Hibbeler R.C. Structural Analysis

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The displacement at the ends of this bridge deck, as it is being constructed,

can be determined using energy methods.

341

In this chapter, we will show how to apply energy methods to solve problems

involving slope and deflection. The chapter begins with a discussion of work

and strain energy, followed by a development of the principle of work and

energy. The method of virtual work and Castigliano’s theorem are then

developed, and these methods are used to determine the displacements at

points on trusses, beams, and frames.

9.1 External Work and Strain Energy

The semigraphical methods presented in the previous chapters are very

effective for finding the displacements and slopes at points in beams

subjected to rather simple loadings. For more complicated loadings or

for structures such as trusses and frames, it is suggested that energy

methods be used for the computations. Most energy methods are based

on the conservation of energy principle, which states that the work done

by all the external forces acting on a structure, is transformed

into internal work or strain energy, which is developed when

the structure deforms. If the material’s elastic limit is not exceeded, the

elastic strain energy will return the structure to its undeformed state

when the loads are removed. The conservation of energy principle can

be stated mathematically as

(9–1)

Before developing any of the energy methods based on this principle,

however, we will first determine the external work and strain energy caused

by a force and a moment. The formulations to be presented will provide a

basis for understanding the work and energy methods that follow.

= U

Deflections Using

Energy Methods

342 CHAPTER 9DEFLECTIONS USING ENERGY M ETHODS

External Work—Force. When a force F undergoes a displacement

dx in the same direction as the force, the work done is If the

total displacement is x, the work becomes

(9–2)

Consider now the effect caused by an axial force applied to the end of a

bar as shown in Fig. 9–1a.As the magnitude of F is gradually increased from

zero to some limiting value the final elongation of the bar becomes

If the material has a linear elastic response, then

Substituting into Eq. 9–2, and integrating from 0 to we get

(9–3)

which represents the shaded triangular area in Fig. 9–1a.

We may also conclude from this that as a force is gradually applied to

the bar, and its magnitude builds linearly from zero to some value P, the

work done is equal to the average force magnitude times the

displacement 1¢2.

(P>2)

P¢

¢,

F = 1P>¢2x.¢.

F = P,

F dx

= F dx.

Fig. 9–1



F 



(a)

9.1 EXTERNAL WORK AND STRAIN ENERGY 343

Suppose now that P is already applied to the bar and that another force

is now applied, so the bar deflects further by an amount Fig. 9–1b.

The work done by P (not ) when the bar undergoes the further

deflection is then

(9–4)

Here the work represents the shaded rectangular area in Fig. 9–1b.In

this case P does not change its magnitude since is caused only

by Therefore, work is simply the force magnitude (P) times the

displacement

In summary, then, when a force P is applied to the bar, followed by

application of the force the total work done by both forces is

represented by the triangular area ACE in Fig. 9–1b. The triangular area

ABG represents the work of P that is caused by its displacement

the triangular area BCD represents the work of since this force

causes a displacement and lastly, the shaded rectangular area

BDEG represents the additional work done by P when displaced as

caused by

External Work—Moment. The work of a moment is defined by

the product of the magnitude of the moment M and the angle through

which it rotates, that is, Fig. 9–2. If the total angle of

rotation is radians, the work becomes

(9–5)

As in the case of force, if the moment is applied gradually to a structure

having linear elastic response from zero to M, the work is then

(9–6)

However, if the moment is already applied to the structure and other

loadings further distort the structure by an amount then M rotates

and the work is

(9–7)U

¿=Mu¿

u¿,

Mdu

= Mdu,

¢¿

¢¿,

F¿

¢,

F¿,

1¢¿2.

F¿.

¢¿

¿=P¢¿

¢¿

F¿

¢¿,F¿

Fig. 9–2

¿



F¿



F¿  P

(b)

¿

Fig. 9–1

Fig. 9–3

Strain Energy—Axial Force. When an axial force N is applied

gradually to the bar in Fig. 9–3, it will strain the material such that the

external work done by N will be converted into strain energy, which is

stored in the bar (Eq. 9–1). Provided the material is linearly elastic,

Hooke’s law is valid, and if the bar has a constant cross-

sectional area A and length L, the normal stress is and the

final strain is Consequently, and the final

deflection is

(9–8)

Substituting into Eq. 9–3, with the strain energy in the bar is

therefore

(9–9)

Strain Energy—Bending. Consider the beam shown in Fig. 9–4a,

which is distorted by the gradually applied loading P and w. These loads

create an internal moment M in the beam at a section located a distance

x from the left support. The resulting rotation of the differential element

dx,Fig.9–4b, can be found from Eq. 8–2, that is,

Consequently, the strain energy, or work stored in the element, is

determined from Eq. 9–6 since the internal moment is gradually

developed. Hence,

(9–10)

The strain energy for the beam is determined by integrating this result

over the beam’s entire length L.The result is

(9–11)U

2EI

du = 1M>EI2 dx.

