BasicPrinciplesAndConceptsOfModelAnalysis Young

EDUCATIONAL LECTURE

Basic Principles and Concepts of Model Analysis

Lecture discusses the techniques and the problems associated wi th the design and use of models

by Donald F. Young

ABSTRACT--The techniques and the problems associated with the design and use of models are covered. Particular emphasis is placed on the development of modeling laws by means of dimensional analysis. Special problems associated with the use of true and distorted models are illustrated by means of selected examples.

Introduction Simulation is widely used in engineering analysis, design and research. In fact, this technique is used whenever a problem is studied by some method other than direct observations on the prototype, which is defined as the actual system of interest. Figure 1 il lustrates schematically the steps commonly taken in the simulation of the prototype. It usually follows that assumptions are made so that the system may be more precisely defined, and this new system may be referred to as an idealized prototype. The assumptions made at this stage are usually, or at least hopefully, not restrictive and are imposed only to the extent that the problem can be well defined.

Following the definition of the problem, a decision must be made with respect to the type of simulation technique to be used. If the problem is to be solved analytically, it is apparent that a mathematical model must be developed and subsequently solved either by mathematical analysis or by an analog. If an analog is used, the system is analyzed experimental ly but with another system, or model, which is not similar in appearance to the original prototype. Thus the term "dissimilar model" is appropriate for the analog. There are a great variety of analogies used, 8,4 although probably the most useful types are ones involving electrical circuits.

To simulate the prototype with a mathematical model, the characteristic equations describing the behavior of the system must be known. This frequently requires additional assumptions with regard to the behavior of the system. For example, in con- sidering the deformation of structures, the common assumption is that the material behaves elastically. Or in dealing with flowing fluids, it may be assumed that the fluid is ideal or nonviscous. Thus, to estab-

Donald F. Young is Professor, Department of Enginee~ng Mechanics and Engineering Research Institute, Iowa State University, Ames, Towa 50010. Lecture was presented at a session sponsored by the Educational Committee at the 19~9 SESA Fall Meeting held in Houston, Tex., on October 14-17. It contains excerpts from two previous papers1, * by the author.

lish a mathematical model, a thorough understanding of the characteristics of the system, and the fundamental equations governing the behavior of the system, must be achieved. There are several advantages in solving problems in this manner. A complete and detailed solution is normally obtained and, if the problem is solved on a completely analytical basis, the expense of building equipment and per- forming tests is eliminated. Probably, the major disadvantage is the required number of assumptions to establish the mathematical model. In many instances, it is difficult to obtain a solution due to the complexity of the governing equations.

The alternative basic simulation technique is one of actually simulating the prototype, or the idealized prototype, with a similar model. A similar model is defined as a system which is similar in appearance to the prototype but not identical to it. In practice, such systems are simply referred to as models and Pre usually smaller in size than the prototype. In some instances, it may be advantageous to have a model that is larger than the prototype. It is frequently possible, through the use of similar models, to study very complex problems with relative ease, and this is one of the major advantages of this technique. It is also true, in many cases, that fewer assumptions are required when similar models are used than for simulation with a mathematical model. One of the major disadvantages of similar models is that the solution is obtained experimental ly and the expense usually associated with experimental work must be incurred. Also results obtained experimen- tally are frequently restrictive and limited in ap- plicability.

It is clear that, regardless of the particular technique chosen to simulate the prototype, a relationship between the model and prototype must be established. Through the development of mathematical models, this relationship evolves natural ly and is readily apparent. However, when similar models are

Fig. 1--Flow chart for problem analysis

MATHEMATICAL

ASSUMPTIONS J IDEALIZED m PROTOTYPE

EDUCATIONAL LECTURE

used, the relationship between model and prototype must also be known. The establishment of this relationship and the fulfi l lment of the various similarity requirements, or model-design conditions, between the two systems is sometimes difficult to achieve. As problems become more complex, the value of similar models increases, and the remainder of this paper is devoted to a discussion of this type of model.

Theory of Similitude and Modeling Two methods commonly used for establishing

similarity relationships between a model and prototype are based on (a) an analysis of the characteristic equations of the system and (b) dimensional analysis. If the former method is used, the system is first described in terms of a mathematical model and then the scaling laws, model-design conditions, or similarity requirements, (these terms are used inter- changeably) are developed from this model. I t is noted that, if this procedure is followed, the same comments made previously, regarding the additional assumptions required to establish a mathematical model, apply. It may be argued that, if the mathematical model is established, why is it necessary to obtain a solution experimental ly using a similar model? The problem is that it may be extremely difficult, if not impossible, to get a closed-form solution, or even a good numerical solution, to certain types of problems. A classical example of this is in fluid mechanics, where the characteristic equations are well known but cannot in general be solved be- cause of their nonlinearity.

Both methods of developing modeling laws are important and the general procedures followed J n their use will be considered in some detail.

Characteristic-equation Method

The characteristic equations describing physical problems are frequently differential equations and these equations, combined with the initial and boundary conditions, describe the problem. Essen- tially, the method of determining similarity requirements from the characteristic equations consists of rewrit ing the characteristic equations in dimensionless form, and determining from the transformed equations the conditions under which the behavior of two systems will be similar. This method is i l lustrated in the following simple example.

