Transcript 4.2 INDETERMINATE TRUSS ANALYSIS

4. APPROXIMATE ANALYSIS OF
INDETERMINATE STRUCTURES
4. APPROXIMATE ANALYSIS OF
INDETERMINATE STRUCTURES
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4.1 INTRODUCTION
4.2 INDETERMINATE TRUSS ANALYSIS
4.3 VERTICAL LOADS ON BUILDING FRAMES
4.4 PORTALS AND TRUSSED FRAME STRUCTURES VERTICAL AND LATERAL LOADS
• 4.5 LATERAL LOADS ON BUILDING FRAMES METHOD I - PORTAL FRAME METHOD
METHOD II - CANTILEVER FRAME METHOD
4. INTRODUCTION
4.1 Introduction to Procedures for Approximate Analysis:
• Conventional design process is normally based on the ‘local’
structural elements (column, beam, floor slabs, wall, etc.)
• But theoretical and experimental studies have shown that
structural systems cannot be considered to be a simple collection
of individual elements
• The responses of the structure is often more than the ‘sum’ of the
responses of individual elements since structural integrity ensures
that the elements work together, producing global responses
through the complex interaction of its elements
• Hence ‘local’ and ‘global’ approaches are necessary for proper
design
4.1. INTRODUCTION
(CONT’D)
• 4.1 Introduction to Procedures for Approximate Analysis
(Cont’d)
• Global analysis carried out on two levels: (i) A ‘numerically exact’
analysis using finite element method or another mathematical
procedure, in which each of the element of the system is described by a
mathematical equation and joined together at various points (or along
edges) by proper boundary or continuity conditions (MATHEMATICA)
- This procedure is quit complex and may have in-built data errors - (ii)
The second procedure is a simplified or approximated procedure,
which reduces the complex structural system to a much simpler system
that could be handled easily by simple calculations; this will be the
subject of our study in this set of lectures
4.1. INTRODUCTION (Cont’d)
• 4.1 Introduction to Procedures for Approximate Analysis
(Cont’d)
• Approximate global structural analysis of a complex structure will
contain:
– Reduction of the complex system into an equivalent simple system
to carry out
• Stability analysis
• Frequency analysis
• Elementary structural analysis - this will form the basis of our
study in these lectures
4.1. INTRODUCTION
(Cont’d)
• 4.1 Introduction to Procedures for Approximate Analysis
(Cont’d)
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A model - which is a determinate structure - must be developed for analysis
Results obtained from this approximated model compares favorably with the
correct results
This study makes the transition from determinate to indeterminate structural
analysis
Preliminary design of all structures is based on this approximate analysis
Structures considered include: Indeterminate Trusses, Portals and Trussed
Frames, Multi-story frames
Loads considered include: Vertical and horizontal loads
4.2 INDETERMINATE TRUSS ANALYSIS
• Indeterminate trusses used for roofs, lateral bracing of building, the vertical,
support deck of a railway bridge, lateral supports of a through-bridge, or the top
and bottom chords of a through bridge
• Two methods have been used for approximate analysis
• Method I: If the diagonals are intentionally designed to be long and slender, it is
reasonable to assume that they cannot support a compressive force; if they
support they will tend to buckle - In this case the panel shear is carried by the
tensile diagonals only; compressive diagonals carry zero forces
• Method II: If the diagonal members are constructed from large rolled steel angle
or channel sections, they may be equally capable of supporting a tensile or a
compressive force. Hence we can assume that the tension and compression
diagonals each carry half the panel shear
4.3 VERTICAL LOADS ON BUILDING FRAMES
• Building frames often consist of girders that are rigidly connected to columns so
that the entire structure is better able to resist the effects of lateral forces
generated due to wind and earthquake forces. Most of the simplifying
assumptions made to reduce a frame from an indeterminate structure to one that
is statically determinate depends on the way that the structure deforms under the
given loads
• Assumptions: (a) There is zero moment in the horizontal beam girder at a
distance of 0.1L from the left and right supports; and (b) The girder does not
support any axial force
4.4 PORTALS AND TRUSSED FRAME STRUCTURES VERTICAL AND LATERAL LOADS
• When a portal is used to span large distances, a truss may be used in place of
the top horizontal girder - Such structures are used for large bridges, large
auditoriums and industrial structures such as mill bents, ware houses, and
others (as transverse frames)
• In all cases, the suspended truss is assumed to be pin connected at its point of
attachment to the columns
• Furthermore, the truss is assumed to keep the columns straight within the
region of attachment, when the portal is subjected to a side-sway .
• For pin-supported columns, assume that the horizontal reactions are equal
• For fully fixed-supported columns, assume that the horizontal reactions are
equal and an inflection point occurs on each column, midway between the base
of the column and the lowest point of truss member connection to the frame For partially fixed-supported columns (at bottom), the inflection points occur
on columns at one-third height from the base
4.5 LATERAL LOADS ON BUILDING FRAMES
• Method I - Portal Frame Method: Inflection points are assumed to occur
at the middle points of beams and columns (earlier assumptions made for
partial fixity at base are also valid) - At any given floor level, interior columns
are assumed to carry twice the horizontal shear carried by the exterior columns.
• Method II - Cantilever frame method: Hinges are placed at the center of
each girder and column (earlier assumptions made for partial fixity at base are
also valid) - 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; since
stress equals force per area, in the case of columns having equal cross-sectional
areas, the force in a column is also proportional to its distance from the
centroid of the column areas.