Pedagogical Approach for the Structural Stability

Pedagogical approach for the structural stability

Walnório Graça Ferreira 1, Diogo Folador Rossi 2, Vitor Folador Gonçalves3 and Augusto Badle Neto4

1Federal University of Espírito Santo, Vitória, Brazil, [email protected] Federal University of Espírito Santo, Vitória, Brazil, [email protected] 3 Federal University of Espírito Santo, Vitória, Brazil, [email protected] Federal University of Espírito Santo, Vitória, Brazil, [email protected]

Abstract

Today the materials used in buildings are becoming more resistant, thus the spans of the structures are getting bigger. This has been used by architects to satisfy the wishes of their customers, and everybody becomes happy because of the large areas without columns. Thus, all the structure that supports the building becomes slender. Consequently, the structural elements in compression are more subject to large lateral deflections, being susceptible to loss of stability, requiring second-order analysis. This article first presents a conceptual approach with regard to the theory of elastic stability, including the terminology involved as a bifurcation, critical load, limit points, jump dynamics and post-critical path. In a second step, this paper presents a didactic computer-numerical procedure for stability analysis of nonlinear systems with one and two degrees of freedom, without loss of generality involved in complex systems that need the finite element method for the solution. This approach will be of great value not only for teaching the theory of elastic stability at the undergraduate level, but also for teaching at graduate level, once it introduces details of computational implementation, concepts of stability, analytical solution of systems geometrically non-linear, as well as incremental-iterative solution based on Newton-Raphson method, using simple mechanical models consisting in rigid bars and rotational and linear springs.

Keywords: Nonlinear analyses, Structural stability , Buckling.

1. Introduction

Making the most economic structures through reduction of weight and consumption of materials, without, however, reduce your safety and durability, has been the main objective of structural engineering.

Weight reduction has been achieved in concrete structures, due to increased use of concrete with high resistance (generally greater than 20 MPa), especially the high-strength concrete (with resistances greater than 50MPa), and the steel

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structures, whose resistances exceeds 250 MPa. It associated with the use of refined analyzes and more accessible and powerful computers have led to bolder projects with thinner structural elements. Consequently, the structural elements compressed are more subject to large lateral deflection, making them susceptible to loss of stability, requiring non-linear analysis. This phenomenon is analyzed with adequate depth in the theory of elastic stability [1]. This work aims to present concepts and terminology relevant to the loss of stability of structural systems, often not approached with adequate depth in the books of mechanics of solids adopted in undergraduate civil engineering and mechanical engineering.

2. Concepts of stability

The stability of the equilibrium is a basic concept of the rigid body mechanics, which can be easily viewed and intuitively assimilated through the classic problem of spherical mass lying in straight or curved surfaces, as illustrated in Figure 1.

Figure 1. Spherical masses in static equilibrium

The points where rest the masses M1, M2 and M3 have zero slope and represent points of static equilibrium, however, the type of equilibrium each of these points is essentially different. Thus, if the mass M1 undergoes a small external disturbance, when removed the cause of trouble, she returns to the starting equilibrium position. In this case, we consider that the original position is a stable equilibrium state. It is observed that in this case, the center of gravity rises, and thus increasing the potential energy of the system (ΔΠ> 0). For the mass M2, unlike what happened with the mass M1, the original position is an unstable equilibrium state, because after a small disturbance, the static forces acting upon the system tend to displace the ball away from of the its equlibrium position. In this case there was a lowering of the center of gravity and, consequently, a decrease in potential energy of the system (ΔΠ <0). In the third case, when the weight rests on a flat surface, we refer to the system as being in a state of neutral stability (or state of indifferent stability), ie, in any position the ball remains in equilibrium. Here the center of gravity of the ball remains in the same level, there is therefore no variation in the potential energy (ΔΠ = 0, [1]).


3. Criteria of stability

To study the stability of structural systems, three criteria can be used. The static criterion of stability, which examines the equilibrium of forces, the energy criterion of stability, which examines the variation of the total potential energy, and the dynamic criterion of stability, which depends on the sign of the natural frequencies of vibration of the system.

4. Equilibrium Paths

The loss of stability is a nonlinear phenomenon, so to understand, accurately,the behavior of the system under this effect, we have to use a nonlinear analysis,although in many cases it is possible to linearize the equations which govern the system with a configuration disturbed, to facilitate analysis.When analyzing the stability of a structural system has a set of control parameters and to know the overall behavior of the system and identify the possible instability phenomena, we consider the variations of the equilibrium position as it varies each control parameter.

