Budynas-Nisbett:Shigley's I.Basics 4.Deflection and Stiffness ©The McGraw-Hfll Mechanical Engineering Companies,2008 Design,Eighth Edition Deflection and Stiffness Chapter Outline 41 Spring Rates 142 4-2 Tension,Compression,and Torsion 143 4-3 Deflection Due to Bending 144 4-4 Beam Deflection Methods 146 4-5 Beam Deflections by Superposition 147 4-6 Beam Deflections by Singularity Functions 150 4-7 Strain Energy 156 4-8 Castigliano's Theorem 158 4-9 Deflection of Curved Members 163 4-10 Statically Indeterminate Problems 168 4-11 Compression Members-General 173 4-12 Long Columns with Central Loading 173 4-13 Intermediate-Length Columns with Central Loading 176 4-14 Columns with Eccentric Loading 176 4-15 Struts or Short Compression Members 180 4-16 Elastic Stability 182 4-17 Shock and Impact 183 4-18 Suddenly Applied Loading 184 141
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition I. Basics 4. Deflection and Stiffness © The McGraw−Hill 145 Companies, 2008 4 Deflection and Stiffness Chapter Outline 4–1 Spring Rates 142 4–2 Tension, Compression, and Torsion 143 4–3 Deflection Due to Bending 144 4–4 Beam Deflection Methods 146 4–5 Beam Deflections by Superposition 147 4–6 Beam Deflections by Singularity Functions 150 4–7 Strain Energy 156 4–8 Castigliano’s Theorem 158 4–9 Deflection of Curved Members 163 4–10 Statically Indeterminate Problems 168 4–11 Compression Members—General 173 4–12 Long Columns with Central Loading 173 4–13 Intermediate-Length Columns with Central Loading 176 4–14 Columns with Eccentric Loading 176 4–15 Struts or Short Compression Members 180 4–16 Elastic Stability 182 4–17 Shock and Impact 183 4–18 Suddenly Applied Loading 184 141
146 Budynas-Nisbett:Shigley's I.Basics 4.Deflection and Stiffness T©The McGraw-Hill Mechanical Engineering Companies,2008 Design,Eighth Edition 142 Mechanical Engineering Design All real bodies deform under load,either elastically or plastically.A body can be suffi- ciently insensitive to deformation that a presumption of rigidity does not affect an analy- sis enough to warrant a nonrigid treatment.If the body deformation later proves to be not negligible,then declaring rigidity was a poor decision,not a poor assumption.A wire rope is flexible,but in tension it can be robustly rigid and it distorts enormously under attempts at compressive loading.The same body can be both rigid and nonrigid. Deflection analysis enters into design situations in many ways.A snap ring,or retain- ing ring,must be flexible enough to be bent without permanent deformation and assembled with other parts,and then it must be rigid enough to hold the assembled parts together.In a transmission,the gears must be supported by a rigid shaft.If the shaft bends too much,that is,if it is too flexible,the teeth will not mesh properly,and the result will be excessive impact,noise,wear,and early failure.In rolling sheet or strip steel to pre- scribed thicknesses,the rolls must be crowned,that is,curved,so that the finished product will be of uniform thickness.Thus,to design the rolls it is necessary to know exactly how much they will bend when a sheet of steel is rolled between them.Sometimes mechanical elements must be designed to have a particular force-deflection characteristic.The suspension system of an automobile,for example,must be designed within a very narrow range to achieve an optimum vibration frequency for all conditions of vehicle loading, because the human body is comfortable only within a limited range of frequencies. The size of a load-bearing component is often determined on deflections,rather than limits on stress. This chapter considers distortion of single bodies due to geometry(shape)and loading,then,briefly,the behavior of groups of bodies. 4-1 Spring Rates Elasriciry is that property of a material that enables it to regain its original configuration after having been deformed.A spring is a mechanical element that exerts a force when deformed.Figure 4-la shows a straight beam of length simply supported at the ends and loaded by the transverse force F.The deflection y is linearly related to the force,as long as the elastic limit of the material is not exceeded,as indicated by the graph.This beam can be described as a linear spring. In Fig.4-1b a straight beam is supported on two cylinders such that the length between supports decreases as the beam is deflected by the force F.A larger force is required to deflect a short beam than a long one,and hence the more this beam is deflected,the stiffer it becomes.Also,the force is not linearly related to the deflection, and hence this beam can be described as a nonlinear stiffening spring. Figure 4-1 (a)A linear spring;(b)a stiffening spring;(da softening spring
