Duke UniversityDepartment of Civil and Environmental Engineering

CEE 421L. Matrix Structural AnalysisFall, 2012

Henri P. Gavin

The Theorems of Castigliano 1

A linear force-displacement relationship between a force, F , and a collocateddisplacement, D, in statically determinate systems can be determined usingthe principle of real work,

W = U (1)12 F ·D = 1

2∫V{σ}T{ε} dV . (2)

The force-displacement relationships for systems with multiple external forcesor distributed loads, or statically indeterminate systems, involve relationshipsbetween multiple forces and displacements. The external work is

W = 12

n∑i=1

FiDi

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D

F

D

Fj

D j

n

1

F1

D

F

n

F

D D

FF

D D

D

F

D

1

1

1

1

i

i

n

Fj

F j

Dj

ji

i

Fn

Fn

Dn

R

R

Ra

c

b

A set of n force-displacement relationships cannot be found with the singleprinciple of real work equation , W = U . Instead, a new method must bedeveloped.

1Castigliano, Carlo Alberto, Intorno ai sistemi elastici, Ph.D. dissertation, Polytechnic of Turin, 1873.

2 CEE 421L. – Matrix Structural Analysis – Duke University – Spring 2012 – H.P. Gavin

Castigliano’s Theorem - Part I (Force Theorem)

The strain energy in any elastic solid subjected to n point forces Fi is afunction of the n collocated displacements, Di.

U(D1, D2, · · · , Dn) =n∑

i=1

∫ Di

0Fi(Di) dDi

∆U ≈ Fj ∆Dj , Fj ≈∆U∆Dj

, ⇒ Fj = ∂U(D)∂Dj

(3)

The force, Fj, on an elastic solid is equal to the partial derivative of the strainenergy, U(D1, D2, · · · , Dn), with respect to the collocated displacement, Dj.

Castigliano’s Theorem - Part II (Deflection Theorem)

The complementary strain energy in any elastic solid subjected to n pointforces Fi is a function of the n forces and is the complement of the strainenergy.

U ∗(F1, F2, · · · , Fn) =n∑

i=1FiDi−U(D1, D2, · · · , Dn) =

n∑i=1

FiDi−n∑

i=1

∫ Di

0Fi(Di) dDi

∆U ∗ ≈ Dj ∆Fj , Dj ≈∆U ∗∆Fj

, ⇒ Dj = ∂U ∗(F )∂Fj

(4)

The partial derivative of the complementary strain energy of an elastic system,U ∗(F ), with respect to a selected force acting on the system, Fj, gives thedisplacement of that force along its direction, Dj.

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0

Dj

∆

jD∆D+j

Dj

U(D)

jF

∆U

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0

∆ jFF∆ jF+j

jF

Dj

U*(F)

∆U*

If the solid is linear elastic, then U ∗(F ) = U(D).

CC BY-NC-ND H.P. Gavin

The Theorems of Castigliano 3

For linear elastic prismatic solids in equilibrium,

U ∗(F ) = U(D) = 12

∫l

N 2

EAdl + 1

2∫l

M 2z

EIzdl + 1

2∫l

M 2y

EIydl +

12

∫l

V 2z

G(A/αz)dl + 1

2∫l

V 2y

G(A/αy) dl + 12

∫l

T 2

GJdl . (5)

y

dl

zM

zx

xN

My

y

dl

zz

x

xT

yV

V

So,

∂U ∗

∂Fj= ∂U

∂Fj=

∫l

N ∂N∂Fj

EAdl +

∫l

Mz∂Mz

∂Fj

EIzdl +

∫l

My∂My

∂Fj

EIydl +

∫l

Vz∂Vz

∂Fj

G(A/αz)dl +

∫l

Vy∂Vy

∂Fj

G(A/αy) dl +∫l

T ∂T∂Fj

GJdl . (6)

Superposition

Superposition is an extremely powerful method for separating a system withmultiple linear force-displacement relationships into multiple systems withsingle linear force-displacement relationships.

The principle of superposition states:

Any response of a linear system to multiple inputs can be represented as thesum of the responses to the inputs taken individually.

By “response” we can mean a displacement, a strain, a stress, an internalforce, a rotation, an internal moment, etc.

By “input” we can mean an externally applied load, a temperature change, asupport settlement, etc.

CC BY-NC-ND H.P. Gavin