Numerical Methods in Structural Dynamics

Numerical Methods in Structural Dynamics

Can. J. Civ. Eng. Downloaded from www.nrcresearchpress.com by 199.201.121.12 on 06/04/13 For personal use only.

Deportment of Civil Engineering, Cnr.letotz University, Ottntc~n,Cnnridrr K I S 5B6 Received November 26, 1973 Accepted September 16, 1974

The dynamic analysis of a structure subjected to a random forcing function from a source such as earthquake, blast, or wind requires the use of a numerical integration technique. The efficiency and accuracy of the technique employed is of great importance for both research and practical design. The more effective methods of numerical integration belong to the category designated as 'predictor-corrector' methods. A systematic method is presented for the derivation of single-point and multiple-point predictor-corrector formulae. I t is shown that most of the methods of numerical integration presently employed in structural dynamics are single-point predictor-corrector methods. A scheme of iteration is usually employed for the solution of the difference equations obtained by the application of these methods. It is shown that for problems in structural dynamics, it is not necessary to use an iterative scheme; a process of elimination is feasible and also gives considerable economy in computation time. It is further shown that the choice of a n appropriate multi-point method for the numerical integration of the equations of motion of an elastic system can lead to a considerable saving in computation time and cost. One such multi-point method is presented, and its truncation error and stability are examined. Dans 1'Ctude dynamique d'une structure soumise B une force perturbatrice aliatoire engendrCe, par exemple, par un tremblement de terre, une explosion ou du vent, on doit recourir B une technique d'intkgration numCrique. L'efficacitC et la prCcision de la mithode utilisCe sont d'une grand? importance tant pour le travail du chercheur que pour des calculs du praticien. Les mCthodes d'intigration numiriqt~e les plus efficaces appartiennent B la catCgorie "essai et correction". Les auteurs prisentent un procCdC systkmatique de dirivation de formules du type "essai et correction" B point simple ou k points multiples. 11s montrent en outre que la pltlpart des mCthodes d'intkgration numtrique prCsentement utilisCes en dynamique des structures sont du type "essai et correction" B point simple. GCnCralement, on recourt ii un mode itCratif dans la solution des equations aux diffCrences rtsultant de l'application de ces mtthodes. Cependant, le mimoire dCmontre que pour les probliimes liCs B la dynamique des structures, il n'est pas nicessaire de recourir B un procCdC ittratif: on peut en effet se contenter d'une mCthode par Climination qui conduit par ailleurs ii des economies considCrables en temps de calcul. Bien plus, on peut aboutir, comme le montrent les auteurs, B des Cconomies analogues de temps et de cotit de calcul en choisissant une mCthode appropriee B points multiples pour llintCgration numtrique des equations du mouvement d'un systime Clastique. L'article se termine par I'exposC d'une mCthode de ce type dont on Ctudie l'erreur de troncature et la stabilitC de la solution. [Traduit par la Revue]

Introduction In practical engineering situations, there is often a need to investigate the behavior of a structure subjected to dynamic excitation from, say, blast, wind, an earthquake, vehicular traffic, or operating machinery. The complexity of dynamic behavior has usually forced engineers in the past to deal with such loads in a very approximate manner, using, for example, the seismic load 'equivalent lateral forces' and live load impact coefficients found in typical structural design specifications. The digital computer, however, is now making it feasible to Can. J. Civ. Eng., 1, 179(1974)

study real life dynamic problems in terms of rather refined mathematical models; these, although still far from ideal, provide a greatly improved picture of the behavior of the structure under study. In the future, as computing costs reduce and the models improve, it will become increasingly practical to use such computerized models, first for the examination of the adequacy of the design codes and then for the direct design of important or unusual structures. To achieve this objective, it is important to maximize the efficiency of the numerical integration process, which is an essential and very time-consuming part of the computerized