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition I. Basics 4. Deflection and Stiffness 146 © The McGraw−Hill Companies, 2008 142 Mechanical Engineering Design All real bodies deform under load, either elastically or plastically. A body can be suffi- ciently insensitive to deformation that a presumption of rigidity does not affect an analysis enough to warrant a nonrigid treatment. If the body deformation later proves to be not negligible, then declaring rigidity was a poor decision, not a poor assumption. A wire rope is flexible, but in tension it can be robustly rigid and it distorts enormously under attempts at compressive loading. The same body can be both rigid and nonrigid. Deflection analysis enters into design situations in many ways. A snap ring, or retaining ring, must be flexible enough to be bent without permanent deformation and assembled with other parts, and then it must be rigid enough to hold the assembled parts together. In a transmission, the gears must be supported by a rigid shaft. If the shaft bends too much, that is, if it is too flexible, the teeth will not mesh properly, and the result will be excessive impact, noise, wear, and early failure. In rolling sheet or strip steel to prescribed thicknesses, the rolls must be crowned, that is, curved, so that the finished product will be of uniform thickness. Thus, to design the rolls it is necessary to know exactly how much they will bend when a sheet of steel is rolled between them. Sometimes mechanical elements must be designed to have a particular force-deflection characteristic. The suspension system of an automobile, for example, must be designed within a very narrow range to achieve an optimum vibration frequency for all conditions of vehicle loading, because the human body is comfortable only within a limited range of frequencies. The size of a load-bearing component is often determined on deflections, rather than limits on stress. This chapter considers distortion of single bodies due to geometry (shape) and loading, then, briefly, the behavior of groups of bodies. 4–1 Spring Rates Elasticity is that property of a material that enables it to regain its original configuration after having been deformed. A spring is a mechanical element that exerts a force when deformed. Figure 4–1a shows a straight beam of length l simply supported at the ends and loaded by the transverse force F. The deflection y is linearly related to the force, as long as the elastic limit of the material is not exceeded, as indicated by the graph. This beam can be described as a linear spring. In Fig. 4–1b a straight beam is supported on two cylinders such that the length between supports decreases as the beam is deflected by the force F. A larger force is required to deflect a short beam than a long one, and hence the more this beam is deflected, the stiffer it becomes. Also, the force is not linearly related to the deflection, and hence this beam can be described as a nonlinear stiffening spring. Figure 4–1 (a) A linear spring; (b) a stiffening spring; (c) a softening spring. l F F y l F y d F y (a) y F (b) y F (c) y
Budynas-Nisbett:Shigley's I.Basics 4.Deflection and Stiffness I©The McGraw-Hil Mechanical Engineering Companies,2008 Design,Eighth Edition Deflection and Stiffness 143 Figure 4-1c is an edge-view of a dish-shaped round disk.The force necessary to flatten the disk increases at first and then decreases as the disk approaches a flat con- figuration,as shown by the graph.Any mechanical element having such a characteristic is called a nonlinear softening spring. If we designate the general relationship between force and deflection by the equation F=F(y) (a) then spring rate is defined as △FdF aymo Ay=dy k(y)=lim (4-1) where y must be measured in the direction of F and at the point of application of F.Most of the force-deflection problems encountered in this book are linear,as in Fig.4-1a.For these,k is a constant,also called the spring constant;consequently Eq.(4-1)is written F k= (4-2) We might note that Eqs.(4-1)and(4-2)are quite general and apply equally well for torques and moments,provided angular measurements are used for y.For linear dis- placements,the units of k are often pounds per inch or newtons per meter,and for angular displacements,pound-inches per radian or newton-meters per radian. 