EDUCATIONAL

LECTURE

Basic Principles and Concepts of Model Analysis Lecture d i s c u s s e s the t e c h n i q u e s and t h e p r o b l e m s associated w i t h the design and use of m o d e l s

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 S i m u l a t i o n is widely used in e n g i n e e r i n g analysis, design and research. I n fact, this technique is used w h e n e v e r a problem is studied b y some method other t h a n direct observations on the prototype, which is defined as the actual system of interest. Figure 1 illustrates schematically the steps commonly taken in the simulation of the prototype. It u s u a l l y follows that assumptions are made so that the system m a y be more precisely defined, and this n e w system m a y 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 e x t e n t that the problem can be well defined. Following the definition of the problem, a decision must be m a d e with respect to the type of simulation technique to be used. If the problem is to be solved analytically, it is apparent that a m a t h e m a t i c a l model m u s t be developed and s u b s e q u e n t l y solved either b y m a t h e m a t i c a l analysis or b y a n analog. If a n analog is used, the system is analyzed experim e n t a l l y b u t with another system, or model, which is not similar in appearance to the original prototype. Thus the t e r m "dissimilar model" is appropriate for the analog. There are a great v a r i e t y of analogies used, 8,4 although p r o b a b l y the most useful types are ones i n v o l v i n g electrical circuits. To simulate the prototype with a m a t h e m a t i c a l model, the characteristic equations describing the behavior of the system must be known. This freq u e n t l y requires additional assumptions with regard to the behavior of the system. For example, in considering the deformation of structures, the common assumption is that the material behaves elastically. Or in dealing with flowing fluids, it m a y be assumed that the fluid is ideal or nonviscous. Thus, to estab-

lish a m a t h e m a t i c a l model, a thorough u n d e r s t a n d i n g of the characteristics of the system, and the f u n d a m e n t a l equations g o v e r n i n g the behavior of the system, m u s t be achieved. There are several advantages in solving problems in this m a n n e r . A complete a n d detailed solution is n o r m a l l y obtained and, if the p r o b l e m is solved on a completely a n a l y t i c a l basis, the expense of b u i l d i n g e q u i p m e n t and p e r f o r m i n g tests is eliminated. Probably, the m a j o r disadvantage is the required n u m b e r of assumptions to establish the m a t h e m a t i c a l model. I n m a n y i n stances, it is difficult to obtain a solution due to the complexity of the governing equations. The a l t e r n a t i v e basic simulation technique is one of actually s i m u l a t i n g 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 b u t not identical to it. I n practice, such systems are simply referred to as models and Pre u s u a l l y smaller in size t h a n the prototype. In some instances, it m a y be advantageous to have a model that is larger t h a n the prototype. It is freq u e n t l y possible, t h r o u g h the use of similar models, to study very complex problems with relative ease, and this is one of the m a j o r advantages of this technique. It is also true, in m a n y cases, that fewer assumptions are r e q u i r e d w h e n similar models are used t h a n for s i m u l a t i o n with a m a t h e m a t i c a l model. One of the m a j o r disadvantages of similar models is that the solution is obtained e x p e r i m e n t a l l y and the expense u s u a l l y associated with e x p e r i m e n t a l w o r k must be incurred. Also results obtained e x p e r i m e n tally are f r e q u e n t l y restrictive and limited in applicability. It is clear that, regardless of the p a r t i c u l a r technique chosen to simulate the prototype, a relationship b e t w e e n the model and prototype m u s t be established. T h r o u g h the d e v e l o p m e n t of m a t h e m a t i c a l models, this relationship evolves n a t u r a l l y a n d is readily apparent. However, w h e n similar models are

Fig. 1--Flow chart for problem analysis MATHEMATICAL

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.

ASSUMPTIONS m

J IDEALIZED PROTOTYPE

EDUCATIONAL

LECTURE EQUILIBRIUM POSITION

used, t h e relationship b e t w e e n model and prototype must also be known. The establishment of this relationship a n d the fulfillment of the various similarity requirements, or m o d e l - d e s i g n conditions, b e t w e e n the two systems is sometimes difficult to achieve. As problems become more complex, the value of similar models increases, a n d the r e m a i n d e r of this paper is devoted to a discussion of this type of model.

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

Theory of Similitude and Modeling Two methods c o m m o n l y used for establishing similarity relationships b e t w e e n a model and prototype are based on (a) a n analysis of the characteristic equations of the system a n d (b) dimensional a n a l y sis. If the former method is used, the system is first described in terms of a m a t h e m a t i c a l model and t h e n the scaling laws, model-design conditions, or similarity requirements, (these terms are used i n t e r 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 r e q u i r e d to establish a m a t h e m a t i c a l model, apply. It m a y be argued that, if the m a t h e matical model is established, w h y is it necessary to obtain a solution e x p e r i m e n t a l l y using a similar model? The p r o b l e m is that it m a y be extremely difficult, if not impossible, to get a closed-form solution, or even a good n u m e r i c a l solution, to certain types of problems. A classical example of this is in fluid mechanics, where the characteristic equations are well k n o w n b u t cannot in general be solved b e cause of their n o n l i n e a r i t y . Both methods of developing modeling laws are i m p o r t a n t and the general procedures followed Jn their use will be considered in some detail.

If we now introduce two dimensionless parameters It

$

y* = - - a n d t * Yo

:--

x

where

eq (1) can be w r i t t e n as d2y *

c

dy *

dt .2

N/mk

dr*

q- y* = 0

(2)

The initial conditions become y*=l and dy*

vo ~ / m

dt*--

Yo V - - k

att* = 0 From a consideration of eq (2), it is seen that for a n y two systems governed by an equation of this form, the solution for y* will be the same, i.e., Y* ~ Ym*

if

Characteristic-equation Method

The characteristic equations describing physical problems are f r e q u e n t l y differential equations and these equations, combined with the initial and b o u n d a r y conditions, describe the problem. Essentially, the method of d e t e r m i n i n g similarity r e q u i r e ments from the characteristic equations consists of r e w r i t i n g the characteristic equations in d i m e n s i o n less form, and d e t e r m i n i n g from the transformed equations the conditions u n d e r which the behavior of two systems will be similar. This method is illustrated in the following simple example. Consider the s p r i n g - m a s s - d a s h p o t system of Fig. 2. The p r o b l e m is to d e t e r m i n e the displacement, y, as a f u n c t i o n of time, t, b y means of a model study. It is well k n o w n that the displacement of the mass is described b y the differential equation d~T dy m "l- c -I- ky ---- 0 dt 2

along with the initial conditions dy Y = Yo a n d - - = v o dt

att----0

326 I July 1971

(1)

e

era

ooV._ -~o

t

k

YOre

~-~tm

km

-~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 k n o w n for the system, the procedure described in this example can be followed to establish the necessary relationships b e t w e e n the prototype and model. However, in m a n y problems, the characteristic equations are more n u m e r o u s and complicated t h a n in the example given, or even u n k n o w n , a n d this method cannot be readily used. A n a l t e r n a t e approach, which does not r e q u i r e such a detailed knowledge of the system, is based on dimensional analysis. Dimensional Analysis

