L5 Statics Theory

8/4/2019 L5 Statics Theory

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ENGINEERING MECHANICS

UGBA 1023

Year 2 Semester 1

Bsc (Hons) EV


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RIGID BODIES MECHANICS

Requirement for statically determinancy


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FRAME STRUCTURE


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SUMMARY

Mostly see one-piece of structural element

3 equations and 3 unknowns to qualify as

STATICALLY DETERMINATE

STRUCTUREException: internal hinge

construction


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DETERMINATE STRUCTURE

For any given loading, it is a straightforward

matter to solve for the unknown reactions usingthe available equations of equilibrium

:0

0:0

structurebeamdimension2:3

0:0:00:0:0

structurebeamdimension3:6

=

==

=

===

===

=

x

yx

zyx

zyx

M

FF

nr

MMMFFF

nr


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STRUCTURES WITH

INTERNAL HINGES

EXCEPTION: TURN INDETERMINATESTRUCTURE TO DETERMINATE STRUCTURE


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STRUCTURES WITH INTERNAL HINGES

Internal hinges are devices that

transmit shear and axial force, but

not bending moments. (2 components

of forces)

In the structural models, generally draw

them as simple pinned devices,


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INTERNAL HINGE

The structure shown has four unknown

reactions (H, R1, R2, and R3) and three

equations of equilibrium

possible to solve for all of the reactionsand internal forces in the structure using

only three equations of equilibrium.


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barofNo:2

forcesunknownofNo:6

3

=

=

=

n

r

nr

FBD


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INTERNAL HINGE

Break off the portion of structure to theright of the hinge as a free body.

Given that moment is zero at the hinge,we have a free body that has threeunknown forces.

These can be solved using the threeavailable equations of equilibrium.

Transfer forces H1 and V (now known)to the free body to the left of the hinge.

This free body now has three unknownforces, which can likewise be calculatedusing the three equations of equilibrium.


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INTERNAL HINGE

Once these reactions have beencalculated, then cutting the beamanywhere along its length yields threeunknown forces N, V, and M, which can

be calculated using the three equations ofequilibrium.

The structure is therefore staticallydeterminate,

Calculate all of the external andinternal forces using the availableequations of equilibrium alone.


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INTERNAL HINGE

Simple structures, i.e., those for which acut produces three unknown forces, arestatically determinate if and only if:

1. The number of unknown reactions

minus the number forces released atinternal hinges is equal to thenumber of equations of equilibrium2.An unknown force applied at anylocation and in any direction creates

nonzero reactions


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(1) DETERMINANCY

(2) STABILITY


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IMPORTANCE OF STABILITY

Stability is of primary importance tostructural designers.

Unstable structures are prone tocatastrophic collapse, often without

regard to the strength of structuralmembers

They can undergo large deformationsunder the action of extremely small loads.

Designers must learn how to (1) identifyinstabilities and (2) eliminate themfrom the structures they design.


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Unstable structures cannot be analyzed.

If proceed to calculate the response of an

unstable structure to load, will eventually

discover that the problem is

mathematically unsolvable,

but often only after investing considerable

time and effort in the analysis.


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STABILITY AND

DETERMINACY OF THE

STRUCTURE

EXAMPLE


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STABILITYCONCEPTS

Structures are unstable when they canundergo large deformations with theslightest application of load.

The methods of structuralanalysis/statics cannot be applied tounstable structures.

Intuitively, an unstable structure is theone that will undergo largedeformations under the slightestload, without the creation of restraining

forces.Simple example on beam


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A simple beam supported at both ends by rollers, neither of

which can resist horizontal force.

A slightest external force F, applied horizontally, will besufficient to induce large horizontal displacements, since the

rollers are assumed to be frictionless.

This structure is therefore unstable.


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undergo large deformationsunder the slightest loadIN THIS CASE TRANSLATION


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GENERAL STATEMENT

Structures for which the number of

unknown reactions is less than the

number of available equations of

equilibrium are unstable

Structures which cannot create nonzero

reactions for all locations and

directions of a given load are

unstable


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EXAMPLES


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NONCONCURRENTAND NONPARALLEL


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STABLE BUT INDETERMINATE to 2nd

degree

2 unknowns more than the equation

r = 8: n = 2

r = 3n:


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UNSTABLE AS 3 REACTIONS ARECONCURRENT AT B


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NO RESTRAINT AT HORIZONTAL AXIS

FBD


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IMPORTANCE OF

STABILITY


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Generally useful and practical to visualizehow the structure will deflect under allpossible directions and points of applicationof load (not just the loads that have beengiven).

To evaluate the degree of staticalindeterminacy, it can be helpful to releaserestraints in the structure by addinginternal hinges or by releasing restraint atsupports, working progressively towards aknown statically determinate arrangement.

The number of released restraints is thedegree of indeterminacy.

Free in choice of which restraints to release


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In some statically indeterminate systems,however, it is possible to create anunstable structure by releasing fewerrestraints than the degree of staticallyindeterminacy plus one.

This can happen, for example, instructures where there is only onereaction providing restraint in a givendirection and several reactions providingrestraint in the other two.


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For example, consider the two-spancontinuous beam

The degree of indeterminacy of thestructure is one.

Expect that by removing two restraints,will make the structure unstable.

This would be the case, for example, ifR2and R3 were removed.

By removing the restraint in the x-

direction H, however, the structurebecomes unstable by removing only onerestraint


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REMOVEDTHENUNSTABLE


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REMOVED:

THENUNSTABLE


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IMPORTANCE OF STABILITY If wish to determine the degree of indeterminacy

of a given structure by progressively removingrestraints, therefore, must never completelyremove restraint in one of the three possible forcedirections (Fx, Fy, or M) when there are stillseveral independent restraints in the other twodirections

Sometimes encounter indeterminate structures forwhich reactions and sectional forces can be solvedusing equilibrium conditions alone, for a specificsubset of load arrangements.

It is clear that the load P must travel from one endof the member to the other, creating only axialforce in the member, and an equal and oppositereaction at the left end of the beam.


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ARE THEY STATICALLYDETERMINATE?


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ARE THEY STATICALLYDETERMINATE?


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TRUSS STRUCTUREDifferent criteria/formula for determinancy


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jointofnumber

reaction)and(momnetreactionofnumber

memberofnumber

Dimension3:3Dimension2:2

=

=

=

=+

=+

j

r

b

jrbjrb

CRITERIA FOR DETERMINACY FOR TRUSS


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SUMMARYSTATICS THEORY


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STATICS THEORY

Building structure requirement:

SAFETY, STRENGTH,

SERVICEABILITY

Structural design: Newtons second law

Equilibrium: (a) translational (2)

rotational

Loading type, free body diagram, joint

types

Determinate and indeterminate structure


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ACTION

LoadsStresses

moments

REACTION

Sizes

Type ofmaterials

Safety factor

Analyses are carried out to determine the

forces/moments/stresses

Design is carried out to determine the sizeand chosen type of material, with

appropriate SF

NEWTONS LAW

LAW OF ELASTICITY


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L5 Statics Theory