Consider the spring-mass-dashpot system of Fig. 2. The problem is to determine the displacement, y, as a function of time, t, by means of a model study. It is well known that the displacement of the mass is described by the differential equation

d~T dy m - "l- -I- ky ---- 0 (1) dt 2 c

along with the initial conditions

dy Y = Yo and- -=vo

dt at t - - - -0

EQUILIBRIUM POSITION

Fig. 2--Simple spring-mass-dashpot system

If we now introduce two dimensionless parameters

It $ y* =- -andt* : - -

Yo x where

eq (1) can be written as

d2y * c dy *

dt .2 N/mk dr*

The initial conditions become

y*=l and

dy* vo ~ / m

dt* - - Yo V - -k att* = 0

q- y* = 0 (2)

From a consideration of eq (2), it is seen that for any two systems governed by an equation of this form, the solution for y* will be the same, i.e.,

if Y* ~ Ym*

e era

ooV._ -~o k YOre km

t ~-~tm -~m

where the subscript m refers to the model. The last condition specifies the time scale for the problem.

It is clear that, if the characteristic equation (s) are known for the system, the procedure described in this example can be followed to establish the necessary relationships between the prototype and model. How- ever, in many problems, the characteristic equations are more numerous and complicated than in the example given, or even unknown, and this method cannot be readily used. An alternate approach, which does not require such a detailed knowledge of the system, is based on dimensional analysis.

Dimensional Analysis

When dealing with physical phenomena, we describe the phenomena in terms of various quantities;

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such as velocity, accelerat ion, density, area, etc., and these so-cal led secondary quantir in turn, are descr ibed and measured in te rms of another set of quantit ies which are considered to be pr imary quantit ies. In mechanics, the pr imary quantit ies, or basic d imensions, are normal ly taken to be length, L, t ime, T, and mass, M. It can be shown 5 that any secondary quan- tity, s~, is expressible in terms of the pr imary quant i - t ies in the form

si = L a T b M e X aye

where X, Y, . . . , are other basic d imensions such as temperature, and elecCrical charge which may be requ i red to descr ibe the secondary quant i ty. Com- mon examples of secondary quant i t ies and their basic dimensions include:

area , A = is vo lume , V : L z veloc i ty , v : LT -1 dens~y , p :ML -3 stress , ~ :MT-ZL -1

To obtain the basic d imensions of stress, use is made of the fact that stress is a force, F, d iv ided by an area, but force and mass are re lated through Newton 's second law of mot ion; i.e., F = MLT -2. I t is thus apparent that an equ iva lent set of basic d imensions to be used in mechanics problems is L, T ~.nd F.

In a g iven problem, there are usual ly several var i - ables, ul, u2, . . . , ub requ i red to descr ibe the phenomenon of interest. A number of d imensionless products of these var iables can be formed by com- bining the var iables in the form

Ul xl U-2~ . . . . . Uk xk

where the exponents x l , xs . . . , Xk are selected so that the resul t ing product is d imensionless. Thus, i f we let any one of Che var iables, say ui, have the basic d imension

zti = L a~ T b~ M c~ X d~ ye~

we can express the product as

(Lm Tb~ Me1 Xa l yel )x~ (La~ Tb, Me2 Xa~ yea)x~ . .

(La~ Tb~ Mc~ Xd~ ye~)xk

In order that the product be dimensionless, the exponents of the var ious basic d imensions must combine to give a zero va lue for each basic dimension. Thus

a lx l q- a2x2 -}- . . . . -{- akxk : 0

blXl "4- b2x2 -I- . . . . -1- bkXk = 0

clxl + csx2 + . . . . + C~Xk : 0 (3)

d lx l + ~xs + . . . . + dkxk : 0

e ix i -~ e2x2 q- . . . . Jr ekxk : 0

We note that there wi l l be as many equar as basic dimensions, say m in number , and k unknown x's, where k is equal to the number of or ig inal var iables in the problem. F rom the theory of equat ions, it is known that there are k -- r l inear ly independent so- lut ions to eqs (3) where r is the rank of the matr ix

of coefficients

al a2 . . . . ak bi b2 . . . . bk e l C2 . . . . ek

dl d2 . . . . dk el e2 . . . . ek

This mat r ix is commonly cal led the d imensional matr ix . S ince the rank of a mat r ix is the h ighest - order nonzero determinant conta ined in the matr ix , it is apparent that the rank cannot exceed the number of equat ions but may be smaller. Thus, the number of independent d imensionless products that can be formed is equal to the number of or ig inal var i - ables, k, minus the rank of the coefficient matr ix . Such a set of d imensionless products is cal led a complete set. Once a complete set of d imensionless products is found, all other possible d imensionless com- binat ions can be formed as products of powers of the products contained in the complete set.

An essent ial postulate of d imensional analysis is that the form of any funct ional re lat ionship between a g iven set of var iables does not depend on the ss ' s tem of units used, i.e., the funct ional re lat ionship is di- mens ional ly homogeneous. If this condit ion of homo- genei ty is uti l ized, ir can be proved 6 that a funct ional re lat ionship between a g iven set of var iab les can be reduced to a re lat ionship among a complete set of d imensionless products of these variables. Thus the we l l -known Buck ingham P i Theorem can b~ stated as fol lows:

I f an equat ion invo lv ing k var iables is d imensional ly homogeneous, it can be reduced to a re lat ionship among k - r independent d imensionless products, where r is the rank of the di- mens iona l matr ix .