2AE

P = N,

¢=

N>A = E1¢>L2,P=¢>L.

s = N>A

s = EP,

344

CHAPTER 9DEFLECTIONS USING ENERGY M ETHODS



x dx

(a)

M M

(b)

Fig. 9–4

9.2 PRINCIPLE OF WORK AND ENERGY 345

9.2 Principle of Work and Energy

Now that the work and strain energy for a force and a moment have

been formulated, we will illustrate how the conservation of energy or the

principle of work and energy can be applied to determine the displacement

at a point on a structure. To do this, consider finding the displacement

at the point where the force P is applied to the cantilever beam in

Fig. 9–5. From Eq. 9–3, the external work is To obtain the

resulting strain energy, we must first determine the internal moment as a

function of position x in the beam and then apply Eq. 9–11. In this case

so that

Equating the external work to internal strain energy and solving for the

unknown displacement we have

Although the solution here is quite direct, application of this method is

limited to only a few select problems. It will be noted that only one load

may be applied to the structure, since if more than one load were

applied, there would be an unknown displacement under each load, and

yet it is possible to write only one “work” equation for the beam.

Furthermore, only the displacement under the force can be obtained, since

the external work depends upon both the force and its corresponding

displacement. One way to circumvent these limitations is to use the

method of virtual work or Castigliano’s theorem, both of which are

explained in the following sections.

¢=

3EI

P¢=

= U

¢,

2EI

1-Px2

2EI

M =-Px,

P¢.

Fig. 9–5

9.3 Principle of Virtual Work

The principle of virtual work was developed by John Bernoulli in 1717

and is sometimes referred to as the unit-load method. It provides a

general means of obtaining the displacement and slope at a specific

point on a structure, be it a beam, frame, or truss.

Before developing the principle of virtual work, it is necessary to make

some general statements regarding the principle of work and energy,

which was discussed in the previous section. If we take a deformable

structure of any shape or size and apply a series of external loads P to

it, it will cause internal loads u at points throughout the structure. It is

necessary that the external and internal loads be related by the equations of

equilibrium. As a consequence of these loadings, external displacements

will occur at the P loads and internal displacements will occur at

each point of internal load u. In general, these displacements do not have

to be elastic, and they may not be related to the loads; however, the

external and internal displacements must be related by the compatibility of

the displacements. In other words, if the external displacements are

known, the corresponding internal displacements are uniquely defined.

In general, then, the principle of work and energy states:

(9–12)

Based on this concept, the principle of virtual work will now be

developed. To do this, we will consider the structure (or body) to be of

arbitrary shape as shown in Fig. 9–6b.* Suppose it is necessary to

determine the displacement of point A on the body caused by the

“real loads” and It is to be understood that these loads cause

no movement of the supports; in general, however, they can strain the

material beyond the elastic limit. Since no external load acts on the body

at A and in the direction of the displacement can be determined by

first placing on the body a “virtual” load such that this force acts in the

same direction as Fig. 9–6a. For convenience, which will be apparent

later, we will choose to have a “unit” magnitude, that is, The

term “virtual” is used to describe the load, since it is imaginary and does

not actually exist as part of the real loading. The unit load does,

however, create an internal virtual load u in a representative element or

fiber of the body, as shown in Fig. 9–6a. Here it is required that and u

be related by the equations of equilibrium.

†

P¿

1P¿2

P¿=1.P¿

¢,

P¿

¢¢,

©P¢=©ud

Work of Work of

External Loads Internal Loads

d¢

346

CHAPTER 9DEFLECTIONS USING ENERGY M ETHODS

*This arbitrary shape will later represent a specific truss, beam, or frame.

†

Although these loads will cause virtual displacements, we will not be concerned with

their magnitudes.

P¿  1

Apply virtual load P¿  1

(a)

Apply real loads P

, P

(b)



Fig. 9–6

9.3 PRINCIPLE OF VIRTUAL WORK 347

Once the virtual loadings are applied, then the body is subjected to the

real loads and Fig. 9–6b. Point A will be displaced an amount

causing the element to deform an amount dL.As a result, the external

virtual force and internal virtual load u “ride along” by and dL,

respectively, and therefore perform external virtual work of on the

body and internal virtual work of on the element. Realizing that

the external virtual work is equal to the internal virtual work done on all

the elements of the body, we can write the virtual-work equation as

virtual loadings

real displacements

(9–13)

where

virtual unit load acting in the direction of

virtual load acting on the element in the direction of dL.

displacement caused by the real loads.

deformation of the element caused by the real loads.

