Structure Analysis I - site.iugaza.edu.pssite.iugaza.edu.ps/marafa/files/2010/02/Chapter_7.pdf ·...

Structure Analysis IStructure Analysis IStructure Analysis IStructure Analysis IChapter Chapter 77Chapter Chapter 77

Approximate Analysis of Statically Approximate Analysis of Statically Indeterminate Structures

IntroductionIntroduction• Using approximate methods to analyse statically g pp y yindeterminate trusses and frames

• The methods are based on the way the structure deforms under the loadunder the load

• Trusses• Portal frames with trusses• Vertical loads on building frames• Lateral loads on building frames

– Portal method– Cantilever method

Approximate AnalysisApproximate Analysis

• Statically determinate structure – the force equilibrium equation is sufficient to find the support reactions

• Approximate analysis – is to develop a simple model of the• Approximate analysis is to develop a simple model of the structure which is statically determinate to solve a statically indeterminate problemTh th d i b d th th t t d f d• The method is based on the way the structure deforms under loads

• Their accuracy in most cases compares favourably with more exact methods of analysis (the statically indeterminate analysis)

D t i TDeterminacy ‐ Trusses

jrb 2=+ Statically determinateStatically determinateStatically determinateStatically determinate

jrb 2>+ Statically indeterminateStatically indeterminateStatically indeterminateStatically indeterminate

b – total number of bars

r total number of external support reactions

b – total number of bars

r total number of external support reactionsr – total number of external support reactions

j – total number of joints

r – total number of external support reactions

j – total number of joints

TrussesTrussesReal Structure

b=b=1616 r=r=33 j=j=88b=b=1616, r=, r=33, j=, j=88b+rb+r = = 19 19 > > 22j=j=1616The truss is statically indeterminate toThe truss is statically indeterminate to the third degree

TrussesTrussesApproximation Method

Assumption 1: If the diagonals are designed to be long and slender, it is assumed that diagonals can g gnot support any compressive force and all panel shear are resisted only by tensile diagonals.Assumption 2: If the diagonals are designed to be large rolled sections such as angles or channels, the are ass med to s pport both tensile andthey are assumed to support both tensile and compressive forces. Each diagonal is assumed to carry half the panel shearcarry half the panel shear.

ExampleExample 11Example Example 11

Vertical Loads on Building FramesVertical Loads on Building Frames

• Building frames often consist of girders that are rigidly connected to columns

If the columns are If the columns are extremely stiffextremely stiffextremely stiffextremely stiff

If the columns areIf the columns are extremely flexible

Average point between the ≈

g ptwo extremes = (0.21L+0)/2 0.1L

Vertical Loads on Building FramesVertical Loads on Building Frames Average point between the two extremes = (0.21L+0)/2

0 1y 0.1L

Building Frames – Vertical LoadsReal structure Approximation

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0.1L 0.1L 0.1L 0.1Lxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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0.1L 0.1L 0.1L 0.1L

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0.1L 0.1L 0.1L 0.1L

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0.1L 0.1L 0.1L 0.1L

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1.There is zero moment (hinge) in the girder 0.1L from the left support2. There is zero moment (hinge) in the girder 0.1L from the right support3.The girder does not support an axial force.

Example 3Example 3

Portal Frames – Lateral LoadsPortal frames are frequently used over the entrance of a bridge or in industrial buildings.Portals can be pin supported fixed supported or supported by partialPortals can be pin supported, fixed supported or supported by partial fixity

Pin Supported Portal FramesPin‐Supported Portal Frames

Pin‐Supported Portal FramesPin Supported Portal Frames

Fixed‐Supported Portal FramesFixed Supported Portal Frames

Partial Fixed‐Supported Portal FramesPartial Fixed Supported Portal Frames

Portal Frames – Lateral Loads

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Pxxxxxxxxxxxxxxxx

Real Structure ApproximationPin-supported

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P A point of inflection is

assumed hinge

A point of inflection is located approximately at the girder’s midpointOne assumption

must be madexxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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fixed -supportedxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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P Points of inflection are located

must be made

h/2assumed

are located approximately at the midpoints of all three membersThree assumption

must be made

Partial fixityxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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P Points of inflection for columns are located approximately at h/3

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assumed hinge

pp yand the centre of the girder

Frames with TrussesFrames with Trusses• When a portal frame is used to span large distance, a truss may be

used in place of the horizontal girderused in place of the horizontal girder• The suspended truss is assumed to be pin connected at its points

of attachment to the columns• Use the same assumptions as those used for simple portal framesUse the same assumptions as those used for simple portal frames

Frames with TrussesR l t t A i tiReal structure Approximation

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Pxxxxxxxxxxxxxxxx

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V VV V

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

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horizontal reactions (shear) are equalxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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fixed supported columnspin connection truss-column

horizontal reactions (shear) are equalthere is a zero moment (hinge) on each column

Building frames – Lateral Loads l h dPortal Method

• A building bent deflects in the same way as a portal frameframe

• The assumptions would be the same as those used for portal framesportal frames

• The interior columns would represent the effect of two pportal columns

Building frames – Lateral loads Portal MethodPortal Method

real structure approximation

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V1 2V1 V1

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V1 2V1 V1

2VV V2VV V

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

The method is most suitable for buildings having low elevation and

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 theuniform framing 3. At the given floor level the shear at the

interior column hinges is twice that at the exterior column hinges

Example 5Example 5

Example 6Determine (approximately) the reactions at h b f h l f h f

Example 6

the base of the columns of the frame.

Building frames – Lateral loads C il h dCantilever method

• The method is based on the same action as a long cantilevered beam subjected to a transverse loadbeam subjected to a transverse load

• It is reasonable to assume the axial stress has a linear variation from the centroid of the column areas

Building frames – lateral loads: Cantilever methodCantilever method

Real structure Approximation

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A B CNA=σA AA NB=σB AB NC=σC AC

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XT = (ΣXi Ai)/A

The method is most suitable if the frame is tall and slender, or has

1. Zero moment (hinge) at the centre of each girder2. Zero moment (hinge) at the centre of each column3. The axial stress in a column is proportional to its ,

columns with different cross sectional areas.

p pdistance from the centroid of the cross-sectional areas of the columns at a given floor level

Structure Analysis I - site.iugaza.edu.pssite.iugaza.edu.ps/marafa/files/2010/02/Chapter_7.pdf ·...

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