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(1)

## Chapter 2

Energy Principles

Principle of Minimum Potential Energy and Principle of Virtual Work

Mg h

P

(2)

Read Chapter 11 (pp.420~ 428) of Elementary Structural Analysis 4th Edition by C .H. Norris et al very carefully. In this note hatted variables ( ˆ) and tilded variables ( ~) denotes virtual and approximated quantities, respectively. The virtual displacement field should satisfy the displacement boundary conditions of supports if specified. For beam and frame problems, displacement boundary conditions include boundary conditions for rotational angles.

Variables with subscript e denote those defined within an element or member.

(3)

## 2.1 Spring-Force Systems

Total Potential Energy: The energy required to return a mechanical system to a reference status

=

 +

=

=

=

=

=

P k

P k

du u k

du u

k 2 ext int ext 2

0 0

int 2

, 1 2

) 1 (

) (

Equilibrium Equation : k =P

Principle of Minimum Potential Energy: ~ =+ˆ

2 2 2

2 2 2 2

2

1 ( ˆ) ( ˆ) 1 ( ) ˆ 1 ( )ˆ ( ˆ)

2 2 2

1 ( ) 1 ( )ˆ ˆ( ) 1 ( ) 1 ( )

2 2 2 2

1 ( )ˆ 2

k P k k k P

k P k k P k P k

k

 =  +  −  +  =  +  +  −  + 

=  −  +  +   − =  −  + 

=  +   

In the above equation, the equality sign holds if and only if  = 0. Therefore the total potential energy of the spring-force system becomes minimum when displacement of spring satisfies the equilibrium equation.

P

(4)

## 2.2 Beam Problems

Boundary conditions: Displacement BCs should be satisfied exactly.

### Potential Energy of a Beam

2 2

2 2

0 0

0

2 2 2

0 0

1 1

( )

2 2

1 ( )

2

l l

int

l ext

l l

int ext

M d w

dx EI dx

EI dx

qwdx

EI d w dx qwdx dx

 = =

 = −

 =  +  = −

 

 

Equilibrium Equation

2 2

dx q M d dx V

dM dx q dV

=





=

=

, or by the moment-curvature relation q dx

w EI d44 =

(5)

Principle of Minimum Potential Energy (w~ = w+ wˆ)

2 2

2 2

0 0

2 2 2 2 2 2

2 2 2 2 2 2

0 0 0 0 0

2 2 2 2

2 2 2 2

0 0

ˆ ˆ

1 ( ) ( )

( ) ( ˆ)

2

ˆ ˆ ˆ

1 1

ˆ

( ) ( ) ( )

2 2

ˆ ˆ ˆ

1 1

( ) ( ) ( )

2

l l

l l l l l

l l

d w w d w w

EI dx w w qdx

dx dx

d w d w d w d w d w d w

EI dx wqdx EI dx EI dx wqdx

dx dx dx dx dx dx

d w d w d w d w

EI dx EI EI dx

dx dx dx EI dx

+ +

 = − +

= − + + −

=  + + − − −

 

    

 

0

2 2

2 2

0 0 0

2 2

2 2

0

ˆ

ˆ ˆ ˆ

1 ( ) ˆ

2

ˆ ˆ

1 ( ) for all virtual ˆ 2

l

l l l

l

wqdx

d w d w MM

EI dx dx wqdx

dx dx EI

d w d w

EI dx w

dx dx

=  + + −

=  +  

  

Since the equation in the box represents the total virtual work in a beam, the total potential energy of a beam becomes a minimum for all virtual displacement fields when the principle of virtual work holds. In the above equation, the equality sign holds if and only if wˆ = 0.

(6)

### Principle of Virtual Work

If a beam is in equilibrium, the principle of the virtual work holds for the beam,.

0 )

ˆ(

0

4

4 − =

l w EI ddxw q dx for all virtual displacement wˆ

2 2 2 3

2 2 2 3

0 0 0 0

ˆ ˆ

ˆ ˆ 0

l l

l l

d w d w dw d w d w

EI dx wqdx EI wEI

dx dx − + dx dxdx =

 

2 2

2 2

0 0 0 0

ˆ ˆ

ˆ ˆ 0

l l l l

d w d w MM

EI dx wqdx dx wqdx

dx dx − = EI − =

   

In case that there is no support settlement, the boundary terms in above equation vanishes identically since either virtual displacement including virtual rotational angle or corresponding forces (moment and shear) vanish at supports. The principle of virtual work yields the displacement of an arbitrary point x~ in a beam by applying a unit load at x0 and by using the reciprocal theorem.

