CIVL3310 STRUCTURAL ANALYSISProfessor CC Chang

Chapter 9: Deflections Using Energy Methods

Work and Energy Principles• Work done by external forces

FF

F

FΔ

FΔF21W

F

FΔ

FΔFW

FΔ

0FF ΔΔFW d

F

Work and Energy Principles• Where does the work go?

Saved in the beam in terms of “Strain energy”

FF

i-th componentif

if

Member force

Member deformation

id

iii df21U

FΔF21W

ii df21UTotal strain energy

Work - Energy

UW

iiF df21ΔF

21

Virtual Work Principle• Under equilibrium, perturb the

structure

FF

F

FΔ

FδΔ

δF

FδΔFδW

FΔδFδW

FΔF21W

Work - Energy

UW

iiF df21ΔF

21

Perturb F by dF

Perturb DF by d DFδUδW

D iiF dδfδF

D iiF δdfδF

FδΔ

δF

ii df iδdiδf

Virtual Work Principle

(Complementary) virtual work principlePerturb F by dF

Virtual work principlePerturb DF by d DF

δUδW

D iiF dδfδF

D iiF δdfδF

FF FδΔ

δF

Virtual Work Principle(Complementary) virtual work principlePerturb F by dF

δUδW

D iiF dδfδF

FF

δF

iδf

The complementary virtual work done by an external virtual force system under the actual deformation of a structure is equal to the complementary strain energy done by the virtual stresses under the actual strains

The complementary virtual work done by an external virtual force system under the actual deformation of a structure is equal to the complementary strain energy done by the virtual stresses under the actual strains

fi di

Virtual Work Principle D iiF dδfδF

FF

δF

iδffi di

Superposition

Actual system Virtual system

Virtual Work Principle

D iiF dδfδF

FF δF

iδffi di

Actual system Virtual system

δUδW

Looking for DF

1

1

fiLi

EiAi

Dii

iiiF AE

Lfδf1

Virtual Work Principle

Dii

iiiF AE

Lfδf1

δUδW

F

F1δF fi

dfi

Virtual Work Principle

D iiF dδf1

δUδW

F1δF

dfiDT

Temperature effect

ii LΔTαd

Virtual Work Principle

D iiF dδf1

δUδW

F1δF

dfiDL

Misfit effect

Virtual Work Principle• For Beams

F

dM(x)21U

M=0, ≠0, k=0

M ≠ 0, ≠0, k ≠ 0M(x)

xd

M(x)

dd dM(x)UIntegrate for the whole beam

LL

dd0

2

0x

EIM

21M(x)

21U

Perturbation

xEI

M(x) dd

d ddLL

dd00

xEIMMMU

DF

δUδW ddL

d0

F xEIMMΔF

dFdM(x)

Virtual Work PrincipleδUδW

ddL

d0

F xEIMMΔF

P

dF

w

Deformation due to actual loads

Moment due to actual loads

DF

M(x) dM(x)

Virtual force corresponds to actual deformation Virtual moment induced by

The virtual force

ddL

d0

AA xEIMMM

AdMA

Virtual Work Principle• For Frames

δUδW

d ddcomponentcomponent

dxEIMM

EIfLfΔF F

Negligible

w

PdF

DF

Castigliano’s Principle• Work-Energy ii FUFW

Fi

Fi

idF

ii dfiδf

Fi

iFΔ

kF

2F

ΔF21W

ii

Fi i

1k

n

n

EALf

2f

df21U

iii

ii ii Ff

i

i

ii

ii

FiF

FiFi

iFi

iiFi

i

iiii

FU

FU

FW

dFFUFUdF

FWFW

dFFUdFFW

D

iFi

iΔ

kF

FW

Castigliano’s Principle• For Trusses

• For Beams

• For Frames

D

D

jj

jj

i

jF

jj

j2

j

FiF AE

LfFf

A2ELf

UFU

i

i

i

dxEIM

FMdx

2EIMU

FU

iF

2

FiF i

i

i

D

D

dxEIM

FM

AELf

Ff

ijj

jj

i

jFi

D

Betti’s Law of Reciprocal Deflections

P

Q

2PD

Location 1 Location 2

1QD

D dxEI

MMQ PQP2

Virtual work principle

D dxEI

MMP Q

PQ1

21 PQ QP DD

Betti’s Law

Q

P

thenQ PIf 12 QP DD

δUδW ddL

d0

F xEIMMΔF

Betti’s Law of Reciprocal Deflections

AD

P

BD

P

BA DD

Betti’s Law of Reciprocal Deflections

A

P

BD

M

AB MP D

AMP

B MdxEIMMP D

thenmagnitude) (inM PIf magnitude) (in AB D

Betti’s Law and Flexibility Coefficients

jiji fD

1

1

jiij ff

Location j Location i

ijij fD

Deflection at j due to a unit load at iFlexibility coefficient

Deflection at i due to a unit load at j

9. Deflections Using Energy Methods

• Work-energy principle• Virtual work principle• Castigliano’s principle• Betti’s law