180

CAN. J. CIV. ENG. VOL. 1, 1974

simulation. This last topic is the subject of this paper. In many real engineering dynamic problems, the forcing function or accelerogram encoun" tered is an arbitrary irregular function of time rather than, say, a smooth sine or cosine curve. Such a function can be handled by either (1) the mode superposition method, which requires an integration of the random forcing function by numerical means, or (2) a direct numerical integration of the equations of motion. In many practical situations, especially with heavily loaded structures, the stiffness properties of the structure do not remain constant and the structural response is not linear elastic. In such cases, the mode superposition method no longer can be used. and reliable results can be obtained only by direct numerical integration of the equations of motion. The analysis of the response of a singledegree-of-freedom structure to such a dynamic excitation involves the numerical integration of a single second-order differential equation over an appropriate time interval; the comparable direct integration analysis for a multi-degreeof-freedom structure involves the simultaneous numerical integration of a set of such equations, one equation for each degree of freedom. The amount of numerical work entailed is often very substantial; hence the efficiency of the computational technique employed may be of considerable practical importance. For the type of problem considered here, the mathematical model of a structure initially at rest is subjected to some form of dynamic excitation and the structural response is computed as a function of time. Such a problem is essentially an 'initial value' problem with respect to the time parameter. The methods available for the numerical integration of the differential equations associated with initial value problems can be divided into two broad categories, the methods based on Taylor series, and the 'predictor-corrector' methods. The methods based on Taylor series, which also include the well known Runge Kutta methods, have received extensive treatment in the literature (Hamming 1962; Norris et al. 1959; Wang 1966). Although the Runge Kutta methods can be applied to problems involving random forcing functions, it is difficult to estimate the errors involved; also the time required

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U

for the solution of a problem by this method may be rather large when compared to other possible means of numerical integration, some of which will be discussed below.

The Predictor-Corrector Methods The predictor-corrector methods are iterative methods that use the information at one or more previous time points to assist in evaluating the dependent variable at each successive time point. These methods offer several advantages over the Taylor series type methods. The Euler predictor-corrector formula (Milne 1953) is a simple example of this class of method. The use of this formula can be illustrated by applying it to the following single first order differential equation: where t is the independent time variable, y is the dependent variable, and j, is the first derivative of y with respect to t. If t, and t,,+l are two successive time intervals, h = t,,+l - t,, is the time step, and y, represents the value of displacement y at t,,, then j,, is approximately given by :

c21

Ji"

=

Yn+l - Yn h

provided the time step h is sufficiently small. A first approximation to Y , , + ~is given by where superscript p signifies a predicted value of Y,,+~.A better estimate of j,+l is obtained by

C41

Y",I

= Y,

where the superscript c indicates a corrected value of Y,,+~. The value of Y , + ~ so obtained can be used to evaluate f ( ~ , , + ~ , t , +for ~ ) the next iteration, which will hopefully give a further improved value of Y , + ~ .The use of the corrector formula, Eq. [4], is repeated as many times as is desired. The accuracy of the final answer will depend upon the degree of convergence of the iteration process and the adequacy of the integration interval h. The method just described is an example of a 'single-point' method,

181

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HUMRR AND WRIGHT: NUMERICAL METHODS

which makes use only of information at location n and not, for example, at location n - l . Numerical methods developed for the dynamic analysis of structures have been variously called the 'Difference Equation Methods' and 'Acceleration Methods' (Chan and Newmark 1952; Tung and Newmark 1954). The latter include the well known Newmark's ,B method (Chan and Newmark 1952; Newmark 1959; Tung and Newmark 1954) and the Acceleration Pulse method developed at M.I.T. (Norris et al. 1959). It will be shown herein that all of these methods, in fact, belong to the category of predictor-corrector methods and can be derived using one unified approach. This approach, though standard in the mathematical theory of numerical analysis, does not seem to have been employed in structural engineering. The authors believe that the identification of the various methods within the general category of the predictor-corrector methods should lead to a better appreciation of their relative merits and to added facility in the derivation of suitable multi-point formulae. Multi-point formulae, which make use of information at more than one previous point in evaluating the dependent variable at a successive location, generally have a smaller truncation error than do single-point formulae. One might therefore expect that their use would result in lower errors in practical numerical solutions. This has not always been found to be the case. On the contrary, it has been observed that, with many multi-point methods, the errors have a tendency to grow with time t (Norris el al. 1959; Wang 1966) ; i.e. the solutions are not stable. An attempt will be made herein to show that it is possible to derive multi-point methods that (1 ) show good performance with respect to stability, and ( 2 ) give more accurate solutions than do the singlepoint methods for the same amount of computation time. One such multi-point formula will be presented in this paper and its truncation error and condition of stability will be derived. It will then be demonstrated by an example that, when used for appropriate problems, properly selected multi-point formulae can give much better accuracy in the computation of the elastic response of structures than do the singlepoint formulae. This study will also show that, for the type