l ˆ = l ˆ = l = = l MˆM
(7)

Approximation using the principle of minimum potential energy - Approximation of displacement:

=

=

n

i

i ig a w

w

1

~

- Total potential energy by the assumed Displacement function



  

= = =

 −

= 

=

l n

i

i i

l n

j

j j n

i

i i l

l

qdx g a dx

g a EI

g a qdx

w dx dx

w EI d

dx w d

0 1

0 1 1

0 0

2 2 2

2

) (

) 2 (

~ 1 )

~

~ 2 (

~ 1

### The first-order Necessary Condition

1 1 1

0 0

1 1

0 0 0

1 0 0 1

1 ( ) ( )

2

1[ ( ) ( ) ]

2

0

l n n l n

i i j j i i

i j i

k k

l n l n l

k j j i i k k

j i

l l

n n

k i i k ki i k

i i

a g EI a g dx a g qdx

a a

g EI a g dx a g EIg dx g qdx

g EIg dxa g qdx K a f

= = =

= =

= =

 

 =     − 

   

   

= + −

 

= − = − =

  

 

 

  

   

or in matrix form f

Ka =

(8)

Examples

i) With one unknown

a w

lx x

a l

x ax

w~ = ( − ) = ( 2 − ) → ~= 2

4 4 2 1 ) 4

2 2 (

) 1 ( 2

) 2 (

~ 1 2 2

2

0

2 0

0

2 l

aP l

a l EI

aP dx

a EI l wdx

x P dx

w EI

l l

l  −  − = + = +

=

  

) 16 (

0 16 4 4

0

~

2 2

xl EI x

w Pl EI

a Pl Pl

a = → aEIl + = → = − → = − −

EI Pl w l

) 64 (2

~ = 3 ,

EI Pl EI

Pl w l

3 3

0208 .

48 0 2)

( = = , Error = 0.25

2) (

2)

~( 2)

( − =

w l w l w l

ii) With two unknowns

bx a

w l

x bx l

x ax

w~ = ( − )+ ( 22) → ~ = 2 + 6

P

(9)

8 ) 3 ( 4

3) 2 36

24 4

2 ( 1

8 ) 3 ( 4

) 6 2

2 (

~ 1 2) (

~ ) 2 (

~ 1

3 2

3 2 2

2

3 2

0

2 0

0

2

b l al

l P l b

ab l

a EI

b l al

P dx bx a

EI dx

l w x P dx

w EI

l l

l

+ +

+ +

=

− +

=

 −

=

  

8 ) 3

12 6

( 0

~

) 4 6 4

( 0

~

3 3

2

2 2

P l b

l a

l b EI

Pl b

l la a EI

= +

 =

= +

 =

0

, 16

1 =

=

b

EI

a Pl (???)

iii) With three unknowns

2 3

3 2

2 ) ( ) ~ 2 6 12

( )

~ ax(x l bx x l cx x l w a bx cx

w = − + − + − → = + +

16 ) 7 8

3 ( 4

4) 3 144

2 48 5 24

3 144 36

4 2 (

1

16 ) 7 8

3 ( 4

) 12 6

2 2 (

1

2) (

) 2 (

~ 1

4 3

2 4

3 2

5 2 3

2 2

4 3

2

0

2 2 0

0

2

c l b l

al l P

l bc l ac

l ab l c

b l

a EI

c l b l

al P dx cx

bx a

EI

l wdx x

P dx

w EI

l

l l

+ +

+ +

+ +

+ +

=

− +

+

=

 −

=

(10)





= +

+

 =

= +

+

 =

= +

+

 =

16 ) 7

5 18 144

8 ( 0

~

8 ) 3

18 12

6 ( 0

~

) 4 8 6

4 ( 0

~

4 5

4 3

3 4

3 2

2 3

2

P l c

l b

l a

l c EI

P l c

l b

l a

l b EI

Pl c

l b l la a EI

EIl P EI

b P EI

a Pl

64 c 5

32 ,

5 , 64

1 = − =

=

) 64 (

) 5 32 (

) 5 64 (

~ 1 2 2 x x3 l3

EIl l P

x EI x l P

x EI x

w= Pl − − − + −

EI Pl EI

Pl EI

Pl EI

Pl w l

3 3

3 3

0205 .

7619 0 .

