CivE 205 9-The Principles of Virtual Work

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CivE 205 – Mechanics of Solids II

Department of Civil & Environmental Engineering © A. Al-MayahUniversity of Waterloo Spring 2014

Part :

The Principle of Virtual Wo



p. 9.2 Part 9: The Principle of Virtual Work

Department of Civil & Environmental Engineering CivE 205 – Mechanics of Solids IIUniversity of Waterloo

Developed by John Bernoulli 1717, laterformalized by J.C. Maxwell 1864, O. Mohr 1874.

Based on the conservation of energy. Used in several mechanics applications.

Applied to calculate displacement and slope atany points on a structure.

1. Consider a particle under the action of three forces,F1, F 2, and F 3.

2. Conservation of energy cannot be applied to calculatedeformation at any points where load is not applied.

3. Therefore, to calculate the displacement ( Δ) of pointA, an imaginary (virtual) load is applied at A in thedirection of the displacement.

4. This external virtual load results in internal virtual load f in an element of the body.

5. In order for the system to be in equilibrium: Externalvirtual work must equal internal work i.e.

(Virtual external force real external displacement= virtual internal loadreal internal displacement )

1=external virtual load ( Q)Δ=external displacement caused by actual load

f = internal virtual load

dL=internal real displacement caused by internal real load

“If a structure is in equilibrium under the action of a set of external forces and issubjected to a set of displacements compatible with the constraints of the structure,the total work done by the external and internal forces during the displacementsmust be zero .”

F 3

F 2

F 1

A Δ

A

Q =1

f

f



Part 9: The Principle of Virtual Work p. 9.3


Notes: The forces can be in a form of moment to calculate the deformation in a form

of rotation. The set of forces need not be related to the set of displacements as cause-and-

effect and either set can be real while the other is imaginary.

Beam Example:

Self-equilibrated forces are applied to the structure in its undeflected position.Then the real deflections due to the real load are allowed to take place. These realdeflections are ( x) and ( Δ) at distance ( a) and ( c) from the hinge, respectively)

Q is virtual external force at a distance (c) from the hinge.

Apply: ( Virtual external force Q real external displacement Δ = virtual

internal load f real internal displacement x )

Q Δ = fx This is a compatibilitycondition

For a linear system

F=Kx (where K=spring stiffness)

Or x=F/K (deflection due to real load)

If Q is a unit load, and f corresponds tothis value of Q:

f = the internal virtual spring force due to unitvirtual applied load.

F = the internal real spring force due to real appliedload. Δ = the required real deflection due to real loads.

P

L

a

f

ΔPx

Force system + Real displacements

Virtual forces f and Q satisfy (fa=Qc)

Δ

ΔPx

F

Qc

Force F

Displacement x

1

K





The more general form is

Where x is a real displacement caused by:

a)

External real forces b) Temperature changesc) Misfit of component partsd) Support settlements etc.

This general form is valid also for nonlinear system.

Note: If a rotation at a point on a linear structure is required, Δ is an angle and Q isa torque (of unit value) applied in the direction of Δ.

TENSILE OR COMPRESSION STRUCTURAL MEMBERS :

Since (real and F)

Let f = virtual member force in equilibrium with external virtual applied load:

To determine the displacement oftruss joints:

Where,

F i= real internal force in truss member caused by applied real loads

f i= virtual internal force in truss member caused by external virtual load

Li=length of a member

Ai= area of a member

E i= modulus of elasticity of a member

Δ L

FF





Example 9.1 Deflection of Trusses_load P is applied to 4-bars truss withgiven areas and lengths. Find verticaldeflections at B and C.

To find : Apply unit vertical force applied at C Calculate the internal virtual memberforces ( f Ci) caused by the applied unitload at C.

To find :o Apply unit vertical force applied at Bo Calculate the internal virtual member

forces ( f B) caused by the applied unitload at B.

Member F i f Ci f Bi L i A i

AB

√ √ √ L A

BC

√ √ 0 L A

0

CD

√ √ 0 L A

0

BD √ 1.5A √ √

Sum

PA, L /√

1

1

/√





Example 9.2 A concentrated load of 9kips is applied at point E in the truss.The internal forces are calculated usingstatics. E=30x10 3 kips/in 2

Areas: Compression members =10 in2

Tension members =5 in 2

Find the deflection at F.

Solution:

Apply unit load at F

Member F i (kips)

f i Li (in) A i (in2)

AC -10 -5/9 270 10 +150

CD -4 -4/9 216 10 +38.4

BD -5 -10/9 270 10 +150

CE +9 0 0

CF -5 +5/9 270 10 -75

DF +3 +2/3 162 5 +64.8

AE +8 +4/9 216 5 +153.6

EF +8 +4/9 216 5 +153.6

FB +4 +8/9 216 5 +153.6

Sum

By virtual work

+8 +8

-5

+4

+3-5

-4

+9

9 kips18ft 18ft 18ft

A B

C D

E F

+4/9 +4/9

-10/9

+8/9

+2/3+5/9

-4/9

0

1

A B

C D

E F





BENDING MOMENT OF BEAMS:

Let m= virtual internal moment due to virtualexternal load (I.V.W.)