To i l lustrate the appl icat ion of the Buck ingham Pi Theorem, we wi l l app ly it to the v ibrat ion prob lem prev ious ly considered. The first step in the analysis is to list the var iab les and the i r d imensions as fol - lows:

y, d isplacement, L m, mass, M c, damping coefficient, MT- 1 k, spr ing constant, MT -2 vo, in it ial velocity, LT -1 Yo, in it ial d isplacement, L t, t ime, T

We now form the product

yzl mR cx8 kx, vox~ yox6 t~

and wi th the subst i tut ion of the basic d imensions for each var iab le we obta in

(L)xl (M)~ (MT-1)zs (MT-~)x , (LT-1)z~ (L)x6 (T )~

wi th the corresponding set of equat ions

L : x1+O+O+Oq-x5+x6+O=O

T: 0 +0- -x3 - - 2x4- -xs+0+xT=0 (4)

M: O- t -x2+xaTx4+O+O+O=O

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The dimensional matr ix is

L T M

y m c k Vo Yo t

I 0 0 0 i i 0 00- - I - -2 - -101 01 1 1 000

Consider now the determinant on the left side of the matrix:

1 0 0 0 0 --i : i 0 1 1

Since this is nonzero, it follows that the rank of the dimensional matrix is three and there are four dimensionless products required to describe this problem.

To find a suitable set of dimensionless products, commonly called "pi terms" we assign values to four of the x's in eq (4) and solve for the remaining three. For example, let x4 = I, xs ---- 0, xs ---- 0, and x7 : 0. The only restriction here is that the determinant of the remaining coefficients must be nonzero so that we can solve for the remaining x's. In this case we have previously shown that this determinant is nonzero. Wi th 3:4 : i, x5 : 0, x6 : 0, andx7 ---- 0 it follows that eqs (4) are satisfied if Xl ---- 0, x2 = 1, and x3 ---- --2. Thus one dimensionless product is

~1 = yo m 1 c -2 k 1 Vo o yo o t ~

km

C 2

Now let x4 ~-- 0, X 5 : 1, X6 = 0, and x 7 = 0 and we find that

Vo~r~ ~2=-

yc

This process can be continued with x4 ---- 0, x5 = 0, x6 ---- 1, andx7 = 0 andx4 ---- 0, x5 = 0, x6 = 0, and x7 ---- 1 to give

Yo

Y and

ct

m

It is apparent that the specific form of the pi terms depends on which of the x's are assigned values and the values themselves. However, it should be emphasized that, once an independent set is determined, all other possible independent sets can be formed as products of powers of the original set. Using this procedure, we can form various combinations to ar- r ive at what we consider to be the most useful set. In the present example, the obvious disadvantage is the fact that, y, the displacement, appears in three of the pi terms. It is usually convenient to have the variable of pr imary interest appearing in only one pi term.

To compare this set of pi terms obtained from dimensional analysis with the dimensionless products developed from a consideration of the differential equation we form the new set

y :~'1 = (~3) - -1 = __

Yo

C s : (~1) - ' /2 _ _ _

N /km

~'4 : (~1) - -1 /2 (~2) (~3) - -1 - - -V~ ~V[/__~ Yo

which contains precisely the same dimensionless products as obtained from the differential equation.

In most problems, the dimensionless products can be obtained by inspection since the required number can easily be determined and the variables can simply be combined into dimensionless groups. An independent set can be assured if each dimensionless product contains one variable not contained in any other product.

In essence, dimensional analysis allows us to study a problem described by the functional relationship

Ul --~ r (U2, U3, U4 . . . . . Uk) (5 )

in terms of a set of dimensionless terms

~1 = f (~2, n3 . . . . . nk--r) (6 )

One obvious advantage is the reduction in the number of variables (from k to k - r) to be controlled in an experiment. In addition, it is usually much easier to control the dimensionless products in an experiment than the original variables.

Modeling laws can be readily developed from eq (6) in the fol lowing manner. We assume that we have two systems, the prototype and the model, each described by the equations

H1 : f (~2, ~3 . . . . . ~k-r) (prototype)

nlm = fm(~2m, ~3m . . . . . ~(k--r)m) (model)

We further assume that the phenomenon with which we are dealing is the same for both the prototype and the model so that the form of the function, f, for the prototype is the same as the function, fro, for the model. It immediately follows that if we let

~2 = ~2m

~3 = g3m

(7) Xfk--r = ~(k--r)m

then

Equations (7) provide us with the required relationships between prototype and model so that we pre- dict ~1 from a measured glm taken on the model. Equations (7) represent the model-design conditions and eq (8) the prediction equation. 7 Application of this procedure to the previously determined pi terms, s n'2, :~'~, ~'4 developed fo r the spring-mass prob-

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IPs --r__,

_l

Fig. 3--Sketch of cantilever beam

lem leads to the same similarity requirements as those obtained from a consideration of the differential equation.

For the remainder of this paper, dimensional analysis will be used to :develop modeling laws. To more clearly i l lustrate the method, the following simple example is given.

Let it be required to establish the similarity relationships for predicting the end deflection of a cantilever beam of rectangular cross section (Fig. 3) due to a concentrated load P. It is assumed that the deformation is small, the material behaves elastically, shearing deflections are negligible, and the beam is loaded in a plane of symmetry so there is no twist. With these conditions, the following variables are applicable:

A, end deflection, L a, length of beam, L b, width of beam, L d, depth of beam, L P, load, F E, modulus of elasticity, FL -2

Application of the Pi Theorem reveals that, since there are six variables expressible in terms of two basic dimensions, four dimensionless parameters are required to describe this problem. One possible set is

a -- Ea 2 a' a ' (9)

It now follows that, if

b bm

a am

d d~ - - : - - (10)

a am

P Pm

Ea 2 Em am 2

for two systems then

A Am - - (11)

a am

Equations (10) are the similarity requirements for this problem, and eq (11) is the prediction equation between the model and prototype. Since there are two basic dimensions in this problem, two "scales" can be arbitrari ly selected; e.g., let

a ~. n a

am

where na is the "length scale", and

P ~- up

Pm

where np is the force scale. The scales for all other variables are then fixed; i.e.,

