CIVL3310 STRUCTURAL ANALYSISProfessor CC Chang

Chapter 9: Deflections Using Energy Methods

Work and Energy Principles• Work done by external forces

FF

F

FΔ

FΔF2

1W

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 df2

1U

FΔF2

1W

ii df2

1UTotal strain energy

Work - Energy

UW

iiF df2

1ΔF

2

1

Virtual Work Principle• Under equilibrium, perturb the

structure

FF

F

FΔ

FδΔ

δF

FδΔFδW

FΔδFδW

FΔF2

1W

Work - Energy

UW

iiF df2

1ΔF

2

1

Perturb F by dF

Perturb DF by d DFδUδW

iiF dδfδF

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

iiF dδfδF

iiF δdfδF

F

F FδΔ

δF

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

δUδW

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

iiF dδfδF

FF

δF

iδffi di

Superposition

Actual system Virtual system

Virtual Work Principle

iiF dδfδF

FF δF

iδffi di

Actual system Virtual system

δUδW

Looking for DF

1

1

fiLi

EiAi

ii

iiiF AE

Lfδf1

Virtual Work Principle

ii

iiiF AE

Lfδf1

δUδW

F

F

1δF fi

d fi

Virtual Work Principle

iiF dδf1

δUδW

F

1δF

d fiDT

Temperature effect

ii LΔTαd

Virtual Work Principle

iiF dδf1

δUδW

F

1δF

d fiDL

Misfit effect

Virtual Work Principle• For Beams

F

dM(x)2

1U

M=0, q≠0, k=0

M ≠ 0, q≠0, k ≠ 0M(x)

xdq

M(x)

dd dM(x)UIntegrate for the whole beam

LL

dd0

2

0x

EI

M

2

1M(x)

2

1U

Perturbation

xEI

M(x)dd

d ddLL

dd00

xEI

MMMU

DF

δUδW ddL

d0

F xEI

MMΔF

dFdM(x)

Virtual Work Principle

δUδW

ddL

d0

F xEI

MMΔF

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 xEI

MMM

qA

dMA

Virtual Work Principle• For Frames

δUδW

d ddcomponentcomponent

dxEI

MM

EI

fLfΔF F

Negligible

w

PdF

DF

Castigliano’s Principle• Work-Energy ii FUFW

Fi

Fi

idF

ii dfiδf

Fi

iFΔ

k

F

2

F

ΔF2

1W

ii

Fi i

1

k

n

n

EA

Lf

2

f

df2

1U

iii

ii ii Ff

i

i

ii

ii

FiF

FiFi

iFi

iiFi

i

iiii

F

U

F

U

F

W

dFF

UFUdF

F

WFW

dFFUdFFW

iFi

iΔ

k

F

F

W

Castigliano’s Principle• For Trusses

• For Beams

• For Frames

jj

jj

i

jF

jj

j2

j

FiF AE

Lf

F

f

A2E

LfU

F

Ui

i

i

dxEI

M

F

Mdx

2EI

MU

F

U

iF

2

FiF i

i

i

dxEI

M

F

M

AE

Lf

F

f

ijj

jj

i

jFi

Betti’s Law of Reciprocal Deflections

P

Q

2P

Location 1 Location 2

1Q

dxEI

MMQ P

QP2

Virtual work principle

dxEI

MMP Q

PQ1

21 PQ QP

Betti’s Law

Q

P

thenQ PIf 12 QP

δUδW ddL

d0

F xEI

MMΔF

Betti’s Law of Reciprocal Deflections

A

P

B

P

BA

Betti’s Law of Reciprocal Deflections

A

P

B

M

AB MP

AMP

B MdxEI

MMP

thenmagnitude) (inM PIf magnitude) (in AB

Betti’s Law and Flexibility Coefficients

jiji f

1

1

jiij ff

Location j Location i

ijij f

Deflection at j due to a unit load at i

Flexibility 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