( ) for an element

( )

Example 9.3 Find the rotation at A due to the concentrated load P applied at C.

I.V.W=E.V.W

Since, the rotation is required at point A,a virtual unit moment is required at A. Draw the bending moment for real andvirtual loads to identify the regions of novirtual moment to simplify the virtualload calculations. In this example, the effect of virtual bending moment is limited to thesegment AB.

M(x)= R A x=-Px/2

∫

is negative (counter clockwise) i.e. opposite from that assumed.

Note: Bending deformations are taken into account here but not shear deformation.

M M dθ

P

2L L2EI IA CB

PL

1

m=1

θA





Example9.4: Find the deflection at the mid span of the cantilever beam. EI isconstant.

M /

V

1

L/2

ωo

L





Torsion of Shafts

Let t= internal virtual torque due to external load

( )

( )

And for a uniform shaft:

( )

System Load Strain Energy U Internal VirtualWok

Uniaxial loading(e.g. Rod)

F

Bending Moment(e.g. beam)

M

Torque (e.g.Shaft)

T





OTHER APPLICATIONS OF THE VIRTUAL WORK METHODTO FIND DEFLECTIONS

1) Deflection due to temperature:

Example 9.5 Determine the deflectiondue to temperature for the cantilever

beams. Temperature at the top of the beam is T+ΔT, and the temperaturein Coefficient of linear expansion = α

Deformation due to temperature gradient

Linear temperature change through depth of beam:(Real element deformation)

To find the total vertical deflection at the tip of the cantilever due to thermaleffects, apply a unit virtual load atthe tip

By virtual work principle

( )

Downward deflection (as assumed).

L

A Bdx

d

Temperature = T+ΔT

Temperature = T

dθ

dxT

T+ΔT

dx. α.ΔT

d

1 L

ΔA

-1.L

(Internal Virtual Bending Moment)

m x =- x





2) Discrepancies in member lengths:

Example 9.6: Suppose some members of a truss were made too long or too short.What is the resulting joint displacement at A.

+ ve sign longer than intended- ve sign shorter than intended

ΔA is the real displacements

Apply a unit load at A

Virtual internal forces will be as shownBy virtual principle:

Virtual deflection at A

e= real length discrepancies

f= virtual internal forces

( )( ) ( )( ) ( )( )

Positive result is referring to the upward deflection (as assumed).

-0.75

A

-1.5

+0.75 0

-1.0

1

+0.2

A

-0.3

+0.4

20’

ΔA





3) Arch Deformation:Example 9.7 Consider bending deformations only (EI is constant)

() ()

() () By prinicple of virtual work

( )

Compare this method with the method of Castigliano’s.

P

θ

dθ

ds=Rdθ

Rsinθ

Δ

1

θ Rsinθ





METHOD OF VIRTUAL WORK APPLIED TO STATICALLYINDETERMINATE STRUCTURES:

Redundancy can be either a support reaction or an internal force.

A statically determinate structure is formed by removing the redundancy: Take away the extra support; OR Cut the redundant member.

Then the primary structure (the statically determinate structure) is analysed bearingin mind the compatibility situation.

Example 9.8 Find the reaction at A(R A)?

( )

( )

|

1

L

ΔA

Internal Virtual Bendin Moment

m x =- x

A B

w

R A x





Example 9.9

Find the forces in the pin-jointed steel structure.

To make the truss a statically determinate structure, cutone of the members (say BD) and calculate the internalforces.

The statically determinate structure with actual forces (F i)caused by actual applied force. Each force causeselongation in each member.

In the statically determinate structure, Δd must be found by applying virtual vertical unit force at D. This unittension causes forces f i.

Member F i f i Li

(in)

Ai

(in 2 )

AB 2.5 0.15 -833

BC -2.5 60 0.25 -500BD 0 40 0.1 0

-1333

A

B

C

D

A=0.1 in 2

L=40 in

3 kips

36 in

36 in

48 in

A

B

C

D

3 kips

0 kips

A

B

C

1

1





But the actual displacement at D is zero in the statically indeterminate structure(provided there is no initial lack of fit (i.e. BD bar is not too long or short).Therefore, a force must be applied at D to return it to its original location.

Assume the actual force in BD is R. This redundant force gives rise to internalforces R.f in the remaining members. The additional elongations in the members

due to R are given by ( )

Apply virtual work method:

External virtual work=Internal virtual work

Then

∑ ∑⁄

Member f i Li

(in)

Ai

(in 2 )

AB 0.15 277.56

BC 60 0.25 166.54

BD 40 0.1 400

845

Based on the calculated R, all other forces can be calculated.

Force at AB=1.18 kips

Force at BC=-1.18 kips

CivE 205 9-The Principles of Virtual Work

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