A b d _ _ - - - - - - na

Am bm dm and

E Ttp'l"ta--2

Em

The foregoing example reveals that there are three basic steps used in establishing modeling laws from a dimensional analysis. These are: (a) the selection of the variables, (b) the application of the Pi Theo- rem, and (c) the development of the similarity requirements by equating pi terms. Although in principle this procedure is straightforward, and relatively simple, two major difficulties are normal ly encountered. The first is in the selection of the pert inent variables. Ir is clear that a good understanding of the problem and the phenomena must be achieved in order to ascertain the pert inent variables. The selection of variables is usual ly based on the experience of the investigator and a knowledge of the fundamental equations which govern the phenomena. This does not imply that a detailed mathematical model of the system must first be established, but simply that certain fundamental laws, such as Newton's laws of motion, are known to be applicable to the system. Common errors encountered at this stage are the inclusion of nonindependent variables and the omission of pert inent variables or parameters such as the acceleration of gravity, on the basis that they are constant. The inclusion of a group of variables that are not independent (such as the beam cross-sectional area, width and depth in the canti lever-beam example) is clearly unnecessary. In addition, it should be emphasized that all pert inent independent variables or parameters must be included and the fact that they may or may not be constant is if no consequence at this stage in the analysis. It should be noted that omissions, or the listing of unnecessary variables, will normally not be detected without the aid of experi- ments.

The second difficulty that frequently arises is in the control of the pi terms. Each pi term yields a requirement between the model and prototype system. In certain instances it is difficult, if not impossible, to satisfy one or more of these requirements. The classical example of this :difficulty is in fluid-flow problems in which both the Reynolds number and the Froude number are important. Similarity requirements arising from these two dimensionless parameters are

V~, Vm~.m - - - - (Reynolds number)

Y Vm

and

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v 2 vm 2 . . . . . . (Froude number) gk gm~.m

where v is a velocity, ~. is a length, v is the kinematic viscosity of the fluid, and g the acceleration of gravity. If the same fluid is used in both model and prototype, and both systems operate in the same gravitational field, it follows that

v ~m - - - - nx -1 (12)

Vm ~.

from the Reynolds number condition, and

- - = = N/nx (13) Vm ~.m

for the Froude number condition. It is apparent that eqs (12) and (13) give different

values for the velocity scale. A conflict therefore exists. It is extremely difficult, in most instances, to find combinations of fluids that allow both of these conditions to be satisfied simultaneously. A model for which at least one of the similarity requirements is not satisfied is said to be distorted. Numerous problems could be cited to show that distorted models are not rare exceptions but may frequently occur. A further discussion of this important topic is given later in this paper.

Typical Applications

Static Elastic Problems

Structural models are widely used for determining stresses, strains and displacements in elastic structures. For these problems, we assume that the material obeys Hooke's law and can be described by Young's modulus, E, and Poisson's ratio, tz. In addition, any stress component, ~, at some point, xl, wil l be a function of the geometry of the system as characterized by some length, l, and other required lengths, ~i. The subscript, i, will be used to designate a set of variables. Thus ~i is equivalent to a set of lengths kt, ~-2, ~-3, 9 9 9

The loading may be specified with the loads, P and P+, and any prescribed boundary displacements by +1+. The stress can therefore be expressed in the functional form

r ---- r l, ~+, P, P+, ~li, E, tz) (14)

We now apply dimensional analysis to this set of variables to obtain

~12 ( x~ Xi Tli P Pi ) - - : f - (15) P l ' l ' l ' El m' P ' ~

Similarity requirements are obtained by making the pi terms on the right side of eq (15) equal between model and prototype. Equal ity of the first three pi terms, xdl, ~Jl, and ~i/l, means that we must maintain geometric similarity between model and prototype; not only with regard to shape but also with respect to prescribed displacements and coordinates.

The loading scale is established from the relation-

ship P Pm

EL 2 Emlm 2 or

P E l u

Pm Em /m ~

Note that the model and prototype materials need not be the same but the elastic moduli scale, E/Em, and the length scale, 1/lm, fix the loading scale. All additional loads, P~, must be in the same ratio, i.e.,

P.. P

Pim Pm

The last pi term in eq (15) imposes the rather stringent similarity requirement that Poisson's ratio must be equal for model and prototype materials. Of course, if the prototype and model are constructed of the same material, this condition is satisfied. For plane-strain or plane-stress problems involving simply connected bodies, for which the body forces are zero, constant, or vary l inearly with position, the stress distribution is known to be independent of Poisson's ratio. 8 Similar problems involving multiply connected bodies containing holes can also be modeled without regard to Poisson's ratio if the resultan.t force acting on the boundary of the hole is zero. However, if these conditions are not met and if different materials are used, the Poisson's ratio condition wil l not, in general, be satisfied and for this case a judgment must be made with respect to the significance of Poisson's ratio for the specific problem under consideration.

If all of the aforementioned design conditions are satisfied, then it follows that

or

al 2 amlm 2

P P,+

P lm 2 E

~rm Pm 12 Em

Since any displacement component, u, or strain component, ~, wil l be a function of the same variables given in eq (15) it follows that the same model- design conditions are required for displacements and strains as for stresses. The corresponding prediction equations become

u Urn

1 Im or

and

u t

Um lm

~ ~m

i.e., the displacements scale as the length scale whereas the strains are equal in model and prototype. It should be noted that these scaling laws for elastic structures are valid for both small and large deformations, as long as the material in both mode] and prototype obeys Hooke's law. Other types of loads; e.g., line, surface and volume loads, can be

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readily incorporated into the analysis. 9 In many problems it is possible to relax, or modify,

the similarity requirements by making use of more detaited information about the phenomena based on theory or experience. Typical examples of this technique follow.