of differential equations involved in structural dynamics, iteration of the corrector is not necessary. On the contrary, a direct solution of the simultaneous difference equations by a process of elimination is not only feasible but also offers the definite advantages of ( 1) conserving computing time, and ( 2 ) eliminating the need for any limitation on the magnitude of the step size h to ensure convergence of the iterative process. The initial part of the paper will deal with methods of solution for a single second-order differential equation related to the analysis of a single-degree-of-freedom system. Later, these methods will be generalized to deal with the solution of a set of simultaneous second-order differential equations of motion related to the analysis of multi-degree-of-freedom dynamic systems. The General Predictor-CorrectorFormula A general corrector formula should give the value of Y , , + ~ in terms of the information at several previous points n - k, n - k 1, . . . , n. It may be written in the form:

+

where R is a remainder term, and A,, Be, C , are constants, some of which may equal zero. The total number of these constants, including possible zero terms, is

Higher order derivatives could be included, but the general principles will remain unchanged. However, when the differential equation being solved is of the second order, as is usually the case with structural dynamics problems, derivatives higher than the second may be difficult to obtain. If Eq. [5] has m free parameters, A,, Be, C , then by a suitable choice of the values for these parameters the equation can be made exact in the special case where y is in the form of a polynomial of order rn - 1. When the formula is exact, the remainder term R will equal zero. Now if Eq. [5] is exact for a polynomial of

182

CAN. J. CIV. E:NG. VOL. 1, 1974

order (m - I ) , it will also be exact when y takes on any one of the following values: Therefore, t o evaluate the m free parameters,

to make the formula capable of representing exactly only a second-order polynomial, one slack parameter is obtained. Selecting n.* as the slack parameter to be used in this latter case, and substituting in turn

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y can be successively set equal to each of these

values in turn, and in each case a substitution made into Eq. [5]. This procedure yields m simultaneous equations involving the set of coefficients A,, B,, C,, which are also m in number. A solution of these equations provides the required values of these parameters. Now if y = tnGs substituted into Eq. [5], R cannot be expected to automatically equal zero. The resulting value of R will be designated as E,,,. It will be evident that, when Eq. [5] is used to estimate the value of y , , + ~ ,a larger number of terms in the equation will have the effect of reducing the magnitude of the estimated truncation error R. Equations having many terms may, however, encounter problems of spurious roots and instability. Therefore it may not be desirable to use all m free parameters to reduce the truncation error. An approach having practical usefulness is to employ Eq. [5] to represent exactly a polynomial of ordcr p - 1, p being an integer smaller than nz by an appropriate number of terms. Then (nz - 11) slack parameters become available, which can be assigncd arbitrary values chosen so as to improve the stability or the convergence characteristics of the resulting predictorcorrector formula. As an example of a solution of a differential equation using a corrector formula, the following second-order differential equation may be considered:

in Eq. [8], one obtains:

Equation [8] then reduces to

where a,$ is an arbitrary constant which can be selected to optimize the performance of Eq. [lo]. Setting a.l = p h2, where /3 is now the arbitrary constant which can be assigned an appropriate value, one obtains:

