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APPLIED STABILITY

Lecture 6.5: Iterative Methods for Solving

Stability ProblemsOBJECTIVE

To present and illustrate the application of methods which can be used to solve stabilityproblems iteratively.

PRE-REQUISITES

None.

RELATED LECTURES

Lecture 6.1: Concepts of Stable and Unstable Elastic Equilibrium

Lecture 6.2: General Criteria for Elastic Stability

Lecture 6.3: Elastic Instability Modes

Lecture 6.4: General Methods for Assessing Critical Loads

RELATED WORKED EXAMPLES

Worked Example 6.3: Application of Vianello's/Newmark's/Vianello-Newmark Methods

SUMMARY

This lecture begins with an introduction which describes the reasons for using iterativemethods to solve stability problems. Then the Vianello Method is introduced. Next, themethod of Newmark for the calculation of internal forces and deflections in transversallyloaded beams is reviewed as a preliminary step to the presentation of the Vianello-Newmark method. This method combines Vianello's method with Newmark's integrationprocedure.

1. INTRODUCTIONEven when deflections are assumed to be small, stability problems are always non-linear,in the sense that the equilibrium equations and boundary conditions must be establishedfor the deformed configuration of the structure. As a result, only in very simple cases, isit possible to obtain analytical solutions of the eigenvalue-eigenfunction problem, leadingto the determination of the critical buckling load and corresponding instability mode (seeLecture 6.3). In general it is necessary to resort to approximate methods. One veryimportant group of such methods - energy methods - was presented in Lecture 6.4.Basically, these methods consist of replacing the original continuous structure by a

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simpler discrete structure. This was achieved by constraining the real structure to deformin a manner which is the superposition of a set of defined shapes with unspecifiedamplitudes. The exact critical buckling load and mode of this simpler structure, which isthe solution of an eigenvalue-eigenvector problem similar to the one addressed inLecture 6.2, are approximate solutions for the original structure. Although the accuracyof these methods (and the effort involved) increases with the number of degrees offreedom considered, very satisfactory approximations can often be obtained using only asmall number. One major drawback of energy methods is that they always lead to upperbounds of the critical buckling load, which is not convenient in design. The discretizationprocedure of a continuous structure may also be achieved by dividing it into several rigidelements connected by springs that provide its stiffness. The deformation of thestructure is a piecewise continuous function which is completely defined by thedisplacements of the nodes connecting the elements. The exact solution of thisdiscretised structure was addressed in Lecture 6.2 and is also an approximate solution ofthe original problem. However, in this case nothing can be said concerning the amount orsign of the error. As before the accuracy also increases with the number of elements.

Thus the determination of the critical buckling load and mode of a structure requires thesolution of a non-linear problem which is either a linear eigenvalue-eigenvector problem(discrete or discretised systems) or, a linear eigenvalue-eigenfunction problem(continuous systems). In the first case an analytical solution is always possible but itrequires the determination of the lowest root of the characteristic equation, which isoften of a relatively high degree. In the second case an analytical solution is possible onlyfor simple problems. An alternative to either of these problems is provided by an iterativemethod first introduced by Vianello and, therefore, designated as Vianello's method. Thebasic idea consists of replacing the solution of the non-linear problem by the solution of asequence of linear problems which can be shown to converge to the critical bucklingmode and enable the calculation of the critical buckling load. A feature of Vianello'smethod which is very convenient in the design and safety checking of structures, is thatit is possible, after each iteration, to calculate upper and lower bounds of the criticalbuckling load and, therefore, to estimate the corresponding error.

Finally, the Vianello-Newmark method combines the concept of Vianello's method withNewmark's numerical integration technique. It is a very efficient alternative for thedetermination of critical buckling loads and modes of axially loaded columns, particularlyif some non-standard features are present in the loads, the column or its supportingconditions. This method can also be used to determine equilibrium configurations ofcolumns acted on by specified axial loads and containing initial geometrical imperfectionsor transverse loads, i.e.beam-columns.

