Belytschko T. - Introduction (779635), страница 67
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Therefore any process in which the loadexceeds the buckling load is unstable. Note that the direction of the divergencedepends on the sign of the initial imperfection.Another example is the flow of a liquid in a pipe. When the flow velocityis below a critical Reynold’s number, the flow is stable. A perturbation of thestate leads to small changes in the evolution of the system. On the other hand,when the flow is above the critical Reynold’s number, a small perturbation leadsto large changes because the flow changes from laminar to turbulent.Stability is usually ascribed to a state, rather than a process. The definitionis then identical: a state is stable if a small perturbation of that state results in asmall differences for all time.
When perturbations lead to large differences in thesubsequent states of the system, the state is unstable. This concept fits within theframework of the definition of stability given by Eq. (6.5.1) with the stateconsidered as the initial condition.A common example of stable and unstable states often given inintroductory dynamics texts is shown in Fig. 6.9. It is clear that state A is stable,since small perturbations of the positionn of the ball will not significantly changethe evolution of the system.
State B is unstable, small perturbations will lead tolarge changes: the ball can roll either to the right or to the left. State C is oftencalled neutral stability in introductory texts. According to the definition of Eq.(6.5.1), state C is an unstable state, since small changes in the velocity will lead tolarge changes in the position as large times.
Thus the definition of stability givenin introductory texts does not completely conform to the one given here.Stability of Equilibrium Solutions. To obtain a good understanding oofthe behavior of a system, its equilibrium paths, or branches, and their stability6-53T. Belytschko & B. Moran, Solution Methods, December 16, 1998must be determined. It is often argued among structural mechanicians that thedifficulties associated with unstable behavior can be circumvented by simplyobtaining a dynamic solution. When a structure is loaded above its limit point ora bifurcation point in a dynamic simulation, the structure passes dynamically tothe vicinity of the next stable branch and the instability is not apparent except forthe onset of a different mode of deformation, such as the lateral deformation in abeam.
However, to understand the behavior of a structure or process thoroughly,its static equilibrium behavior should be carefully examined. Many vagaries ofstructural behavior may be hidden by dynamic simulations. For example, whenthe fundamental path bifurcates with an asymmetric branch as shown in Fig.6.10, the structure becomes very sensitive to imperfections. The theoreticalbifurcation point is not a realistic measure of the strength; an actual structure willbuckle at a much lower load than the theoretical value because imperfections areunavoidable.
A single numerical simulation could miss this sennsitivitycompletelly. This sensitivity to imperfections for cylindrical shells was analyzedby Koiter(??) and is a classical example of imperfection sensitivity.As a first step in studying the equilibrium behavior of a system, the loadand any other parameters of interest, such as the temperature or an active control,must be parametrized.
Up to now we have parametrized the load by the time t,which is convenient in many practical problems. However, a single parameterdoes not always suffice in the study of equilibrium problems. We will nowparametrize the load by nγ parameters γ a , so the load is then given by γ a qa ,where qa represent a distributed loading such as a pressure or concentrated loads.We use the convention that repeated indices are summed over the range, in thiscase nγ . For distributed loads, the parameter γ a should not be applied directly tothe external nodal forces, since the external nodal forces will depend on the nodaldisplacements. The discrete loads can be parametrized by γ a faext , where faext arecolumn matrices of nodal external forces associated with a loading mode a.Our intention is to trace the equilibrium behavior of the model as afunction of the parameters γ α The problem then is then is to find d( γ a ) suchthat()r d( γ a ) = 0(6.5.2b)For purposes of characterizing the nonlinear system, the solutions are usuallygrouped into branches, which are continuous lines describing the response for onechange of one parameter.
Branches along which the solution is in equilibrium, i.e.satisfies Eq(6.5.2b), are called equilibrium branches, regardless of whether theyare stable or unstable.Nonlinear systems exhibit three types of branching behavior:1. turning points, usually called limit points in structural analysis, inwhich the slope of the branch changes sign;2.
stationary bifurcations, often called simply bifurcations, in which twoequilibrium branches intersect.6-54T. Belytschko & B. Moran, Solution Methods, December 16, 19983. Hopf bifurcations, in which an equilibrium branch intersects with abranch on which there is periodic motion.The behavior of the shallow truss exhibits a limit (or turning) point, as canbe seen from Fig. 6.11. Subsequent to a turning point, a branch can be eitherstable or stable. In this case, as shown in the analysis of the problem in Example6.4, the branch after the first limit point, point a, is unstable, while the branchafter thhe second limit point, point b, is stable.The beam problem shown in Fig.