Thus we obtain the so-called equilibrium paths. Along these paths, the equilibrium configurations may be qualitative changes with regard to their stability. The border points are called critical points that can be of two types: bifurcation points or limit points [1] (Figure 2). The bifurcation points are points of sudden change in path of equilibrium. The limit points can be load or displacement, corresponding, respectively, maximum or minimum values for the load around a point (maximumor local minima) in the path of equilibrium and maximum or minimum relativedisplacements around a point (local maximum or minimum values) in the path ofof equilibrium. The paths shown in Figure 2 are nonlinear paths of equilibrium and they are associated with large displacements of structural systems.

Figure 2. Limits points of bifurcation and load [2]

5. Nonlinear behavior of mechanical systems

In This section will be presented mechanical systems with single degree of freedom with purpose of continue showing the basic concepts of elastic stability (non


linear behavior ). All systems in this work are composed of rigid bar associated withlinear or rotational springs, both with linear behavior, ie, linear correlation betweeneffort (moment or force) and displacement (rotation or linear displacement).

5.1. Stable-symmetric bifurcation system

5.1.1. Nonlinear analysis without approximations

The system is composed of a vertical bar, no weight, with a vertical load Papplied at its free upper end and lower extremity free to rotate, but witha circular spring coupled torsional stiffness K, as illustrated in Figure 3.

Figure 3. Rigid bar with circular spring

If the bar is vertical, the system is in equilibrium regardless of the value P of the load,equilibrated by the reaction with R (R = P). However, in this situation, the spring will be free of stress. To study the stability is necessary to analyze this system subjected to a small disturbance and use the criteria of stability ever seen. It is observed that the application of small disturbance is a procedure that will always follow a stability analysis. Only by applying small disturbance can be established the equations governing the problem and obtain more accurate information of the analysis, because it is non-linear behavior. The Table 1 summarizes the equationsrelevant to the perturbed system under study.

Table 1. Total potential energyPotential energy of elastic spring Potential of load P Total potential energy

U = ½ Kθ² V = - P∆ = - PL.(1 - cosθ) Π = ½ Kθ² - PL.(1 - cosθ)

It is observed that the total potential energy depends on θ. By the energy criterion ofstability the system will be stable if the potential energy of the perturbed system is minimal. Consequently, as Π = Π (θ), this condition is fulfilled if

dΠ/d θ = 0 (1)


This condition would also be achieved if the potential energy of the disturbed system had a maximum value, but in this case, the equilibrium would be unstable. It will be verified later. From equation (1) is

K θ – PL sen θ = 0 (2)

That provides two solutions, θ = 0, It means vertical bar (the original equilibrium position). This solution is called fundamental solution of equilibrium, and its graph coordinates in PL / K x θ called fundamental path or primary path [3]. Or P = Kθ/Lsenθ. It means inclined bar, which corresponds to the equilibrium condition at the equilibrium position disturbed.

This solution is called postcritical solution, and its graph coordinates in PL / K x θcalled the secondary path of equilibrium or postcritical path. It is observedthat the second solution is a nonlinear function of θ, derived from the fact that the problem is inherently nonlinear due to its geometry, because of this, it considers that occurred nonlinearity geometric.

To find out if the two solutions of equation (2) represent the stable or unstable systemyou need to verify the sign of the second differential of the function Π = Π (θ). This second differential is:

j²Π/jθ² = K – P.L. cosθ (3)

Table 2. Stability studySolution θ = 0 Solution P = Kθ/Lsenθ

[j²Π/jθ²]θ=0 = K - PL

[j²Π/jθ²]θ=0 = K(1 – θ.cotanθ)

K – PL > 0 ou P < K/L

Stable equilibrium

Fundamental Path

K – PL < 0 ou P > K/L

Unstable equilibrium

Fundamental Path

For θ < |p|, j²Π/jθ² = > 0

Stable equilibrium

Post-critical Path

From the study of the stability of the mechanical system it is concluded that if the load is less than the critical load (P <Pcr, where Pcr = K / L), the system is stable (in the vertical bar, or θ = 0 , fundamental path), otherwise the fundamental path will be unstable (vertical bar, or θ = 0). In this case P> K / L, the total potential energy is minimal and any disturbance in the vertical bar will lead to an inclined position seeking a stable situation, standing in the path post-critical,which is stable for θ < |π| . Laminar structures such as plates under specific conditions, may have stable post-critical paths with high curvature and larger loads of collapse than the critical load.


