Eliminating the Body-force Effect in Stress-frozen Models As extremely small forces are employed for loading stress-frozen photoelastic models, the weight of the model often contributes a disproportionately large share of the total model loading. This technical note examines the problem for a typical model analysis and several practical methods of eliminating the body-force effect are presented

by Robert Mark

Introduction In the design of a model analysis, it is usually impossible to maintain the ratio of externally ap- plied force to body force that exists in the pro- totype. This is true particularly where the model is to be subjected to discrete loadings. The effect caused by this disparity is best described by considering the model-proto type strain ratio (a) resulting from (1) the applied force acting alone, and (2) the body force acting alone. A simple derivation using principles of dimensional anal- ysis 1 gives*

where Q = externally applied force E = Young's modulus k = characteristic length p = density e = strain

and the subscripts m and p refer to the model and prototype.

Complete similitude would of course require tha t a~ = a2. The impossibility of achieving this is best illustrated by considering a typical model analysis: an aluminum plate subjected to discrete static forces is to be analyzed using a one-quarter scale stress-frozen epoxy model (Fig. 1). The strain ratio (a) is generally chosen to give maxi- mum sensitivity to the experiment. The maximum fiber strain in a well-designed aluminum proto type will be of the order of 10 -3 . This will usually correspond to a strain in the photoelastic model of about 10 -3 . Hence a ~ 10. Substi tut ing this value and data from Table 1 into eq (1) yields:

Q'~ = 1.25 (10)~4 q,

Obviously there will be no difficulty in maintaining a~ = 10 for the applied force. However, sub- sti tuting the tabulated data into eq (2) indicates a strain ratio due to the body-force of as = 590, which is entirely incompatible with the desired

Robert M a r k is Lecturer, Department of Civil Engineering, Princeton University, Princeton, N . J .

* Complete geometric similarity between model and prototype is assumed. The effect of any difference in Poisson" s ratio is not considered.

TABLE 1--CHARACTERISTICS OF MODEL AND PROTOTYPE

Model (Hysol 4290 Parameter Prototype (a luminum) epoxy at 270 ~ F)

p 165 Ib/ f t 3 78 Ib/ f t 3 E 107 psi 2(10) 3 psi X 1 1/4

strain ratio of 10 for the applied force. This problem is almost always present with thin-plate and shell-segment models, where the unsupported spans tend to be long compared with their thick- ness. Setting as = 0 by eliminating the body-force effect is generally the best solution. Several possible methods to accomplish this are discussed in the following sections.

Analytical Correction I f the configuration of the nodel is relatively

simple, and the model weight is not large compared to the applied loading, the body-force effect may be estimated analytically. This procedure is often used in design applications.

Superposition As stress-frozen models are sliced after loading,

it is impossible to separate the effect of the body force experimentally. The only way to achieve this is to employ two models, one loaded by dead weight only, and the other with the applied load and dead weight. In addition to the disadvantage of requiring two models to be fabricated, a further

Fig. } - -Epoxy model of an a luminum plate

Experimental Mechanics ] 239

difficulty can arise in tha t the second model may be subjected to large nonlinear deflections because of the excessive total loading. For these reasons the experimental approaches described below are more expeditious.

TABLE 2--PROPERTIES OF PURE GLYCERINE 4

Specific gravity 1.26 Color Water-white Boiling point 220 ~ C Heat of vaporation 21,060 cal. /mole at 55 ~ C

Rotating the Gravity Field Figure 2 shows the plate model of Fig. 1 in a test

fixture which places the gravi ty field in the plane of the model. Hence, very small compressive stresses are substi tuted for the large bending stresses tha t result when the field is perpendicular to the plate surface. (Small bending stresses are also produced as the model deflects; however, these were judged to be negligible.)

Concentrated forces were simulated by wires passing through very small holes acting on small circular plates at the model surface. A uniform line loading about the circumference of the large opening was applied by a rigid circular plate of slightly larger diameter than the opening, bearing against a thick silicone-sponge-rubber pad. Hori- zontal forces were applied using long 45-deg re- acting wires as shown in the figure so tha t hori- zontal force components were equal to vertically acting weights.

Because of the high rate of thermal expansion of epoxy, the model supports were a t tached to a second epoxy base plate. This allowed the model and its supports to expand together when the assembly was subjected to elevated temperature. Thus, interface sliding caused by differential ex- pansion took place between the base plate and the metal frame.

Predicted results based on the model tests were compared with measurements made on several of a

Fig. 2--Vertical-support test fixture

series of aluminum pro to types . t The general agreement was rather good (8-12 percent) and verified the soundness of the approach in this case. 2

Flotation This is the least elaborate of the experimental

approaches and hence may be the most attractive. The specific gravity of cast epoxy 3 is 1.25 while U.S.P. grade glycerine has a specific gravi ty of 1.26. Other properties of the glycerine are given in Table 2.

The model is supported and loaded while im- mersed in glycerine in a container placed in the controlled-circulation stress-freezing oven. Be- tween 200-300 ~ F, a fair amount of nontoxic, almost odorless vapor is given off from the sur- face of the bath so tha t it is best to cover the con- tainer during the test. However, even without covering, fluid losses from a large container were not observed to be significant.

Our experience indicates tha t U.S.P. grade glycerine is inert with respect to the epoxy. To keep excessive quantities of water vapor from being dissolved into the glycerine, models are removed from the bath at 150 o F, and the glycerine is re- turned to a closed storage container. We have seen no indication of deleterious edge activi ty after subjecting immersed epoxy specimens to re- peated stress-freezing cycles.

The glycerine bath has also been used for an- nealing complex flexible epoxy-model assemblies after they have been cemented together so as to measure and verify their stress-free state.

Acknowledgments The reported results were obtained during the

course of a shell-research program being carried on in the Depar tment of Civil Engineering at Prince- ton, supported by a grant from the National Science Foundation, and from investigations per- formed for the Plasma Physics Laboratory, For- restal Research Center. The author wishes to ex- press appreciation to the Foundat ion and to Jorge Riera, graduate student in the Depar tment of Civil Engineering, for his contributions to this work.

References 1. Goodier, J . N . , "Dimensional Analysis ," Hdbk. ExptL Str. Anal.,

ed. by M. Hetdnyi, John Wiley and Sons, 1036 (1952). 2. Mark, R., "Elastic Model Analys is of a Thin Plate Structure Sub-

jected to Static Lateral Loads," Princeton Univ., Plasma Physics Lab., Tech. Memo. No. 192 (March 1964).

3. Hysol Data Sheet 6000-OP-1, Houghton Labs. (Hysol Corp.), Olean, N . Y . (1958).

4. Encyclopedia of Chemical Technology, ed. by K i rk and Othmer, The ]nterscience Encyclopedia, Inc., New York, 7, 216-222 (1951).

t The prototype design was determined by the model tests.