The above formula is basic to Newmark's ,f3 method (Newmark 1959). Table 1 shows the formulae obtained by substituting into Eq. [ I l l the following values of /3: 0, 1/4, 1/6, 1/8. Equation [ l l ] can be made capable of representing a third-order polynomial exactly by assigning the appropriate value to p, or in effect eliminating the slack parameter. This is donc by setting y = t3 in the equation, with the result that p becomes 1/6. Equation [ l l ] then reduces to

A possible second-order single-point correcter formula is: which is Newmark's linear acceleration formula. The predictor to be used with Eq. [12] is where lower case letters are used here for the constant coefficients to distinguish from the convention used for Eq. [5]. In this formula, the constant coefficient of jr,,+l has in effect obtained by setting y,,+l = jj,,. Newmark (1959) has presented the criterion for stability been set equal to zero. Since there are four free parameters, the and convergence of Eq. [I 11 for various values formula can be used to represent exactly a of p. Corresponding to Eq. [8], the general secpolynomial of the third order. If one chooses

183

HUMAR AND WRIGHT: NUMERICAL METHODS

TABLE1. A comparison of some of the methods of numerical analysis used in structural dynamics Truncation Error Term For Displacement

Truncation Error Term For Velocity

Method

Criterion for Stability of Displacement

Remarks

Constant Acceleration Method

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hw = { Y ) - 4 1 ) An r degree of freedom system requires the solution of 3r simultaneous equations at each time step, instead of the three for a single degree of freedom system. If, for example, the system is a linear elastic frame subject to a base motion and it is decided to use corrector formulae [19] and [20] for computing the response, these formulae would then be used in their vector form:

.,

Starting from known values of vectors

{z,,-I), {z,,-I), {z,,), (ilr), and {%), the three sets of equations [33], [31A], and [32A] can be solved simultaneously to obtain {z,,+ I}, {i,,.;. 11, and {i;,.t I). The problem ultimately reduces to the solution of the following matrix equation: C341

C~c*l{~,,= + ~{f,,+l*l )

where [35]

[k*] = [rn]

11 5h +ic] + - [lc] 12 12

and C361 { f , , + ~ * )= -Xt,+~Cml{l)

- Ccl{A) - CkI{B)

where, in turn

C371 { A ) = -

h

..

{z,,-

1

J+

8 11

{%,I + {ill)

and i381 { B ) = - {z,,- I 1 + 2{z1,1

+ h {""-'I + 101z2 Ci',) Cholesky's mcthod can be used very effectively to solve the matrix Eq. [34]. Since [k"] is a symmetric, positive definite matrix, it can be decomposed into the product of a lower triangular matrix and its transpose. C391

CLICLIT{~ll+ 1)

=

if,,+1":)

HUMAR AND WRIGHT: NUMERICAL METHODS

This equation is solved by Cholesky's method in two steps. First the equation C401

CLl{C> =

{A,+,*)

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is solved for C and then Ed11 CLJT{.i?,+l>= { C ) is solved for {?,+ l ) . It will be observed that matrix [k*] does not change with time provided the time interval lz is constant. The more time-consuming decomposition of matrix [Ic*] has, therefore, to be carried out only once. For each time interval, the vector {f:" is calculated and Eqs. [40] and [41] are solved in a segment of the Cholesky subroutine to obtain { i ; , + l ) . The advantage of a solution by elimination over that by an iterative process will now be evident. The fact that the decomposition of matrix [k*] has to be carried out only once makes the elimination process much less time consuming than an iteration method where the iterations have to be repeated at each time step. The elimination procedure is also more accurate than the iterative scheme and its use removes any problem of convergence of an iteration process. The conservation of computation time is by far the most important of the advantages mentioned above. This has become very evident in comparative test runs carried out using both the elimination and iteration methods of solution. The solution of the differential equations of an inelastic system follows the same lines as for an elastic analysis, except that only singlestep methods can be used and all the equations must be used in incremental form. The equations of motion are now E421

+ [c]{Ai) + [ k ] { A z ) = -Ajt[m]{l), {z,,+,) = {z,,) + { A z ) , and so on and

[m]{Az)

where the A's are used to indicate that the particular relations given are valid only over a small interval of the variables { z ) . Ultimately, the problem reduces to the solution of an equation of the form: The Cholesky method can again be used to advantage to solve the above matrix equation.