When dealing with physical phenomena, we describe the p h e n o m e n a in terms of various quantities;

EDUCATIONAL

s u c h as v e l o c i t y , a c c e l e r a t i o n , density, area, etc., a n d t h e s e s o - c a l l e d s e c o n d a r y quantir in t u r n , a r e d e s c r i b e d a n d m e a s u r e d i n t e r m s of a n o t h e r set of q u a n tities w h i c h a r e c o n s i d e r e d to be p r i m a r y q u a n t i t i e s . In m e c h a n i c s , t h e p r i m a r y q u a n t i t i e s , or basic d i m e n sions, a r e n o r m a l l y t a k e n to be length, L, time, T, a n d mass, M. It c a n be s h o w n 5 t h a t a n y s e c o n d a r y q u a n tity, s~, is e x p r e s s i b l e in t e r m s of t h e p r i m a r y q u a n t i ties in t h e f o r m si = L a T b M e X a y e

w h e r e X, Y, . . . , a r e o t h e r basic d i m e n s i o n s such as t e m p e r a t u r e , and elecCrical c h a r g e w h i c h m a y be r e q u i r e d to d e s c r i b e t h e s e c o n d a r y q u a n t i t y . C o m m o n e x a m p l e s of s e c o n d a r y q u a n t i t i e s a n d t h e i r basic dimensions include: area volume velocity dens~y stress

, ,

A V

=is :L z

,

v

:LT

, ,

p ~

: M L -3 :MT-ZL

-1

-1

To o b t a i n t h e basic d i m e n s i o n s of stress, use is m a d e of t h e f a c t t h a t stress is a force, F, d i v i d e d b y an area, b u t f o r c e and m a s s a r e r e l a t e d t h r o u g h N e w t o n ' s second l a w of m o t i o n ; i.e., F = M L T - 2 . It is t h u s a p p a r e n t t h a t an e q u i v a l e n t set of basic d i m e n s i o n s to b e used in m e c h a n i c s p r o b l e m s is L, T ~.nd F. In a given problem, there are usually several variables, ul, u2, . . . , u b r e q u i r e d to d e s c r i b e t h e p h e n o m e n o n of i n t e r e s t . A n u m b e r of d i m e n s i o n l e s s p r o d u c t s of t h e s e v a r i a b l e s c a n b e f o r m e d b y c o m b i n i n g t h e v a r i a b l e s in t h e f o r m Ul xl U-2~

.....

Uk xk

w h e r e t h e e x p o n e n t s x l , x s . . . , Xk a r e s e l e c t e d so t h a t t h e r e s u l t i n g p r o d u c t is d i m e n s i o n l e s s . T h u s , if w e let a n y o n e of Che v a r i a b l e s , say ui, h a v e t h e basic dimension zti = L a~ T b~ M c~ X d~ ye~

w e c a n e x p r e s s t h e p r o d u c t as (Lm Tb~ Me1 X a l y e l ) x ~ (La~ T b , Me2 Xa~

yea)x~

of coefficients al

a2

ak

bk

el

C2

dl el

d2 . . . . e2 . . . .

....

ek

dk ek

If an e q u a t i o n i n v o l v i n g k v a r i a b l e s is d i m e n s i o n a l l y h o m o g e n e o u s , it can be r e d u c e d to a relationship among k- r independent dimensionless p r o d u c t s , w h e r e r is t h e r a n k of t h e d i mensional matrix. To i l l u s t r a t e t h e a p p l i c a t i o n of t h e B u c k i n g h a m P i T h e o r e m , w e w i l l a p p l y it to t h e v i b r a t i o n p r o b l e m p r e v i o u s l y c o n s i d e r e d . T h e first step in t h e a n a l y s i s is to list t h e v a r i a b l e s a n d t h e i r d i m e n s i o n s as f o l lows: y, m, c, k,

. .

In o r d e r t h a t t h e p r o d u c t be d i m e n s i o n l e s s , t h e e x p o n e n t s of t h e v a r i o u s basic d i m e n s i o n s m u s t c o m b i n e to g i v e a z e r o v a l u e f o r e a c h basic d i m e n s i o n . T h u s -{- a k x k :

....

bi b2 . . . .

T h i s m a t r i x is c o m m o n l y c a l l e d t h e d i m e n s i o n a l m a t r i x . S i n c e t h e r a n k of a m a t r i x is t h e h i g h e s t o r d e r n o n z e r o d e t e r m i n a n t c o n t a i n e d in t h e m a t r i x , it is a p p a r e n t t h a t t h e r a n k c a n n o t e x c e e d t h e n u m b e r of e q u a t i o n s b u t m a y b e s m a l l e r . Thus, t h e n u m b e r of i n d e p e n d e n t d i m e n s i o n l e s s p r o d u c t s t h a t c a n be f o r m e d is e q u a l to t h e n u m b e r of o r i g i n a l v a r i ables, k, m i n u s t h e r a n k of t h e coefficient m a t r i x . S u c h a set of d i m e n s i o n l e s s p r o d u c t s is c a l l e d a c o m p l e t e set. O n c e a c o m p l e t e set of d i m e n s i o n l e s s p r o d ucts is found, all o t h e r possible d i m e n s i o n l e s s c o m b i n a t i o n s c a n b e f o r m e d as p r o d u c t s of p o w e r s of t h e p r o d u c t s c o n t a i n e d in t h e c o m p l e t e set. A n e s s e n t i a l p o s t u l a t e of d i m e n s i o n a l a n a l y s i s is t h a t t h e f o r m of a n y f u n c t i o n a l r e l a t i o n s h i p b e t w e e n a g i v e n set of v a r i a b l e s does n o t d e p e n d o n t h e s s ' s t e m of u n i t s used, i.e., t h e f u n c t i o n a l r e l a t i o n s h i p is dim e n s i o n a l l y h o m o g e n e o u s . If this c o n d i t i o n of h o m o g e n e i t y is utilized, ir c a n b e p r o v e d 6 t h a t a f u n c t i o n a l r e l a t i o n s h i p b e t w e e n a g i v e n set of v a r i a b l e s c a n b e r e d u c e d to a r e l a t i o n s h i p a m o n g a c o m p l e t e set of d i m e n s i o n l e s s p r o d u c t s of t h e s e v a r i a b l e s . T h u s t h e well-known Buckingham P i T h e o r e m c a n b~ s t a t e d as f o l l o w s :

(La~ Tb~ Mc~ Xd~ y e ~ ) x k

a l x l q- a2x2 -}- . . . .

LECTURE

vo, Yo,

t,

displacement, mass, d a m p i n g coefficient, s p r i n g constant, initial velocity, initial displacement, time,

L M MT- 1 MT -2

LT -1 L T

0

We now form the product blXl "4- b2x2 -I- . . . . -1- bkXk = 0 y z l m R cx8 k x , vox~ yox6 t ~ clxl + csx2 + . . . .

