CE 382 L9 - Intro to Indeterminate Analysis

8/9/2019 CE 382 L9 - Intro to Indeterminate Analysis

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Introduction to StaticallyIndeterminate Analysisndeterminate Analysisu

forces of statically determinate

using only the equations of

equilibrium. However, theanalysis of statically indeter-minate structures requires

the geometry of deformation of

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Additional equations come fromcompatibility relationships,

displacements throughout the

structure. The remaininequations are constructed frommember constitutive equations,.e., re at ons ps etween

stresses and strains and the

over the cross section.

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Design of an indeterminates ruc ure s carr e ou n aniterative manner, whereby the

members are initially assumed

and used to analyze the structure.Based on the computed results(displacements and internalmem er orces , e mem er

sizes are adjusted to meet.iteration process continues untilthe member sizes based on the

results of an analysis are close tothose assumed for that analysis.

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Another consequence of

structures is that the relativevariation of member sizesinfluences the magnitudes ofthe forces that the member

.another way, stiffness (largemember size and/or hi hmodulus materials) attracts

force.

Despite these difficulties withstatically indeterminate

structures, an overwhelmingmajority of structures being

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u o ay are s a ca yindeterminate.


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Advantages Statically

n e erm na e ruc ures

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Statically indeterminate

structures typically result insmaller stresses and greater

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s ness sma er e ec onsas illustrated for this beam.


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u

if middle support is removed

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ystructures introduce redundancy,

one part of the structure will notresult in catastrophic or collapse

8failure of the structure.


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Disadvantages of

a ca y n e erm na eStructures

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Statically indeterminate structure-

settlement, which producesstresses as illustrated above.

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Statically indeterminate struc-

tures are also self-strained dueto temperature changes and

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a r ca on errors.


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Indeterminate Structures:

n uence nesInfluence lines for staticall

indeterminate structuresprovide the same informationas influence lines for staticallydeterminate structures, i.e. it

response function at aarticular location on the

structure as a unit load movesacross the structure.

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Our goals in this chapter are:

1.To become familiar with theshape of influence lines for thesupport react ons an nternaforces in continuous beams

.2.To develop an ability to sketch

influence functions for

indeterminate beams andframes.

3.To establish how to positiondistributed live loads oncontinuous structures to

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values.


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Qualitative Influence

nes or a ca y n e-terminate Structures:

-

In many practical applications, itis usually sufficient to draw onlythe qualitative influence lines toec e w ere o p ace e ve

loads to maximize the response.Muller-Breslau Principle pro-vides a convenient mechanism

to construct the qualitativeinfluence lines, which is stated

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:


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The influence line for a force (ormoment res onse function is

given by the deflected shape of

the released structure by

remov ng e sp acemenconstraint corresponding to the

from the original structure and

giving a unit displacement (orrotation) at the location and in

the direction of the response

unc on.

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Procedure for constructing

qualitative influence lines forindeterminate structures is: (1)remove rom e s ruc ure erestraint corresponding to the

,apply a unit displacement orrotation to the released structureat the release in the desired

response function direction, andw u v

shape of the released structure

support and continuityconditions.

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Notice that this rocedure is

identical to the one discussed forstatically determinate structures.

However, unlike staticallydeterminate structures, theinfluence lines for staticallyindeterminate structures arey y u v .

Placement of the live loads tomaximize the desired response

function is obtained from thequalitative ILD.

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Uniforml distributed live

loads are placed over thepositive areas of the ILD tomaximize the drawn responsefunction values. Because the

diminish rapidly with distancefrom the res onse functionlocation, live loads placed more

than three span lengths awaycan e gnore . nce t e veload pattern is known, an

structure can be performed todetermine the maximum value of

18the response function.


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19QILD for RA


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20QILDs for RC and VB


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QILDs for (MC)-,M + and R

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Live Load Pattern to

Maximize Forces inMultistory Buildings

Building codes specify thatmembers of multistorybuildings be designed tosupport a uniformly distributed

load of the structure. Deadn liv l r n rm ll

considered separately sincethe dead load is fixed inposition whereas the live loadmust be varied to maximize a

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of the structure. Such


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maximum forces are

typically produced bypatterned loading.

Qualitative Influence Lines:

. displacement at the desiredres onse function location.

2. Sketch the displacement

dia ram alon the beam orcolumn line (axial force incolumn) appropriate for theun sp acemen an

assume zero axial

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.


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3. Axial column force (do notconsider axial force in beams):

(a) Sketch the beam linequalitative displacementdiagrams.

b Sketch the column linequalitative displacement

diagrams maintaining equalityo e connec on geome rybefore and after deformation.

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4. Beam force:

(a) Sketch the beam lineualitative dis lacement

diagram for which the releasehas been introduced.

(b) Sketch all column linequalitative displacement

connection geometry before.

the column line qualitativedisplacement diagrams from

the beam line diagram of (a).

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c Sketch remainin beam

line qualitative displacementdiagrams maintaining con-nection geometry before andafter deformation.

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VerticalReaction F

Load Pattern toMaximize F

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M

Maximize M


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QILD and Load Pattern for

Center Beam Moment M

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M

QILD and Load Pattern for

End Beam Moment M

Ex anded Detailfor Beam EndMoment

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Envelope Curves

Design engineers often usen uence nes o cons ruc s earand moment envelope curves for

for bridge girders. An envelopecurve defines the extremeboundary values of shear or

bending moment along the beamue o cr ca p acemen s o

design live loads. For example,-

continuous beam.