48 21 1024

) 21 16

7 64

5 8 3 32

5 4 1 64 ( 1 2)

~( = − + − = = = , Error = 0.0144

iv) With one sin function

l x a l

w l x

a

w~ = sin =→ ~= ()2sin

2 2 2

0 0 0

1 1

( ) ( ) ( ( ) sin )

2 2 2

1 1

l l l

l

EI w dx P x l wdx EI a x dx aP

l l

l

 

 =  −  − = −

  

  

(11)

l x EI

x Pl l EI

w Pl EI

a Pl l P

EIa l a

= 

= 

 →

=

=

 −

 =

 sin

7045 .

48 sin 1

2 0 2

) 2 ( 0

~ 3 3

4 3

4 4

EI Pl w l

3

7045 .

48 ) 1

(2

~ = ,

EI Pl w l

) 48 (2

= 3 , Error =0.0145

v) With two sin functions

l x b l

l x a l

w l x

b l x

a

w=  +  =→ =   +  3

sin 3 )

( sin

)

~ ( sin3

~ sin 2 2

2

0 0

2 2 2

0

2 4 2 2 2 2 4 2

0 0 0

2 4 2 4

1 ( ) ( )

2 2

1 3 3

( ( ) sin ( ) sin )

2

1 3 3 1 3 3

( ) sin ( ) ( ) sin sin ( ) sin

2 2

1 1 3

( ) ( )

2 2 2

l l

l

l l l

EI w dx P x l wdx

EI a x b x dx aP bP

l l l l

EIa xdx EIab xdx EIb xdx

l l l l l l l l

aP bP

l l

EIa EIb

l l

 =  −  −

   

= + − +

       

= + +

− +

 

= +

 

  

2 −aP bP+

(12)

3 ) 81sin (sin 1

~ 2 )

3 ( 0 2

) 2 (3 0

~

0 2 ) 2

( 0

~

3 3 4

4 4

3 4 4

l x l x

EI w Pl

EI b Pl

l P EIb l

b

EI a Pl

l P EIa l

a  − 

= 





− 

=

=

 +

 =

= 

=

 −

 =

EI Pl EI

Pl w l

3 3

0208 .

0 )

0123 .

0 1 ( 0205 .

0 2)

~( = + = ,

EI Pl w l

3

0208 .

0 2)

( = , Error 0

(13)

## 2.3 Beams with Axial Loads

### Total Potential Energy

dx wq Q

dx dx w EI d

dx w

d l

l

=

0 0

2 2 2

2

2 1

dx w

dx w

ds L

L L

L

) ) 2( 1 1 ( )

( 1

0

2 0

2 0

= + +

=

L dx

w dx

w dx

w L

L

L L

L  →  =     

+

=

21( )

21( ) 21

( ) for

0

2 0

2 0

2

dx wq dxdx

dw dx Q dw dx dx

w EI d

dx w

d l l

l

 

=

0 0

0

2 2 2

2

2 1 2

1

Q

(14)

Principle of the Minimum Potential Energy

− +

=

− +

− +

+

=

− +

− +

=

+ + −

− + +

= +

l

l l

l

l l

l l

l l

l

dx dx w Qd dx

w d dx

w EI d

dx w d

dx dx w Qd dx

w d dx

w EI d

dx w dx d

dx q w Q d

dx w EI d

w

dx dx w Qd dx

w d dx

w EI d

dx w d

qdx w dx dx

w Q d dx dw dx

w EI d

dx w wqdx d

dx dx Qdw dx dw dx

w EI d

dx w d

qdx w w dx dx

w w Q d dx

w w dx d

dx w w EI d

dx w w d

0

2 2 2

2

0

2 2 2

2

0

2 2 4

4 0

2 2 2

2

0 0

2 2 2

2

0 0

2 2 2

2

0 0

0

2 2 2

2

ˆ) ˆ

ˆ ( ˆ

2 1

ˆ) ˆ

ˆ ( ˆ

2 ) 1

ˆ(

ˆ) ˆ

ˆ ( ˆ

2 1

ˆ ˆ)

( ˆ )

2 ( 1

ˆ) ) (

ˆ (

ˆ) (

2 1 ˆ)

( ˆ)

( 2

~ 1

The principle of minimum potential energy holds if and only if

2 2

2 2

0

ˆ ˆ ˆ ˆ

( ) 0 for all possible ˆ

l d w d w dw dw

EI Q dx w

dx dxdx dx

The principle of the minimum potential energy is not valid for the following cases.