6.12 is a classical example of abifurcation. The point b where the two branches intersect is the point ofbifurcation. Subsequent to the bifurcation point, the continuation of thefundamental branch ab, becomes unstable. Point b, the bifurcation point,corresponds to the classical buckling load of the Euler beam. This type ofbranching is often called a pitchfork, (do you see the hay on the end?)Hopf bifurcations are quite uncommon in passive structures. They arefound in general nonlinear behavior and can be seen in structures under activecontrol.
In a Hopf bifurcation, stable equilibtrium solutions become impossible atthe end of a branch. Instead, there are two branches with periodic solutions. Anexample of a Hopf bifurcation is given in Example ??.Methods of continuation and arclength methods. The tracing ofbranches is called a continuation method. The tracing of equilibrium branches isoften quite difficult and robust, automatic procedures for continuation are notavailable.
In the following, we describe continuation methods base onparametrization, such as the arc length method. In the arc length method, the arclength along the equilibrium path replaces the load as the incremental parameter.It enables branches to be followed around turning points, which is critical to thesuccesful continuation of equilibrium branches..We first consider continuation with the arc length method for the case of a singleload parameter. In tracing the branches in a model with a single load parameter,the load parameter is usually started at zero and incremented. For each loadincrement, an equilibrium solution is computed, i.e.
we find dn +1 , a solution tor d n +1, γ n+1 = 0 orf int dn+1 − γ n+1f ext = 0(6.5.3)where n is the step number and f ext is the load distribution chosen for tracing thebranch. We assume that the loads are prescribed discretely so that the distributionof nodal external forces does not change with the deformation of the model. Theinertial term is not included in the above because continuation methods areapplicable only to equilibrium problems. One of the most widely usedcontinuation methods in structural mechanics is the arc length method. Instead ofincrementing the load parameter γ to trace the branch, a measure of the arclengthis incremented. This is accomplished by adding a parametrization equation to theequilibrium equations6-55T.
Belytschko & B. Moran, Solution Methods, December 16, 1998()p d, ∆d,γ = ∆s2 ,∆γ = γ n +1 − γ n , ∆d = dn +1 − dn(6.5.4)The parametrization equations may be written in terms of the displacements orincrements in the displacements or both. For example, in the arclength methodthe parametrizationn equations are written directly in terms of the displacementincrements∆dT ∆d +∆γ 2 = ∆s2(6.5.4b)Many other types of parametrization equations can be devised, and some aredescribed at the end of this section. DEESCRIBED FISH PARAMETRIZATIONLATER WHEN SCALINNG IS DESCRIBEDThe total system of equations then consists of the equilibrium equationsand the parametrization equation, so we have( ) r d, γ p d,∆d,γ( 0 = 2 ∆s )(6.5.5)The load parameter γ is now treated as an additional unknown of the system andthe arclength s is now incremented instead of the parameter γ .This procedure is most easily explained for a one degree-of-freedom problemsuch as the shallow truss shown in Fig.
6.13. The fundamental branch is shown inFig. 6.13 and we assume that a solution has been obtained at point n. Thearclength equation when viewed in the γ , dy is the circle about point n; in the 3-()space γ , dx , dy it would be a sphere about the point. In solving the parametrizedequations, (6.5.5), we seek a solution which is the intersection of the equilibriumbranch with the circle about the last solution point, which gives the solutionshown as point n+1 in Fig. 6.13. Thus, while incrementing the load parameterwould be fruitless at point n, the problem has been restated in terms of thearclength along the branch so that a solution with a lower load can be found.The parametrized equations for the truss with symmetry can then be posed asfollows: find a solution to()()2 2r1( d1, γ ) = − f1( d1 ,γ ) = 0 subject to γ s − γ n + d1 s − d1n = ∆s2 (6.5.6)Alternatively, we can write the above in terms of increments in the displacementsand the load parameters as: find a solution tof1 = 0 subject to ∆γ 2 + ( ∆d1) =∆s226-56(6.5.7)T.
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