189

This time, however, the matrix [Ic:~:] is not constant, but varies with the number of plastic hinges that form in the structure. Every time [lc:':]changes, the decomposition into two triangular matrices has to be carried out afresh. But since [ I C ' ~ ]does not change at each step, and in fact the number of changes during an entire response history may not be very large, there is still considerable economy in computation timc if the elimination process is used instead of an iterative scheme.

Selection of Time Step Size The step size to be adopted for a numerical solution using the predictor-corrector method depends upon the following: (1) The convergence characteristics of the corrector. (2) Truncation errors. (3) The stability of the solution. In general, an upper limit is imposed upon the possible time step size by the need to ensure convergence and stability and to reduce the truncation errors. However, as discussed above, an iterative solution of the corrector is not necessary for problems in the dynamic analysis of structures. Therefore, in structural dynamics, convergence need not be a criterion for limiting the step size. The step size can then be selected on the basis of the other two considerations. In all the formulae presented in this paper, the truncation error term is of the form: c h p Y(P) (e), where c is a constant and p is the degree of the formula. Since the pth derivative of y will contain a factor wP, the magnitude of truncation error depends on the value of (oh)P. It is evident that, to keep the truncation error low, the time step size h should be chosen so that w h is less than 1. For systems with only a few degrees of freedom, the criteria for stability are not usually as restrictive as those for truncation error. For systems with a larger number of degrees of freedom, the truncation error term is usually important only for the first few low frequency modes, since the contribution from the higher modes is comparatively small. However, the stability criterion must be based on the highest mode frequency, because instability even in the highest mode may result in a progressively

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190

CAN. J. CIV. ENG. VOL. 1, 1974

dominating contribution from that mode, ultimately rendering the solution unstable. In such situations, the stability criterion may become more restrictive. For example, in vibraticn problems involving shells or flat plate structures modelled by finite elements, the number of degrees of freedom may be very large. Consequently the highest mode period may be so small that it is impracticable to meet the criterion of stability imposed by the highest mode. In such situations, the formulae that are unconditionally stable, such as No. 2 in Table 1, may be the only one that can be successfully used. On the other hand, for multistory buildings that can be modelled by a finite number of degrees of freedom, the highest mode period will usually still be large enough to permit the formulae with conditional stability but lower truncation errors to be used. An Example of Response Calculations To provide an indication of the comparative accuracy of some of the predictor-corrector methods described in this paper, an example of a response calculation problem is now presented. The single degree of freedom oscillator shown in Fig. 1 has the following properties: Natural frequency = w/2.7r = 1.0 cycles per second. Ratio of damping to critical damping = 0.0. The response of this system to a sinusoidal base excitation is calculated using the linear acceleration method and the formulae of Eqs. [18], [19], and [20]. The oscillator, initially at rest, is excited by a base motion given by: x = Q sin Rt where Q = amplitude = 1.00 mm R = exciting frequency = 1.5708 radians per second t = time in seconds

The results of the calculations are compared in Table 4 with an accurate solution given by: W

sin at - - sin z=Q

(;I2

a -1

REST

FIG.1. Single degree-of-freedom oscillator subject to base motion.

less than 3.46. This limits the maximum value of h to 0.55 s. The corresponding limit for the multi-point corrector formula of Eq. [20] is 0.39 s. However, for both methods to reduce the truncation errors, wh should be limited to 1, or h should be limited to 0.16 s. In fact, a considerably smaller value of 1%should be used to ensure reasonable accuracy. Even with h = 0.04 s the errors in the results obtained by the linear acceleration method are considerable. A value of h = 0.02 s gives much better accuracy. The multi-point methods, on the other hand, give comparabIe or better accuracy even with a step size of h = 0.04 s.