2 ˆ 2 ˆ ˆ ˆ

l d w d w dw dw

(15)

The critical status of a structure is defined as

0 ˆ)

ˆ ˆ

( ˆ

0

2 2 2

2 − =

l ddxwEI ddxw ddxwQddxw dx

Approximation using the principle of minimum potential energy - Approximation of displacement:

=

= n

i

i ig a w

1

ˆ - Critical Status

1 1 1 1

0 0

1 1 0 1 1 0 1 1 1 1

( ) ( ) ( )( )

( ) ( )( ) 0 ( ) 0

l n n l n n

i i j j i i j j

i j i j

l l

n n n n n n n n

G

i i j j i i j j i ij j i ij j

i j i j i j i j

T G G

a g EI a g dx Q a g a g dx

a g EIg dxa Q a g g dxa a K a Q a K a

a Q a Det Q

= = = =

= = = = = = = =

  −   =

  −   = − =

− = → − =

   

 

     

K K K K

(16)

### Example - Simple Beam

- with a parabola: wˆ = ax(xl) → g1 = 2xl , g1= 2

0 0

2 2 2 3 3

0 0 0

2 3

2 2 2

2 2 4

4 1

(2 ) (4 4 ) ( 2 1)

3 3

1 12

(4 ) 0 (exact : 9.86 , error 22%)

3

l l

i j

l l l

i j

cr

g EIg dx EI dx EIl

g g dx x l dx x xl l dx l l

EI EI EI

Det EIl Q l Q

l l l

  = =

  = − = − + = − + =

− = → =  = =

 

  

- with one sine curve: ˆ sin 1 cos , 1 ( )2sin l

x g l

l x g l

l a x

w=  →  =   =  

4 2 4 2 2 2

0 0 0 0

2

( ) sin ( ) , ( ) cos ( )

2 2

l l l l

i j i j

x l x l

g EIg dx EI dx EI g g dx dx

l l l l l l

     

  = =   = =

  

   

Q

(17)

Example – Cantilever Beam

- with one unknown:

2

1 1

2 3

0 0 0 0

2 3

2 2 2

ˆ 2 , 2

2 2 4 , 4 4

3

4 3

(4 ) 0 (exact : 2.46 , error 22%)

3 4

l l l l

i i i i

cr

w ax g x g

g EIg dx EI dx EIl g g dx x dx l

EI EI EI

Det EIl Q l Q

l l l

 

= → = =

  = =   = =

− = → =  = =

   

- with two unknowns:

3 0

2 22

2 0

21 12

0 11

5 0

4 22

4 0

3 21

12 3

0 2 11

2 1

2 2

1 3

2

12 36

, 6 12

, 4 4

5 9 9

4 , 6 6

3 , 4 4

6

, 2

, 3 ,

ˆ 2

l dx

x K

l xdx K

K l dx K

l dx

x K

l dx

x K

K l

dx x K

x g

g x

g x g

bx ax

w

l l

l

l G l

G G

l G

=

=

=

=

=

=

=

=

=

=

=

=

=

=

 =

=

 =

 =

→ +

=

Q Q

(18)

32.181 or

487 . 0727 2

. or 1 0829 .

0 45

27 . 22 26

45

180 26

26

0 45

52 4

0 )

45 6

( ) 54 12

)(

40 4

(

30 , 1

54 0 12

45 6

45 6

40 0 4

54 ) 45

45 40

30 1 12

6

6 ( 4

2 2

2 2

2 8

12 12

2 6

2 8 6

4 2

4 2

5 3

3

5 3

4 2

4 2

3 5

4

4 3

3 2

2

l EI l

Q EI l

l l

l

l l

l

l l

l l

l l

l l

l

EI Q l

l l

l

l l

l l

l l

l Q l

l l

l EI l

Det

=

 =

− =

= 

=

 +

=

=

 =

→ −

 =

 

− 



 

2.47 2 exact

Q EI

= l (error = 1.2%) or 22.19 2 l

Qexact = EI (error = 45%)

(19)

Example – Beam-Column

- with one sine curve: w asin x g1 cos x , g1 ( ) sin2 x

l l l l l

      

= → = =

4 2 4 2 2 2

0 0 0 0

0 0

( ) sin ( ) , ( ) cos ( )

2 2

sin 2

l l l l

i i i i

l l

i

x l x l

g EIg dx EI dx EI Q g g dx Q dx Q

l l l l l l

x l

g qdx q dx q

l

     