Reduced-simi lar i ty Requirements

If we again consider the problem of predicting stresses in elastic structures under static loading conditions and impose the additional restriction that the deformations are small, the modeling problem can be considerably simplified. It is well known from small- deformation theory of elastic materials that stress, strain and displacement are linear functions of the app!ied loads. Thus, from eq (15), we noCe that al2/p must be independent of the pi term, P/E l ~, since must depend l inearly on P, and the appropriate equation becomes

al2 , ( xi ~i ~li Pi ) - - : , , , ~ (16) P l ' l l P

The design condition relating the loading scale to the modulus-of-elasticity scale has been el iminated and, thus, one is free to arbitrari ly select the model load P as long as the imposed condition of small deformations is maintained. If we apply the same argument to displacements and strains, we obtain

u P (x l ~,i Tll P, ) _ _ l -- E1 -T I1 ' l ' l ' P ' ~ (17)

and

: f2 . . . . ~ (18) El ~ l l P

which gives much more flexibility in the model design.

The principle of superposition applies for small- deformation problems and, if desired, the model may be tested with a series of loads rather than applying all loads simultaneously.

Although it is generally true that the model must be geometrically similar to the prototype, there are exceptions to this rule. For instance, in the cantilever-beam example, we know from theoretical con- siderations that the beam deflection is actually a function of the moment of inertia, I, of the beam cross-sectional area rather than the beam width and depth individually. Thus eq (9) could be written as

a -- a4 , Ea 2 (19)

and eqs (10) replaced with

a 4 am 4

Thus, the cross section of the model is not required to be of the same shape as that of the prototype. Since it is frequently difficult to fabricate a geo- metrical ly similar small model, for example, structural elements such as I-beams are not readily available in an assortment of small sizes, the possibility

of util izing another geometry in the model is an important consideration.

It should be emphasized that models designed on the basis of reduced similarity requirements of the type described in this section are not considered to be distorted models since all similarity requirements deemed necessary are satisfied.

Dynamic Elastic Prob lem

As a further example of the development of similarity relationships by means of dimensional analysis, we will consider the problem of predicting the strain in an elastic structure under the influence of a dynamic-pressure loading over some part of the surface of the structure. As before, we let e represent one component of strain at the position xi, and let the geometry of the system be described by a set of characteristic lengths, l and ~i, where the ~'s also include the required spatial coordinates of the loading. We continue to impose the condition that the material obeys Hooke's law so that only two elastic constants E and ~ are required. However, since the structure is under dynamic loading, an additional material property, the mass density, p, must be included in the analysis.

We assume that the pressure at any point can be described in dimensionless form as

- - = , I , - - (20)

Po p - l

where l N/p/E has the dimension of time and can be combined with t to form a dimensionless t ime var i - able. The functional relationship for the strain can, therefore, be wr i t ten as

e = 9 (xi, l, ~, E, ~, p, Po, t) (21)

where it is tacitly implied that the form of the pressure function, ,I,, is the same in both the model and prototype system. In dimensionless form, eq (21) can be written as

"----f l ' "T' "' "E" (22)

The three pi terms xi/l, Xi/l, and ~, yield the similarity requirements previously considered, i.e., we must maintain geometric similarity, and Poisson's ratio for model and prototype materials must be the same. The pressure scale is established from the condition

Po Pom

E E~ or

Po E

Pore Em

We note that if the same material is used in both model and prototype, the pressures at corresponding locations and times must be the same.

The time scale for the problem is established from the condition

t ~ Em tm -Z-f= pm

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d

P I I

- --I-

4-1

____t

(a)

I L | /jE, /!

(b) Fig. 4~Sketch of tensile specimen and stress-strain diagrams

%

0 I /

~] i i III

/ I ~o

UNIT STRAIN ( ( )

(c)

or t . /Em p l

V tm E pm Im This relation indicates that, in general, corresponding times in the model aI~d prototype will differ. If the same materials are used in model and prototype, the time scale will equal the length scale. Since the length scale is generally greater than unity, it follows that when modeling with similar materials, corresponding times in the model will be shorter than for the prototype. Thus, in a problem of this type, the loading pressure-t ime relationship, when expressed in terms of p/p~, and the dimensionless t ime variable must be identical, whereas, in terms of real time, the model and prototype pressure-time relationship must be different. From a practical point of view, this is a common problem encountered in dynamic testing, i.e., it is difficult to generate a properly scaled model loading. If all similarity requirements are met, it follows tha~t

e = e m

where the strains in the model and prototype are measured at corresponding times based on the time scale.

Modeling of Inelastic Behavior

In all the examples of structural models considered thus far, the material was assumed to obey Hooke's law and, thus, the constitutive equations are completely characterized by the modulus of elasticity, E, and Poisson's ratio, g. Either of these two properties could be replaced by the modulus of rigidity, G. We will now consider the problem in which the loading is such that the material is sCrained beyond the proportional limit. To i l lustrate several important ideas related to the development of similarity requirements

for this type of problem, the following simple example will be considered.