+ C~Xk :

0

d l x l + ~ x s + . . . . + dkxk :

0

e i x i -~ e2x2 q- . . . .

0

Jr e k x k :

(3)

W e n o t e t h a t t h e r e w i l l b e as m a n y equar as basic dimensions, s a y m in n u m b e r , a n d k u n k n o w n x's, w h e r e k is e q u a l to t h e n u m b e r of o r i g i n a l v a r i a b l e s in t h e p r o b l e m . F r o m t h e t h e o r y of e q u a t i o n s , it is k n o w n t h a t t h e r e a r e k -- r l i n e a r l y i n d e p e n d e n t sol u t i o n s to eqs (3) w h e r e r is t h e r a n k of t h e m a t r i x

a n d w i t h t h e s u b s t i t u t i o n of t h e b a s i c d i m e n s i o n s f o r each v a r i a b l e w e o b t a i n (L)xl (M)~ (MT-1)zs

(MT-~)x,

( L T - 1 ) z ~ (L)x6 ( T ) ~

w i t h t h e c o r r e s p o n d i n g set of e q u a t i o n s L:

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

T:

0 +0--x3--

M:

O-t-x2+xaTx4+O+O+O=O

2x4--xs+0+xT=0

(4)

E x p e r i m e n t a l M e c h a n i c s 1 327

EDUCATIONAL

LECTURE

y

The dimensional m a t r i x is y L T M

m

c

k

I0 0 0 00--I--2--101 01 1 1

Vo Yo

:~'1 =

(~3)--1

= __ Yo

s

:

(~1)-'/2

_

~'4 :

(~1)--1/2

(~2) (~3)--1

t

C

ii0 000

_ _ N/km

Consider n o w the d e t e r m i n a n t on t h e left side of the matrix: 1 0 0

0 0 1

0 --i 1

:i

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" w e 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. T h e only restriction here is that the determinant of the remaining coefficients m u s t be nonzero so that w e can solve for the remaining x's. In this case w e have previously s h o w n that this determinant is nonzero. W i t h 3:4 : i, x5 : 0, x6 : 0, a n d x 7 ----0 it follows that eqs (4) are satisfied if Xl ----0, x2 = 1, and x3 ------2. T h u s one dimensionless product is

w h i ch contains precisely t h e same dimensionless products as o b t a i n e d from the differential equation. In most problems, the dimensionless products can be obtained by inspection since the r e q u i r e d n u m b e r can easily be d e t e r m i n e d and the variables can simply be combined into dimensionless groups. An indep en d en t set can be assured if each dimensionless product contains one variable not contained in any o t h er product. In essence, dimensional analysis allows us to study a p r o b l e m described by t h e functional r e l a tionship U l --~ r

C2

No w let x4 ~-- 0, X5 : find that

1, X6 = 0, and x 7 = 0 and we Vo~r~ ~2=-

yc

This process can be continued w i t h x4 ---- 0, x5 = 0, x6 ---- 1, a n d x 7 = 0 a n d x 4 ---- 0, x5 = 0, x6 = 0, and x7 ---- 1 to give Yo Y

and ct m

It is a p p a r e n t that t h e specific form of t h e pi t er m s depends on w h i c h of t h e x's are assigned values and the values themselves. H o w e v e r , it should be e m p h a sized that, once an i n d e p e n d e n t set is determined, all o t h er possible i n d e p e n d e n t sets can be f o r m e d as products of powers of the original set. Using this procedure, w e can f o r m various combinations to arr i v e at w h a t we consider to be the most useful set. In t h e p r e s e n t example, t h e obvious d i s a d v a n t a g e is the fact that, y, the displacement, appears in t h r e e of the pi terms. It is usually c o n v e n i e n t to h a v e t he v a r i a b l e of p r i m a r y interest a p p e a r i n g in only one pi term. To c o m p a r e this set of pi t e r m s obtained from d i m en s i o n al analysis w i t h the dimensionless products d e v e l o p e d f r o m a consideration of the differential equation we form t h e n e w set

328 I ] u l y 1 9 7 1

(U2, U3, U4 . . . . .

Uk)

(5)

in t er m s of a set of dimensionless t e r m s ~1 =

~1 = y o m 1 c - 2 k 1 Vo o y o o t ~ km

--V~ ~V[/__~ Yo

-

f ( ~ 2 , n3 . . . . .

nk--r)

(6)

One obvious a d v a n t a g e is the r ed u ct i o n in t he n u m b er of v ar i ab l es (from k to k - r) to be controlled in an e x p e r i m e n t . In addition, it is usually m u c h easier to control the dimensionless products in an e xp e r i m e n t than t h e original variables. Modeling laws can be r ead i l y d e v e l o p e d from eq (6) in the f o l l o w i n g manner. W e assume that we h a v e two systems, the prototype and t h e model, each described by t h e equations H1 :

f ( ~ 2 , ~3. . . . .

~k-r)

nlm = fm(~2m, ~3m . . . . .

(prototype) ~(k--r)m)

(model)

We f u r t h e r assume that the p h e n o m e n o n with w hi c h w e are d eal i n g is the same for both t h e prototype and the m o d el so t h a t the f o r m of the function, f, for the p r o t o t y p e is t h e same as the function, fro, for t he model. It i m m e d i a t e l y follows t h a t if w e let ~2 = ~ 2 m ~3 = g 3 m

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

then Equations (7) p r o v i d e us with the r e q u i r e d relationships b e t w e e n p r o t o t y p e and m o d e l so that we p r e dict ~1 f r o m a m e a s u r e d glm t a k e n on the model. Equations (7) r e p r e s e n t the m o d e l - d e s i g n conditions and eq (8) the prediction equation. 7 Application of this procedure to the previously d e t e r m i n e d pi terms, s n'2, :~'~, ~'4 developed f o r the spring-mass pr ob-

EDUCATIONAL

IPs

LECTURE

a ~. n a am

--r__,

w h e r e na is the " l e n g t h scale", and P

~- up

Pm

_l

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

Fig. 3--Sketch of cantilever beam

A

_

_

--

Am

lem leads to the same similarity r e q u i r e m e n t s as those obtained from a consideration of the differential equation. For the r e m a i n d e r of this paper, dimensional a n a l ysis will be used to :develop modeling laws. To more clearly illustrate 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 r e c t a n g u l a r cross section (Fig. 3) due to a concentrated load P. It is assumed that the deformation is small, the m a t e r i a l behaves elastically, shearing deflections are negligible, and the beam is loaded in a p l a n e of s y m m e t r y so t h e r e 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, F L - 2 Application of the P i Theorem reveals that, since there are six variables expressible i n terms of two basic dimensions, four dimensionless p a r a m e t e r s are required to describe this problem. One possible set is a' a'

a --

Ea 2

(9)