  = =   = =

=  =

   

 

4 2 2 2 4 2

4 4

4 2 2 5

1 1

( ) ( ) 2 ( ) ( ) 2 0

2 2 2 2 2 2

4 / 4 1 1 5 384 4 1 1.0038

( ) 384 5 1 / 1

( ) ( ) 1 ( )

st cr

l l l l l l

EI a Q a q a EI a Q a q

l l a l l

q q l ql

a EI Q l EI Q Q t

EI Q

l l EI

    

 = − − → = − − =

  

 

=  −  =   − =  − =  −

Q Q

q

(20)

0.0 2.0 4.0 6.0 8.0 10.0

0 0.2 0.4 0.6 0.8 1

Exact Solution

Energy Method with one sine function



st

P/P

Q/Qcrcr
(21)

## 2.4. 2-D Framed Structures

Internal Potential Energy Neglecting the Shear Deformation 2 )

1 2

(1

2 2

int

int =

=

 

+

e

e l

e

e l

e e

e dx

EI dx M

EA Q

Global (structural) Coordinate System Local (member) Coordinate System x

y x

y

(22)

### External Potential Energy in case No Distributed Moment in Members

  

  

=

 +

 +

− +

=

 +

=

j

j T

j e l

e T e

j

z j z

j y

j y

j x

j x

j

e l

y e y e l

x e x e

j

ext j e

ext e ext

e

e e

dx

M P

P dx

p u dx

p u

P U p

u

) (

) (

where uTe = (uex ,uey ,ez) , pTe = (pex ,pey ,0) , UTj = (xj ,yj ,zj) , PTj = (Pjx ,Pjy ,Mzj) Total Potential Energy

  

 

+

=

 +

=

j

j T

j e l

e T e

l z

e

e l

e ext

e e

e

dx EI dx

dx M EA

Q ) u p U P

2 1 2

(1

2 2

int

Errors

e e

e u u

u ˆ

~ = + , Q~e = Qe +Qˆe, M~e = Me + Mˆe, U~ j = Uj +Uˆ j

(23)

Total Potential Energy with an Approximation Solution

 +

 +

=

− +

+

+ +

− +

=

 +

 +

+ − + +

=

  

  

  

  

  

  

  

ˆ ˆ

ˆ ˆ ˆ )

( ˆ

ˆ ) ( ˆ

2 ) 1

2 ( 1

ˆ ) (

ˆ ) (

) ) ( ˆ

ˆ ) ( (

2

~ 1

int

ˆ

ˆ

2 2

2 2

2 2

int

 

 

 

 

 

 

j

j T

j e l

e T e

l z

e e e l

e e

l z

e e l

e j

j T

j e l

e T e

l z

e e l

e

j

j T j j

e l

e T e e

l z

e e

e l

e e

e e

e

e e

e e

e

e e

e

dx EI dx

M dx M

EA Q Q

EI dx dx M

EA dx Q

EI dx dx M

EA Q

dx EI dx

M dx M

EA Q Q

P U p

u

P U p

u

P U

U p

u u

Strain-Displacement Relationship in a Member

2 2

,

dx

u EI d dx M

y e e

x e

e = = − , 2

2ˆ ˆ

, ˆ ˆ

dx u EI d dx M

Qe = x e = − y

(24)

Integration by Parts of the Virtual Work Expression

j e

j

j T

j e

l e z e l

e y e l

e x e

e l l

y e y e

e x

e x e

e

j

j T

j e l

e T e e

l e y e l

y e e l

e x e

e l l

y e e x e

e

j

j T

j e l

e T e

e l l

e y e e

x e

j

j T

j e l

e T e

e l z

e e l

e e

e e

e

e e

e e e

e

e e

e

e e

e e

e

M V

u Q

u dx

dx p M u d

dx dx p

u dQ

dx dx M

u d dx

u dM Q

u

dx dx M u d

dx dx u dQ

dx dx

dx M u dx d

dx Q u d

dx EI dx

M dx M

EA Q Q

 +

=

− +

+ +

+ +

=

− +

+

+

=

=

− +

=

   

  

  

  

  

  

  

ˆ ˆ

) ˆ ˆ ˆ

ˆ ( )

) ˆ (

) ˆ (

(

ˆ ˆ ˆ )

ˆ ˆ

(

ˆ ) ˆ

(

ˆ ˆ ˆ )

( ˆ

ˆ ˆ ˆ )

( ˆ ˆ

0 0

2 0 2 0 0

0

2 2 2 2

P U P

U p

u

P U p

u

P U p

u

ˆ e should vanish for all possible errors in displacement as long as a given structure is in equilibrium because it represents the equilibrium of each member.

