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CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method
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CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Dec 15, 2015

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Page 1: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

CIVL3310 STRUCTURAL ANALYSISProfessor CC Chang

Chapter 10: Analysis of Statically Indeterminate Structures by the Force

Method

Page 2: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Determinate or Indeterminate ?

StructureEquilibrium

Determinate

Indeterminate

Yes

No

Page 3: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Why Indetermiante ?• Advantages

Smaller stresses and deflections

Page 4: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Why Indetermiante ?• Advantages

Fail safe

1995Oklahoma City bombing

Page 5: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Prices• Disadvantages

Stresses due to support settlement

Page 6: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Prices• Disadvantages

Stresses due to temperature changes

Page 7: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Indeterminate Structures

Page 8: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Symmetric Structures

Structure Reflection

Axis of symmetry

Identical in geometry, supports and material properties

Page 9: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Symmetrical Structures

Page 10: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Symmetrical Structures

Page 11: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Symmetrical Loadings

Page 12: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Symmetrical Loadings

Page 13: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Anti-Symmetrical Loadings

Anti-Symmetrical

Page 14: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Anti-Symmetrical Loadings

Page 15: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Decomposition of Loadings

(A)

(B) = (A)/2

(C) = Reflection of (B)

(B)+(C)

(B)-(C)

Symmetrical

Anti-symmetrical

sum

Page 16: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Decomposition of Loadings• Loadings = Symmetrical + Anti-symmetric Loads

+

=

Page 17: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Decomposition of Loadings

Page 18: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Decomposition of Loadings

Page 19: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Analysis of Symmetrical Structures

Loading

Anti-symmetricalLoading

SymmetricalLoading

SymmetricalStructure

Response 1 Response 2+

Response

Page 20: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Symmetrical Structures under Symmetrical Loads

La a

P P

Moment & vertical displacement ≠ 0Slope & axial displacement = 0

P

V ≠ 0

slope = 0

M ≠ 0

Page 21: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Symmetrical Structures under Symmetrical Loads

Page 22: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

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

Page 23: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Symmetrical Structures under Anti-symmetrical Loads

Page 24: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Analysis

Page 25: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Analysis6 degrees of indeterminacy

4 degrees of indeterminacy

4 degrees of indeterminacy

Page 26: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Analysis

Page 27: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Analysis of Statically Indeterminate Structures

• Force methods This chapter

• Displacement methods Next two chapters

Page 28: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Compatibility

0 BByB fB

By

DB=0

0' BB

Page 29: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Compatibility

0fM AAA0A

Page 30: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Compatibility

0fB BBy0B

Page 31: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Compatibility

0fD DDx0D

0D

DDf

Page 32: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Compatibility

A B

PC DC

D

P

AD

11

=

+

ADF

ADf

0Ff ADADAD

Page 33: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Compatibility

0

0

0

0

YDY,DYXDX,DYDY

YDY,DXXDX,DXDX

DfDf

DfDf

Page 34: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Compatibility• Settlement

CYCCyCBCO

BYBCyBBBO

CfBf

CfBf

Page 35: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Compatibility• Settlement

Page 36: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

Least Work Method• Castigliano’s theory

P

=?D

D

PF

U(P,F) dxEI2M2

M(P,F)

dxF

M

F

UEIM

F 0

Page 37: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

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

Page 38: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

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 !

Page 39: CIVL3310 STRUCTURAL ANALYSIS Professor CC Chang Chapter 10: Analysis of Statically Indeterminate Structures by the Force Method.

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