the angleθ needed so that the stress element will have no shear stress.Ans.θ= 0.
y
x
θ σx
σy
Sketchee, used in Problem 2.70
Quantitative Problems 51 2.71 A stress tensor is given by
S=
200 40 0 40 25 0
0 0 0
where all values are in megapascals. Calculate the prin- cipal normal stresses and the principal shear stresses.
Ans.σ1= 208.7MPa,σ2= 16.29MPa,σ3= 0.
2.72 A thin, square steel plate is oriented with respect to the x- and y-directions. A tensile stress σ acts in the x-direction, and a compressive stress−σacts in they- direction. Determine the normal and shear stresses on the diagonal of the square.Ans.σ45◦=0,τ45◦=−σ.
2.73 A thin, rectangular brass plate has normals to the sides in thex- andy-directions. A tensile stressσacts on the four sides. Determine the principal normal and shear stresses.Ans.σ1=σ2=σ,τ=0.
2.74 Given the thin, rectangular brass plate in Problem 2.73, but with the stress in they-direction beingσy=−σin- stead of+σ, determine the principal normal and shear stresses and their directions. Ans. σ1 = −σ2 = σ, τ =±σ.
2.75 For the following stress states, sketch the stress element, draw the appropriate Mohr’s circle, determine the prin- cipal stresses and their directions, and sketch the princi- pal stress elements:
(a) σx = 8,σy= 14, andτxy = 4.Ans.σ1 = 16MPa, σ2= 6MPa.
(b) σx=−15,σy= 9, andτxy= 5.Ans.σ1= 10MPa, σ2=−16MPa.
(c) σx = 12,σy = 28, andτxy = 15. Ans. σ1 = 35 MPa,σ2= 5MPa.
(d) σx =−54,σy= 154, andτxy=−153. Ans. σ1 = 235MPa,σ2=−135MPa.
All stresses are in megapascals.
2.76 Repeat Problem 2.75 for
(a) σx=σy=−10, andτxy= 0.Ans.τxy= 0MPa.
(b) σx= 0,σy= 30, andτxy= 20.Ans.σ1= 40MPa, σ2=−10MPa.
(c) σx =−20,σy= 40, andτxy =−40.Ans.σ1 = 60 MPa,σ2=−40MPa.
(d) σx = 30,σy = 0, andτxy =−20. Ans. σ1 = 40 MPa,σ2=−10MPa.
All stresses are in megapascals.
2.77 Given the state of stresses shown in the two parts of Sketchf fdetermine the principal stresses and their di- rections by using Mohr’s circle and the stress equations.
Show the stress elements. All stresses in Sketchf fare in megapascals.Ans.(a)σ1= 34MPa,σ2=−38MPa.
(a) 45°
20 28
30
(b) 30°
60
20
40
Sketchff, used in Problem 2.77
2.78 A certain loading on a machine element leads to a stress state ofσx=a,σy=a/2, andτxy=a/4. What value of aresults in the maximum allowable shear stress of 100 MPa?Ans.a= 282.8MPa.
2.79 Given the normal and shear stressesσx= 66MPa,σy= 34MPa, andτxy =− −63MPa, draw the Mohr’s circle diagram and the principal normal and shear stresses on thex-yaxis. Determine the triaxial stresses and give the corresponding Mohr’s circle diagram. Ans. σ1 = 115 MPa,σ2= 0,σ3=−15MPa.
2.80 Given the normal and shear stressesσx = 0, σy = 10 MPa, andτxy = 12MPa, draw the Mohr’s circle dia- gram and the principal normal and shear stresses on the x-yaxis. Determine the triaxial stresses and give the cor- responding Mohr’s circle diagram. Ans. σ1 = 18MPa, σ2= 0,σ3 =−8MPa.
2.81 Given the normal and shear stressesσx= 72MPa,σy=
−72MPa, andτxy =−65MPa, draw the Mohr’s circle diagram and the principal normal and shear stresses on thex-yaxis. Determine the triaxial stresses and give the corresponding Mohr’s circle diagram. Ans. σ1 = 97 MPa,σ2= 0,σ3=−97MPa.
2.82 A stress element in plane stress encounters σx = 20 MPa,σy=−10MPa, andτxy= 13MPa. (a) Determine the three principal stresses and maximum shear stress.
(b) Using a Mohr’s circle diagram, explain the effect of superimposing a hydrostatic pressurepon the principal stresses and maximum shear stress.Ans.(a)σ1= 24.85 MPa,σ2= 0,σ3=−14.85MPa.
2.83 In a three-dimensional stress field, the stresses are found to be σx = 40 MPa, σy = 20 MPa, σz = 60 MPa, τxy = −20MPa, τyz = 0, andτxz = 20MPa. Draw the stress element for this case. Determine the princi- pal stresses and sketch the corresponding Mohr’s cir- cle diagram. Ans. σ1 = 70.64MPa,σ2 = 46.94MPa, σ3= 2.412MPa.
2.84 Given the normal and shear stresses σx = −36MPa, σy= 60MPa, andτxy= 20MPa, determine or draw the following:
(a) Two-dimensional Mohr’s circle diagram.
(b) Normal principal stress element in thex-yplane.
(c) Shear principal stress.
(d) Three-dimensional Mohr’s circle diagram and cor- responding principal normal and shear stresses.
Ans.σ1= 64MPa,σ2= 0,σ3 =−40MPa.
2.85 The strain tensor in a machine element is
T=
0.0012 −0.0001 0.0007
−0.0001 0.0003 0.0002 0.0007 0.0002 −0.0008
Find the strain in thex-,y-, andz-directions, in the di- rection of the space diagonal
√1 3;√1
3;√1
3
, and in the directionx,y, andz.Ans.In the direction of the diag- onal,= 0.0005.
2.86 A strain tensor is given by
T=
0.0023 0.0006 0 0.0006 0.0005 0
0 0 0
Calculate the maximum shear strain and the principal strains.Ans.1= 0.00248,2= 0.00032.
Design and Projects
2.87 Without using the words “stress” or “strain,” defineelas- tic modulus.
2.88 A bookshelf sees a uniform distributed load across its entire length, and is supported by two brackets. Where should the brackets be located?Hint:See Problem 2.48.
2.89 For the beam shown in Sketchgg, determine the force Pso that the maximum bending moment in the beam is as small as possible. What is the value of|M|max? Ans.
|Mmax|= 12wol2
wo
P l
Sketchgg, used in Problem 2.89
2.90 Three- and four-point bending tests are common tests used to evaluate materials.
(a) Sketch the shear and bending moment diagrams for each test.
(b) Is there any difference in the stress state for the two tests?
(c) Which test will cause specimens to fail at a lower bending moment? Why?
2.91 Sketchhhshows a beam that is fixed on both ends with a central load. Can you determine the shear and mo- ment diagrams for this case? If not, explain what addi- tional information you would need and how you would go about solving this problem.
P
l/2 l/2
Sketchhh, used in Problem 2.91
2.92 For a0◦–60◦–120◦strain gage rosette with one base on thex-axis, derive the strainsx,y, andγxyas a function of0,60, and120. Ans. x =0,y = 23(60+120)−
1 30.
2.93 A stress tensor is given by
S=
200 40 −30 40 25 10
−30 10 −25
where all values are in megapascals. Calculate the prin- cipal normal stresses and the principal shear stresses.
Ans. σ1 = 211.8MPa,σ2 = 21.46MPa,σ3 = −33.31 MPa.