Statically Inde. Structures

8/12/2019 Statically Inde. Structures

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ANALYSIS OF INDETERMINATESTRUCTURES

1.IntroductionA structure of any type is classified as statically

indeterminate when the number of unknown reaction or

internal forces exceeds the number of equilibrium

equations available for its analysis.

What is statically DETERMINATE structure? (Fig.1)

No. of unknown = 3

No. of equilibrium equations = 3

3 = 3 thus statically determinate

Fig.1

No. of unknown = 4


43 thus statically Indeterminate

Fig.2

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Fig.3

No. of unknown = 6


6 5 thus statically Indeterminate

2.Why we study indeterminate structure ? Most of the structures designed today are statically indeterminate

Reinforced concrete buildings are considered in most cases as

statically indeterminate structures since the columns & beams are

poured as continuous member through the joints & over the supports

More stable compare to determinate structure or in another wordsafer.

In many cases more economical than determinate.

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Contrast

Determinate Structure Indeterminate Structure

PL3 /48EIP

PL3 /192EIP

Considerable compared to

indeterminate

structure

DEFLECTION

PL/4

P PL/8 PSTRESS

High moment caused thicker

member & more material needed

Less moment, smaller cross

section& less material needed

TEMPERATUREP

No effect & no stress would be

Developed in the beam

Serious effect and stress would

be developed in the beam

P

3

Generally smaller than

determinate structure


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4

EQUILIBRIUM

Equilibrium issatisfied when the

reactive forces

hold the structure

at rest

The force -

displacement

requirement depend

upon the way the

material responds,

which assumedlinear-elastic

response

FORCE-DISPLACEMENT

COMPATIBILITY

Compatibility is

satisfied when thevarious segments

of the structure fit

together without

intentional breaks

or overlaps

Methods of Analyses


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Methods of Analysis

Two different Methods are available

Force method

known as consistent deformation, unit load method,

flexibility method, or the superposition equations method.

The primary unknowns in this way of analysis are forces

Displacement method Known as stiffness matrix method

The primary unknowns are displacements

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Force method of analysisThe deflection or slope at any point on a structure as a result of a

number of forces, including the reactions, is equal to the algebraic

sum of the deflections or slopes at this particular point as a result

of these loads acting individually

General Procedure Indeterminate to thefirst degree

1 Compatibility equation

is needed

Choosing one of the

support reaction as a redundant

The structure become statically

determinate & stable

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Downward displacementB at B calculated (load action)

BB upward deflection per unit force at

B

Compatibility equation0 =B + ByBB

Reaction By known

Now the structure is statically

determinate

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General Procedure

Indeterminate to the first

degree

1 Compatibility equation isneeded

Choosing MA at A as a

redundant

The structure become statically

determinate & stable

Rotation A at A caused by load P is

determined

AA rotation caused by unit couple

moment applied at A

Compatibility equation

0 = A + MA AA

Moment MA known

Now the structure is statically determinate 8


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General Procedure Indeterminate to the 2nd degree

2 Compatibility equations needed

Redundant reaction B & C

Displacement B &C caused by load P1 & P2 are determined

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BB & BC Deflection per unit

force at B are determined

CC &CB Deflection per unit

force at C are determined

1

Compatibility equations

0 =B + ByBB + CyBC

0 =C + ByCB + CyCC

Reactions at B & C are known

Statically determinate structure


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Maxwells Theorem

The displacement of a point B on a structure due to a unit load

acting at a point A is equal to the displacement of

point A when the unit load is acting at point B is:

fBC=fCB

The rotation of a point B on a structure due to a unit moment

acting at a point A is equal to the rotation of point A when the unitmoment is acting at point B the is

BC=CB

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Procedure for Analysis

Determine the degree of statically indeterminacy

Identify the redundants, whether its a force or a moment, that

would be treated as unknown in order to form the structurestatically determinate & stable

Calculate the displacements of the determinate structure at the

points where the redundants have been removed

Calculate the displacements at these same points in thedeterminate structure due to the unit force or moment of each

redundants individually

Workout the compatibility equation at each point where there is

a redundant & solve for the unknown redundants

Knowing the value of the redundants, use equilibrium to

determine the remaining reactions

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Determine the reaction at B

Indeterminate to the 1st degree

thus one additional equationneeded

Lets take B as a redundant

Determine the deflection at point B

in the absence of support B. Using

the momentarea method

B

= 1/EI[300x6/2{(x6)+6}]

B = 9000 kNm3/EI

Determine the deflection caused by

the unit load at point B

Example

CA

B

50kN300kNm


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BB =1/EI[12x12/2xx12]

BB = 576m3/EI Compatibility equation

0=B

+BY

BB

0= 9000 /EI + BY

576 /EI

BY

=15.6kN

The reaction at B is known now so

the structure is staticallydeterminate & equilibrium

equations can be applied to get

the rest of the unknowns

AB

1

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15

90kN

C

ED

BA

60kN

C

BA

CC

C ED

A B

1.671.34

0.440.56

1.0

2.67

60kN90kN

68.33 81.67

C ED BA

245230205

C

60kN 90kN

EDAB

C


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A

2.5

20kN8kN / m

B C

3 2.5

B

8kN / m20kN

A CB

B1 =RB BBA C

B

RB

Compatibility Equation

B =B1 =RBBB

RB = B / BB

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Statically Inde. Structures