Design Procedure 5.2: Procedure for Using Castigliano’s Theorem
5.7 Summary
The three main failure modes for machine elements are (a) from being overstressed, (b) from excessive elastic deforma- tions, and (c) from the lack of a tribological film. This chap- ter described the deformations that machine elements, es- pecially beams, may experience. Deformations due to dis- tributed and concentrated loads were both considered. For a distributed load four major approaches to describing the de- formations were presented: the moment-curvature relation, singularity functions, the method of superposition, and Cas- tigliano’s theorem. Each has its particular strengths and lim- itations. The type of load being applied (normal, bending, shear, or transverse shear) determines the approach. Cas- tigliano’s theorem is the most versatile of the four approaches considered, since it can be applied to a wide range of deflec- tion problems.
Key Words
Castigliano’s theorem theorem that when a body is elasti- cally deformed by a system of loads, deflection at any point in any direction is equal to the partial derivative of strain energy (for the system of loads) with respect to load in the direction of interest
method of superposition principle that deflection at any point in beam is equal to sum of deflections caused by each load acting separately
moment-curvature relation relationship between beam cur- vature and bending moment
radius of curvature distance from center to inside edge of beam in bending
singularity function function that permits expressing in one equation what would normally be expressed in several separate equations with boundary conditions
strain energy internal work that was converted from exter- nal work done by applying load
Qualitative Problems 127
Summary of Equations
Beam equations:
Radius of curvature: d2y dx2 = M
EI Load intensity: q
EI = d4y dx4
Shear force:−V EI = d3y
dx3 =Rx 0 q(x)dx Moment: M
EI = d2y dx2 =−Rx
0 V(x)dx Slope:θ=dy
Strain energy: dx
Normal stress:U = P2l 2AE Bending stress:U =Rl
0
M2 2EIdx Torque:U= T2l
2GJ
Transverse shear:U= 3V2l
5Gbh(rectangular cross section) Castigliano’s Theorem:yi= ∂U
∂Qi
Recommended Readings
Beer, F.P., Johnson, E.R., DeWolf, J., and Mazurek, D. (2011) Mechanics of Materials, 6th ed., McGraw-Hill.
Budynas, R.G., and Nisbett, J.K. (2011),Shigley’s Mechanical Engineering Design, 9th ed., McGraw-Hill.
Craig, R.R. (2001)Mechanics of Materials, 2nd ed., Wiley.
Hibbeler, R.C. (2010)Mechanics of Materials, 8th ed. Prentice- Hall.
Juvinall, R.C., and Marshek, K.M. (2012)Fundamentals of Ma- chine Component Design, 5th ed., Wiley.
Norton, R.L. (2011)Machine Design, 4th ed., 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.
Ugural, A.C. (2007)Mechanics of Materials, Wiley.
Questions
5.1 How are bending moment and deflection related in beams?
5.2 What is the moment-curvature relation?
5.3 How can one obtain deflection in a beam?
5.4 How does stress depend on the radius of curvature in a beam?
5.5 What are singularity functions?
5.6 What is the Method of Superposition?
5.7 What is the difference between strain and strain energy?
5.8 What is Castigliano’s Theorem?
5.9 Why was a fictitious load used in Example 5.9?
5.10 What are the units of slope in Eq. (5.6)?
Qualitative Problems
5.11 It was mentioned in the text that the radius of curvature in a beam is measured from the center of curvature to the inside surface of the beam. Can the radius of curvature ever equal zero? Explain.
5.12 Design Procedure 2.1 discussed singularity functions.
Which of the rules are useful for the material presented in this chapter?
5.13 In general, what method for calculating beam deflection would you use for an impact loading?
5.14 Can the Method of Superposition be used for impact loads? Explain.
5.15 Could you use Castigliano’s Theorem in Example 5.9 if the bars are replaced by cables? Why or why not?
5.16 List the strengths and weaknesses of singularity func- tions compared to direct integration in order to obtain beam deflection.
5.17 How can Castigliano’s Theorem be used for statically in- determinate beams?
5.18 Define Castigliano’s Theorem without the use of equa- tions.
5.19 Can Castigliano’s Theorem be used for viscoelastic ma- terials? Explain.
5.20 Assume that the summary of the chapter is not present and write a suitable one- or two-page summary.
Quantitative Problems
5.21 A beam is loaded by a concentrated bending momentM at the free end. Find the vertical and angular deforma- tions along the beam by using the equation of the elastic line, Eq. (5.3).Ans.y=−M x
2
2EI.
