Hi,
In this post I will introduce you with the approximate analysis methods for the indeterminate structures.
Introduction: In the approximate analysis we make some assumptions to make the structure determinate and so the analysis is based on those assumptions, so the final results are approximate. This method is different from the exact methods of structural analysis.
While designing any structure there are already a numbers of assumptions used even in the exact analysis, like the homogeneity of material, elasticity, the external loads are assumed but they can never be pre-determined 100% accurately. So talking in general even exact analysis of structure is approximate because there are a numbers of assumptions used.
Trusses:
Generally trusses are of similar geometry. One common truss used is as shown in the figure.
Suppose the external reactions are known, then structure is internally indeterminate to third degree. While designing with the exact analysis the geometry of the members must be known but generally geometry is known to us so we have to use the approximate method. For trusses we can use the following methods:
Method 1: Suppose the diagonal member is long/slender enough, so that they will buckle under the compressive loads. In that case we assume that one of the diagonal member will carry a tensile load, and other carries zero force, and we will design it accordingly.
Method 2: In case we use such members which can carry the compressive as well as tensile force effectively then we can assume that both the diagonal members will carry forces of equal magnitude but one will be tensile and another compressive to take up the shear V at the section as shown in the figure below.
As the two diagonal forces are equal, we can easily determine all the forces at the cut section.
Similarly we can cut a section at other panels of the truss and we can get the forces in the members desired.
Rigid Frame members:
Rigid frames can also be analyzed using the approximate analysis. Rigid frame are generally analyzed for the localized members such as the shown in the figure(a) below. Member AB is fixed to the rigid columns on both ends, so member has a redundancy of degree 3.
So in order to analyze this member locally using the approximate analysis we have to form three assumptions.
If we assume the two columns to be perfectly rigidly fixed to the member AB, then with the earlier experience of the exact analysis, we can say that the inflection points are located at a distance of 0.2L from either point, so the moment values are zero at these points.
Second if the member AB has flexible ends, then beam AB can be analyzed as a simply supported beam.
In actual practice, joints A and B are neither perfectly rigid nor flexible, so we take the average of the two, so we will assume that the inflection points are located at distance of 0.1L from either (i.e. (0.2L+0)/2 = 0.1L).
Further for the vertical loads this can be assumed that the axial force in the member is zero. So up to length of 0.1L from either end beam will be analyzed at the cantilevers and middle part can be considered as the simply supported. Finally the required three assumptions as shown in figure(d) are:
(1) Moment is zero at a distance of 0.1L from left end,
(2) Moment is zero at a distance of 0.1L from right end,
(3) Axial force is zero in the member AB.
using these three assumptions beam can be easily analyzed to find out the bending moment and shear force diagram in the beam member AB.
Portal frames with lateral loads:
(Reference:- Structure Analysis by R.C.Hibbler.. wonder fully explained in this book)
In this post I will introduce you with the approximate analysis methods for the indeterminate structures.
Introduction: In the approximate analysis we make some assumptions to make the structure determinate and so the analysis is based on those assumptions, so the final results are approximate. This method is different from the exact methods of structural analysis.
While designing any structure there are already a numbers of assumptions used even in the exact analysis, like the homogeneity of material, elasticity, the external loads are assumed but they can never be pre-determined 100% accurately. So talking in general even exact analysis of structure is approximate because there are a numbers of assumptions used.
Trusses:
Generally trusses are of similar geometry. One common truss used is as shown in the figure.
Suppose the external reactions are known, then structure is internally indeterminate to third degree. While designing with the exact analysis the geometry of the members must be known but generally geometry is known to us so we have to use the approximate method. For trusses we can use the following methods:
Method 1: Suppose the diagonal member is long/slender enough, so that they will buckle under the compressive loads. In that case we assume that one of the diagonal member will carry a tensile load, and other carries zero force, and we will design it accordingly.
Method 2: In case we use such members which can carry the compressive as well as tensile force effectively then we can assume that both the diagonal members will carry forces of equal magnitude but one will be tensile and another compressive to take up the shear V at the section as shown in the figure below.As the two diagonal forces are equal, we can easily determine all the forces at the cut section.
Similarly we can cut a section at other panels of the truss and we can get the forces in the members desired.
Rigid Frame members:
So in order to analyze this member locally using the approximate analysis we have to form three assumptions.
If we assume the two columns to be perfectly rigidly fixed to the member AB, then with the earlier experience of the exact analysis, we can say that the inflection points are located at a distance of 0.2L from either point, so the moment values are zero at these points.
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| Rigid member subjected to vertical load- approximate analysis |
In actual practice, joints A and B are neither perfectly rigid nor flexible, so we take the average of the two, so we will assume that the inflection points are located at distance of 0.1L from either (i.e. (0.2L+0)/2 = 0.1L).
Further for the vertical loads this can be assumed that the axial force in the member is zero. So up to length of 0.1L from either end beam will be analyzed at the cantilevers and middle part can be considered as the simply supported. Finally the required three assumptions as shown in figure(d) are:
(1) Moment is zero at a distance of 0.1L from left end,
(2) Moment is zero at a distance of 0.1L from right end,
(3) Axial force is zero in the member AB.
using these three assumptions beam can be easily analyzed to find out the bending moment and shear force diagram in the beam member AB.
Portal frames with lateral loads:
 |
| Portal frame with lateral loads(ends pinned)- approximate analysis |
 |
| Portal frames with lateral loads(ends fixed)- approximate analysis |
(Reference:- Structure Analysis by R.C.Hibbler.. wonder fully explained in this book)
1 comment:
Thanks for a very clear explanation! We recently had an article on types of truss and how to analyse and calculate truss using the method of joint as well. If it’s something you are interested in, please check it out
Types of Truss Structures
Truss Analysis and Calculation using Method of Joints
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