It remains to study the stability of the bifurcation point where P = K / L. In this case θ = 0, and equation (3) results equal to zero. Assessing the upper differentials it is found that the third differential of Π is zero, in θ =0, and the fourth differential is PL, in θ =0.Therefore, the bifurcation point the system will be stable.

5.1.2. Nonlinear analysis with approximations

The analysis in the previous section was done exactly, without approximations. Now the effects will be shown in the solution when using approximations. For this, is done a Taylor series expansion of cosine nearby of θ = 0, leading to cosθ = 1- (θ2/2!) + (θ4/4!) - (θ6/6!) + ...

Adopting only the first two terms and substituting in the equation of total potential energy (Table 1), it has Π = ½ Kθ2 - ½ PLθ2. Using the energy criterion of stability (equation (1)), it obtains (K-PL).θ = 0 (equation 4).

This equation provides two solutions, θ = 0 (fundamental path) and P = K / L (post-critical path), which is the critical load. The first one is the same as obtained in the previous analysis, but the second is very different because it represents a post-critical path independent of θ (P =K/L for any θ). The homogeneous equation represented by equation (4), a trivial solution and other non-trivial corresponding to the critical load is known mathematically as eigenvalue problem. The second diferential of the total potential worth K - PL, that for the fundamental solution, provides the same information from the previous nonlinear analysis. However for the second solution of equation (4), there is the second diferential is zero, ie, the equilibrium is neutral. This often happens in the theory of the stability when using a linearized solution, it only determines the critical load, and it loses information about post-critical paths.

5.1.3. Effect of initial geometric imperfections

In real cases, the structures often have imperfections, there are no perfectly straight bars, thus, a way to consider these defects is to allow, in case of the model of Figure 3, the bar is initially at an angle θ0 in respect to its vertical position, as illustrated in Figure 4.

Figure 4. System showing a no weight rigid bar with imperfections and circular spring


Table 3. Total potential energy of system with imperfectionsPotential energy of elastic spring Potential of load P Total potential energy

U = ½ K∅² = ½ K(θ – θo)²

V = - P∆ = - PL.(cosθo - cosθ) Π = ½ K(θ – θo)² - PL.(cosθo - cosθ)

For the energy criterion of stability, the system is in equilibrium (stable, unstable or indifferent) when the equation (1) is satisfied, or when:

K(θ − θ0) - PL sen θ = 0 (5)

This equation shows that θ = 0 is no longer the problem solution. Isolating P in equation (5), we have

P = K(θ − θ0)/ L sen θ (6)

The following figure shows the critical paths for the system of Figure 4.

Figure 5. Nonlinear solution of system of Figure 4

The Figure 5 shows the critical paths of the imperfect system illustrated in Figure 4,along with the those of perfect system illustrated in Figure 3. It is observed that there is no bifurcation, the solutions of the nonlinear imperfect system bordering the solutions of the perfect system (there is an asymptotic approximation.) The paths above the curve of the perfect system are called complementary equilibrium paths and can only be achieved indynamic problems.In both previous examples was presented the equations for the total potential energy and the equations resulting from applying the energy criterion of stability. Below we present other examples of mechanical stability, but without the deduction of the equations,because these procedures are similar deductions, not bringing anything new under theconceptual point of view. In the following examples we will discuss information aboutconcepts and terminology, the true objective of this work.


5.2. Unstable-symmetric bifurcation system

Figure 6 shows a system consisting of vertical bar with compression load P in the upper end associated to a linear spring of stiffness K. Figure 6 (a) shows the system in its initial state, and Figure 6 (b) present it in the disturbed state.

Figure 6. Rigid bar with spring linear, variation of total potential energy and the analogy with spherical mass

The analysis of this system shows that for P <KL, the equilibrium is stable, but when P>KL, the path is the postcritical and it is always unstable (Figure 6 (c)), with symmetricbifurcation. This case is known as unstable-symmetric bifurcation system [4].

5.2.1. Effect of initial geometric imperfections

The imperfections in this system are treated similarly to the case of rigid bar with circular spring (Figure 4). Figure 7 shows its behavior.

Figure 7. Nonlinear solution of the system of figure 6 with imperfections and variation of limit load with imperfections

In this system, is verified that starting from the P = 0 (when θ0> 0), the critical path is stable until it reaches a maximum value represented by PL, from which it becomes unstable. The load is called buckling load or limit load of the imperfect structure and the corresponding point is called a limit point (point limit load). From the limit point there is instability with the strains growing indefinitely (Figure 7


(a)). This process of loss of stability is called dynamic jump. As it increases θ0 imperfection of the system, the limit load becomes lower (Figure 7 (b)).