Let it be required to establish the similarity relationships for predicting the elongation, 4, in a length, ~, of a simple tensile specimen of diameter, d, that is loaded with a load, P [Fig. 4(a)] . The hypothetical stress-strain characteristics of the material are given in Fig. 4(b). For this material, a proportional limit ~o exists and the stress-strain relationship is also l inear beyond the proportional l imit to the fracture stress, ~. If unloading occurs prior to fracture, the slope of the unloading curve is given by El. Based on experience with elastic materials, a logical list of variables for this problem is as follows:

4, elongation, L ~, gage length, L P, applied load, F d, specimen diameter, L El, modulus of elasticity, FL -2 E2, modulus of elasticity, FL -2

However, further consideration reveals that the stress-strain curve is not completely defined by the parameters, E1 and E2, since the stress, or strain, at which the slope of the curve changes is not defined in the list of variables. Two additional parameters, r and Cs, are required. In essence, the constitutive equation for the material must be defined. This is done by: (a) specifying the form of the equation, and (b) by defining the parameters that appear in the equation. For elastic materials under simple ten- sion or compression, the form of the constitutive equation is

i.e., stress and strain are l inearly related, and the required parameter is E. It should be clearly under- stood that, whenever a material property such as a modulus of elasticity, viscosity, etc., is listed as a

332 I July 1971

EDUCATIONAL LECTURE

pertinent variable, the form of the relationship in which the property appears is tacitly implied.

With the inclusion of the variables ~o and af, dimensional analysis gives

A ( d P ~o E2 ~, ) (23) -~- = ] ~.' Eld 2' E1 El ' E1

The similarity requirements from the last three pi terms in eq (23) indicate that the dimensionless stress-strain diagram for the model and prototype specimens [Fig. 4(c)] must be identical.

Another factor that must be considered in this example is the significance of the similarity requirement,

P Pm - - - - - - (24) Eld 2 Elmdm 2

It can be seen from Fig. 4(b) that different values of elongation can be obtained depending on the loading path. If the load is applied monotonically to point (a), a certain elongation wil l be obtained; whereas, if the specimen is strairmd to point (b), then returned to point (c), where the toad is of the same magnitude as at (a), but the elongation wil l be of a different magnitude. This is due to the fact that the strain above the elastic limit is not a single-valued function of stress. It is clear that eq (24) must be interpreted to relate not only the magnitudes of the applied load in the model and prototype, but also to require that the pattern of loading be similar.

For the more general case, we may assume that any stress component can be expressed in the functional form

during any monotonically increasing or decreasing loading phase, where E is some characteristic modulus, having dimension of stress, ei, the strain com- ponents, and ~/~ a set of dimensionless coefficients. In this case, the characteristic properties of the material will be the moduli, E and G (moduli for normal and shearing stresses), and a set of dimensionless parameters, -~. As far as dimensional analysis is concerned, the use of this set of material properties wil l not alter the form of the design conditions from those obtained for l inearly elastic materials, but the additional similarity requirements wil l be

E Em

G Gm and

"Yi ~ ~' im

with the tacitly implied condition that the form r of the constitutive relationships are identical for model and prototype materials. The obvious way to satisfy these conditions is to use the same materials in model and prototype systems. Thus, we may conclude that the similarity requirements for modeling l ineari ly elastic system can be applied to systems involving inelastic behavior if the same materials are used in both model and prototype systems, and if the loading history is similar. In principle, the same materials are not required but it is virtual ly impossible to

satisfy the required conditions related to the non- l inear constitutive equations if different materials are used. This same conclusion has been shown in a more rigorous fashion to be true by Baker I~ and additional discussion of this point can be found in Refs. 6 and i i . If strain-rate effects are important, the problem is much more complex and a further discussion of this point is given in the last section of the paper.

Distorted Models As discussed previously, a common difficulty en-

countered in model studies is the experimentor's in- ability to satisfy all similarity requirements. For example, in the list of model-design conditions, eqs (7), if

then

~1 ~ ~tlm

and the model is said to be distorted. Unfortunately, distorted models are commonplace and, in general, predictions of prototype behavior based on distorted model data must be made with caution.

Possible procedures for handling distorted models include:

(a) Neglect certain variables that may be only slightly significant but lead to the distortion.

(b) Determine the effect of the distortion, analytically.

(c) Determine the effect of the distortion, empirically.

Although frequently not recognized as such, the first of these methods is probably the most common one for handling distortion. In the previous example from the field of fluid mechanics, it was noted that a common problem arises when both the Reynolds number and the Froude number are considered to be important. If the same fluid is used in model and prototype, distortion is encountered. In this type of problem, it is common practice to neglect one or the other of these numbers, which in effect means that either viscosity or the acceleration of gravity is neglected, and to base the model design on the remaining parameters. In many instances this has been a suc- cessful treatment. Also, it is well known that, for three-dimensional photoelastic models, Poissons' ratio, ~, is an important material property. And for proper scaling ~ must be the same for model and prototype materi, als. Since this condition is seldom satisfied this type of model is distorted. However, it is recognized that, in many problems of this type, Poisson's ratio is not highly significant and its effect is simply neglected, and this procedure is thus another example of method (a) for handling distort ionJ 2

It is apparent that, if the neglected parameter has any significance (and if it hasn't, it shouldn't be included), perfect correlation wil l not be achieved between the model and prototype so that the val idity of this technique depends on how accurate the results must be in order for them to be of value. Also, it is highly desirable to have some way of estimating the amount of error introduced by neglecting the effect

Experimental Mechanics [ 333

EDUCATIONAL LECTURE

of the distortion. Although it wil l probably not be feasible to make such an estimate with a high degree of precision, it may be possible to obtain some insight by solving analytically a similar, but simpler, problem.