5.22 A simply supported beam of lengthlcarries a forceP. Find the ratio between the bending stresses in the beam whenP is concentrated in the middle of the beam and evenly distributed along it. Use the moment-curvature relation given in Eq. (5.3). Also, calculate the ratio of the deformations at the middle of the beam. Ans. yconc
ydist = 1.6.
5.23 A simply supported beam with length l is centrally loaded with a forceP. How large a moment needs to be applied at the ends of the beam
(a) To maintain the slope angle of zero at the sup- ports?Ans.Mo =1
8P l.
(b) To maintain the midpoint of the beam without de- formation when the load is applied? Use the equa- tion of the elastic line, Eq. (5.3).Ans.Mo=1
6P l
P wo wo
2l l
2 l
2 l
2 Sketcha, for Problem 5.24
5.24 Find the relation betweenP andwoso that the slope of the deflected beam is zero at the supports for the loading conditions shown in Sketcha. Assume thatEandAare constant.Ans.P =2
3wol.
5.25 Given a simply supported beam with two concentrated forces acting on it as shown in Sketchb, determine the expression for the elastic deformation of the beam for anyxby using singularity functions. Assume thatEand Iare constant. Also determine the location of maximum deflection and derive an expression for it.
l
a a
P P
Sketchb, for Problem 5.25
5.26 For the loading condition described in Sketchcobtain the internal shear force V(x)and the internal moment M(x)by using singularity functions. DrawV(x),M(x), q(x), andy(x)as a function ofx. Assume thatwo = 9 kN/m andl= 3m.
y
x wo
3l l
3 l
3 Sketchc, for Problem 5.26
5.27 A simply supported beam is shown in Sketch dwith wo= 4kN/m andl= 12m.
(a) Draw the free-body diagram of the beam.
(b) Use singularity functions to determine shear force, bending moment, slope, and deflection.
(c) Construct diagrams of shear force, bending mo- ment, slope, and deflection.
y
x wo
l/2 l/4 l/4
Sketchd, for Problem 5.27
5.28 The simply supported beam in Problem 5.25 is altered so that instead of a concentrated forceP, a concentrated moment M is applied at the same location. The mo- ments are positive and act parallel with each other. De- termine the deformation of the beam for any positionx along it by using singularity functions. Assume thatE andI are constant. Also, determine the location of the maximum deflection.
5.29 The simply supported beam considered in Problem 5.28 has moments applied in opposite directions so that the moment atx=aisMoand atx=l−athe moment is
−Mo. Find the elastic deformation of the beam by using singularity functions. Also, determine the location and size of the maximum deflection.
5.30 Given the loading condition shown in Sketche, find the deflection at the center and ends of the beam. Assume thatEI= 750 kNm2. Ans. yend =−0.05477m,ymid = 0.01359m.
2 m
2 m 3 m
5 kN
1 kN/m
5 kN
Sketche, for Problem 5.30
5.31 Given the loading condition shown in Sketchf obtain an expression for the deflection at any location on the beam. Assume thatEIis constant.
a a 2a k
P = woa
wo
Sketchf, for Problem 5.31
5.32 Given the loading condition and spring shown in Sketch gdetermine the stiffness of the spring so that the bend- ing moment at point B is zero. Assume thatEIis con- stant.Ans.k= 2EI
l3 .
Quantitative Problems 129
B
l l l l
k P
Sketchg, for Problem 5.32
5.33 When there is no load acting on the cantilevered beam shown in Sketchh, the spring has zero deflection. When there is a spring and a force of 20 kN is applied at point C, a deflection of 25 mm occurs at the spring. If a 50- kN load is applied at the location shown in Sketchh, what will be the deflection of the beam? Assume that the stiffness of the spring is 450 kN/m.
C 50 kN
k = 450 kN/m
2 m 1 m 1 m
Sketchh, for Problem 5.33
5.34 Determine the deflection at point A and the maximum moment for the loading shown in Sketchi. Consider only bending effects and assume that EI is constant.
Ans.δH=P r3π 8EI .