5.3. Asymmetric bifurcation system

We have the system illustrated in Figure 8 (a), a rigid bar, no weight, initially vertical, with inclined linear spring 45º when discharged.

Figure 8. Asymmetric bifurcation system

The Figure 8 (b) shows the behavior of this system. Disturbances negative (θ <0) causes the system to stable equilibrium. Disturbances positive (θ> 0) lead to a critical path with limit point, the limit load being the highest load that the system can support. In civil engineering, the non-symmetric plane frames may exhibit this behavior.

5.4. System with no bifurcation

For now, it has the structure shown in Figure 9 (a) consisting of two rigid bars freely hinged to each other and with two supports, the support C is attached to a linear spring of stiffness K. Both the bar, when discharged, have θ0 slope.

Figure 9. System with no bifurcation

Comments will be made only for θ> 0. By increasing the load P from zero, the angle θ, which is downloaded to the system value θ0, will decrease until it reaches a critical point θcr '. An infinitesimal increment greater than this value there will be a sudden


change in system configuration, passing the configuration I to configuration II (Figure 9(b)). This sudden change of configuration is called a dynamic jump.

5.5. Systems with multiple degrees of freedom

In this study, using the energy criterion of stability, a structural model with two degrees of freedom, consisting of three rigid bars, with no weight and length L each.The bars are freely hinged to each other . At the lower end it has a support pinned and at the top a support roller.The model is supported by two linear springs with stiffness K and it is subject to a compressive vertical load.All concepts that were seen for systems with single degree of freedom are valid for systems with multiple degrees of freedom. Thus, it will only make a compact representation of this system of equations for two degrees of freedom.

(a) (b)

Figure 10. Proposed structure for analysis and configuration mechanism after it loses stability

For structural system shown, defines the potential energy from values y1, y2 and ∆, as follows

U(θ1, θ2) = (1/2)K.y1² + (1/2)K.y2² (7)

V(θ1, θ2)= - P . ∆ (8)

Thus,

∏(θ1, θ2) = (1/2)K.(Lsen(θ1))² + (1/2)K.(Lsen(θ2))² + [-P.[3.L-[Lcos(θ1) + Lcos(θ2) + L.[1- [(sen(θ2) - sen(θ1))²]1/2]]]]

Now, using the Principle of Stationary Potential Energy, it gets the pair of equations of nonlinear equilibrium:

dΠ(θ1, θ2)/dθ1 = 0 Thus,


KL2.sen(θ1). cos(θ1) – P.[ L sen(θ1) – [L. (sen(θ2) - sen(θ1)). cos(θ1)]/[1- (sen(θ2) - sen(θ1))²](1/2)] = 0

And dΠ(θ1, θ2)/dθ2 = 0

Thus,

KL2.sen(θ2). cos(θ2) – P.[ L sen(θ2) – [L. (sen(θ2) - sen(θ1)). cos(θ2)]/[1- (sen(θ2) - sen(θ1))²](1/2)] = 0

So, the procedure follows as seen to single-degree-of-freedom systems. The Figure 11 shows the nonlinear behavior of system shown in Figure 10.

Figure 11. Nonlinear solution of the system of figure 10

6. Final considerations

This article aims to present relevant concepts and terminology of the theory of structural stability. To achieve this, we used the mechanical models with rigid bars weightless associated with circular or linear springs . By studying the stability of these simple models have been discussed all the concepts and behaviors commonly used to study the stability of real structures, such as plane frames, arches and others. This text is an important teaching in this subject, for students, also serving as a pedagogical support to teachers in the basic disciplines and professional courses in the civil and mechanical engineering, not familiar with the geometric nonlinear analysis.


7. Acknowledgements

The authors acknowledge CNPq, CAPES and FAPES the support received for this work.

References

[1] W. G. Ferreira, Design of Elements of Laminates and Welded Steel sections (in portuguese). With Numerical Examples (in portuguese), Grafer, Vitória, 2004.

[2] R. A. M. Silveira, Notes Class for Advanced Topics in Steel Structures I, Graduate Program in Steel Construction (in portuguese), PROPEC/UFOP, 2008.

[3] J. G. A. Croll and A. C. Walker, Elements of Structural Stability, Macmillan, London, 1972.

[4] H. G. Allen and P. S. Bulson, Background to Buckling, McGraw-Hill, Berkshire, 1980.


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Pedagogical Approach for the Structural Stability