PPT: Energy Methods of Structural Analysis | Structural Analysis - Civil Engineering (CE) PDF Download

 Page 1


CASTIGLIANO’S FIRST THEOREM
? “When forces act on elastic systems subject to small
displacements, the displacement corresponding to any force,
in the direction of the force, is equal to the partial derivative
of the total strain energy w.r.t. that force.”
2
i
i
F
U
?
?
? ?
Where:
F
i
= Force at i-th application point
d
i
= Displacement at i-th point in the direction of F
i
U = Total strain energy
Page 2


CASTIGLIANO’S FIRST THEOREM
? “When forces act on elastic systems subject to small
displacements, the displacement corresponding to any force,
in the direction of the force, is equal to the partial derivative
of the total strain energy w.r.t. that force.”
2
i
i
F
U
?
?
? ?
Where:
F
i
= Force at i-th application point
d
i
= Displacement at i-th point in the direction of F
i
U = Total strain energy
CASTIGLIANO’S THEOREM 
? We can also use this method to find the angle of rotation ( ?).
3
Where:
M
i
= Moment at i-th application point
?
i
= Slope at i-th point resulting from M
i
U = Total strain energy
i
i
M
U
?
?
? ?
Page 3


CASTIGLIANO’S FIRST THEOREM
? “When forces act on elastic systems subject to small
displacements, the displacement corresponding to any force,
in the direction of the force, is equal to the partial derivative
of the total strain energy w.r.t. that force.”
2
i
i
F
U
?
?
? ?
Where:
F
i
= Force at i-th application point
d
i
= Displacement at i-th point in the direction of F
i
U = Total strain energy
CASTIGLIANO’S THEOREM 
? We can also use this method to find the angle of rotation ( ?).
3
Where:
M
i
= Moment at i-th application point
?
i
= Slope at i-th point resulting from M
i
U = Total strain energy
i
i
M
U
?
?
? ?
CASTIGLIANO’S THEOREM 
4
U stored in 
structure
F
1
F
2
F
3
F
n
d
1
d
2
d
3
d
n
1
1
F
U
?
?
? ?
2
2
F
U
?
?
? ?
3
3
F
U
?
?
? ?
n
n
F
U
?
?
? ?
Page 4


CASTIGLIANO’S FIRST THEOREM
? “When forces act on elastic systems subject to small
displacements, the displacement corresponding to any force,
in the direction of the force, is equal to the partial derivative
of the total strain energy w.r.t. that force.”
2
i
i
F
U
?
?
? ?
Where:
F
i
= Force at i-th application point
d
i
= Displacement at i-th point in the direction of F
i
U = Total strain energy
CASTIGLIANO’S THEOREM 
? We can also use this method to find the angle of rotation ( ?).
3
Where:
M
i
= Moment at i-th application point
?
i
= Slope at i-th point resulting from M
i
U = Total strain energy
i
i
M
U
?
?
? ?
CASTIGLIANO’S THEOREM 
4
U stored in 
structure
F
1
F
2
F
3
F
n
d
1
d
2
d
3
d
n
1
1
F
U
?
?
? ?
2
2
F
U
?
?
? ?
3
3
F
U
?
?
? ?
n
n
F
U
?
?
? ?
EXAMPLE:
? CALCULATE         AND         FOR THE BEAM SHOWN IN 
FIGURE. TAKE E=                AND I=
B
?
B
?
5
2 10 ?
8
5 10 ?
5
Page 5


CASTIGLIANO’S FIRST THEOREM
? “When forces act on elastic systems subject to small
displacements, the displacement corresponding to any force,
in the direction of the force, is equal to the partial derivative
of the total strain energy w.r.t. that force.”
2
i
i
F
U
?
?
? ?
Where:
F
i
= Force at i-th application point
d
i
= Displacement at i-th point in the direction of F
i
U = Total strain energy
CASTIGLIANO’S THEOREM 
? We can also use this method to find the angle of rotation ( ?).
3
Where:
M
i
= Moment at i-th application point
?
i
= Slope at i-th point resulting from M
i
U = Total strain energy
i
i
M
U
?
?
? ?
CASTIGLIANO’S THEOREM 
4
U stored in 
structure
F
1
F
2
F
3
F
n
d
1
d
2
d
3
d
n
1
1
F
U
?
?
? ?
2
2
F
U
?
?
? ?
3
3
F
U
?
?
? ?
n
n
F
U
?
?
? ?
EXAMPLE:
? CALCULATE         AND         FOR THE BEAM SHOWN IN 
FIGURE. TAKE E=                AND I=
B
?
B
?
5
2 10 ?
8
5 10 ?
5
For deflection at B
Segment Origin Limits Mx
BC B 0 to 1 -P.x -x
CA C 0 to 3 -P(x+1)-40x -(x + 1)
x
M
P
?
??
??
?
??
6