A second method for handling distortion is the de- termination of the effect of the distorted parameter analyt ical ly so that this effect can be taken into account. This procedure can best be i l lustrated with a simple example. Consider a phenomenon that is governed by three pi terms so that

=i : $ (=~, =3)

Assume that the value of the model pi term ~3m is distorted by an amount ~ so that

It then follows that ~I : ~ ;~lm

where 5 is a prediction factor required to correct for the distortion of :t3m. If the manner in which ~3 in- fluences ~l can be determined, the relationship between 8 and ~ can be determined. Unfortunately, in many instances, 8 and /~ wil l not be simply related since

=i i (=2, =3) =2= f (=2m, =3m)

which shows that 8 can be a function not only of but also of ~t2 and n3. However, in certain special cases, where the distorted pi term is separable, and expressible in the form

~i : =~ f~ (=~) it follows that

A detailed discussion of this method for handling distortion can be found in Murphy. 7

A third method for handling distortion, which is perhaps the most practical for problems in which distortion must be taken into account, is one in which the effect of the distortion is determined empirically. Consider a problem in which four pi terms are in- volved, i.e.,

and the required design cortdition ~2 = ~t2m cannot be met although the other two design conditions

~r~ =~lm

mi = #r4m

are satisfied. If sufficient control over the model ex- periments is available, we can run a series of model tests in which ~2,~ is varied while holding ~t3rn and ~etm constant at the required prototype values as illustrated in Fig. 5.

Ideally, the series of model tests would be run so that the prototype value of n~ would fal l between the actual model values, as i l lustrated at point A in Fig. 5. If this is not possible, as is frequently the case, then extrapolation is required as i l lustrated a~ point B in Fig. 5. Of course, extrapolation is not a desirable procedure and the predicted value of nl could be

grossly in error. The usefulness of this approach wil l depend on the particular problem and the degree of accuracy required.

This method can be extended to model studies for which two pi terms, such as ~2 and n3, are distorCed. For this case, a series of model tests are required in which n4m is held constant at the prototype value while ~2,~ and ~3m are varied. The pi term, ~lm, can be represented by a point on a surface when Jq,, is plotted versus ~2m and =3rn (Fig. 6). A sufficient number of points must be determined from the model tests so that this surface can be reasonably defined. As in the previous example, it would be desirable to span the prototype values of n2 and ~8 so that the prediction can be made at some point, such as A, on the surface. Otherwise, the surface would have t.o be extrapolated to obtain the predicted value of nl. Extension to systems with higher degrees of distortion is possible but not feasible. Although, in principle, this method of empirically handling distortion is simple, there are many practical difficulties. Frequently, to obtain a series of model tests, it is necessary to vary material properties or the size of the models. Since the range of materials available for the model system is usually quite limited, varying material properties is a difficult, if not impossible, technique. It may be possible to construct a series of models of different sizes, but this is usually expensive and t ime consuming. Thus, we must conclude that this method is not the final answer to all distorted-model problems and, in general, the use of distorted models remains a difficult problem.

Modeling of Complex Coupled Systems As the complexity of the prototype system in-

creases, the more appealing and, perhaps, necessary a model study becomes. Some of the most complex problems involve the interaction, or coupling, between different environments. For example, there are important problems in which we have coupling between thermodynamic and structural phenomena, hydrody- namic and structural phenomena, magnetic fields and

~'lm

~1 (PREDICTED) B

~1 (PREDICTED) A ~ ~'~'~'~-'1t

/ ~3rn = ~3 = CONSTANT ~4m = ~r4 = CONSTANT O EXPERIMENTAL POINTS FROM MODEL TESTS

VALUE ~.~ REQUIRED VALUE ~REQUIRED ~Zm

Fig. S--Prediction technique with one distorted pi term

334 l inty 19rl

EDUCATIONAL LECTURE

fluid flow, and soils and structures, to mention but a few. To i l lustrate the use of models for the study of complex coupled systems, an example of a model study of a soil-structure system is considered in some detail./z This type problem is rather unique in that the pert inent material properties of the soil are not well defined. However, as demonstrated in this example, it is still possible to obtain useful results from a model test.

The interest in this problem stems from a desire to establish modeling relationships for the study of the response of underground structures to blast load- ings. The approach taken in this example was to determine whether or not data could be correlated between small-scale structures of different sizes when tested under laboratory conditions. This procedure is recommended whenever there is some doubt with regard to the validity of the model design, since it provides a necessary condition for the establishment of similarity requirements. For the particular study under discussion, the dynamic load was applied by means of a weight dropped onto the surface of the soil in which a hollow cylinder was buried. The pert inent variable of interest was taken to be the circumferential strain measured on the inner wall of the buried cylinder (Fig. 7).

The variables considered in this study were:

e, circumferential strain, FoL~ ~ D, cylinder diameter, L ~, all other pert inent lengths, L p~, density of cylinder, FT2L -4 E, modulus of elasticity of cylinder, FL -~ ~, Polsson's ratio of cylinder, F~ ~ M, mass of impacting weight, FT2L-Z V, impact velocity of weight, LT - 1 t, time, T p, init ial density of soil, FT2L -4 ~11, property of soil, FL -2 ~1~, other soil properties, FL -2 ~ other dimensionless soil properties, F~ ~

For the purpose of ~his study, it was assumed that the soil could be characterized by a set of properties that had dimensions, FL -2, or were dimensionless. As far as dimensional analysis is concerned, this is all that is required. A suitable set of pi terms is

e =1~ (Z,i ,opDa ,o, ED3 Vt~111~i ) ' M ' p~, MV2' ~' --D" E ' E ' "Y~

If the same combination of materials is used in the model and prototype systems, then all s imilarity requirements arising from pi terms consisting solely of material properties are immediately satisfied. Other similarity requirements are