It now follows that, if b

bm

a

am

d

d~

--

:

a

- -

(10)

am P

Pm

Ea 2

E m am 2

for two systems then A

Am --

a

(11)

am

Equations (10) are the similarity r e q u i r e m e n t s 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 a r b i t r a r i l y selected; e.g., let

b

--

bm

d

--

na

dm

and E

Ttp'l"ta--2

Em

The foregoing example reveals that there are three basic steps used i n e s t a b l i s h i n g modeling laws from a dimensional analysis. These are: (a) the selection of the variables, (b) the application of the Pi Theorem, and (c) the d e v e l o p m e n t of the s i m i l a r i t y req u i r e m e n t s by e q u a t i n g pi terms. A l t h o u g h in p r i n c i ple this procedure is straightforward, a n d r e l a t i v e l y simple, two m a j o r difficulties are n o r m a l l y e n c o u n tered. The first is i n the selection of the p e r t i n e n t variables. Ir is clear t h a t a good u n d e r s t a n d i n g of the problem and the p h e n o m e n a m u s t be achieved in order to ascertain the p e r t i n e n t variables. The selection of variables is u s u a l l y based on the experience of the investigator a n d a knowledge of the f u n d a m e n tal equations which govern the phenomena. This does not i m p l y that a detailed m a t h e m a t i c a l model of the system must first be established, b u t simply that certain f u n d a m e n t a l laws, such as Newton's laws of motion, are k n o w n to be applicable to the system. C o m m o n errors e n c o u n t e r e d at this stage are the i n clusion of n o n i n d e p e n d e n t variables and the omission of p e r t i n e n t variables or p a r a m e t e r s such as the acceleration of gravity, on the basis that they are constant. T h e inclusion of a group of variables t h a t are not i n d e p e n d e n t (such as the b e a m cross-sectional area, w i d t h and d e p t h in the c a n t i l e v e r - b e a m e x a m ple) is clearly unnecessary. I n addition, it should be emphasized that all p e r t i n e n t i n d e p e n d e n t variables or p a r a m e t e r s must be i n c l u d e d and the fact that they m a y or m a y not be c o n s t a n t is if no consequence at this stage i n the analysis. It should be noted that omissions, or the listing of u n n e c e s s a r y variables, will n o r m a l l y not be detected w i t h o u t the aid of experiments. The second difficulty that f r e q u e n t l y arises is in the control of the pi terms. Each pi t e r m yields a r e q u i r e m e n t b e t w e e n the model and prototype system. I n certain instances it is difficult, if not impossible, to satisfy one or m o r e of these requirements. The classical e x a m p l e of this :difficulty is in fluid-flow problems i n which both the Reynolds n u m b e r and the F r o u d e n u m b e r are important. S i m i l a r i t y r e q u i r e m e n t s arising from these two dimensionless p a r a m e t e r s are V~,

Vm~.m

--

Y

- -

(Reynolds n u m b e r )

Vm

and

Experimental Mechanics

I 329

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LECTURE

v2

vm 2

gk

gm~.m

. . . . . .

ship

(Froude number)

w h e r e v is a velocity, ~. is a length, v is the k i n em at i c viscosity of the fluid, and g the acceleration of g r a v ity. If the same fluid is used in both model and prototype, and both systems o p e r a t e in t h e same g r a v i t a t i o n a l field, it follows that v

Vm

~m --

~.

--

nx

-1

(12)

P

Pm

EL2

Emlm2

or P

E

Pm

Note that the m o d el and p r o t o t y p e materials need not be the same but the elastic moduli scale, E/Em, and the length scale, 1/lm, fix t h e loading scale. All additional loads, P~, must be in t h e same ratio, i.e.,

f r o m the Reynolds n u m b e r condition, and - -

=

= N/nx

Vm

(13)

~.m

for the F r o u d e n u m b e r condition. It is a p p a r e n t that eqs (12) and (13) give different values for the v e l o c i t y scale. A conflict t h e r e f o r e exists. It is e x t r e m e l y difficult, in most instances, to find combinations of fluids t h a t allow both of these conditions to be satisfied simultaneously. A m o d e l for wh i ch at least one of the similarity r e q u i r e m e n t s is not satisfied is said to be distorted. N u m e r o u s problems could be cited to show that distorted models are not r a r e exceptions but m a y f r e q u e n t l y occur. A f u r t h e r discussion of this i m p o r t a n t topic is g i v en l at er in this paper.

Typical Applications Static Elastic Problems S t r u c t u r a l models a r e w i d e l y used f o r d e t e r m i n i n g stresses, strains and displacements in elastic structures. F o r these problems, w e assume that the m a terial obeys Hooke's l a w and can be described by Young's modulus, E, and Poisson's ratio, tz. In addition, any stress component, ~, at some point, xl, w i l l be a function of the g e o m e t r y of t h e system as ch ar acterized by some length, l, and o t h e r r e q u i r e d lengths, ~i. T h e subscript, i, will be used to designate a set of variables. Thus ~i is e q u i v a l e n t to a set of lengths kt, ~-2, ~-3, 9 9 9 T h e loading m a y be specified w i t h the loads, P and P+, and any prescribed b o u n d a r y displacements by +1+. The stress can t h e r e f o r e be expressed in the f u n ctional f o r m r ---- r l, ~+, P, P+, ~li, E, tz) (14) We now ap p l y d im e n s i o n a l analysis to this set of v ari ab l es to obtain

~12 - -

P

( x~ Xi Tli :

f

l'

l'

P

Pi -

l ' El m' P ' ~

330 [ ]uly 1971

P..

P

Pim

Pm

The last pi t e r m in eq (15) imposes the r a t h e r stringent s i m i l a r i t y r e q u i r e m e n t that Poisson's ratio must be eq u al for model and prototype materials. Of course, if t h e p r o t o t y p e and model are constructed of the same material, this condition is satisfied. For p l a n e - s t r a i n or plane-stress problems i n v o l v i n g simply connected bodies, for w h i c h the body forces are zero, constant, or v a r y l i n e a r l y with position, the stress distribution is k n o w n to be i n d ep en de nt of Poisson's ratio. 8 S i m i l a r p r o b l e m s involving m u l t i p l y connected bodies containing holes can also be m o d e l e d without r e g a r d to Poisson's ratio if the resultan.t force acting on the b o u n d a r y of the hole is zero. However, if these conditions are not m e t and if different m a t e rials are used, the Poisson's ratio condition will not, in general, be satisfied and for this case a j u d g m e n t must be m a d e w i t h respect to the significance of Poisson's ratio for the specific p r o b l e m under consideration. If all of t h e a f o r e m e n t i o n e d design conditions are satisfied, then it follows that al 2

amlm 2

P

P,+

or

~rm

P

lm2

E

Pm

12

Em

Since any displacement component, u, or strain component, ~, w i l l be a function of the same variables given in eq (15) it follows that the same m o d e l design conditions are r e q u i r e d for displacements and strains as for stresses. Th e corresponding prediction equations b e c o m e u Urn

1

Im

or

) (15)

S i m i l a r i t y r e q u i r e m e n t s are obtained by m a k i n g t h e pi t e r m s on t h e r i g h t side of eq (15) equal b e t w e e n model and prototype. E q u a l i t y of the first t h r ee pi terms, xdl, ~Jl, and ~i/l, m e a n s t h a t w e must m a i n tain g eo m et r ic s i m i l a r i t y b e t w e e n m o d e l and p r o t o type; not only w i t h r e g a r d to shape but also w i t h r e spect to prescribed displacements and coordinates. The loading scale is established f r o m the r e l a t i o n -

lu

Em /m~

u

t

Um

lm

and ~

~m

i.e., the displacements scale as t h e length scale w h er eas the strains are e q u a l in model and prototype. It should be noted t h a t these scaling laws for elastic structures are valid for both small and large deformations, as long as t h e m a t e r i a l in both mode] and p r o t o t y p e obeys Hooke's law. O t h er types of loads; e.g., line, surface and v o l u m e loads, can be

EDUCATIONAL

readily incorporated into the analysis. 9 In m a n y problems it is possible to relax, or modify, the similarity r e q u i r e m e n t s by m a k i n g use of m o r e detaited i n f o r m a t i o n about the p h e n o m e n a based on theory or experience. Typical e x a m p l e s of this t e c h nique follow.

