This book enables the student to master the methods of analysis of isostatic and hyperstatic structures. To show the performance of the methods of analysis of the hyperstatic structures, some beams, gantries and reticular structures are selected and subjected to a comparative study by the different methods of analysis of the hyperstatic structures. This procedure provides an insight into the methods of analysis of the structures.
Salah KHALFALLAH, National Polytechnic School of Constantine, Algeria.
Introduction to Statically Indeterminate Structural Analysis
The teaching objectives of this chapter are as follows:
- - the importance and usefulness of statically indeterminate structures;
- - calculating the degree of external and internal static indeterminacy of the structures;
- - analyzing kinematic static indeterminacy;
- - illustrating the strengths and weaknesses of statically indeterminate structures.
In the first part, we give a general introduction to the methods of analyzing statically indeterminate structures. In this context, we describe the external, internal and kinematic static indeterminacies of the structures. In the second part, we illustrate the analysis methods for statically indeterminate structures. Lastly, we list the advantages and disadvantages of statically indeterminate structures.
Structures are grouped into two categories: (1) statically determinate structures and (2) statically indeterminate structures. The static equations are not sufficient for analyzing statically indeterminate structures. In this case, the number of unknowns is strictly greater than the number of independent equilibrium equations.
The primary role of analysis of a statically indeterminate structure is to remove the static indeterminacy of the given structure. This removal means that we can calculate the support reactions and the internal actions when the structure is solicited by mechanical loads, or subjected to deflections, and/or undergoing a support settlement. The analysis methods for statically indeterminate structures are used here to make the number of unknowns equal to the number of equations, which allows the problem to be solved.
This book is particularly devoted to the analysis of statically indeterminate structures. To present the differences between the analysis methods for statically indeterminate structures, the problems we consider generally have a common object across the analysis methods. In this context, each chapter illustrates the theoretical foundation of the analytical method, presented in detail and accompanied by a series of numerical examples.
1.2. External static indeterminacy
The purpose of structural analysis is to determine the support reactions and the variation of internal actions in the elements of a statically indeterminate structure. The static indeterminacy of a structure can be internal, external or internal and external simultaneously. It is called externally statically indeterminate if the number of support reactions exceeds the number of independent equations. The plane structures are externally statically indeterminate if the number of support reactions is greater than 3 (Figure 1.1) and it is greater than 6 if the structure is spatial (Figure 1.2).
From this explanation, we define the degree of static indeterminacy of a system by the difference between the number of support reactions and the number of independent equations that can be constructed. The degree of external static indeterminacy f of a plane structure [1.1] or a space structure [1.2] is deduced by [1.1] [1.2]
We calculate the degree of static indeterminacy of the beam and frame (Figure 1.1).
Beam, f = 5 - 3 =2
Frame, f = 10 - (3 + 1) = 6
Figure 1.1. Statically indeterminate externally of plane structures1
Figure 1.2. Statically indeterminate externally of space structures
For space structures (Figure 1.2), the degree of static indeterminacy is
Frame (1), f = 12 - 6 = 6
Beam (2), f = 10 - 6 = 4
Frame (3), f = 24 - (6 + 12) = 6
The degree of static indeterminacy of trusses is calculated by using relationships [1.1] and [1.2]. Figure 1.3 presents plane and space trusses.
Figure 1.3. Externally statically indeterminate plane structures
The degree of external static indeterminacy of plane structures (Figure 1.3) is
Structure (1): f = 4 - 3 = 1
Structure (2): f = 6 - 3 = 3
In the same way, space truss structures (Figure 1.4) are the most used in the construction of large exhibition halls and sports halls, etc.
Figure 1.4. Statically indeterminate space truss
The degree of static indeterminacy of the structure is
f = 12 - 6 = 6
1.3. Internal static indeterminacy
In this section, we describe how to calculate the degrees of static indeterminacy of trusses, frames, beams and crossbeams.
1.3.1. Truss structures
Consider a truss structure with r support reactions, b bars and n joints including support joints. The number of unknowns (b + r) of the problem is the support reactions and the forces in the bars of the truss.
At each joint of the truss, it is possible to write the following equations: [1.3a] [1.3b]
So, the total number of independent equations is 2n.
We define the degree of internal static indeterminacy by [1.4]
Calculate the degree of internal static indeterminacy of the structure (Figure 1.5).
Figure 1.5. Given truss
Applying relationship [1.4] allows us to calculate the degree of static indeterminacy.
r = 5, b = 14, n = 8
f = (14 + 5) - 2 × 8 = 3
The given structure is 3 times statically indeterminate internally.
In the case of a space truss, the equations of static are written as [1.5a] [1.5b] [1.5c]
In the relationship [1.4], we can calculate the degree of internal static indeterminacy by [1.6]
Calculate the degree of internal static indeterminacy of the truss (Figure 1.6).
Figure 1.6. Space truss
n = 4, b = 3, r = 9
f = (3 + 9) - 3.4 = 0
The structure is statically determinate internally.
Calculate the degree of internal static indeterminacy of the structure (Figure 1.7).
Figure 1.7. Space truss
n = 8, b = 13, r = 12
f = (12 + 13) - 3.8 = 1
The structure is once statically indeterminate internally.
1.3.2. Beam and frame structures
Relationships [1.4] and [1.6] can be applied to frames and beams with rigid joints to calculate the degree of internal static indeterminacy. For each rigid joint, it is possible to write three equations: [1.7a] [1.7b] [1.7c]
Note that each end of the bar on a beam or a frame has three unknowns. So we define the degree of internal static indeterminacy by [1.8]
where n is the number of rigid joints including the support joints. If the frame or beam contains k hinges, the relationship [1.6] is written as [1.9]
Determine the degree of internal static indeterminacy of the structures (Figure 1.8).
Figure 1.8. Static indeterminacy of frames and beams
For space structures, we can write six equilibrium equations per joint and each bar has six unknowns. The degree of internal static indeterminacy can be deduced by [1.10]
If the frame or beam contains k hinges, the degree of internal static indeterminacy is written as [1.11]
Determine the degree of internal static indeterminacy of the structures (Figure 1.9).
Figure 1.9. Static indeterminacy of space beams and frames
There is a layer of orthogonal beams linked together at the levels of the rigid joints. At each joint, we can write the following three equations: [1.12a] [1.12b] [1.12c]
At each end of a bar, we consider a vertical force along the axis (zz), bending and torsion moments (Mxx, Myy, Mxy) (Figure 1.10).
The internal forces at any section can be determined when three out of six actions of a beam element are known. Therefore, each member presents three unknowns and the degree of internal static indeterminacy is obtained by [1.13]
Figure 1.10. Crossbeams
Especially when the links between the bars are joints, the degree of internal static indeterminacy...