2. METHOD OF VIANELLOVianello's method is an iterative procedure which may be used to determineapproximately the critical buckling load and mode of continuous or discrete structuralsystems acted on by a set of loads that may be expressed in terms of a single loadparameter l (proportional loading). The method is based directly on the differentialequation (system of simultaneous equations) of equilibrium of the system, which meansthat it does not involve energy concepts. The application of the method consists of thefollowing steps:

(i) Make an initial estimate of the deflected configuration associated with the criticalbuckling mode of the structure, which must satisfy the kinematic boundary conditions.This initial estimate is a vector (discrete systems) or a function (continuous systems).

(ii) Based upon this assumed configuration calculate the internal forces in terms of the


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unknown buckling loading parameter l. These internal forces are concentrated forcesand/or moments (discrete systems) or bending moments (continuous systems).

(iii) Using a standard linear analysis determine the deflected configuration produced bythe internal forces calculated in (ii). This new deflected configuration, which depends onl, constitutes an improved approximation of the critical buckling mode of the structure.The linear analysis involves the solution of a system of simultaneous equilibriumequations (discrete or discretized systems) or of a differential equation (continuoussystems).

(iv) Equate the assumed and calculated deflections mentioned in (i) and (iii) to obtainupper and lower bounds and an estimate of the critical value of the load parameter lcr. Indiscrete systems the upper (lower) bound of lcr is the larger (small) value of l required toequalize a pair of corresponding non-zero components of the vectors defining theassumed and calculated deflections. A possible estimate of l is required to equalize thevalues of the functions defining the assumed and calculated deflections at a point ofnon-zero value. These bounds are often rather difficult to calculate and only an estimateof lcr is determined, which is the value of l required to equalize the functions at aspecified point.

(v) Repeat the process using as initial estimate the shape of the deflection calculated inthe previous iteration. Stop whenever the desired accuracy is achieved. It is oftenconvenient, for numerical reasons, to normalize the calculated deflection before using itas the initial estimate in the next iteration. The accuracy of the solution is measuredeither by the difference between the upper and lower bounds or by the proximity of theconsecutive estimates of lcr.

It can be shown that the process converges to the critical instability mode, thereforeallowing the calculation of the critical buckling load parameter lcr.

3. REVIEW OF NEWMARK'S METHODMathematically, the essence of Newmark's method is a numerical integration technique

for solving differential equations of the type = f (x). It leads to a rapid andsystematic calculation of shears and moments in arbitrary statically determinate beamsacted on by transverse loads. Through the combination of Newmark's integration schemewith the conjugate beam method, it is possible to calculate also slopes and deflectionsdue to bending. Statically indeterminate beams may be analysed by the force method,with Newmark's method providing a straightforward way of determining the flexibilitymatrix.

3.1 Sign Conventions

The sign conventions are chosen so that quantities may be added when proceeding fromleft to right across the beam and subtracted in the opposite case. Then, the axial force(N) is positive if it is compressive, the shear force (V) is positive if it tends to turnclockwise, the bending moment (M) and curvature (c) are positive when the top fibres

are compressed, the slope (q) is positive upwards to the right, the lateral deflection (y)and applied loads (q, Q) are positive upwards, and the axial applied load (P, p) arepositive from left to right.

3.2 Concepts


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In order to apply Newmark's method it is necessary to divide the beam into several equalsegments. Each division point is referred to as a station. The number of stations mustenable a good description of the beam, loads and supporting conditions. When theloading consists of concentrated loads acting at the stations, the method determines theshears in the segments and the moments at the stations exactly. The shears aredetermined by summing algebraically the loads along the beam and the bendingmoments are found by adding or subtracting the product of successive shears and thelengths of the segments over which they act. When the value of the shear or moment isnot known at any point along the beam, the calculations may be continued on the basisof some arbitrary chosen value (usually zero), with a linear or constant correction addedlater to the resulting moments (shears).