LECTURE

of utilizing a n o t h e r g e o m e t r y in the m o d e l is an i m p o r t an t consideration. It should be e m p h a s i z e d that models designed on the basis of r e d u c e d s i m i l a r i t y r e q u i r e m e n t s of the type described in this section are not considered to be distorted models since all similarity r e q u i r e m e n t s d e e m e d necessary are satisfied.

Reduced-similarity Requirements If we again consider the p r o b l e m of p r e d i c t i n g stresses in elastic structures u n d e r static loading conditions and impose the additional restriction that the deformations are small, the modeling p r o b l e m can be considerably simplified. It is well k n o w n f r o m smalldeformation theory of elastic m a t e r i a l s that stress, strain and displacement are linear functions of the app!ied loads. Thus, f r o m eq (15), w e noCe t h a t al2/p must be i n d e p e n d e n t of the pi term, P / E l ~, since must depend l i n e a r l y on P, and the a p p r o p r i a t e e q u a tion becomes

al2 - -

P

, ( xi :

~i ~li

l'

l

,

l

Pi ,

P

) ,

~

(16)

The design condition relating the loading scale to the m o d u l u s - o f - e l a s t i c i t y scale has been e l i m i n a t e d and, thus, one is free to a r b i t r a r i l y select t h e m o del load P as long as t h e imposed condition of small d e f o r m a tions is maintained. If we apply the same a r g u m e n t to displacements and strains, we obtain

D y n a m i c Elastic P r o b l e m As a f u r t h e r e x a m p l e of the d e v e l o p m e n t of s i mi larity relationships by means of dimensional analysis, we will consider the p r o b l e m of p r e d i c t i n g the strain in an elastic s t r u c t u r e u n d e r the influence of a d y n a m i c - p r e s s u r e loading o v e r some part of the surface of t h e structure. As before, w e let e r e p r e s e n t one co m p o n en t of strain at the position xi, and let the g e o m e t r y of the system be described by a set of characteristic lengths, l and ~i, w h e r e t h e ~ 's also include the r e q u i r e d spatial coordinates of the loading. We continue to impose t h e condition t h a t t he m a t e r i a l obeys Hooke's l aw so that only t w o elastic constants E and ~ are required. H o w e v e r , since the st r u ct u r e is u n d e r d y n am i c loading, an additional m a t e r i a l property, the mass density, p, must be i ncluded in the analysis. We assume t h a t t h e pressure at any point c a n be described in dimensionless f o r m as --

u P l -- -E1 TI1

(xl

~,i Tll )P,_ _ ' l ' l' P ' ~

(17)

and :

El ~

f2

.

.

l

.

l

.

P

~

(18)

which gives m u c h m o r e flexibility in t h e m od el design. The principle of superposition applies for s m a l l d e f o r m a t i o n problems and, if desired, the m o d el m a y be tested w i t h a series of loads r a t h e r t h a n applying all loads simultaneously. A l t h o u g h it is g e n e r a l l y true that the m o del must be g e o m e t r i c a l l y similar to the prototype, t her e are exceptions to this rule. For instance, in the cantil e v e r - b e a m example, w e k n o w f r o m t h e o r e ti c al considerations t h a t the b e a m deflection is actually a function of the m o m e n t of inertia, I, of the beam cross-sectional area r a t h e r t h a n t h e b e a m w i d t h and depth individually. Thus e q (9) could be w r i t t e n as a4 , Ea 2

a --

=

,I,

--

Po

(20)

p -l

w h e r e l N/p/E has the dimension of time and can be combined with t to f o r m a dimensionless t i m e v a r i able. T h e f u n c t i o n a l r e l a t i o n s h i p for t h e strain can, therefore, be w r i t t e n as

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

(21)

w h e r e it is t aci t l y i m p l i e d that t h e f o r m of t h e pressure function, ,I,, is the same in both the m o d e l and p r o t o t y p e system. In dimensionless form, eq (21) can be w r i t t e n as "----f

l'

"T' "' "E"

(22)

The t h r e e pi t e r m s xi/l, Xi/l, and ~, yield the s i m i l a r ity r e q u i r e m e n t s p r e v i o u s l y considered, i.e., we must m a i n t a i n g eo m et r i c similarity, and Poisson's ratio for m o d el and p r o t o t y p e m a t e r i a l s m u s t be t h e same. T he pressure scale is established f r o m t h e condition

(19)

Po

Pom

E

E~

or and eqs (10) replaced w i t h

a4

am4

Thus, the cross section of t h e m o d e l is not r e q u i r e d to be of the same shape as that of the prototype. Since it is f r e q u e n t l y difficult to fabricate a geom e t r i c a l l y similar small model, for example, structural e l e m e n t s such as I - b e a m s are not r e a d i l y a v a i l able in an assortment of small sizes, t h e possibility

Po

E

Pore

Em

We note that if t h e same m a t e r i a l is used in both m o d el and prototype, the pressures at corresponding locations and times must be t h e same. The t i m e scale for t h e p r o b l e m is established from the condition

t

-Z-f=

~

Em tm pm

Experimental Mechanics I 331

i

EDUCATIONAL

I

LECTURE

P

I

- --I%

d

0

I| L/jE,

4-1

I

~]

/

iiIII

/!