A P r
Sketchi, for Problem 5.34
5.35 Determine the maximum deflection of the beam shown in Sketchj.Ans.ymax= wol2
24EI 9l2
+ 20al+ 12a2. wo
a
B
l
Sketchj, for Problem 5.35
5.36 Determine the deflection at any point in the beam shown in Sketchkusing singularity functions.
A B k
P 3l 2l3
Sketchk, for Problem 5.36
5.37 Determine the deformation of a cantilevered beam with loading shown in Sketchlas a function ofx. Also de- termine the maximum bending stress in the beam and the maximum deflection. Assume that E = 207 GPa, I = 2.50×10−6 m4,P = 1000N,wo = 3000N/m, a= 0.5m,b= 0.15m,c= 0.45m. The distance from the neutral axis to the outermost fiber of the beam is 0.040 m.
Ans.ymax=−0.5826mm,σmax= 33.44MPa.
wo P
E, I
x
a b c
Sketchl, for Problem 5.37
5.38 Given the loading shown in Sketchm, let a = 0.6m, b = 0.7m,M = 6500N-m, andwo = 20,000N/m.
The beam has a square cross section with sides of 75 mm and the beam material has a modulus of elasticity of 207 GPa. Determine the beam deformation by using the method of superposition. Also, calculate the maxi- mum bending stress and maximum beam deformation.
Ans.ymax=−3.988mm,σmax= 96.71MPa.
wo
x y
a b
M
Sketchm, for Problem 5.38
5.39 The cantilevered beam shown in Sketch nhas both a concentrated force and a moment acting on it. Leta= 1 m,b = 0.7m,P = 8700N, andM = 4000N-m. The beam cross section is rectangular with a height of 80 mm and a width of 35 mm. Also,E = 207GPa. Calculate the beam deformation by using the method of superpo- sition. Find how largeM has to be to give zero defor- mation atx=a.Ans.M = 5800Nm.
x y
a b
M P
Sketchn, for Problem 5.39
5.40 A simply supported beam has loads as shown in Sketch o. Calculate the beam deflection by using the method of superposition. Also, calculate the maximum bend- ing stress and the deflection at mid-span. Assume that E = 207GPa and that the beam has a rectangular cross section with a height of 30 mm and a width of 100 mm.
Also,P= 1200 N,wo= 10,000N/m,a= 0.2m,b= 0.1 m,c= 0.4m, andd= 0.2m. Ans. σmax = 74.47MPa, ymax=−0.7277mm.
a b c d
x
y P
wo
RA RB
Sketcho, for Problem 5.40
5.41 The beam shown in Sketchpis fixed at both ends and center loaded with a force of 2300 N. The beam is 3.2 m long and has a square tubular cross section with an outside width of 130 mm and a wall thickness of 10 mm. The tube material is AISI 1080 high-carbon steel.
Calculate the deformation along the beam by using the method of superposition. What is the deformation at mid-span?Ans.y(x= 1.6m) = 0.1593mm.
l = 3.2 m P
Sketchp, for Problem 5.41
5.42 Beam A shown in Sketch q is a 13-mm-diameter alu- minum beam; beam B is an 8-mm-diameter steel beam.
The lower member is of uniform cross section and is as- sumed to be rigid. Find the distancexif the lower mem- ber is to remain horizontal. Assume that the modulus of elasticity for steel is three times that for aluminum.Ans.
x= 0.564m.
450 mm 450 mm
150 mm
x
A B
W
Sketchq, for Problem 5.42
5.43 An aluminum rod 20 mm in diameter and 1.2 m long and a nickel steel rod 10 mm in diameter and 0.8 m long are spaced 1.5 m apart and fastened to a horizontal beam
that carries an 8000-N load, as shown in sketchr. The beam is to remain horizontal after load is applied. As- sume that the beam is weightless and absolutely rigid.
Find the locationxof the load and determine the stresses in each rod.Ans.x= 0.821 m,σA= 11.5 MPa,σB= 55.76 MPa.
1.2 m
0.8 m 1.5 m
x
8000 N 20 mm-diam aluminum rod
10 mm-diam steel rod
Sketchr, for Problem 5.43
5.44 Find the force on each of the vertical wires shown in sketchs. The weight is assumed to be rigid and horizon- tal, implying that the three vertical bars are connected to the weight in a straight line. Also, assume that the sup- port at the top of the bars is rigid. The bar materials and its circular cross-sectional area are given in the sketch.
Ans.Ps= 3419N,PB = 3136N.
250 mm
10,000 N Steel
(12 mm dia.) Steel
(12 mm dia.) Al-Bronze
(15 mm dia.)