4-2 Tension,Compression,and Torsion The total extension or contraction of a uniform bar in pure tension or compression, respectively,is given by 6= (4-3) Ae This equation does not apply to a long bar loaded in compression if there is a possibil- ity of buckling (see Secs.4-11 to 4-15).Using Egs.(4-2)and (4-3),we see that the spring constant of an axially loaded bar is k=AE (4-4 The angular deflection of a uniform round bar subjected to a twisting moment T was given in Eq.(3-35),and is TI 0= GJ (4-5 where 0 is in radians.If we multiply Eq.(4-5)by 180/and substitute J=d4/32 for a solid round bar,we obtain 583.6Tl 0= (4-61 Gd4 where 0 is in degrees. Equation (4-5)can be rearranged to give the torsional spring rate as T GJ k= =7 (4-7刀
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition I. Basics 4. Deflection and Stiffness © The McGraw−Hill 147 Companies, 2008 Deflection and Stiffness 143 Figure 4–1c is an edge-view of a dish-shaped round disk. The force necessary to flatten the disk increases at first and then decreases as the disk approaches a flat con- figuration, as shown by the graph. Any mechanical element having such a characteristic is called a nonlinear softening spring. If we designate the general relationship between force and deflection by the equation F = F(y) (a) then spring rate is defined as k(y) = lim y→0 F y = d F dy (4–1) where y must be measured in the direction of F and at the point of application of F. Most of the force-deflection problems encountered in this book are linear, as in Fig. 4–1a. For these, k is a constant, also called the spring constant; consequently Eq. (4–1) is written k = F y (4–2) We might note that Eqs. (4–1) and (4–2) are quite general and apply equally well for torques and moments, provided angular measurements are used for y. For linear displacements, the units of k are often pounds per inch or newtons per meter, and for angular displacements, pound-inches per radian or newton-meters per radian. 4–2 Tension, Compression, and Torsion The total extension or contraction of a uniform bar in pure tension or compression, respectively, is given by δ = Fl AE (4–3) This equation does not apply to a long bar loaded in compression if there is a possibility of buckling (see Secs. 4–11 to 4–15). Using Eqs. (4–2) and (4–3), we see that the spring constant of an axially loaded bar is k = AE l (4–4) The angular deflection of a uniform round bar subjected to a twisting moment T was given in Eq. (3–35), and is θ = T l G J (4–5) where θ is in radians. If we multiply Eq. (4–5) by 180/π and substitute J = πd4/32 for a solid round bar, we obtain θ = 583.6T l Gd4 (4–6) where θ is in degrees. Equation (4–5) can be rearranged to give the torsional spring rate as k = T θ = G J l (4–7)
148 Budynas-Nisbett:Shigley's I.Basics 4.Deflection and Stiffness T©The McGraw-Hil Mechanical Engineering Companies,2008 Design,Eighth Edition 144 Mechanical Engineering Design 4-3 Deflection Due to Bending The problem of bending of beams probably occurs more often than any other loading problem in mechanical design.Shafts,axles,cranks,levers,springs,brackets,and wheels, as well as many other elements,must often be treated as beams in the design and analy- sis of mechanical structures and systems.The subject of bending,however,is one that you should have studied as preparation for reading this book.It is for this reason that we include here only a brief review to establish the nomenclature and conventions to be used throughout this book. The curvature of a beam subjected to a bending moment M is given by 1 M D-EI (4-8) where p is the radius of curvature.From studies in mathematics we also learn that the curvature of a plane curve is given by the equation dy/dx2 1+(dy/dx) (4-91 where the interpretation here is that y is the lateral deflection of the beam at any point x along its length.The slope of the beam at any point x is o= dx (a) For many problems in bending,the slope is very small,and for these the denominator of Eq.(4-9)can be taken as unity.Equation(4-8)can then be written El=dx 6 Noting Eqs.(3-3)and(3-4)and successively differentiating Eq.(b)yields v d3y El=dx3 (c g dy EI=dxi (d) It is convenient to display these relations in a group as follows: 品 dy (4-10) v dy EI=dx3 (4-11) M dy EIdx (4-12) dy (4-13) y=f(x) (4-14
