L9 - Intro to Indeterminate Analysis

8/8/2019 L9 - Intro to Indeterminate Analysis

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Introduction to StaticallyIndeterminate AnalysisSupport reactions and internal

forces of statically determinatestructures can be determined

using only the equations of

equilibrium. However, the

anal sis of staticall indeter-

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minate structures requires

additional equations based on

the geometry of deformation ofthe structure.

Additional equations come from

compatibility relationships,

which ensure continuity ofdisplacements throughout the

structure. The remaining

equations are constructed from

member constitutive equations,

i.e., relationships betweenstresses and strains and the

integration of these equations

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over e cross sec on.

Design of an indeterminate

structure is carried out in aniterative manner, whereby the

(relative) sizes of the structural

members are initially assumed

and used to analyze the structure.Based on the computed results

(displacements and internalmember forces), the member

sizes are adjusted to meet

governing design criteria. This

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the member sizes based on the

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

Another consequence of

statically indeterminatestructures is that the relative

variation of member sizes

influences the magnitudes of

will experience. Stated in

another way, stiffness (large

member size and/or highmodulus materials) attracts

force.

4

esp e ese cu es w

statically indeterminate

structures, an overwhelmingmajority of structures being

built today are statically

indeterminate.


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Advantages StaticallyIndeterminate Structures

5 6

Statically indeterminate

structures typically result insmaller stresses and greater

stiffness (smaller deflections)

as illustrated for this beam.

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Determinate beam is unstable

if middle support is removed

or knocked off!

Statically indeterminate

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structures introduce redundancy,

which may insure that failure in

one part of the structure will not

result in catastrophic or collapsefailure of the structure.


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Disadvantages ofStatically Indeterminate

Structures

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Statically indeterminate structureis self-strained due to support

settlement, which produces

stresses, as illustrated above.

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Statically indeterminate struc-

tures are also self-strained dueto temperature changes and

fabrication errors.

Indeterminate Structures:

Influence Lines

Influence lines for staticallyindeterminate structures

prov e e same n orma on

as influence lines for statically

determinate structures, i.e. itrepresents the magnitude of a

response function at a

particular location on the

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across the structure.


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Our goals in this chapter are:

1.To become familiar with theshape of influence lines for the

support reactions and internalforces in continuous beamsand frames.

2.To develop an ability to sketch

the appropriate shape ofinfluence functions for

indeterminate beams and

frames.

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3.To establish how to position

distributed live loads oncontinuous structures to

maximize response function

values.

Qualitative InfluenceLines for Statically Inde-

terminate Structures:

Muller-Breslaus Principle

In many practical applications, itis usually sufficient to draw only

the qualitative influence lines to

decide where to place the live

loads to maximize the responsefunctions of interest. The

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- -

vides a convenient mechanism

to construct the qualitativeinfluence lines, which is statedas:

The influence line for a force (or

moment) response function is

given by the deflected shape of

removing the displacement

constraint corresponding to the

response function of interestfrom the original structure and

giving a unit displacement (or

rotation at the location and in

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the direction of the response

function.

Procedure for constructingqualitative influence lines for

indeterminate structures is: (1)remove from the structure the

restraint corres ondin to the

response function of interest, (2)

apply a unit displacement orrotation to the released structure

at the release in the desiredresponse function direction, and

(3) draw the qualitative deflected

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shape of the released structureconsistent with all remaining

support and continuity

conditions.


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Notice that this procedure is

identical to the one discussed forstatically determinate structures.

However, unlike statically

determinate structures, theinfluence lines for statically

indeterminate structures are

typically curved.

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maximize the desired response

function is obtained from thequalitative ILD.

Uniformly distributed live

loads are placed over thepositive areas of the ILD to

maximize the drawn response.

influence line ordinates tend todiminish rapidly with distance

from the response function

location, live loads placed morethan three span lengths away

can be ignored. Once the live

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load pattern is known, an

indeterminate analysis of the

structure can be performed todetermine the maximum value of

the response function.

19QILD for RA 20QILDs for RC and VB


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QILDs for (MC)-,

(MD)+ and RF

Building codes specify that

Live Load Pattern to

Maximize Forces inMultistory Buildings

mem ers o mu s ory

buildings be designed tosupport a uniformly distributed

live load as well as the dead

load of the structure. Dead

and live loads are normally

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the dead load is fixed in

position whereas the live loadmust be varied to maximize a

particular force at each section

of the structure. Such

maximum forces aretypically produced by

patterned loading.

Qualitative Influence Lines:

1. Introduce appropriate unit

displacement at the desiredresponse function location.

2. Sketch the displacement

diagram along the beam or

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column) appropriate for the

unit displacement and

assume zero axial

deformation.

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

(a) Sketch the beam line

qua a ve sp acemendiagrams.

(b) Sketch the column linequalitative displacement

diagrams maintaining equality

of the connection geometry

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before and after deformation.


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4. Beam force:

(a) Sketch the beam line

qualitative displacement

diagram for which the release.

(b) Sketch all column line

qualitative displacement

diagrams maintainingconnection geometry before

and after deformation. Start

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displacement diagrams from

the beam line diagram of (a).

(c) Sketch remaining beam

line qualitative displacementdiagrams maintaining con-

nection geometry before and.

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27

Vertical

Reaction F

Load Pattern toMaximize F

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Column MomentM

Load Pattern to

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

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Expanded Detail

for Beam EndMoment

Design engineers often use

influence lines to construct shearand moment envelo e curves for

Envelope Curves

continuous beams in buildings or

for bridge girders. An envelope

curve defines the extremeboundary values of shear or

bending moment along the beam

due to critical placements of

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design live loads. For example,

consider a three-span

continuous beam.