____t

/

I ~o

UNIT STRAIN ( ( )

(a)

(c)

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

or t

./Em

tm

V

E

p

l

pm

Im

This r e l a t i o n indicates that, i n general, corresponding times i n 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 t h a n unity, it follows that w h e n modeling with similar materials, corresponding times in the model will be shorter t h a n for the prototype. Thus, i n a p r o b l e m of this type, the loading p r e s s u r e - t i m e relationship, w h e n expressed in terms of p/p~, a n d the dimensionless t i m e variable must be identical, whereas, i n terms of real time, the model and prototype p r e s s u r e - t i m e relationship must be different. F r o m a practical point of view, this is a c o m m o n p r o b l e m e n c o u n t e r e d in d y n a m i c testing, i.e., it is difficult to generate a properly scaled model loading. If all similarity r e q u i r e m e n t s are met, it follows tha~t e =

em

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

Modeling of Inelastic Behavior I n all the examples of s t r u c t u r a l models considered thus far, the material was assumed to obey Hooke's law and, thus, the constitutive equations are completely characterized b y the modulus of elasticity, E, a n d Poisson's ratio, g. Either of these two properties could be replaced b y the m o d u l u s of rigidity, G. We will now consider the p r o b l e m in which the loading is such that the m a t e r i a l is sCrained beyond the proportional limit. To illustrate several i m p o r t a n t ideas related to the d e v e l o p m e n t of similarity r e q u i r e m e n t s

332 I July 1971

for this type of problem, the following simple example will be considered. Let it be r e q u i r e d 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 a n d the stress-strain relationship is also linear b e y o n d the proportional l i m i t to the fracture stress, ~. If u n l o a d i n g occurs prior to fracture, the slope of the u n l o a d i n g curve is g i v e n b y El. Based on experience with elastic materials, a logical list of variables for this problem is as follows: 4, ~, P, d, El, E2,

elongation, gage length, applied load, specimen diameter, modulus of elasticity, modulus of elasticity,

L L F L

FL -2 FL -2

However, f u r t h e r consideration reveals that the stress-strain curve is not completely defined by the parameters, E1 a n d E2, since the stress, or strain, at which the slope of the curve changes is not defined i n the list of variables. Two additional parameters, r and Cs, are required. I n essence, the constitutive equation for the material m u s t b e defined. This is done by: (a) specifying the form of the equation, a n d (b) by defining the p a r a m e t e r s that appear in the equation. For elastic materials u n d e r simple t e n sion or compression, the form of the constitutive equation is i.e., stress a n d s t r a i n are l i n e a r l y related, and the required p a r a m e t e r is E. It should be clearly u n d e r stood that, w h e n e v e r a m a t e r i a l property such as a modulus of elasticity, viscosity, etc., is listed as a

EDUCATIONAL

p e r t i n e n t variable, the f o r m of t h e relationship in which the p r o p e r t y appears is tacitly implied. With the inclusion of the variables ~o and af, d i m e n sional analysis gives

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

(23)

The s i m i l a r i t y r e q u i r e m e n t s from the last t h r e e pi terms in eq (23) indicate that the dimensionless stress-strain d i a g r a m for t h e m o d e l and p r o t o t y p e specimens [Fig. 4 ( c ) ] must be identical. A n o t h e r factor t h a t m u s t be considered in this e x a m p l e is the significance of the s i m i l a r i t y r e q u i r e ment,

P Eld 2 - -

- -

Pm Elmdm2 -

-

during any m o n o t o n ic a l ly increasing or decreasing loading phase, w h e r e E is some characteristic m o d u lus, having dimension of stress, ei, the strain components, and ~/~ a set of dimensionless coefficients. In this case, the characteristic properties of the m a t e r i a l will be the moduli, E and G (moduli f o r n o r m a l and shearing stresses), and a set of dimensionless p a r a m e ters, -~. As far as dimensional analysis is concerned, the use of this set of m a t e r i a l properties w i l l not alter the form of the design conditions from those obtained for linearly elastic materials, but the additional similarity r e q u i r e m e n t s w i l l be Em

G

Gm

and "Yi ~

satisfy the r e q u i r e d conditions r e l a t e d to the n o n l i n e a r constitutive equations if different m a t e r i a l s are used. This same conclusion has b e e n s h o w n in a m o r e rigorous fashion to be t r u e b y B a k e r I~ and additional discussion of this point can be f o u n d in Refs. 6 and i i . If s t r a i n - r a t e effects are important, the p r o b l e m is m u c h m o r e c o m p l e x and a f u r t h e r discussion of this point is given in the last section of the paper.

Distorted Models As discussed previously, a c o m m o n difficulty co u n t er ed in m o d e l studies is t h e e x p e r i m e n t o r ' s ability to satisfy all s i m i l a r i t y r e q u i r e m e n t s . example, in the list of m o d e l - d e s i g n conditions, (7), if

e ninFor eqs

(24)

It can be seen f r o m Fig. 4(b) that different values of elongation can be obtained depending on the loading path. If t h e load is applied m o n o t o n i c a l l y to point (a), a c e r t a i n elongation will be obtained; whereas, if the specimen is strairmd to point (b), t h e n r e t u r n e d to point (c), w h e r e the toad is of the same m a g n i t u d e as at (a), but the elongation will be of a different magnitude. This is due to the fact t h a t the strain above the elastic limit is not a s i n g l e - v a l u e d function of stress. It is clear that eq (24) must be i n t e r p r e t e d to relate not only the m a g n i t u d e s of the applied load in t h e model and prototype, but also to r e q u i r e that the p a t t e r n of loading be similar. F o r the m o r e general case, we m a y assume that any stress component can be expressed in the functional form

E

LECTURE

~'im

with t h e tacitly i m p l ie d condition that the f o r m r of the constitutive relationships are identical for m o d e l and p r o t o t y p e materials. T h e obvious w a y to satisfy these conditions is to use th e same m a t e r i a l s in m o d e l and p r o t o t y p e systems. Thus, we m a y conclude that the similarity r e q u i r e m e n t s for m o d e l i n g li n ear i l y elastic system can be applied to systems i n v o l v i n g inelastic b e h a v i o r if the same m a t e r i a l s a r e used in both model and p r o t o t y p e systems, and if t h e loading history is similar. In principle, the same m a t e r i a l s ar e not r e q u i r e d but it is v i r t u a l l y impossible to

then ~1 ~ ~tlm

and the m o d e l is said to be distorted. U n f o r t u n a t e l y , distorted models are c o m m o n p l a c e and, in general, predictions of p r o t o t y p e b e h a v i o r based on distorted m o d el data m u s t be m a d e w i t h caution. Possible procedures for h a n d l i n g d i s t o r t e d models include: (a)

