CIVL3310 STRUCTURAL ANALYSISProfessor CC Chang
Chapter 10: Analysis of Statically Indeterminate Structures by the Force
Method
Determinate or Indeterminate ?
StructureEquilibrium
Determinate
Indeterminate
Yes
No
Why Indetermiante ?• Advantages
Smaller stresses and deflections
Why Indetermiante ?• Advantages
Fail safe
1995Oklahoma City bombing
Prices• Disadvantages
Stresses due to support settlement
Prices• Disadvantages
Stresses due to temperature changes
Indeterminate Structures
Symmetric Structures
Structure Reflection
Axis of symmetry
Identical in geometry, supports and material properties
Symmetrical Structures
Symmetrical Structures
Symmetrical Loadings
Symmetrical Loadings
Anti-Symmetrical Loadings
Anti-Symmetrical
Anti-Symmetrical Loadings
Decomposition of Loadings(A)
(B) = (A)/2
(C) = Reflection of (B)
(B)+(C)
(B)-(C)
Symmetrical
Anti-symmetrical
sum
Decomposition of Loadings• Loadings = Symmetrical + Anti-symmetric Loads
+
=
Decomposition of Loadings
Decomposition of Loadings
Analysis of Symmetrical StructuresLoading
Anti-symmetricalLoading
SymmetricalLoading
SymmetricalStructure
Response 1 Response 2+
Response
Symmetrical Structures under Symmetrical Loads
La a
P P
Moment & vertical displacement ≠ 0Slope & axial displacement = 0
P
V ≠ 0
slope = 0
M ≠ 0
Symmetrical Structures under Symmetrical Loads
Symmetrical Structures under Anti-symmetrical Loads
L/2
a
a
P
P
Slope ≠ 0Moment & vertical displacement = 0
P
L/2
Slope ≠ 0
M = 0V = 0
Symmetrical Structures under Anti-symmetrical Loads
Analysis
Analysis6 degrees of indeterminacy
4 degrees of indeterminacy
4 degrees of indeterminacy
Analysis
Analysis of Statically Indeterminate Structures• Force methods
This chapter• Displacement methods
Next two chapters
Compatibility
0 BByB fB
By
B=0
0' BB
Compatibility
0fM AAA0A
Compatibility
0fB BBy0B
Compatibility
0fD DDx0D
0D
DDf
Compatibility
A B
PC DC
D
P
AD
11
=
+
ADF
ADf
0Ff ADADAD
Compatibility
0
0
0
0
YDY,DYXDX,DYDY
YDY,DXXDX,DXDX
DfDf
DfDf
Compatibility• Settlement
CYCCyCBCO
BYBCyBBBO
CfBf
CfBf
Compatibility• Settlement
Least Work Method• Castigliano’s theory
P
?
PF
U(P,F) dxEI2M2
M(P,F)
dxFM
FU
EIM
F 0
Least Work MethodP
F
FM(P,F)
U(P,F) dxEI2M2
dxFM
FU
EIM
F 0
0FU
Obtain F
The magnitude of redundant force must be such that the strain energy stored in the structure is a minimum
Least Work Method• Virtual work principle
P
FUW
P
P
FM(P,F)
F,PU
PPW FFUP
PUU
FFUP
PUP P
0FFUP
PU
P
0FUPU
P
Castigliano’s theorem
Least work principle
Note: F does not do any work !
Least Work MethodP1
FnF2F1
PmP2
Strain energy
n21m21 F,,F,F,P,,P,PU
0FUPU
i
Pi
i
Forces that do not do work
Forces that do work