Civil Engineering ► Structural Analysis ► Approximate Analysis of Indeterminate Structures
Approximate Analysis of Statically Indeterminate Structures 

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Introduction to Procedures for Approximate Analysis

 Global analysis carried out on two levels:
 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 inbuilt data errors
 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
 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
 Reduction of the complex system into an equivalent simple system to carry out
 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, Multistory frames
 Loads considered include: Vertical and horizontal loads
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:
 There is zero moment in the horizontal beam girder at a distance of 0.1L from the left and right supports; and
 The girder does not support any axial force
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 sidesway.
 For pinsupported columns, assume that the horizontal reactions are equal
For fully fixedsupported 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 fixedsupported columns (at bottom), the inflection points occur on columns at onethird height from the base
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 crosssectional areas of the columns at a given floor level; since stress equals force per area, in the case of columns having equal crosssectional areas, the force in a column is also proportional to its distance from the centroid of the column areas.
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