Neglect cer t ai n v ar i ab l es t h a t m a y be only slightly significant but lead to the distortion. (b) D e t e r m i n e the effect of t h e distortion, a n a lytically. (c) D e t e r m i n e t h e effect of t h e distortion, e m pirically. A l t h o u g h f r e q u e n t l y not r e c o g n i z e d as such, t he first of these m et h o d s is p r o b a b l y t h e most c o m m o n one for h a n d l i n g distortion. In the p r e v i o u s e x a m p l e f r o m t h e field of fluid mechanics, it was n o t ed t h a t a c o m m o n p r o b l e m arises w h e n both the Reynolds n u m ber and the F r o u d e n u m b e r ar e considered to be i m portant. If the same fluid is used in m o d el and p r o t o type, distortion is encountered. In this t y p e of p r o b lem, it is c o m m o n practice to neglect one or t h e ot he r of these numbers, w h i c h in effect m ean s t h a t e i t h e r viscosity or t h e acceleration of g r a v i t y is neglected, and to base the m o d el design on the r e m a i n i n g parameters. In m a n y instances this has been a successful t r e a t m e n t . Also, it is w e l l k n o w n that, for t h r e e - d i m e n s i o n a l photoelastic models, Poissons' ratio, ~, is an i m p o r t a n t m a t e r i a l property. A n d for p r o p e r scaling ~ m u s t be the s a m e for m o d e l and p r o t o t y p e materi, als. Since this co n d i t i o n is seldom satisfied this type of m o d e l is distorted. H o w e v e r , it is recognized that, in m a n y p r o b l e m s of this type, Poisson's ratio is not highly significant and its effect is s i m p l y n e glected, and this p r o c e d u r e is thus a n o t h e r e x a m p l e of m e t h o d (a) for h a n d l i n g d i s t o r t i o n J 2 It is a p p a r e n t that, if t h e n eg l ect ed p a r a m e t e r has any significance (and if it hasn't, it sh o u l d n 't be i nc l u d e d ) , p er f ect co r r el at i o n w i l l not be achieved be t w e e n the m o d el and p r o t o t y p e so that the v a l i d i t y of this t e c h n i q u e depends on h o w accurate the results m u s t be in o r d e r for t h e m to be of value. Also, it is h i g h l y desirable to h a v e some w a y of e s t i m a t i n g the a m o u n t of e r r o r i n t r o d u c e d b y n e g l e c t i n g t h e effect

Experimental Mechanics [ 333

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LECTURE

of the distortion. A l t h o u g h it will p r o b a b l y not be feasible to m a k e such an estimate w i t h a high degree of precision, it m a y be possible to obtain some insight by solving analytically a similar, but simpler, problem. A second m e t h o d for h a n d l i n g distortion is the det e r m i n a t i o n of t h e effect of the distorted p a r a m e t e r a n a l y t i c a l l y so that this effect can be t a k e n into account. This p r o c e d u r e can best be illustrated w i t h a simple example. Consider a p h e n o m e n o n t h a t is g o v e r n e d by t h r e e pi t e r m s so that

=i : $ (=~, =3) Assume that the v a l u e of t h e model pi t e r m ~3m is distorted by an a m o u n t ~ so that

It t h e n follows t h a t ~I :

~ ;~lm

w h e r e 5 is a prediction factor r e q u i r e d to correct for the distortion of :t3m. If the m a n n e r in w h i c h ~3 influences ~l can be determined, the relationship b et w e e n 8 and ~ can be determined. U n f o r t u n a t e l y , in m a n y instances, 8 and /~ will not be s i m p l y r el at ed since

=i =2=

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

w h i c h shows that 8 can be a function not only of but also of ~t2 and n3. H o w e v e r , in certain special cases, w h e r e the distorted pi t e r m is separable, and expressible in the f o r m ~i :

= ~ f~

(=~)

it follows t h a t A detailed discussion of this m e t h o d for handling distortion can be found in Murphy. 7 A t h i r d m e t h o d for h a n d l i n g distortion, w h i ch is perhaps the most practical for problems in w h i ch distortion m u s t be t a k e n into account, is one in w h i ch the effect of the distortion is d e t e r m i n e d empirically. Consider a p r o b l e m in w h i c h four pi t e r m s are i n volved, i.e.,

grossly in error. T h e usefulness of this approach will depend on the p a r t i c u l a r p r o b l e m and the degree of accuracy required. This m e t h o d can be e x t e n d e d to m o d e l studies for w h i c h two pi terms, such as ~2 and n3, are distorCed. F o r this case, a series of m o d el tests ar e r e q u i r e d in which n4m is held constant at t h e prototype v a l u e w h i l e ~2,~ and ~3m are varied. Th e pi term, ~lm, can be r ep r esen t ed b y a point on a surface w h e n Jq,, is plotted versus ~2m and =3rn (Fig. 6). A sufficient n u m ber of points must be d e t e r m i n e d f r o m t h e m o d e l tests so that this surface can be r e a s o n a b l y defined. As in the previous example, it would be desirable to span the p r o t o t y p e values of n2 and ~8 so that the prediction can be m a d e at some point, such as A, on t h e surface. Otherwise, the su r f ace would h a v e t.o be e x t r a p o l a t e d to obtain t h e p r e d i c t e d v al u e of nl. E x t e n s i o n to systems w i t h h i g h e r degrees of distortion is possible but not feasible. Although, in principle, this m e t h o d of e m p i r i c a l l y h a n d l i n g distortion is simple, t h e r e are m a n y practical difficulties. F r e q u e n t l y , to obtain a series of m o d e l tests, it is necessary to v a r y m a t e r i a l p r o p e r t i e s or t h e size of the models. Since t he r a n g e of m at er i al s a v a i l a b l e for t h e m o d e l system is usually quite limited, v a r y i n g m a t e r i a l properties is a difficult, if not impossible, technique. It m a y be possible to construct a series of models of different sizes, but this is usually e x p e n s i v e and t i m e consuming. Thus, w e must conclude that this m e t h o d is not the final answer to all d i s t o r t e d - m o d e l p r o b l e m s and, in general, the use of distorted models r e m a i n s a difficult problem.

Modeling of Complex Coupled Systems As the c o m p l e x i t y of t h e p r o t o t y p e system increases, the m o r e appealing and, perhaps, necessary a model study becomes. Some of the most complex problems i n v o l v e t h e interaction, or coupling, b e t w e e n different en v i r o n m en t s. Fo r example, t h e r e are i m portant problems in which we h a v e coupling b e t w e e n t h e r m o d y n a m i c and structural phenomena, h y d r o d y namic and s t r u c t u r a l phenomena, magnetic fields and

and t h e r e q u i r e d design cortdition ~2 = ~t2m cannot be m et although the o t h e r tw o design conditions ~1 (PREDICTED)

~r~ = ~ l m

B

mi = #r4m are satisfied. If sufficient control o v e r the m o d e l e x p e r i m e n t s is available, w e can r u n a series of m o d e l tests in w h i c h ~2,~ is v a r i e d w h i le h o l d i ng ~t3rn and ~etm constant at the r e q u i r e d p r o t o t y p e values as illustrated in Fig. 5. Ideally, the series of m o d e l tests w o u ld be r u n so that the p r o t o t y p e v a l u e of n~ w o u l d f a l l b e t w e e n the actual m o d e l values, as i l l u s t r a t e d at point A in Fig. 5. If this is not possible, as is f r e q u e n t l y the case, t h e n e x t r a p o l a t i o n is r e q u i r e d as i ll u s t r a t e d a~ point B in Fig. 5. Of course, e x t r a p o l a t i o n is not a desirable p ro ced u re and t h e p r e d i c t e d v a l u e of nl could be

334 linty 19rl

~1 (PREDICTED) ~'lm

/

A

~

~'~'~'~-'1t ~3rn= ~3 = CONSTANT ~4m= ~r4= CONSTANT O EXPERIMENTAL POINTS FROM MODELTESTS

~REQUIREDVALUE ~.~REQUIREDVALUE ~Zm Fig. S--Prediction technique with one distorted pi term