By choosing it can be seen that the solution for follows directly,

since

In a similar manner, if the rotational displacement or slope of the tangent

at a point on a structure is to be determined, a virtual couple moment

having a “unit” magnitude is applied at the point. As a consequence, this

couple moment causes a virtual load in one of the elements of the body.

Assuming that the real loads deform the element an amount dL, the

rotation can be found from the virtual-work equation

virtual loadings

real displacements

(9–14)

where

virtual unit couple moment acting in the direction of

virtual load acting on an element in the direction of dL.

rotational displacement or slope in radians caused by the

real loads.

deformation of the element caused by the real loads.

This method for applying the principle of virtual work is often referred

to as the method of virtual forces, since a virtual force is applied resulting

in the calculation of a real displacement. The equation of virtual work in

this case represents a compatibility requirement for the structure.

Although not important here, realize that we can also apply the principle

dL = internal

u = external

= internal

u.M¿=1 = external

u =©u

M¿

¢=©udL.

¢P¿=1,

dL = internal

¢=external

u = internal

¢.P¿=1 = external

¢=©u

¢P¿

¢,

of virtual work as a method of virtual displacements. In this case virtual

displacements are imposed on the structure while the structure is

subjected to real loadings. This method can be used to determine a force

on or in a structure,* so that the equation of virtual work is then

expressed as an equilibrium requirement.

9.4 Method of Virtual Work: Trusses

We can use the method of virtual work to determine the displacement

of a truss joint when the truss is subjected to an external loading,

temperature change, or fabrication errors. Each of these situations will

now be discussed.

External Loading. For the purpose of explanation let us consider

the vertical displacement of joint B of the truss in Fig. 9–7a. Here a

typical element of the truss would be one of its members having a length

L,Fig.9–7b. If the applied loadings and cause a linear elastic

material response, then this element deforms an amount

where N is the normal or axial force in the member, caused by the loads.

Applying Eq. 9–13, the virtual-work equation for the truss is therefore

(9–15)

where

virtual unit load acting on the truss joint in the stated

direction of

virtual normal force in a truss member caused by the

external virtual unit load.

joint displacement caused by the real loads on the truss.

normal force in a truss member caused by the real loads.

of a member.

area of a member.

of elasticity of a member.

The formulation of this equation follows naturally from the development

in Sec. 9–3. Here the external virtual unit load creates internal virtual

forces n in each of the truss members. The real loads then cause the truss

joint to be displaced in the same direction as the virtual unit load, and

each member is displaced in the same direction as its respective

n force. Consequently, the external virtual work equals the internal

virtual work or the internal (virtual) strain energy stored in all the truss

NL>AE

E = modulus

A = cross-sectional

L = length

N = internal

¢=external

n = internal

¢.

1 = external

¢=

nNL

¢L = NL>AE,

348

CHAPTER 9DEFLECTIONS USING ENERGY M ETHODS

Fig. 9–7

*It was used in this manner in Sec. 6–3 with reference to the Müller-Breslau principle.

(a)

Apply virtual unit load to B



Apply real loads P

, P

(b)

9.4 METHOD OF VIRTUAL WORK: TRUSSES 349

Temperature. In some cases, truss members may change their

length due to temperature. If is the coefficient of thermal expansion

for a member and is the change in its temperature, the change in

length of a member is Hence, we can determine the

displacement of a selected truss joint due to this temperature change

from Eq. 9–13, written as

(9–16)

where

external virtual unit load acting on the truss joint in the stated

direction of

internal virtual normal force in a truss member caused by the

external virtual unit load.

external joint displacement caused by the temperature change.

coefficient of thermal expansion of member.

change in temperature of member.

length of member.

Fabrication Errors and Camber. Occasionally, errors in fabricating

the lengths of the members of a truss may occur.Also, in some cases truss

members must be made slightly longer or shorter in order to give the

truss a camber. Camber is often built into a bridge truss so that the

bottom cord will curve upward by an amount equivalent to the downward

deflection of the cord when subjected to the bridge’s full dead weight. If a

truss member is shorter or longer than intended, the displacement of a

truss joint from its expected position can be determined from direct

application of Eq. 9–13, written as

(9–17)

where

external virtual unit load acting on the truss joint in the stated

direction of

internal virtual normal force in a truss member caused by the

external virtual unit load.

external joint displacement caused by the fabrication errors.

difference in length of the member from its intended size as

caused by a fabrication error.

A combination of the right sides of Eqs. 9–15 through 9–17 will be

necessary if both external loads act on the truss and some of the members

undergo a thermal change or have been fabricated with the wrong

dimensions.

¢L =

¢=

n =

¢.

1 =

¢=©n ¢L

L =

¢T =

a =

¢=

n =

¢.

1 =

¢=©na ¢T L

¢L = a ¢T L.

¢T