X~ ~im = - -

n

where the length scale, n, is equal to D/Dm,

M Mm = - -

n 3

and Vm= V

t tm=- -

n

If these conditions are met, then it follows that

at corresponding times. To determine the validity of this model design, a

series of tests was run with 1-in., 2-in., and 4-in.- diam hollow cylinders embedded in dry Ottawa sand. Figures 8 and 9 show typical results of these model tests. The results are reasonably satisfactory con- sidering the difficulty in obtaining data of this type. It should be noted that strain-rate effects were neglected in this analysis. It can be shown that, if material properties describing strain-rate effects are added to the list of variables, then a distorted model wi]l result. Tests of the same type as those run with the dry sand were also performed with a sand-oi l

~'lm ,e4 m = ,e 4 = CONSTANT J O EXPERIMENTAL POINTS ~ O FROM MODEL TESTS O ~

0 0 ~. \>

/ .'" pREDICTED / ' / pREDIoTED 0 / o l - / /

/ o ! cr //~E,~u!,,p / ~ E D VALUE / / ~ALUE

/ I / / / ,2. /" . . . . . . . . . .Z- . . . . ~/ 8 / / i~,%~f"~ ~ . . . . . _~.X/ , / / _~- (PREmCrEO)~ / I ../-// ~ / I . / /7 I / L / I I / /

REQUI ;E ' J I / VALUE I //

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . J

~3m

Fig. 6--Prediction technique with two distorted pi terms

/ - FALLING

I IV ~ WEIGHT

~ " J ~-BURIED CYLINDER CIRCUMFERENTIAL

SOIL TEST CYLINDER

\ \ \ \ \~ \ \ \ \ \ \ \ \ \ \ \

Fig. 7 - -Sketch of drop-weight loader and cy l inder

Experimental Mechanics I 335

E D U C A T I O N A L L E C T U R E

1000

500 .=_ I .o u

i =<

.<

1 O0 0.I

I I I ~l~ i I I I J I I I

x 9

O l - in . CYLINDER 9 2- in . CYLINDER X 4- in. CYLINDER

0.5 1 .0

DEPTH DIAMETER

DRY SAND

I , , ,

I , I , 5.0 10.0

Fig. 8--Comparison of peak strain data for three model cylinders

I F ...... i I I d = DEPTH OF BURIAL : 1 "D CYLINDER DIAMETER

800 D = 2 in.

DRY SAND = i .

.e_" 600 /,

20(I

L~ I _ _1 I _ _1 [ 0.0 1.0 2 .0 3.0 4.0 5.0

SCALED TIME, msec

Fig. 9--Comparison of average strain-time curves for three model cylinders

6.0

8o0 1 I I I

~ - 4-1n. CIRCULAR CYLINDER ~ ~ OIL - SAND

OF BURIAL

CIRCULAR

r I ( I _ _ O.O 1.0 2.0 3.0 4.0

SCALED TIME, msec

Fig. lO--Distortion due to strain-rate effects

mixture and typical results are shown in Fig. 10. Considerable distortion is present, and numerous other tests uti l izing highly cohesive soils reveal the same kind of distortion, x4

Problems of this type are typical of those for which model studies can be extremely useful, i.e., problems not readily amen,able to other methods of study, due to their extreme complexity. The example also illus- trates one of the most common difficulties in modeling, that of being able to adequately describe, and control, material properties of the model and prototype systems. Much additional work is needed in this important area.

References 1. Young, D. F., "'Simulation and Modeling Techniques," Trans.

ASAE, 11, 590 (1968). 2. Young, D. F., "'Similitude of Soil Machine Systems," Trans.

ASAE, U, 653 (1968). 3. Karplns, W. 1. and Soroka, W. W., Analog Methods: Com-

putation and Simulation, McGraw-HiU Book Co., Inc., New York, 2nd ed. (1959).

4. Murphy, G., Shippy, D. ]. and Luo, H. L., Engineering Anal- ogies, Iowa State University Press, Ames, lowa (1963).

5. Bridgman, P. W., Dimensional Analysis, Chap. 2, Yale Uni- versity Press, New Haven (1931).

0. Langhaar, H. L., Dimensional Analysis and Theory of Models, ]ohn Wiley & Sons, Inc., New York (1951).

7. Murphy, G., Similitude in Engineering, Ronald Press Co., New York (1950).

8. Dally, 1. W. and Riley, W. F., Experimental Stress Analysis, McGraw-Hill Book Co., Inc., New York, 247 (1965).

9. Durelli, ,4. 1., Phillips, E. A. and Tsao, C. H., Introduction to the Theoretical and Experimental Analysis of Stress and Strain, McGraw-HiU Book Co., Inc., New York, Chap. 12 (1958).

10. Baker, W. E., "'Modeling of Large Elastic and Plastic De- formations of Structures Subieeted to Transient Loading," Proc. of Colloquium on the Use of Models and Scaling in Shock and Vibra- tion, ASME, 71 (1963).

11. Goodier, ]. N., "'Dimensional Analysis," Handbook of Experi- mental Stress Analysis, M, Hetenyi, Ed., 1ohn Wiley and Sons, Inc., New York (1950).

12: Mdnch, E., "'Similarity and Model Laws in Photoelastle Ex- periments," EXPElaX~,Z~NT.~r, MEC~Z~CS, 4 (5), 141-150 (May 1964).

13. Young, D. F. and Murphy, G., "'Dynamic Similitude of Un- derground Structures," ]nl. Engrg. Mech. Div., Proe. ASCE, 90, 111 (1964).

14. Murphy, G., Young, D. F. and MeConnell, K. G., "'Similitude of Dynamically Loaded Buried Structures," U.S.A.F. Weapons Lab- oratory Bpt. WL TR-64-142 (1965).

336 I 1uly 1971

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