Structural Analysis 2

Statically Indeterminate Structures
 
 
Standards Information Network (Verlag)
  • 1. Auflage
  • |
  • erschienen am 8. Oktober 2018
  • |
  • 396 Seiten
 
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978-1-119-55793-7 (ISBN)
 
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.
1. Auflage
  • Englisch
  • Newark
  • |
  • USA
John Wiley & Sons Inc
  • Für Beruf und Forschung
  • 15,00 MB
978-1-119-55793-7 (9781119557937)
weitere Ausgaben werden ermittelt
Salah KHALFALLAH, National Polytechnic School of Constantine, Algeria.
  • Cover
  • Half-Title Page
  • Dedication
  • Title Page
  • Copyright Page
  • Contents
  • Preface
  • 1. Introduction to Statically Indeterminate Structural Analysis
  • 1.1. Introduction
  • 1.2. External static indeterminacy
  • 1.3. Internal static indeterminacy
  • 1.3.1. Truss structures
  • 1.3.2. Beam and frame structures
  • 1.3.3. Crossbeams
  • 1.4. Kinematic static indeterminacy
  • 1.5. Statically indeterminate structural analysis methods
  • 1.6. Superposition principle
  • 1.7. Advantages and disadvantages of statically indeterminate structures
  • 1.7.1. Advantages of statically indeterminate structures
  • 1.7.2. Disadvantages of statically indeterminate structures
  • 1.8. Conclusion
  • 1.9. Problems
  • 2. Method of Three Moments
  • 2.1. Simple beams
  • 2.2. Continuous beam
  • 2.3. Applying Clapeyron's theorem
  • 2.3.1. Beam with two spans
  • 2.3.2. Beam with support settlements
  • 2.3.3. Beam with cantilever
  • 2.4. Focus method
  • 2.4.1. Left focus method
  • 2.4.2. Right focus method
  • 2.4.3. Focus method with loaded bays
  • 2.5. Conclusion
  • 2.6. Problems
  • 3. Method of Forces
  • 3.1. Beam with one degree of static indeterminacy
  • 3.2. Beam with many degrees of static indeterminacy
  • 3.3. Continuous beam with support settlements
  • 3.4. Analysis of a beam with two degrees of static indeterminacy
  • 3.5. Analysis of a beam subjected to a moment
  • 3.6. Analysis of frames
  • 3.6.1. Frame with two degrees of static indeterminacy
  • 3.6.2. Frame with cantilever
  • 3.6.3. Frame with many degrees of static indeterminacy
  • 3.6.4. Frame with oblique bars
  • 3.7. Analysis of truss
  • 3.7.1. Internally statically indeterminate truss
  • 3.7.2. Externally statically indeterminate truss
  • 3.7.3. Internally and externally statically indeterminate truss
  • 3.8. Conclusion
  • 3.9. Problems
  • 4. Slope-Deflection Method
  • 4.1. Relationship between deflections and transmitted moments
  • 4.2. Fixed-end moments
  • 4.2.1. Bi-hinged beam
  • 4.2.2. Simply supported beam
  • 4.3 Rigidity factor and transmission coefficient
  • 4.4. Beam analysis
  • 4.4.1. Single span beam
  • 4.4.2. Continuous beam
  • 4.4.3. Continuous beam with cantilever
  • 4.4.4. Beam with support settlements
  • 4.4.5. Beam subjected to a moment
  • 4.5. Analysis of frames
  • 4.5.1. Frame without sidesway
  • 4.5.2. Frames with sidesway
  • 4.6. Conclusion
  • 4.7. Problems
  • 5. Moment-Distribution Method
  • 5.1. Hypotheses of the moment-distribution method
  • 5.2. Presentation of the moment-distribution method
  • 5.2.1. Distribution of a moment around a rigid joint
  • 5.2.2. Distribution procedure
  • 5.3. Continuous beam analysis
  • 5.3.1. Beam with support settlement
  • 5.3.2. Beam with cantilever
  • 5.3.3. Beam subjected to a moment
  • 5.4. Analysis of frames
  • 5.4.1. Frame without sidesway
  • 5.4.2. Frame with sidesway
  • 5.5. Conclusion
  • 5.6. Problems
  • 6. Influence Lines of Statically Indeterminate Structures
  • 6.1. Introduction
  • 6.2. Influence lines of beams
  • 6.2.1. Beam with one degree of static indeterminacy
  • 6.2.2. Beam with two degrees of static indeterminacy
  • 6.3. Influence lines of frames
  • 6.4. Influence lines of trusses
  • 6.4.1. Internally statically indeterminate truss
  • 6.4.2. Externally statically indeterminate truss
  • 6.5. Conclusion
  • 6.6. Problems
  • 7. Statically Indeterminate Arch Analysis
  • 7.1. Introduction
  • 7.2. Classification of arches
  • 7.3. Semicircular arch under concentrated load
  • 7.4. Parabolic arch under concentrated load
  • 7.5. Semicircular arch under distributed load
  • 7.6. Parabolic arch under distributed load
  • 7.7. Semicircular arch fixed under concentrated load
  • 7.8. Statically indeterminate tied arch
  • 7.9. Arch with many degrees of freedom
  • 7.10. Influence lines of statically indeterminate arch
  • 7.10.1. Influence lines of bi-hinged arch
  • 7.10.2. Influence line of fixed-end arch
  • 7.11. Conclusion
  • 7.12. Problems
  • Appendix
  • A.1. Standard structural deflections
  • A.2. Fixed-end moments
  • Bibliography
  • Index
  • Other titles from iSTE in Civil Engineering and Geomechanics
  • EULA

1
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.

1.1. Introduction


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]

EXAMPLE 1.1.-

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]

EXAMPLE 1.2.-

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.

EXAMPLE 1.3.-

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]

EXAMPLE 1.4.-

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]

EXAMPLE 1.5.-

Determine the degree of internal static indeterminacy of the structures (Figure 1.9).

Figure 1.9. Static indeterminacy of space beams and frames

1.3.3. Crossbeams


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...

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