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Structure Analysis IChapter 7

Approximate Analysis of StaticallyIndeterminate Structures

Introduction Using approximate methods to analyse staticallyindeterminate trusses and frames The methods are based on the way the structure deformsunder the load Trusses Portal frames with trusses Vertical loads on building frames Lateral loads on building frames– Portal method– Cantilever method

Approximate Analysis Statically determinate structure – the force equilibriumequation is sufficient to find the support reactions Approximate analysis – is to develop a simple model of thestructure which is statically determinate to solve a staticallyindeterminate problem The method is based on the way the structure deforms underloads Their accuracy in most cases compares favourably with moreexact methods of analysis (the statically indeterminate analysis)Dr. Mohammed ArafaStructural Analysis II

Determinacy - Trussesb r 2jStatically determinateb r 2jStatically indeterminateb – total number of barsr – total number of external support reactionsj – total number of joints

TrussesReal Structureb 16, r 3, j 8b r 19 2j 16The truss is statically indeterminate tothe third degree

TrussesApproximation MethodsMethod 1: If the diagonals are designed to belong and slender, it is assumed that diagonals cannot support any compressive force and all panelshear is resisted by tensile diagonal onlyMethod 2: If the diagonals are designed to belarge rolled sections such as angles or channels,they are assumed to support both tensile andcompressive forces. Each diagonal is assumed tocarry half the panel shear.

Example 1Dr. Mohammed ArafaStructural Analysis II

Example 2

Vertical Loads on Building Frames Building frames often consist ofgirders that are rigidly connected tocolumnsIf the columns areextremely stiffIf the columns areextremely flexibleAverage point between thetwo extremes (0.21L 0)/20.1L

Vertical Loads on Building FramesAverage point between thetwo extremes (0.21L 0)/2y 0.1LDr. Mohammed ArafaStructural Analysis II

Building Frames – Vertical LoadsReal structureApproximation1.There is zero moment (hinge) in the girder 0.1Lfrom the left support2. There is zero moment (hinge) in the girder 0.1Lfrom the right support3.The girder does not support an axial force.

Example 3

ProblemDetermine (approximately ) the internal moments at A andB caused by the vertical loading.

Portal Frames – Lateral Loads Portal frames are frequently used over the entrance of a bridge or inindustrial buildings. Portals can be pin supported, fixed supported or supported by partialfixity

Pin-Supported Portal Frames

Pin-Supported Portal Frames

Fixed-Supported Portal Frames

Fixed-Supported Portal Frames

Partial Fixed-Supported Portal Frames

Portal Frames – Lateral LoadsReal StructureApproximationPin-supportedOne assumptionmust be madeassumedhingefixed -supportedThree assumptionmust be madeassumedhingePartial fixityassumedhingeA point of inflection islocatedapproximately atthe girder’s midpointPoints of inflectionare locatedapproximately at themidpoints of all threemembersPoints of inflection forcolumns are locatedapproximately at h/3and the centre of thegirder

Frames with Trusses When a portal frame is used to span large distance, a truss may beused in place of the horizontal girder The suspended truss is assumed to be pin connected at its pointsof attachment to the columns Use the same assumptions as those used for simple portal framesDr. Mohammed ArafaStructural Analysis II

Frames with TrussesReal structurepin supported columnspin connection truss-columnfixed supported columnspin connection truss-columnApproximationthe horizontal reactions (shear) are equalhorizontal reactions (shear) are equalthere is a zero moment (hinge) on eachcolumn

Example 4

Building frames – Lateral LoadsPortal Method A building bent deflects inthe same way as a portalframe The assumptions would bethe same as those used forportal frames The interior columns wouldrepresent the effect of twoportal columns

Building frames – Lateral loadsPortal Methodreal structureThe method is most suitable forbuildings having low elevation anduniform framingapproximation1. A hinge is placed at the centre of eachgirder, this is assumed to be a point of zeromoment.2. A hinge is placed at the centre of eachcolumn, this to be a point of zero moment.3. At the given floor level the shear at theinterior column hinges is twice that at theexterior column hinges

Example 5

BMD for the Girder

Example 6Determine (approximately) the reactions atthe base of the columns of the frame.

Building frames – Lateral loadsCantilever method The method is based on the same action as a long cantileveredbeam subjected to a transverse load It is reasonable to assume the axial stress has a linear variation fromthe centroid of the column areas

Building frames – lateral loads:Cantilever methodReal structureThe method is most suitable if theframe is tall and slender, or hascolumns with different crosssectional areas.Approximation1. Zero moment (hinge) at the centre of each girder2. Zero moment (hinge) at the centre of eachcolumn3. The axial stress in a column is proportional to itsdistance from the centroid of the cross-sectionalareas of the columns at a given floor level

Example 7

Example 8

Using approximate methods to analyse statically indeterminate trusses and frames The methods are based on the way the structure deforms 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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