Indeterminate StructureSession 23-26

Subject : S1014 / MECHANICS of MATERIALSYear : 2008

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Indeterminate Structure

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What is Indeterminate ?a structure is statically indeterminate when the static equilibrium equations are not sufficient for determining the internal forces and reactions on that structure …..

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What is Indeterminate ?

“Statically Indeterminate” means the # of unknowns exceeds the number of available equations of equilibrium.

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What is Indeterminate ?Statics (equilibrium analysis) alone cannot solve the problemnR = # of reactions (or unknowns)nE = # of equilibrium equations

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What is Indeterminate ?

• If nR > nE: statically indeterminate- too many unknowns, must invoke a constraint suchas a deformation relation.

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What is Indeterminate ?

• If nR = nE: statically determinate- forces in each member only depend on equilibrium.

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Free body diagram

Statically Indeterminate Examples

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Statically Indeterminate Examples

Free body diagram

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STATISTICALLY INDETERMINATE BEAMS AND SHAFTS (CONT.)

• Strategy:

The additional support reactions on the beam or shaft that are not needed to keep it in stable equilibrium are called redundants. It is first necessary to specify those redundant from conditions of geometry known as compatibility conditions.

Once determined, the redundants are then applied to the beam, and the remaining reactions are determined from the equations of equilibrium.

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METHOD OF SUPERPOSITION

• Necessary conditions to be satisfied:1. The load w(x) is linearly related to the deflection v(x), 2. The load is assumed not to change significantly the

original geometry of the beam of shaft.

Then, it is possible to find the slope and displacement at a point on a beam subjected to several different loadings by algebraically adding the effects of its various component parts.

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STATISTICALLY INDETERMINATE BEAMS AND SHAFTS

• Definition: A member of any type is classified statically indeterminate if the number of unknown reactions exceeds the available number of equilibrium equations.

e.g. a continuous beam having 4 supports

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USE OF THE METHOD OF SUPERPOSITION

Elastic Curve• Specify the unknown redundant forces or moments that must be removed from the beam in order to make it statistically determinate and stable.

• Using the principle of superposition, draw the statistically indeterminate beam and show it equal to a sequence of corresponding statistically determinate beams.

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USE OF THE METHOD OF SUPERPOSITION

Elastic Curve•The first of these beams, the primary beam, supports the same external loads as the statistically indeterminate beam, and each of the other beams “added” to the primary beam shows the beam loaded with a separate redundant force or moment.

• Sketch the deflection curve for each beam and indicate the symbolically the displacement or slope at the point of each redundant force or moment.

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USE OF THE METHOD OF SUPERPOSITION

Compatibility Equations

• Write a compatibility equation for the displacement or slope at each point where there is a redundant force or moment.

• Determine all the displacements or slopes using an appropriate method

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USE OF THE METHOD OF SUPERPOSITION

Compatibility Equations

• Substitute the results into the compatibility equations and solve for the unknown redundants.

• If the numerical value for a redundant is positive, it has the same sense of direction as originally assumed. Similarly, a negative numerical value indicates the redundant acts opposite to its assumed sense of direction.

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USE OF THE METHOD OF SUPERPOSITION

Equilibrium Equations

Once the redundant forces and/or moments have been determined, the remaining unknown reactions can be found from the equations of equilibrium applied to the loadings shown on the beam’s free body diagram.

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Buckling• Buckling is a mode of failure that does not depend on stress

or strength, but rather on structural stiffness• Examples:

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More buckling examples…

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Buckling• The most common problem involving buckling is

the design of columns – Compression members

• The analysis of an element in buckling involves establishing a differential equation(s) for beam deformation and finding the solution to the ODE, then determining which solutions are stable

• Euler solved this problem for columns

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Euler Column Formula•

2

2

LEIcPcrit

2

2

ecrit L

EIP

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Euler Column Formula• Where C is as follows:

C = ¼ ;Le=2LFixed-free

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Euler Column Formula• Where C is as follows:

C = 2; Le=0.7071LFixed-pinned

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Euler Column Formula• Where C is as follows:

C = 1: Le=LRounded-rounded

Pinned-pinned

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Euler Column Formula• Where C is as follows:

C = 4; Le=L/2Fixed-fixed

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Buckling• Geometry is crucial to correct analysis

– Euler – “long” columns– Johnson – “intermediate” length columns– Determine difference by slenderness ratio

• The point is that a designer must be alert to the possibility of buckling

• A structure must not only be strong enough, but must also be sufficiently rigid

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Buckling Stress vs. Slenderness Ratio

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Johnson Equation for Buckling

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Solving buckling problems

Find Euler-Johnson tangent point with

y

e

SEL 22

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Solving buckling problemsFor Le/ < tangent point (“intermediate”), use Johnson’s

Equation

2

2

e

crL

ES

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Solving buckling problemsFor Le > tangent point (“long”), use Euler’s

equation:

2

2

2

4

ey

ycrL

ES

SS

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Solving buckling problems

For Le < 10 (“short”)

Scr = Sy

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Solving buckling problems

If length is unknown, predict whether it is “long” or “intermediate”, use the appropriate equation, then check using the Euler-Johnson tangent point once you have a numerical solution for the critical strength

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Special Buckling Cases• Buckling in very long Pipe

2

2

LEIcPcrit

Note Pcrit is inversely related to length squaredA tiny load will cause bucklingL = 10 feet vs. L = 1000 feet:

Pcrit1000/Pcrit10 = 0.0001

•Buckling under hydrostatic Pressure

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Pipe in Horizontal Pipe Buckling Diagram•

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Far End vs. Input Load with Buckling•

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Buckling Length: Fiberglass vs. Steel•