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 Structures

Loading

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

DB=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

=?D

D

PF

U(P,F) dxEI2M2

M(P,F)

dxF

M

F

UEIM

F 0

Least Work MethodP

DF

FM(P,F)

U(P,F) dxEI2M2

dx

F

M

F

UEIM

F 0

0F

U

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

DP

+dP

+dFM(P,F)

F,PU

PPW FF

UP

P

UU

FF

UP

P

UP P

0FF

UP

P

UP

0F

UP

UP

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

0F

UP

U

i

Pi

i

Forces that do not do work

Forces that do work