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Qualitative influence lines for

positive moments are given,shear influence lines arepresented later. Based on thequalitative influence lines, critical

determined and a structuralanal sis com uter ro ram canbe used to calculate the member

end shear and moment valuesor t e ea oa case an t ecritical live load cases.

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a b c ed

1 2 3 4

-

a b c ed

QILD for (Ma)+

a b c ed

1 2 3 4

+

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a b c ed

1 2 3 4

+ c

a b c ed

QILD for (M )+

a b c ed

1 2 3 4

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e


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a b c ed

1 2 3 4

Critical Live Load Placement

+ a

a b c ed

1 2 3 4

Critical Live Load Placementfor (Ma)-

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a b c ed

1 2 3 4


+ b

a b c ed

1 2 3 4

Critical Live Load Placementfor (Mb)-

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a b c ed

1 2 3 4


+ c

a b c ed

1 2 3 4

Critical Live Load Placementfor (Mc)-

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a b c ed

1 2 3 4


+ d

a b c ed

1 2 3 4

Critical Live Load Placementfor (Md)-

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a b c ed

1 2 3 4


+ e

a b c ed

1 2 3 4

Critical Live Load Placementfor (Me)-

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Calculate the moment envelope

curve for the three-spancontinuous beam.

a b c ed

1 2 3 4

L L L

L = 20 = 240,

A = 60 in2

= 4

wDL = 1.2 k/ft dead load

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LL = .


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Shear and Moment Equations

for a Loaded SpanqMi

xi

ie

V = V x

Viie

Mie = -Mi + Vi xi 0.5q (xi)

2

Shear and Moment Equationsfor an Unloaded Span

=

Vie = Vi

41Mie = -Mi + Vi xi


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Load Cases

wDL

a b c ed

wLL wLL

a b c ed

wLL

a b c ed

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wLL

a b c ed

LC4

a b c ed

wLL

1 2 3 4wLLLC5

a b c ed

LL

1 2 3 4

w

LC6

a b c ed

431 2 3 4LC7


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A summar of the results from

the statically indeterminate beamanalysis for each of the sevenload cases are given in yourclass notes.

----- RESULTS FOR LOAD SET: 1***** M E M B E R F O R C E S *****

MEMBER AXIAL SHEAR BENDINGMEMBER NODE FORCE FORCE MOMENT

(kip) (kip) (ft-k)

1 1 0.00 9.60 0.002 -0.00 14.40 -48.00

. . .

3 -0.00 12.00 -48.00

3 3 0.00 14.40 48.00

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4 -0.00 9.60 0.00


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The equations for the internal

shear and bending moments foreach span and each load caseare:

Load Case 1

V = 9.6 1.2xM12 = 9.6x1 0.6(x1)

2

= M23 = -48 + 12x2 0.6(x2)

2

34 . . 3

M34 = -48 + 14.4x3 0.6(x3)2

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Load Case 2

12 = . . x1M12 = 43.2x1 2.4(x1)2

V23 = 0M23 = -96

V34 = 52.8 4.8x3M34 = -96 + 52.8x3 2.4(x3)

2

Load Case 3

-12 .M12 = -4.8x1

23 = . 2M23 = -96 + 48x2 2.4(x2)2

4634 = .

M34 = -96 + 4.8x3


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Load Case 4

12 = . . x1M12 = 41.6x1 2.4(x1)2

V23 = 8M23 = -128 + 8x2

V34 = -1.60M34 = 32 - 1.6x3

Load Case 5

12 .M12 = 1.6x1

23 = -

M23 = 32 - 8x2

4734 = . . x3

M34 = -128 + 54.4x3 2.4(x3)2


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Load Case 6

V12 = 36.8 4.8x1M12 = 36.8x1 2.4(x1)2

V23 = 56 4.8x2M23 = -224 + 56x2 2.4(x2)

2

V34 = 3.2M34 = -64 + 3.2x3

Load Case 7

-12 .M12 = -3.2x1

23 = . 2M23 = -64 + 40x2 2.4(x2)2

4834 = . . x3

M34 = -224 + 59.2x3 2.4(x3)2


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Bending Moment Diagram LC1

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Live Load E-Mom (+) Live Load E-Mom (-)


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A spreadsheet program listing is

gives the moment values alongthe span lengths and is used tograph the moment envelopecurves.

In the spreadsheet:

ve oa - om +

= max (LC2 through LC7)- -= min (LC2 through LC7)

- =

+ Live Load E-Mom (+)Total Load E-Mom - = LC1

54+ Live Load E-Mom (-)


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Total Load E-Mom (+) Total Load E-Mom (-)


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Construction of the shearenve ope curve o ows e sameprocedure. However,just as is

envelope, a complete analysisshould also load increasing/decreasing fractions of the span

where shear is being considered.

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1

1 2 3 4

QILD (V1)+

a b c ed

-1

QILD (V2L

)+

a b c ed

1 2 3 4

ILD V R +

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a b c ed

1 2 3 4

-1 ILD V L +

1

a c e

1 2 3 4

QILD (V3R)+

a b c ed

1 2 3 4

-1QILD (V )+

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Shear ILD Notation:Su erscri t L = ust to the left of

the subscript point

Su erscri t R = ust to the ri htof the subscript point

To obtain the negative shear

qualitative influence line dia-grams s mp y p t e rawnpositive qualitative influence line

.

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,

exact shear envelope is usuallyunnecessary since an approximateenvelope obtained by connectingthe maximum possible shear at the

w x upossible value at the center of the

.course, the dead load shear must

be added to the live load shearenvelope.

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CE 382 L9 - Intro to Indeterminate Analysis

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