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition I. Basics 4. Deflection and Stiffness 148 © The McGraw−Hill Companies, 2008 144 Mechanical Engineering Design 4–3 Deflection Due to Bending The problem of bending of beams probably occurs more often than any other loading problem in mechanical design. Shafts, axles, cranks, levers, springs, brackets, and wheels, as well as many other elements, must often be treated as beams in the design and analysis of mechanical structures and systems. The subject of bending, however, is one that you should have studied as preparation for reading this book. It is for this reason that we include here only a brief review to establish the nomenclature and conventions to be used throughout this book. The curvature of a beam subjected to a bending moment M is given by 1 ρ = M E I (4–8) where ρ is the radius of curvature. From studies in mathematics we also learn that the curvature of a plane curve is given by the equation 1 ρ = d2 y/dx2 [1 + (dy/dx)2]3/2 (4–9) where the interpretation here is that y is the lateral deflection of the beam at any point x along its length. The slope of the beam at any point x is θ = dy dx (a) For many problems in bending, the slope is very small, and for these the denominator of Eq. (4–9) can be taken as unity. Equation (4–8) can then be written M E I = d2 y dx2 (b) Noting Eqs. (3–3) and (3–4) and successively differentiating Eq. (b) yields V E I = d3 y dx3 (c) q E I = d4 y dx4 (d) It is convenient to display these relations in a group as follows: q E I = d4 y dx4 (4–10) V E I = d3 y dx3 (4–11) M E I = d2 y dx2 (4–12) θ = dy dx (4–13) y = f (x) (4–14)
Budynas-Nisbett:Shigley's I.Basics 4.Deflection and Stiffness T©The McGraw-Hil 149 Mechanical Engineering Companies,2008 Design,Eighth Edition Deflection and Stiffness 145 I Figure 4-2 1=20in -x Loading.w w=80 Ibf/in (a) R1= R,=号 Shear,V V=+800 lbf V=-8001bf (b) M—x Moment M (c) M=M=0 Ele Slope,El0 82=0 Ely Deflection,Ely y0=%=0 (e) The nomenclature and conventions are illustrated by the beam of Fig.4-2.Here,a beam of length /=20 in is loaded by the uniform load w=80 Ibf per inch of beam length. The x axis is positive to the right,and the y axis positive upward.All quantities- loading,shear,moment,slope,and deflection-have the same sense as y:they are pos- itive if upward,negative if downward. The reactions R1=R2=+800 lbf and the shear forces Vo=+800 Ibf and Vi=-800 Ibf are easily computed by using the methods of Chap.3.The bending moment is zero at each end because the beam is simply supported.For a simply- supported beam,the deflections are also zero at each end. EXAMPLE 4-1 For the beam in Fig.4-2,the bending moment equation,for 0sxsl.is M=- Using Eq.(4-12).determine the equations for the slope and deflection of the beam,the slopes at the ends,and the maximum deflection. Solution Integrating Eq.(4-12)as an indefinite integral we have E1 =∫Md=x2-”x+C1 4 1) dx 6 where Ci is a constant of integration that is evaluated from geometric boundary conditions. We could impose that the slope is zero at the midspan of the beam,since the beam and
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition I. Basics 4. Deflection and Stiffness © The McGraw−Hill 149 Companies, 2008 Deflection and Stiffness 145 The nomenclature and conventions are illustrated by the beam of Fig. 4–2. Here, a beam of length l = 20 in is loaded by the uniform load w = 80 lbf per inch of beam length. The x axis is positive to the right, and the y axis positive upward. All quantities— loading, shear, moment, slope, and deflection—have the same sense as y; they are positive if upward, negative if downward. The reactions R1 = R2 = +800 lbf and the shear forces V0 = +800 lbf and Vl = −800 lbf are easily computed by using the methods of Chap. 3. The bending moment is zero at each end because the beam is simply supported. For a simplysupported beam, the deflections are also zero at each end. Figure 4–2 (a) (b) (c) (d) (e) l = 20 in R1 = wl 2 R2 = wl 2 y Loading, w w = 80 lbf/in x + + – V0 M0 Ml Vl V M x – – + EI EI0 EIl x x Shear, V V0 = +800 lbf Vl = –800 lbf EIy x Deflection, EIy y0 = yl = 0 Moment, M M0 = Ml = 0 Slope, EI l/2 = 0 w EXAMPLE 4–1 For the beam in Fig. 4–2, the bending moment equation, for 0 ≤ x ≤ l, is M = wl 2 x − w 2 x2 Using Eq. (4–12), determine the equations for the slope and deflection of the beam, the slopes at the ends, and the maximum deflection. Solution Integrating Eq. (4–12) as an indefinite integral we have E I dy dx = M dx = wl 4 x2 − w 6 x3 + C1 (1) where C1 is a constant of integration that is evaluated from geometric boundary conditions. We could impose that the slope is zero at the midspan of the beam, since the beam and