EDUCATIONAL

fluid flow, and soils and structures, to m e n t i o n b u t a few. To illustrate the use of models for the s t u d y of complex coupled systems, a n e x a m p l e of a model study of a soil-structure system is considered in some detail./z This type p r o b l e m is r a t h e r u n i q u e in that the p e r t i n e n t m a t e r i a l properties of the soil are not well defined. However, as d e m o n s t r a t e d i n this e x a m ple, it is still possible to obtain useful results from a model test. The interest i n this problem stems from a desire to establish modeling relationships for the s t u d y of the response of u n d e r g r o u n d structures to blast loadings. The approach t a k e n in this e x a m p l e was to det e r m i n e w h e t h e r or not data could be correlated b e t w e e n small-scale structures of different sizes w h e n tested u n d e r laboratory conditions. This procedure is recommended w h e n e v e r 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 p a r t i c u l a r study u n d e r discussion, the d y n a m i c load was applied by means of a weight dropped onto the surface of the soil in which a hollow cylinder was buried. The p e r t i n e n t variable of interest was t a k e n to be the c i r c u m f e r e n t i a l strain measured on the i n n e r wall of the buried cylinder (Fig. 7). The variables considered i n this s t u d y were: e, c i r c u m f e r e n t i a l strain, FoL~ ~ D, c y l i n d e r diameter, L ~, all other p e r t i n e n t lengths, L p~, density of cylinder, FT2L -4 E, m o d u l u s of elasticity of cylinder, F L -~ ~, Polsson's ratio of cylinder, F~ ~ M, mass of impacting weight, FT2L-Z V, impact velocity of weight, L T - 1 t, time, T p, i n i t i a l density of soil, FT2L -4 ~11,p r o p e r t y of soil, F L -2 ~1~,other soil properties, F L -2 ~ other dimensionless soil properties, F~

~'lm J ~

O

,e4m= ,e4=CONSTANT O EXPERIMENTAL POINTS FROMMODELTESTS O 0

pREDIC TED 0 pREDI oTED / o l / / o ! cr / ~ E D VALUE / I/ /"......... .Z-. . . . ~/ i~,%~f"~ ~ ..... / / _~I ../-// I .//7

L/ REQUI;E'J

VALUE

.............................

For the purpose of ~his study, it was assumed t h a t the soil could be characterized by a set of properties that had dimensions, F L -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 '

M

,o, ED3 ' p~,

Vt~111~i)

MV2' ~' --D"

E'

E ' "Y~

If the same c o m b i n a t i o n of m a t e r i a l s is used i n the model a n d prototype systems, t h e n all s i m i l a r i t y r e q u i r e m e n t s arising from pi terms consisting solely of m a t e r i a l properties are i m m e d i a t e l y satisfied. Other similarity r e q u i r e m e n t s are X~ ~im = - n where the l e n g t h scale, n, is equal to D/Dm,

Mm =

M - n 3

and

Vm= V t tm=-n

If these conditions are met, t h e n it follows that

at corresponding times. To d e t e r m i n e the validity of this model design, a series of tests was r u n with 1-in., 2-in., a n d 4-in.diam hollow cylinders embedded i n d r y Ottawa sand. Figures 8 and 9 show typical results of these model tests. The results are reasonably satisfactory considering the difficulty in o b t a i n i n g data of this type. It should be noted that s t r a i n - r a t e effects were n e glected in this analysis. It can be s h o w n that, if m a t e r i a l properties describing s t r a i n - r a t e effects are added to the list of variables, t h e n a distorted m o d e l wi]l result. Tests of the same type as those r u n w i t h the d r y sand w e r e also p e r f o r m e d w i t h a s a n d - o i l

~

~

IV

0

LECTURE

/ - FALLING ~ WEIGHT

I

~.\>

/

.'" /'/

~

/ //~E,~u!,,p // ~ALUE

8 // (PREmCrEO)~ ~ _~.X/ I

I I

I // J

/ / //

,

//

,2.

"

J

CIRCUMFERENTIAL

TESTCYLINDER

/

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

CYLINDER

SOIL

/

~3m

~-BURIED

\

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

Fig. 7 - - S k e t c h

of drop-weight loader and c y l i n d e r

Experimental Mechanics I 335

E D U C A T I O N A L

LECTURE

1000

I

I

I

~l~

i

I

I

I

x .=_ I .o u

J

I

I

I

I

,

, ,

9

500 O 9 X

i =<

l - i n . CYLINDER 2 - i n . CYLINDER 4 - i n . CYLINDER

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

.<

DRY SAND

I

1 O0

0.I

0.5

5.0

1.0

,

I

,

10.0

DEPTH DIAMETER

I

F ......

i

I d =

"D 800

m i x t u r e and typical results are s h o w n in Fig. 10. Considerable distortion is present, and n u m e r o u s other tests utilizing highly cohesive soils reveal the same kind of distortion, x4 Problems of this type are typical of those for which model studies c a n be e x t r e m e l y useful, i.e., problems not readily amen,able to other methods of study, due to their extreme complexity. The example also illustrates one of the most common difficulties i n modeling, that of being able to adequately describe, and control, m a t e r i a l properties of the model and prototype systems. Much additional w o r k is needed i n this i m p o r t a n t area.

I

DEPTHOF BURIAL : 1 CYLINDERDIAMETER

D = 2 in.

DRY SAND = .e_" /,

i .

600

20(I

References L~

I

0.0

__1

1.0

I

2.0

3.0

__1

[

5.0

4.0

6.0

SCALED TIME, msec

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

8o0

1

I

I

I

~ - 4-1n.CIRCULAR CYLINDER ~

~

OIL - SAND

OF BURIAL

CIRCULAR

r

I O.O

1.0

(

2.0 3.0 SCALED TIME, msec

I 4.0

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

336 I 1uly 1971

__

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: Computation and Simulation, McGraw-HiU Book Co., Inc., New York, 2nd ed. (1959). 4. Murphy, G . , Shippy, D. ]. and Luo, H. L., Engineering Analogies, Iowa State University Press, Ames, lowa (1963). 5. Bridgman, P. W., Dimensional Analysis, Chap. 2, Yale University 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 Deformations of Structures Subieeted to Transient Loading," Proc. of Colloquium on the Use of Models and Scaling in Shock and Vibration, ASME, 71 (1963). 11. Goodier, ]. N., "'Dimensional Analysis," Handbook of Experimental Stress Analysis, M, Hetenyi, Ed., 1ohn Wiley and Sons, Inc., New York (1950). 12: Mdnch, E., "'Similarity and Model Laws in Photoelastle Experiments," EXPElaX~,Z~NT.~r, M E C ~ Z ~ C S , 4 (5), 141-150 (May 1964). 13. Young, D. F. and Murphy, G., "'Dynamic Similitude of Underground Structures," ]nl. Engrg. Mech. Div., Proe. ASCE, 9 0 , 111 (1964). 14. Murphy, G., Young, D. F. and MeConnell, K. G., "'Similitude of Dynamically Loaded Buried Structures," U.S.A.F. Weapons Laboratory Bpt. W L TR-64-142 (1965).