Qualitative influence lines forpositive moments are given,

shear influence lines arepresented later. Based on the

ualitative influence lines critical

live load placement can be

determined and a structuralanalysis computer program can

be used to calculate the member

end shear and moment values

for the dead load case and the

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critical live load cases.


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a b c ed

1 2 3 4

Three-Span Continuous Beam

a b c ed

1 2 3 4

QILD for (Ma)+

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a b c ed

1 2 3 4

QILD for (Mb)+

a b c ed

1 2 3 4

QILD for (Mc)+

a b c ed

1 2 3 4

QILD for (Md)+

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a b c ed

1 2 3 4

QILD for (Me)+

a b c ed

1 2 3 4

a b c ed

for (Ma)+

35

1 2 3 4

Critical Live Load Placementfor (Ma)

-

a b c ed

1 2 3 4

a b c ed

for (Mb)+

36

1 2 3 4

Critical Live Load Placementfor (Mb)

-


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a b c ed

1 2 3 4

a b c ed

for (Mc)

+

37

1 2 3 4

Critical Live Load Placementfor (Mc)

-

a b c ed

1 2 3 4

a b c ed

for (Md)

+

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1 2 3 4

Critical Live Load Placementfor (Md)

-

a b c ed

1 2 3 4

a b c ed

for (Me)+

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1 2 3 4

Critical Live Load Placementfor (Me)

-

a b c ed

Calculate the moment envelopecurve for the three-span

continuous beam.

1 2 3 4

L L L

L = 20 = 240

E = 3,000 ksi

40

A = 60 in

I = 500 in4

wDL = 1.2 k/ft dead load

wLL = 4.8 k/ft live load


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Shear and Moment Equations

for a Loaded Span

qMiMie

Vie = Vi q x i

Mie = -Mi + Vi xi 0.5q (xi)2

xi

ViVie

Shear and Moment E uations

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for an Unloaded Span(set q = 0 in equations above)

Vie = Vi

Mie = -Mi + Vi xi

a b c ed

Load Cases

wDL

1 2 3 4

a b c ed

1 2 3 4

wLL wLL

LC1

LC2

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a b c ed

1 2 3 4

wLL

LC3

a b c ed

1 2 3 4wLL

wLL

LC4

a b c ed

1 2 3 4

a b c ed

wLL

wLLLC5

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a b c ed

1 2 3 4

1 2 3 4

wLL

LC7

A summary of the results fromthe statically indeterminate beam

analysis for each of the sevenload cases are given in your

class 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 0.00

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2 -0.00 14.40 -48.00

2 2 0.00 12.00 48.003 -0.00 12.00 -48.00

3 3 0.00 14.40 48.00

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 case

are:

Load Case 1

V12 = 9.6 1.2x1M12 = 9.6x1 0.6(x1)

2

V23 = 12 1.2x22

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23 = - 2 . 2

V34 = 14.4 1.2x3M34 = -48 + 14.4x3 0.6(x3)2

Load Case 2

V12 = 43.2 4.8x1M12 = 43.2x1 2.4(x1)

2

V23 = 0= -

Load Case 3

V12 = -4.8

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

2

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12 = - . x1

V23

= 48 4.8x2M23 = -96 + 48x2 2.4(x2)

2

V34 = 4.8

M34 = -96 + 4.8x3

Load Case 4

V12 = 41.6 4.8x1M12 = 41.6x1 2.4(x1)

2

V23 = 8

= -

Load Case 5

V12

= 1.6

V34 = -1.60M34 = 32 - 1.6x3

47

12 = . x1

V23 = -8M23 = 32 - 8x2

V34 = 54.4 4.8x3M34 = -128 + 54.4x3 2.4(x3)

2

Load Case 6

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

2

V23 = 56 4.8x22

Load Case 7

V12

= -3.2

23 = - 2 . 2

V34 = 3.2

M34 = -64 + 3.2x3

48

12 = - . x1

V23 = 40 4.8x2M23 = -64 + 40x2 2.4(x2)

2

V34 = 59.2 4.8x3M34 = -224 + 59.2x3 2.4(x3)

2


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Bending Moment Diagram LC1

49 50

51 52


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

A spreadsheet program listing is

included in your class notes thatgives the moment values along

the span lengths and is used to

graph the moment envelope

.

In the spreadsheet:

Live Load E-Mom (+)

= max (LC2 through LC7)

Live Load E-Mom (-)

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=

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

Total Load E-Mom (-) = LC1

+ Live Load E-Mom (-)

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

Construction of the shearenvelope curve follows the same

rocedure. However, ust as isthe case with a bending moment

envelope, a complete analysis

should also load increasing/decreasing fractions of the span

where shear is being considered.

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a b c ed

1 2 3 4

1

QILD (V1)+

a b c ed

1 2 3 4

-1QILD (V2

L)+

1

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a b c ed

1 2 3 4

QILD (V2R)+

a b c ed

1 2 3 4

-1

QILD (V3L

)+

1

a b c ed

1 2 3 4

QILD (V3R)+

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a b c ed

1 2 3 4

-1QILD (V4)

+

Shear ILD Notation:

Superscript L = just to the left of

the subscript point

Superscript R = just to the rightof the subscript point

To obtain the negative shear

qualitative influence line dia-

grams simply flip the drawn

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positive qualitative influence line

diagrams.

In practice, the construction of theexact shear envelope is usuallyunnecessary since an approximate

envelope obtained by connecting

reactions with the maximumpossible value at the center of the

spans is sufficiently accurate. Of

course, the dead load shear mustbe added to the live load shear

envelo e.

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L9 - Intro to Indeterminate Analysis

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