wt

For Newmark's linear acceleration method, the condition of stability requires that oh be

Summary and Conclusions In many practical engineering situations, the analysis of structures subjected to dynamic load requires the use of a numerical method of computation. The more effective methods of

HUMAR AND WRIGHT: NUMERICAL METHODS

TABLE4. Response of single-degree-of-freedom oscillator

Displacement, mm. 1/6 Formulae (18), (19)

Relative Newnark's Method

Time, S

Exact

h = 0.02 s

B

=

h = 0.04 s

h = 0.04 s

Forinulae

(19), (20)

?, = 0.04 s -

Value

Value

Error

Value

Error

Value

Error

.06622

+ 45 + 74

.06616

-.I9036

+ 51 + 15 54

1.2

-.I9021

-.l9010

- 60 - 11

-07056 -.la934

- 87

-.l9095

1.4 1.6

-.lo281

-.lo362

+ 81

-.lo763

+ 482

-.lo39

.19593

+ 80

.27422

.19690 .27485

.00129

-129

.l9114 .27690 .00786

+ 479

1.8

.I9513 .27465

.00643

-642

.03919 -.25959

-3918

-203

-.20133

+539

-.22741

-.19578

.09732

+547

1.0

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Error

2.0

10.0

.06667

- .Ooooo .00001

10.2

-.27422

10.4

-.19594 .lo279

.06727

-.27219

-

43

-

-

-

389

268 786

-1463

-.00091 -.00411 -.28078

- 2 - 97

-.lo227

-

-

- 63

.19648 .27388

+ 34

+ 81

-.00094

+ 94 +473 +I37

+412

-.00472

+656 - 16 -696 -425 +450

-.27559 -.l9188

55

-406

10.8

.l9022

.l9240

-218

.ZOO71

11.0

-.Of5665

-.05955

-710

-.02355

+3147 +3509 -1049 -4310

14.0 14.2

-899

.05468

-5466

-.00661

+663

-288

-2207

+920

-.27614

+I93

14.4

-.I9595

-.25214 -.23825

-.00579 -.28341

+581

-.27421

+4230

-.I9572

-

-.19033

-562

.ll243 .l9606

.I0832

-553

10.6

14.6 14.8 15.0

.00002

,10279 ,19022

-.06665

.00901

-.27133 -.20337

.09519

+742 +760

A5770

.053@

.I9312

-290

.20268

+by31 -1246

-.05697

-968

-.00810

-5855

numerical analysis belong to the general category designated as 'Predictor-corrector Methods'. With some problems in structural dynamics, it is not necessary to use the predictor. In this paper, several existing predietor-corrector methods were reviewed and compared with reference to the errors involved in the solution. Ncxt a general development of the higher order methods was presented; one such method was examined in detail for its truncation error and stability when applied to the solution of problems in structural dynamics. An example of the analysis of structural response for dynamic loads was given, in which the effectiveness of properly selected higher order formulae was demonstrated. The following conclusions can be drawn from this study: 1. For the dynamic analysis of structures not strained into the inelastic range, i.e. where the stiffness matrix remains constant, the choice of an appropriate multi-point numerical method will give a definite improvement in the accuracy of the solution beyond what can bc expected from single-point methods. For comparable accuracy, the time step size that can be used

.lo975 .I9447 -.07115

-.07285

23

-964 -584 +620

.10682'

-403

.I6857

1.165

-.07182

+517

.18794 -.07370

1278

+705

with a properly selected multi-point formula is considerably larger than that which is possible with a single-point formula. The main consideration that should weigh in the selection of a multi-point method is the requirement of stability. Multi-point methods that give rise to extraneous roots or partial instability should be used with caution. Methods that do not suffer from these short-comings can be used with advantage. The formula presented in Eq. [20] of this paper is very suitable for use in analysis of structures subject to dynamic forces such as wind, blast, or earthquake, provided conditions of stability can be met, i.e, provided lz can be selected so that W?Z"S less than 6. 2. For most problems encountered in structural dynamics, the iteration of the corrector is not necessary. Instead a process of elimination described herein under the heading 'Solution of the Differential Equations', can be used for the solution, resulting in considerable economy in computation time. Although this is true of both elastic and inelastic systems, the economy will be most evident in the case of elastic systems. 3. For the reason stated in conclusion 2,

