CE 382 L10 - Approximate Analysis

8/9/2019 CE 382 L10 - Approximate Analysis

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Approximate Analysis ofStatically Indeterminate

Structures

Every successful structure mustbe capable of reaching stableequilibrium under its appliedloads, regardless of structuralbehavior. Exact analysis of indeterminate structures involvescomputation of deflections andsolution of simultaneousequations . Thus, computer programs are typically used.


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To eliminate the difficultiesassociated with exact analysis,preliminary designs of indeter-minate structures are often basedon the results of approximateanalysis.

Approximate analysis is basedon introducing deformationand/or force distribution

assumptions into a staticallyindeterminate structure, equalin number to degree of indeter-minacy, which maintains stable

equilibrium of the structure .


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No assumptions inconsistentwith stable equilibrium areadmissible in any approximateanalysis .

Uses of approximate analysisinclude:

(1) planning phase of projects,when several alternative designs

of the structure are usuallyevaluated for relative economy;

(2) estimating the various

member sizes needed to initiatean exact analysis;


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(3) check on exact analysisresults;

(4) upgrades for older structuredesigns initially based onapproximate analysis; and

(5) provide the engineer with a

sense of how the forcesdistribute through the structure.


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In order to determine the reac-tions and internal forces for indeterminate structures usingapproximate equilibrium me-

thods, the equilibrium equationsmust be supplemented byenough equations of conditionsor assumptions such that the

resulting structure is stable andstatically determinate .


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The required number of suchadditional equations equalsthe degree of static indeter-minacy for the structure , witheach assumption providing anindependent relationshipbetween the unknownreactions and/or internalforces .

In approximate analysis, theseadditional equations are basedon engineering judgment of appropriate simplifying assump-tions on the response of thestructure.


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Approximate Analysis of

a Continuous Beam forGravity Loads

Continuous beams and girdersoccur commonly in building floor systems and bridges. In theapproximate analysis of con-tinuous beams, points ofinflection or inflection point(IP) positions are assumedequal in number to the degreeof static indeterminacy .


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For continuous beam struc-tures, the degree of staticindeterminacy in bending ( Ib)equals

number of bendingreactions (vertical andmoment support reactions)

C = number of equationsof condition in bending

b bRI N C 2=

bRN =


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Each inflection point positionintroduces one equation of

condition to the static equilibri-um equations . Three strategiesare used to approximate thelocation of the inflection points:

1. qualitative displacementdiagrams of the beamstructure,

2. qualitative bending momentdiagrams ( preferred methodfor students ), and

3. location of exact inflectionpoints for some simplestatically indeterminatestructures.


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Approximate analysis of con-tinuous beams using the quali-tative deflection diagram isbased on the fact that the elasticcurve ( deflected shape ) of acontinuous beam can generallybe sketched with a fair degree of accuracy without performing anexact analysis . When the elasticcurve is sketched in this manner,the actual magnitudes of deflec-tion (displacements and rota-tions) are not accurately por-trayed, but the inflection pointlocations are easily estimatedeven on a fairly rough sketch.


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Qualitative bending momentdiagrams can also be used tolocate inflection points .Bending moments in spans withno loading are linear or constant;with point loading the spanbending moment equations arepiecewise linear; and withuniform loading the moments arequadratic. Remember, internalbending moments at interior support locations adjacent to oneor two loaded spans is negative .

Recall that zero momentlocations correspond to theinflection point locations .


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From the total set of inflectionpoints, select the needednumber to achieve a solutionby statics .

In the case of beams , there willnormally be enough inflectionpoints to reduce the structureto a statically determinatestructure and typically thereare more inflection points thanthe degree of indeterminacy .


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With the inflection points located

(equal in number to the degreeof static indeterminacy), theanalysis can proceed on thebasis of statics alone. Since an

inflection point is a zero momentlocation , it may be thought ofas an internal hinge forpurposes of analysis .

Some examples to guide thelearning and practice aregiven on the following pages.

Both the elastic curve andbending moment diagrams aregiven.


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Fixed-Fixed Beam Subjectedto a Uniform Load


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Fixed-Fixed Beam Subjectedto a Central Point Force


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Propped Cantilever BeamSubjected to a Uniform Load


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Propped Cantilever BeamSubjected to aCentral Point Force


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When considering problems thatdo not match the exact valuesgiven, some useful guides are:

Inflection points move

towards positions of reducedstiffness,

No more than one inflectionpoint can occur in an un-loaded span, and

No more than two inflectionpoints will occur in a loaded

span .


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Example Approximate Analysisof a Continuous Beam

L L

qEI = constant

IP1

IP2

L/3L

0.1 < 0.25

QualitativeDeflection Diagram


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Qualitative BendingMoment Diagram

-M

0.5 M

L/3

L

L/3 2L/3 L (1- )LV1 V2R1

M1

R 2 R3

FBD through IPs

q q


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3

2 3

q(1 )LR2

q(1 )LV R

2

=

= =

From the last FBD:

From the middle FBD:

2

1 2

1

2L q( L)V LV

3 2

3q LV4

= + +

=

2M 0=


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From the first FBD:

y 2 1 2F 0 R V q L V= = +

2qL

R (2 5 )4

= +

2

1 1 1L q L

M 0 M V3 4

= = =

y 1 1 3q LF 0 R V 4= = =


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Shear Force Diagram

R1

R2

-R 3

R2 + R 1

M1

-2M1

0.125q[(1- )L]2

(1- )L/2

Bending Moment Diagram


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Frame Example


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Trusses with Double

DiagonalsTruss systems for roofs, bridgesand building walls often containdouble diagonals in each panel ,which makes each panelstatically indeterminate .

Approximate analysis requiresthat the number of assump-tions introduced must equalthe degree of indeterminacy sothat only the equations of equi-librium are required to performthe approximate analysis.


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Since one extra diagonal exists

in each double diagonal panel,one assumption regarding theforce distribution between thetwo diagonals must be made in

each panel . If the diagonals areslender , it may be assumed thatthe diagonal members are onlycapable of resisting tensile

forces and that diagonalssubjected to compression canbe ignored since they aresusceptible to buckling , i.e.,

assume very small bucklingload and ignore post-bucklingstrength .


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Such an assumption is illustratedin Fig. DD.1(a). In Fig. DD.1(a),the total panel shear is assumedto be resisted by the tensiondiagonal as shown. Compres-sion diagonals are assumed toresist no loading . With thisassumption, the truss of Fig.DD.1 is statically determinate.


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28Fig. DD.1 Truss with Double

Diagonal Panels


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The assumption discussed in the

previous paragraph is generallytoo stringent , i.e., the compres-sion diagonals can resist aportion of the panel shear .

Figures DD.1(b) and (c) showtwo different assumptionsregarding the ability of the

compression diagonals to resistforce.


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Figure DD.1(b) shows the shear (vertical) components of thediagonal members assuming thatthe compression and tensiondiagonals equally resist thepanel shear .

Figure DD.1(c) shows thevertical force distribution amongthe compression and tensiondiagonals based on the tensiondiagonal resisting twice the forceof the compression diagonal or two-thirds of the panel shear .


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Any reasonable assumptioncan be made .

The compression diagonalassumptions for double diagonaltrusses can be mathematicallysummarized as:

C = T; 0 1

If P S = Panel Shear , then

T(1+ ) = P S


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Once the diagonal memberforces are determined, theremaining member forces inthe truss can be calculatedusing simple statics , i.e., themethod of sections and/or themethod of joints.

CE 382 L10 - Approximate Analysis

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