When the beam is acted on by distributed loads they must be replaced by equivalentconcentrated loads acting at the stations. Physically, these loads represent the reactionsof a series of hypothetical weightless stringers coinciding with the segments andinterposed between the loads and the beam (see Figure 1). The stringer reactions areequivalent to the distributed loads in the sense that they produce the same shears andbending moments at the stations. The formulae for computing the equivalentconcentrated loads are exact respectively for linear and parabolic loading distributions,and approximate for higher order distributions. The formulae for end stations must alsobe used whenever there is a jump in the magnitude or slope of the applied load.

For linear discretization (Figure 1), the formulae are:

End stations: Ri1 = Dx (2pi1 + pi) / 6Intermediate stations:


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Rii1

= Dx (2pi + pi1) / 6

Ri = Rii+1 + Rii-1 = Dx (pi-1+ 4pi + pi+1) / 6

For parabolic descretization (Figure 1), the formulae are:

Ri1 = Dx (7pi1 + 6pi - pi-1) / 24

Rii1

= Dx (3pi1 + 10pi - pi-1) / 24

Ri = Rii+1

+ Rii-1

= Dx (pi-1 + 10pi + pi+1) / 12When the loading contains distributed loads the method determines directly the averageshears in the segments and the bending moments at the stations. A simple addition givesthe shears at the stations. All these values are exact provided that no error is introducedby the load discretisation.

Once the bending moments are known it is possible to compute the curvatures bydividing by the bending stiffness EI. Since the load (p), shear (V) and bending moment(M) bear the same relations to each other as the curvature (c = M/EI) , slope (q) anddeflection (y), it may be concluded that the procedure used to compute bendingmoments from loads can also be used to compute deflections from curvatures, as longas account is taken for the different boundary conditions. In order to repeat theprocedure mentioned above, the first step is to replace the curvatures (a continuouslydistributed quantity) by equivalent "concentrated curvatures". Physically, these quantitiesrepresent the sudden changes in slope that take place at the nodes of an hypotheticalbeam formed by rigid segments hinged to each other and with a bending stiffnessprovided exclusively by rotational springs placed at the hinges. The changes in slope areequivalent to the distributed curvatures in the sense that they produce the same slopesand deflections at the stations. The formulae for computing the equivalent concentratedcurvatures are the ones used for the loads and shown in Figure 1b. Next the procedureyields successively average slopes in the segments and deflections at the stations. Itshould be noted that these quantities are precisely the equivalent concentrated loads,average shears and bending moments of the "conjugate beam" when acted on bydistributed loads which coincide with the curvature diagram of the original beam (theconcept of "concentrated curvature" is replaced by the definition of the "conjugatebeam").

Finally, in the case of statically indeterminate beams, Newmark's method is well suited tothe use of the force method, since it provides a straightforward way of determining theflexibility matrix and the deflections in the basic system.

4. METHOD OF VIANELLO-NEWMARKWhenever Vianello's method is applied to axially loaded columns and step (iii) isperformed by means of Newmark's method, one has the method of Vianello-Newmark.Concerning step (ii), i.e. the computation of the values of the bending moments at thestations in terms of the load parameter and on the basis of the initial estimate of thebuckling mode, the following procedure is applicable, which is exact provided that all theaxial loads are concentrated at the stations:

(i) Calculate the axial forces (N) in the segments in terms of the axial loads (P) whichmay be expressed in terms of a single load parameter l. If the column is staticallyindeterminate in the axial direction the values of N must be determined by means of a


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suitable method (e.g. force method).

(ii) Calculate the values of deflection minus the increment in deflection taking place ineach segment, on the basis of the initial estimate (Dyij = yi - yj). This sign convention isadopted so that all the quantities may still be added when proceeding from left to rightacross the beam and subtracted in the opposite case.

(iii) Calculate the increment in bending moment due to the axial force taking place ineach segment (DMij = Nij Dyij).

(iv) Calculate the bending moments, due to the axial forces, at the stations (M) byadding or subtracting the values of DM. These bending moments do not include theinfluence of the support reactions, and therefore, need to be corrected whenever thisinfluence is present.

