Chapter 4 Deflection and Stiffness

Omit 4.6, 4.9, 4.16, 4.17Omit 4.6, 4.9, 4.16, 4.17 Rigid Body :

A body is said to be rigid if it exhibits no change in size or shape under the influence of forces or couples; the distance between any two points within the body remains constant under the application of forces.

All real bodies deform under load, either elastically or plastically. Classification of a real body as a rigid is an idealization.

Deflection analysis enters into design situations in many ways:In a transmission, the gears must be supported by a rigid shaft. If the shaft bends too much (too flexible) the teeth will not mesh properly, and the result will be excessive impact, noise, wear, and early failure.

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.

4.14.1 Spring Rates Spring Rates Elasticity is that property of a material that enables it to regain its original

configuration after having been deformed. Stiffness is the rigidity of an object — the extent to which it resists deformation

in response to an applied force. Flexibility is the ability of a body to distort. The complementary concept is

stiffness. The more flexible an object is the less stiff it is. A spring is a mechanical element that exerts a force when deformed; figures

shown:(a) A straight beam of length l simply supported at ends loaded by force F. The

deflection y is linearly related to the force, as long as the elastic limit of the material is not exceeded, this beam can be described as a linear spring.

(b) A straight beam is supported on two cylinders; beam is shorter. A larger force is required to deflect a short beam, it becomes stiffer. The force is not linearly related to the deflection; the beam can be described as a nonlinear stiffening spring.

(c) 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 configuration. Any mechanical element having such a characteristic is called a nonlinear softening spring .

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Consider that force and deflection are related F =F(y); then the spring rate is defined as

where y is measured in the direction of F and at the point of application of F.

For linear spring:

Here k is called spring constant. Units lbf/in ; or N/m. Above Eqns. are quite general and apply equally well for torques and

moments.

4.24.2 Tension, Compression, and Torsion Tension, Compression, and Torsion Consider a uniform bar subjected to axial (or compressive) force F

where δ is the linear deformation.This equation does not apply to a long bar loaded in compression if there is a possibility of buckling.The spring constant of an axially loaded bar is

The angular deflection of a uniform solid or hollow round bar subjected to a twisting moment T was given in Eq. (3–35), and is

where θ is the angular deformation expressed in radians.Equation (4–5) can be rearranged to give the torsional spring rate (kθ) as

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Above equations apply only to circular cross sections. For noncircular cross section refer to chapter 3.

4.34.3 Deflection Due to BendingDeflection Due to Bending Beams deflect great deal more than axially loaded members. 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.

Consider a beam segment of length L. After deformation, the length of the neutral surface remains L. At other sections:

where σm denotes the maximum absolute value of the stress. So, the curvature of a beam subjected to a bending

moment M is given by

where ρ is the radius of curvature. From studies in Calculus the curvature of a plane

curve is given by:

where y is the deflection of the beam at any point x along its length.

The slope of the beam at any point x is given by

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

Noting that Eqs. (3–3) and (3–4)

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and successively differentiating Eq. (b) yields

It is convenient to display these relations in a group as follows:

The nomenclature and conventions are illustrated by the beam shown.

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IntegrationIntegration

Example 4-1 (see textbook)

4.44.4 Beam Deflection Methods Beam Deflection Methods Beams have intensities of loading that can be q = constant, variable intensity q(x), or

concentrated loads. There are many techniques employed to solve the integration problem for beam

deflection. Some of the popular methods include:1. Integration of moment equation (example 4-1)2. Superposition (section 4-5)3. Moment-area method4. Singularity functions (section 4-6) –Omitted.5. Strain energy with Castigliano’s theorem (sections 4-7, 4-8)6. Numerical integration.

4.54.5 Beam Deflections by Superposition Beam Deflections by Superposition Table A-9 provides some cases for results of beams subjected to simple loads and

boundary conditions. Superposition resolves the effect of combined loading on a structure by determining the

effect of each load separately and adding the results algebraically. In using the superposition principle, the followings are required:

1. Each effect is linearly related to the load that produces it.2. A load does not create a condition that affects the results of another load.3. The deformations resulting from any specific load are not large enough to

appreciably alter the geometric relations of the parts of the structure.Example 4-2

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Superposition:

Example 4-3

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Example 4-4 (see textbook)

4.64.6 Beam Deflections by Singularity Functions (Omitted) Beam Deflections by Singularity Functions (Omitted)

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4.74.7 Strain Energy Strain Energy It is the potential energy stored in a body by virtue of an elastic deformation.

The total work U done by the load as the rod undergoes a deformation x1 is thus

and is equal to the area under the load-deformation diagram between x = 0 and x = x1.The work done by the load P as it is slowly applied to the rod must result in the increase of some energy associated with the deformation of the rod. This energy is referred to as the strain energy of the rod. We have, by definition,

In the case of a linear and elastic deformation, the portion of the load-deformation diagram involved can be represented by a straight line of equation P = kx (Fig. 11.4). Substituting for P in Eq. (11.2), we have

OR If a member is deformed a distance y, and if the force deflection

relationship is linear, this energy is equal to the product of the average force F and the deflection y, or, force F and the deflection y, or,

This equation is general in the sense that the force F can also mean torque, or moment, provided, of course, that consistent units are used for k.

