structure i lecture18

8/12/2019 Structure I Lecture18

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Approximate Analysis ofStatically Indeterminate

Structures


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Introduction

Using approximate methods to analyse staticallyindeterminate trusses and frames

The methods are based on the way the structuredeforms under the load

Trusses

Portal frames with trusses

Vertical loads on building frames

Lateral loads on building frames

Portal method

Cantilever method


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Approximate Analysis

Statically determinate structure the forceequilibrium equation is sufficient to find the supportreactions

Approximate analysisto develop a simple model ofthe structure which is statically determinateto solve astatically indeterminate problem

The method is based on the way the structuredeforms under loads

Their accuracy in most cases compares favourablywith more exact methods of analysis (the staticallyindeterminate analysis)


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Determinacy - truss

jrb 2 Statical ly determ inate

jrb 2 Stat ical ly indeterm inate

btotal number of bars

rtotal number of external support reactions

jtotal number of joints


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Trusses

real structure approximationMethod 1: Design long, slender

diagonals - compressive diagonals are

assumed to be a zero force member and

all panel shear is resisted by tensile

diagonal only

Method 2: Design diagonals to support

both tensile and compressive forces -

each diagonal is assumed to carry half

the panel shear.

b=16, r=3, j=8

b+r = 19 > 2j=16The truss is stat ical ly

indeterminate to the third

degree Three assumptions regarding the

bar forces will be required


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Example 1 - trusses

Determine (approximately) the forces in themembers. The diagonals are to be designed to

support both tensile and compressive forces.

FFB= FAE= F

FDB= FEC= F


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Portal frames lateral loads

Portal frames are

frequently used over

the entrance of a bridge

Portals can be pinsupported, fixed

supported or supported

by partial fixity


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Portal frames lateral loadsreal structure approximation

Pin-supported

fixed -supported

Partial fixity

assumed

hinge

assumed

hinge

assumed

hinge

A point of inflection

is located

approximatelyat

the girders

midpoint

Points of inflection

are located

approximatelyat

the midpoints of all

three members

Points of inflection forcolumns are located

approximatelyat h/3

and the centre of the

girder

One assumption

must be made

Three assumption

must be made


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A point of inflection where the moment changesfrom positive bending to negative bending.

Bending moment is zero at this point.

Pin-Supported Portal Frames

The horizontalreactions (shear) at

the base of each

column are equal


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A point of inflection where the moment changesfrom positive bending to negative bending.

Bending moment is zero at this point.

Fixed-Supported Portal Frames

The horizontalreactions (shear) at

the base of each

column are equal


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Frames with trusses

When a portal is used to spanlarge distance, a truss may beused in place of the horizontalgirder

The suspended truss isassumed to be pin connectedat its points of attachment tothe columns

Use the same assumptions asthose used for simple portal

frames


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Frames with trussesreal structure approximation

pin supported columns

pin connection truss-column the horizontal reactions (shear) are equal

fixed supported columns

pin connection truss-column

horizontal reactions (shear) are equal

there is a zero moment (hinge) on each column


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Example 2 Frame with trusses

Determine by approximate methods the forcesacting in the members of the Warren portal.


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Example 2 (contd)


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Building frames vertical loads

Building frames often consistof girders that are rigidlyconnected to columns

The girder is statically

indeterminate to the thirddegree require 3assumptions

If the columns are

extremely stiff

If the columns are

extremely flexible

Average point

between the twoextremes =

(0.21L+0)/2 0.1L


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Building frames vertical loadsreal structure approximation

1.There is zero moment (hinge) in the girder 0.1L

from the left support

2. There is zero moment (hinge) in the girder 0.1L

from the right support

3.The girder does not support an axial force.


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Example 3 Vertical loads

Determine (approximately) the moment at the jointsE and C caused by members EF and CD.


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Building frames lateral loads: Portal

method

A building bent deflectsin the same way as aportal frame

The assumptions wouldbe the same as thoseused for portal frames

The in ter ior columnswould represent theeffect of two po r talco lumns

B ildi f l t l l d


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Building frames lateral loads:

Portal method

real structure approximation

The method is most suitable for

buildings having low elevation

and uniform framing

1.A hinge is placed at the centre of each girder,

since this is assumed to be a point of zero

moment.

2. A hinge is placed at the centre of each column

this to be a point of zero moment.

3. At the given floor level the shear at the interior

column hinges is twice that at the exterior

column hinges


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Example 4 Portal method

Determine (approximately) the reactions atthe base of the columns of the frame.


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Cantilever method

The method is based on the

same action as a long

cantilevered beam subjected to a

transverse load

It is reasonable to assume theaxial stress has a linear variation

from the centroid of the column

areas



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Cantilever method

real structure approximation

The method is most suitable if

the frame is tall and slender, or

has columns with different

cross sectional areas.

1 zero moment (hinge) at the centre of each girder

2. zero moment (hinge) at the centre of each column

3. The axial stress in a column is proportional to its distance

from the centroid of the cross-sectional areas of the columns at

a given floor level

AAxx i /)(


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Example 5 Cantilever method

Show how to determine (approximately) thereactions at the base of the columns of the

frame.


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Example 5 Cantilever method

structure i lecture18

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