(v) Perform the appropriate corrections on the bending moments calculated in (iv).These corrections are identical to the ones discussed in the previous chapter and lead toexact values in the case of statically determinate (in the transverse direction) columns. Ifthe column is statically indeterminate and assuming that the force method is used, theprocedure described above is performed in the basic system chosen. The compatibility isenforced during step (iii) of Vianello's method, which also uses Newmark's technique,and enables the determination of the bending moments and deflections at the stations ofthe original column.

If distributed axial loads are present, they must be replaced by equivalent concentratedaxial loads (pdisc) using the formulae given in Figure 1b. The procedure of calculating the

bending moments at the stations just mentioned then becomes approximate (the errorcan be reduced by increasing the number of segments).

It should be noted that the calculation of the equivalent concentrated curvatures is nowalways approximate. Thus the method of Vianello-Newmark will lead to a value of thecritical buckling load slightly different from the exact one. This error will be decreased byan increase in the number of segments.

5. EQUILIBRIUM CONFIGURATIONSThe methods of Vianello and Vianello-Newmark may also be used to determineequilibrium configurations of geometrically imperfect or transversally loaded columnsunder the action of specified axial loads. Only the method of Vianello-Newmark isdiscussed below. Vianello's method can be applied only in very simple cases.

For instance, the behaviour of a beam-column is given by the solution of the followingdifferential equation (N piecewise constant):

(5)

The application of the method of Vianello-Newmark consists of an iterative procedurewhich requires an initial guess of the deflected shape of the beam-column. It convergesto the corresponding exact shape y(x). Each iteration involves the solution of thefollowing two equations:

= q (6)


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= (7)

Equation (6) is a standard linear analysis and only needs to be solved once, since yI (x)

is the same in all iterations. Equation (7) strongly resembles the eigenvalue-eigenfunction problem dealt with before, the difference residing in the fact that the axialforces are now due to known applied forces. The amplitude of the initial estimates of thedeflected shape must, therefore, be controlled by a factor D, determined at the end ofeach iteration by the condition

(8)

where n is the number of stations. This condition imposes a similarity between the initialand calculated deflected shapes, in the sense that the sum of their station values mustbe the same.

If the initial imperfection consists of an eccentricity e0 of all the applied loads, then yI (x)

is the solution of (N piecewise constant):

Finally, it should be mentioned that the method will diverge if the axial loading parameterl is larger than the corresponding critical value lcr.

6. CONCLUDING SUMMARYThis lecture dealt with the use of iterative methods to solve stability problems,namely the determination of critical loads and equilibrium configurations.The basic idea of these methods was introduced by Vianello and consists ofreplacing the solution of a non-linear problem by the solution of a convergentsequence of linear problems.The method of Vianello is used to calculate critical buckling loads of discrete andcontinuous systems. However, in the case of continuous systems the method isonly applicable to rather simple problems.Combining the method of Vianello and Newmark's integration technique, it ispossible to establish an efficient method to calculate critical loads and determineequilibrium configurations of axially loaded columns.The Vianello-Newmark method is particularly useful in the presence of non-standardfeatures such as distributed axial loads, variable bending stiffness, or complexboundary conditions.

7. ADDITIONAL READINGNewmark, N.M. - "Numerical Procedures for Computing Deflections, Moments andBuckling Loads", Transactions ASCE, Vol. 108, 1943.

1.

Timoshenko, S.P. and Gere, J.M. - "Theory of Elastic Stability", McGraw-Hill, NewYork, 1961.

2.

Bleich, F. - "Buckling Strength of Metal Structures", McGraw-Hill, 1952.3.Allen, A.G. and Bulson, P.E. - "Background to Buckling", McGraw-Hill (UK), 1980.4.Lind. N.C. - "Numerical Analysis of Structural Elements", Solid Mechanics Division,University of Waterloo Press, Canada, 1982.

5.


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Chen, W.F. and Lui, E.M. - "Structural Stability-Theory and Implementation",Elsevier Science Publishing Co, New York, 1987.

6.

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