Strain Energy For tension and compression , we employ Eq. (4–4) and obtain

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where the first equation applies when all the terms are constant throughout the length, and the more general integral equation allows for any of the terms to vary through the length.

Strain energy for torsion , we employ (4–7) and get

Strain energy for direct shear , consider the element with one side fixed in Fig. 4–8a. The force F places the element in pure shear, and the work done is U = Fδ/2. Since the shear strain is γ = δ/l = τ/G = F/AG, we have

Strain energy stored in a beam or lever by bending may be obtained by referring to Fig.b. Here AB is a section of the elastic curve of length ds having a radius of curvature ρ. The strain energy stored in this element of the beam is dU = (M/2) dθ.Since ρ dθ = ds, we have

We can eliminate ρ by using Eq. (4–8), ρ = EI/M. Thus

For small deflections, ds = dx. Then, for the entire beam

Summarized to include both the integral and nonintegral form, the strain energy for bending is

Equations (4–22) and (4–23) are exact only when a beam is subject to pure bending. Even when transverse shear is present, these equations continue to give quite good results, except for very short beams.

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Strain energy due to shear in bending is a complicated problem. An approximate solution can be obtained by using Eq. (4–20) with a correction factor whose value depends upon the shape of the cross section. If we use C for the correction factor and V for the shear force, then the strain energy due to shear in bending is

Values of the factor C are listed in Table 4–1.

Example 4-8 (see textbook)4.84.8 Castigliano’s Theorem Castigliano’s Theorem It is an unusual, powerful, and often surprisingly simple approach to deflection analysis. It is a unique way of analyzing deflections and is even useful for finding the reactions of

indeterminate structures. Castigliano’s theorem states that when forces act on elastic systems subject to small

displacements, the displacement corresponding to any force, in the direction of the force, is equal to the partial derivative of the total strain energy with respect to that force.

Mathematically, for linear displacement, the theorem of Castigliano is

where δi is the displacement of the point of application of the force Fi and in its direction.

For rotational displacement Eq. (4–26) can be written as

where θi is the rotational displacement, in radians, of the beam where the moment Mi exists and in its direction.

As an example, apply Castigliano’s theorem using Eqns. (4–16) and (4–18) to get the axial and torsional deflections. The results are

Compare Eqns. (a) and (b) with Eqns. (4–3) and (4–5).

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Example 4-9 (see textbook) The relative contribution of transverse shear to beam deflection decreases as the length-to-

height ratio of the beam increases, and is generally considered negligible for l/d > 10. Note that the deflection equations for the beams in Table A–9 do not include the effects of transverse shear.

Castigliano’s theorem can be used to find the deflection at a point even though no force or moment acts there. The procedure is:

1. Set up the equation for the total strain energy U by including the energy due to a fictitious force or moment Q acting at the point whose deflection is to be found.

2. Find an expression for the desired deflection δ, in the direction of Q, by taking the derivative of the total strain energy with respect to Q.

3. Since Q is a fictitious force, solve the expression obtained in step 2 by setting Q equal to zero. Thus, the displacement at the point of application of the fictitious force Q is

In cases where integration is necessary to obtain the strain energy, it is more efficient to obtain the deflection directly without explicitly finding the strain energy, by moving the partial derivative inside the integral. For the example of the bending case,

This allows the derivative to be taken before integration, simplifying the mathematics. The expressions for the common cases in Eqs. (4–17), (4–19), and (4–23) are rewritten

as

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Example

Example

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Example 4-10 (see textbook)

Example 4-11 (see textbook)

Stiffness (k) is the rigidity of an object — the extent to which it resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.[2]

Young's modulus (tensile modulus or elastic modulus) is a measure of the stiffness of an elastic material and is a quantity used to characterize materials. It is defined as the ratio of the stress along an axis over the strain along that axis in the range of stress in which Hooke's law holdsDeflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance.In materials science, deformation is a change in the shape or size of an object due to an applied force or a change in temperature. The first case can be a result of tensile forces, compressive forces, shear, bending or torsion. In the second case, the most significant factor, which is determined by the temperature, is the mobility of the structural defects such as grain boundaries, point vacancies, line and screw dislocations, stacking faults and twins in both crystalline and non-crystalline solids. Deformation is often described as strain.In materials science, ductility is a solid material's ability to deform under tensile stress;Malleability, a similar property, is a material's ability to deform under compressive stress;

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A material is brittle if, when subjected to stress, it breaks without significant deformation (strain). Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Brittle materials include most ceramics and glasses (which do not deform plastically) and some polymers, such as PMMA and polystyrene. Many steels become brittle at low temperatures, depending on their composition and processing.A fracture is the separation of an object or material into two, or more, pieces under the action of stress.In materials science, toughness is the ability of a material to absorb energy and plastically deform without fracturing; Material toughness is defined as the amount of energy per volume that a material can absorb before rupturing. It is also defined as the resistance to fracture of a material when stressed.Resilience is the ability of a material to absorb energy when it is deformed elastically, and release that energy upon unloading. The modulus of resilience is defined as the maximum energy that can be absorbed per unit volume without creating a permanent distortion.The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed.Ultimate tensile strength ( UTS ), often shortened to tensile strength (TS) or ultimate strength,[1][2] is the maximum stress that a material can withstand while being stretched or pulled before failing or breaking. Tensile strength is the opposite of compressive strength and the values can be quite different.

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