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192

CAN. J. CIV. ENG. VOL. 1, 1974

convergence of the corrector usually need not be a criterion for limitingthe time s t Ie ~size. U To reduce the truncation error, the step size should be chosen so that h w is less than 1 ; in fact it should be considerably less than this limit to ensure adequate accuracy in the solution. The need to maintain stability of the solution will impose a different liniitation on !he step size depending on the formula selected. Some formulae are, in fact, unconditionally stable. 4. For systems with a few degrees of freedom, the limitation imposed on the step size by the need to minimize the truncation error is more restrictive than that imposed by the requirement to maintain stability of the solution. However, for systems with a large number of degrees of freedom, the stability criterion may become more restrictive. When the number of degrees of freedom are very large, for example in the case of flat plate or shell structures modelled by finite elements, the stability criterion may be so restrictive that it becomes impracticable to select a step size to meet the stability requirement. In such situations, corrector formulae with unconditional stability may be the only one that can be successfully used. In cases where it is possible to select a reasonable step size and still meet the criterion of stability multi-point formulae like the one in Eq. [20] can be used with advantage. CHAN,S. P. and NEWMARK, N. M. 1952. Comparison of numerical methods for analysing the dynamic response of structures. Civ. Eng. Stud. Struct. Res. Ser. No. 36, Univ. Illinois, Urbana, Ill. HAMMING, R. W. 1962. Numerical methods for scientists and engineers. McGraw-Hill Book Co., Inc., New York. MILNE,W. E. 1953. Numerical solution of differential equations. John Wiley and Sons, Inc., New York. NEWMARK, N. M. 1959. A method of computation for structural dynamics. J. Eng. Mech. Div. Am. Soc. Civ. Eng., 85, pp. 67-94. R. J., HOLLEY, M. J., BIGGS,J. M., NORRIS,C., HANSEN, NAMYET,S., and MINAMI,J. K. 1959. Structural design for dynamic loads. McGraw-Hill Book CO., Inc., New York. N. M. 1954. A review of TUNG,T. P. and NEWMARK, numerical integration methods for dynamic response of structures. Civ. Eng. Stud., Struct. Res. Ser. No. 69, Univ. Illinois, Urbana, Ill. WANG,P. C. 1966. Numerical and matrix methods in structural mechanics. John Wiley and Sons, Inc., New York.

Notations Constants of integration Coefficients in corrector formula Coefficients in corrector formulae for displacement Coefficients in corrector formulae for velocity Damping coefficient Damping matrix Error term in corrector formula Force which varies as a function of time Function of time and variable y Interval of time used in numerical integration Stiffness Stiffness matrix Triangular matrix used with Cholesky's method of solving simultaneous equations Mass Mass matrix Subscript indicating the location or interval of time at which the subscripted variable is evaiuated Q = Amplitude of exciting sinusoidal motion R = Remainder term in integration formula t = Time x = Displacement of foundation of a structure y = Absolute displacement z = Relative displacen~ent i ,j , i = Velocities corresponding to displacements x, y, z 2,y, z = Accelerations corresponding to displacements x, y, z yc = Corrected value of displacement yP = Predicted value of displacement p = Parameter used in Newmark's method

B, C, D, E, A,, Be, Ce

= =

HUMAR AND WRIGHT: NUMERICAL METHODS

Incremental values of x, y and z 19 = A particular value of time t located somewhere in the interval of length lz

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Ax, Ay, Az

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between t,, and t,,, Frequency of exciting base motion Natural frequency