Maxwell's Reciprocal Theorem

flexibility and stiffness matrices are symmetric

d d= T or d dij ji=

k k= T or k kij ji=

1

2

F1

F2

Δ1

Δ2

1 11 12 1

2 21 22 2

d d Fd d F

Δ⎧ ⎫ ⎡ ⎤ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥Δ⎩ ⎭ ⎣ ⎦ ⎩ ⎭

1

2

F1

Δ11

Δ21

Δ12

Δ22

1

2

F2

11 11 12 1

21 21 22 0d d Fd d

Δ⎧ ⎫ ⎡ ⎤ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥Δ ⎩ ⎭⎩ ⎭ ⎣ ⎦

12 11 12

22 21 22 2

0d dd d F

Δ⎧ ⎫ ⎡ ⎤ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎢ ⎥Δ⎩ ⎭ ⎣ ⎦ ⎩ ⎭

W F F F1 2 11 1 12 1 22 2

12

12, = + +Δ Δ Δ

W F F F2 1 22 2 21 2 11 1

12

12, = + +Δ Δ Δ

W W1 2 2 1, ,=

Δ Δ12 1 21 2F F=

d d12 21=

andij ji ij jid d k k= =

Example I

BA

a a

PB

vAB

A

a a

B

vBA

PA

2 3AB B

[3 (2 ) ] 56 6

BP a a a av PE I E I× −

= + =

2 3

BA A[3 (2 ) ] 56 6

AP a a a av PE I E I× −

= + =

AB BAB A

v vP P

=

Example II

PBA

a

θAB

a

B

A

a a

Μ v

B

BAA

2AB B

[2 (2 ) ] 32 2

BP a a a a PE I E I× −

θ = + =

2

BA A[2 (2 ) ] 3

2 2AM a a a av M

E I E I× −

= + =

AB BAB A

vP Mθ

=

Example III

PBA

a

θAB

a a

B

A

a a a

Μ v

B

BAA

a)

3 3B

end14 9 0

3P a R avE I E I

= − + = , B1427

R P=

2 2B

AB3 5

2 2P a R a

E I E Iθ = −

2B

AB1154

P aE I

θ =

b)

2 3A

end5 9 0

2M a R av

E I E I= − + = , A5

18M

Ra

=

2 3A

BA3 14

2 3M a R av

E I E I= −

2A

BA1154

M av

E I=

c)

2AB BA

B A

1154

v aP M E Iθ

= =

Example IV

x

y, v

/2v2

P1

/2

E I v P b x

L L b x= − − −62 2 2( )

P P v x v L a b x a= = = = = = ≤1 2

12 2 3

212, ( ) , , , ,

E I v P= − − −24 4 1

42 2 2( )

vPE I21

31148= −

x

y, v

/2P2

v1

/2

E I v P b x

L L b x= − − −62 2 2( )

P P v x v L a b x a= − = − = = = = ≤2 2 1

22 1, ( ) , , , ,

E I v P= − − −24 4 14

2 2 2( )

vPE I12

31148= −

vP

vP E I

21

12

31148= = −

Electrical Reciprocity

Voltage Generator

V1

RA Ammeter

I2RB

RC

2 1B

A B A C B C

RI V

R R R R R R=

+ +

RAAmmeter Voltage Generator

V2I1 RB

RC

1 2B

A B A C B C

RI V

R R R R R R=

+ +

V1

I2

V2

I1

1 11 12 1

2 21 22 2

I a a VI a a V⎡ ⎤ ⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

12 21a a=