(25)

### Sign Conventions

Old sign Convention for the Differential Equations New sign Convention for Stiffness Method

new z e old

z e new

y e old

y e new

x e old

x

e u u u

uˆ ) (ˆ ) , (ˆ ) (ˆ ) , (ˆ ) (ˆ )

( = = −  = − 

1 2

1 1 2 3

2 4 5 6

( ) ( (0) , (0) , (0)) , ( ) ( ( ) , ( ) , ( )) ( ) ( (0) , (0) , (0)) ( , , )

( ) ( ( ) , ( ) , ( )) ( , ,

T x y z T x e y e z e

e e e e e e e e

T x y z

e e e e e e e

T x e y e z e

e e e e e e e

u u u l u l l

u u d d d

u l u l l d d d

=  = 

= − −  =

= − −  =

u u

d

d )

)) ) ( , ) ((

) ( , ) ) ( , ) ((

)

(ue T = u1e T ue2 T de T = d1e T de2 T y

x

(26)

) 0 ( )

0 ( , ) 0 ( )

0 ( , ) 0 ( )

0

( ex e ey e em

e f V f M f

Q = − = = −

) ( )

( , ) ( )

( , ) ( )

( e ex e e e ey e e e em e

e l f l V l f l M l f l

Q = = − =

)) 0 (

, ) 0 ( , ) 0 ( (

)) 0 ( , ) 0 ( , ) 0 ( ( )

(fe1 T = fex fey fem = −Qe VeMe )) ( , ) ( ,

) ( ( )) ( , ) ( , ) ( ( )

(fe2 T = fex le fey le fem le = Qe leVe le Me le )

) ( , ) ((

)

(fe T = fe1 T fe2 T

The Virtual Work Expression at Joints with the Old Sign Conventions

− +

+

 +

=

− +

=

j

j T

j e

e e e z e e

e e y e e

e e x e e

z e e

y e e

x e

j

j T

j e

l e y e l

e z e l

e x e j

l M l l

V l u l

Q l u M

V u

Q u

V u M

Q u

e e

e

P U

P U

ˆ

)) ( ) ˆ (

) ( ) ˆ (

) ( ) ˆ (

) 0 ( ) 0 ˆ ( ) 0 ( ) 0 ˆ ( ) 0 ( ) 0 ˆ ( (

) ˆ ˆ ˆ

ˆ ˆ (

0 0

0

### The Virtual Work Expression at Joints with the New Sign Conventions

+ + + + +

=

j

j T

j e

e m e e e

y e e e e

x e e m

e e y

e e x

e e

j (d f (0) d f (0) d f (0) d f (l ) d (l ) f (l ) d f (l )) Uˆ P

ˆ 1 2 3 4 5 6

(27)

Transformation matrix

v

V T

y x y

x y

x y

y x

x

v v V

V v

v V

v v

V

] [ 1

0 0

0 cos

sin

0 sin

cos cos

sin

sin cos

=

 →





 





 =









=

 +

=

=

1 1

1

1 [ ] ˆ ˆ [ ]ˆ

ˆ e e e

T

e d d D

D =  → =  , ˆ 2 [ ] ˆ2 ˆ2 [ ]ˆ 2

e e

e T

e d d D

D =  → = 

1 1

1

1 [ ]T e e [ ] e

e f f F

F =  → =  , Fe2 =[]Tfe2fe2 =[]Fe2 1

2

(28)

Joint Equilibrium (vide the next page)

0 ) ˆ (

ˆ ˆ

ˆ ˆ

ˆ ˆ

) ˆ ˆ )

( ˆ )

((

) ˆ ] [ ] [ ˆ ) ( ]

[ ] [ ˆ ) ˆ ((

2 2

1 1

2 2

1 1

=

=

=

=

=

− +

=

 +

=

P F

U P U F

C U

P U F

C U

P U F

D P

U F

D F

D

P U F

D F

D

S T

T e

e T e T

T e

e T e T

j

j T

j e

e T e j

j T

j e

e T e e

T e

j

j T

j e

e T

T e e

T T e j

ˆ j

 should vanish for all possible errors in displacement as long as a given structure is in equilibrium bec

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