approximate analysis of statically indeterminate structures

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Approximate Analysis of Statically Indeterminate Structures

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Page 1: Approximate Analysis of Statically Indeterminate Structures

Approximate Analysis of Statically Indeterminate

Structures

Approximate Analysis of Statically Indeterminate

Structures

Page 2: Approximate Analysis of Statically Indeterminate Structures

IntroductionIntroduction

• 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

Page 3: Approximate Analysis of Statically Indeterminate Structures

Approximate AnalysisApproximate Analysis

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

• Approximate analysis – to develop a simple model of the structure which is statically determinate to solve a statically indeterminate problem

• 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)

Page 4: Approximate Analysis of Statically Indeterminate Structures

Determinacy - trussDeterminacy - truss

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

j – total number of joints

b – total number of bars

r – total number of external support reactions

j – total number of joints

Page 5: Approximate Analysis of Statically Indeterminate Structures

TrussesTrussesreal structure approximation

Method 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=8b=16, r=3, j=8

b+r = 19 > 2j=16b+r = 19 > 2j=16

The truss is statically indeterminate to the third degree Three assumptions regarding the

bar forces will be required

Page 6: Approximate Analysis of Statically Indeterminate Structures

Example 1 - trussesExample 1 - trusses Determine (approximately) the forces in the

members. The diagonals are to be designed to support both tensile and compressive forces.

FFB = FAE = F

FDB = FEC = F

Page 7: Approximate Analysis of Statically Indeterminate Structures

Portal frames – lateral loadsPortal frames – lateral loads

• Portal frames are frequently used over the entrance of a bridge

• Portals can be pin supported, fixed supported or supported by partial fixity

Page 8: Approximate Analysis of Statically Indeterminate Structures

Portal frames – lateral loadsPortal frames – lateral loadsreal structure approximation

Pin-supported

fixed -supported

Partial fixity

assumed hinge

assumed hinge

assumed hinge

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

Points of inflection are located approximately at the midpoints of all three members

Points of inflection for columns are located approximately at h/3 and the centre of the girder

One assumption must be made

Three assumption must be made

Page 9: Approximate Analysis of Statically Indeterminate Structures

• A point of inflection – where the moment changes from positive bending to negative bending.

• Bending moment is zero at this point.

Pin-Supported Portal FramesPin-Supported Portal Frames

The horizontal reactions (shear) at the base of each column are equalequal

Page 10: Approximate Analysis of Statically Indeterminate Structures

• A point of inflection – where the moment changes from positive bending to negative bending.

• Bending moment is zero at this point.

Fixed-Supported Portal FramesFixed-Supported Portal Frames

The horizontal reactions (shear) at the base of each column are equalequal

Page 11: Approximate Analysis of Statically Indeterminate Structures

Frames with trussesFrames with trusses

• When a portal is used to span large distance, a truss may be used 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 frames

Page 12: Approximate Analysis of Statically Indeterminate Structures

Frames with trussesFrames 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

Page 13: Approximate Analysis of Statically Indeterminate Structures

Example 2 – Frame with trussesExample 2 – Frame with trusses Determine by approximate methods the forces

acting in the members of the Warren portal.

Page 14: Approximate Analysis of Statically Indeterminate Structures

Example 2 (contd)Example 2 (contd)

Page 15: Approximate Analysis of Statically Indeterminate Structures

Building frames – vertical loadsBuilding frames – vertical loads

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

• The girder is statically indeterminate to the third degree – require 3 assumptions

If the columns are If the columns are extremely stiffextremely stiff

If the columns are extremely flexible

Average point between the two extremes = (0.21L+0)/2 0.1L

Page 16: Approximate Analysis of Statically Indeterminate Structures

Building frames – vertical loadsBuilding 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.

Page 17: Approximate Analysis of Statically Indeterminate Structures

Example 3 – Vertical loadsExample 3 – Vertical loads Determine (approximately) the moment at the joints

E and C caused by members EF and CD.

Page 18: Approximate Analysis of Statically Indeterminate Structures

Building frames – lateral loads: Portal method

Building frames – lateral loads: Portal method

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

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

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

Page 19: Approximate Analysis of Statically Indeterminate Structures

Building frames – lateral loads: Portal method

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

Page 20: Approximate Analysis of Statically Indeterminate Structures

Example 4 – Portal methodExample 4 – Portal method Determine (approximately) the reactions at

the base of the columns of the frame.

Page 21: Approximate Analysis of Statically Indeterminate Structures

Building frames – lateral loads: Cantilever method

Building frames – lateral loads: 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 the axial stress has a linear variation from the centroid of the column areas

Page 22: Approximate Analysis of Statically Indeterminate Structures

Building frames – lateral loads: Cantilever method

Building frames – lateral loads: 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 /)(

Page 23: Approximate Analysis of Statically Indeterminate Structures

Example 5 – Cantilever methodExample 5 – Cantilever method Show how to determine (approximately) the

reactions at the base of the columns of the frame.

Page 24: Approximate Analysis of Statically Indeterminate Structures

Example 5 – Cantilever methodExample 5 – Cantilever method