100 mm 250 mm
100 mm 1 m
Sketchs, for Problem 5.44
5.45 Two solid spheres, one made of aluminum alloy 2014 and the other made of AISI 1040 medium-carbon steel, are lowered to the bottom of the sea at a depth of 10,000 m. Both spheres have a diameter of 0.3 m. Calculate the elastic energy stored in the two spheres when they are at the bottom of the sea if the density of water is 1000 kg/m3 and the acceleration of gravity is 9.807 m/s2. Also, calculate how large the steel sphere has to be to have the same elastic energy as the 0.3-m-diameter alu- minum sphere. Ans. Ual = 960.6Nm,Us = 393.1Nm.
5.46 Use Castigliano’s approach instead of singularity func- tions to determine the maximum deflection of the beam considered in Problem 5.25. Assume that transverse shear is negligible.
5.47 Using Castigliano’s Theorem, find the maximum deflec- tion of the two-diameter cantilevered beam shown in Sketcht. Neglect transverse shear.Ans.y= 3P l3
16EI.
Quantitative Problems 131
l/2 l/2
P
I x
2I
Sketchtfor Problem 5.47
5.48 The right-angle-cantilevered bracket shown in Sketch u is loaded with force P in the z-direction. Derive an expression for the deflection of the free end in the z-direction by using Castigliano’s Theorem. Neglect transverse shear effects.
h y
Solid round rod
l
z x
b c
a P
Sketchufor Problem 5.48
5.49 A triangular cantilevered plate is shown in Sketch v.
Use Castigliano’s Theorem to derive an expression for the deflection at the free end, assuming that transverse shear is neglected.Ans.δ= 6P l3
Ebh3.
h
x l
y
b P 2
b2
Sketchvfor Problem 5.49
5.50 A right-angle-cantilevered bracket with concentrated load and torsional loading at the free end is shown in Sketchw. Using Castigliano’s Theorem, find the deflec- tion at the free end in thez-direction. Neglect transverse shear effects.
y
Solid round rod of properties E, G, A,
I, and J
z x
b a
P T
Sketchwfor Problem 5.50
5.51 A cantilevered I-beam has a concentrated load applied to the free end as shown in Sketchx. What upward force at point S is needed to reduce the deflection at S to zero?
Use Castigliano’s Theorem. Transverse shear can be ne- glected.Ans.Sy= 10kN.
500 mm
300 mm S x
5 kN
Sketchxfor Problem 5.51
5.52 Using Castigliano’s Theorem calculate the horizontal and vertical deflections at point A shown in Sketchy.
Assume thatEandAare constant.
P θ r
A
Sketchyfor Problem 5.52
5.53 Calculate the deflection at the point of load application and in the load direction for a load applied as shown in Sketchz. Assume thatEandIare constant.
l
P
l/4
Sketchz, for Problem 5.53
5.54 Using Castigliano’s Theorem determine the horizontal and vertical deflections at point A of Sketchaa. Assume thatEandIare constant.
r A
45°
Sketchaafor Problem 5.54
5.55 For the structure shown in Sketch bbfind the force in each member and determine the deflection at point A.
Assume thatEandAare the same in each member.
P
P
θ θ
θ θ
A
Sketchbbfor Problem 5.55
5.56 Obtain the maximum deflection of the beam given in Problem 2.59 as a function of the loadP, span l, mo- ment of inertia,I, and elastic modulus,E. Ans.ymax =
−0.2292P l3.
Design and Projects
5.57 Construct a Design Procedure for the Method of Super- position.
5.58 Design an experiment to verify the energy stored in a beam is as given in Table 5.1.
5.59 Sketchccshows a split ring used as a compression ring on an automotive piston (see also Fig. 12.31). To install the ring, it is necessary to open a gapδas shown. IfEI of the cross section is constant, derive the required force Pas a function of ring radius.Ans.P = δEI
3πr3.
r
P P
δ +
Sketchcc, for Problem 5.59
5.60 Consider the situation of a golf club striking a golf ball.
Set up a system of equations to describe the deformation and motion of the golf ball.
5.61 Table 5.1 has an entry for the strain energy associated with transverse shear for a rectangular beam. Why is there no entry for circular cross-sections? Explain.
5.62 Assume you are the instructor of a course covering the subject matter in this chapter. Prepare two qualitative and two quantitative problems for the chapter and pro- vide solutions.