APPLICATION OF 3-D PHOTO- ELASTICITY TO AXI-SYMMETRICAL

PROBLEMS

bY CARLOS DEL RIO

Universidad Nacional de Ingenieria, Lima, Peru

Notation r, 0, cylindrical co-ordinates

o r 9 0 6 0,

Trg, Tn, TI32

%emax, % h y 782-

t,

4

direct stresses shear stresses

maximum shear stresses thickness of meridional

thickness of transverse Slit%

n, fringe order observed at any point of meridional slice.

"t fringe order observed at any point of transverse Slice. Model fringe/stress value for meridional slice Model fringelstress value for transverse slice

qrn Isoclinic parameter in meridional slice

R Radial body force referred to unit volume.

The experimental analysis of stresses and strains in three-dimensional problems, and particularly three- dimensional photoelasticity 1-5, finds many applica- tions in modem industry. Many practical problems are characterised by axial symmetry, for instance atomic reactors6, rocket propulsion 7, cylindrical vessels of complicated shape8.9. This Note concerns a special approach to these problems by means of photoelasticity.

Consider a loaded structural component with axial symmetry of shape and loading in a cylindrical co- ordinate system (Fig. 1). The state of stress is to be determined.

A model is made of a material suitable to three- dimensional photoelasticity4s.5 and tested in the frozen- stress technique7. Two types of slices are obtained, meridional and transverse (Fig. 2). The fringes in all the slices and the isoclinics in the meridional slices then permit an analysis of the stress distribution.

Slice

F%

F'a

Fig. 1 Typical axially symmetrical problem.

i B

A

Fig. 2 Dissection after stress freezing. A=transverse slices of thickness 1,.

B=meridional slices of thickness t,.

The equilibrium condition for the radia1:directionlis

For axial symmetry

so that with R=O

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The second term can now be referred to the maxi- mum shear stress in the meridional plane,

7, = ’nmax sin 29,

‘Sn- = n, Fm, (4)

f (a8-cr,)=2n,F 1, (5)

(3) Photoelastic observation gives

The positive sign here and in the subsequent equations indicates oe > 0,. For equal thickness of meridional and transverse slices (t,=b=t; Fm,= F,,=F,) the equations (2) to (5 ) lead to

asin29m az “1 r + n,.--f--‘ =O (6)

Integration in the radial direction from r =r, to r= rl at a given axial position z=zi yields in terms of fringe units

or also

where v.p. indicates the average value of the respective function in the region defined by z = zi, r = r,, r = rl.

Hence if or is known at the point r=r,, it can be calculated for any other point r, by means of (8) where the value of nt is obtained by analysis of the fringes in the transverse slice whilst n,, qm, an,/az and a (sin 29,)/az are determined by the analysis of fringes and isoclinics in the meridional slice.

References

(1) HET~NYI, M., “The Fundamentals of Three- dimensional Photoelasticity”, J. Appl. Mech. 1938, V O ~ . 5, NO. 4, pp.149-155.

(2) DRUCKER, D. and MINDLIN, R., “Stress Analysis by Three-dimensional Photoelastic Methods”, J. Appl. Phys., 1940, V O ~ . 1 1, pp.724-732.

(3) MESNAGER, M., “Sur la dktermination optique des tensions inthieures dans les solides A trois di- mensions,” Compt. Rend., 1930, vol. 190, p.1249.

(4) LEVEN, M. M. and WAHL, A. M., “Three-Dimen- sional Photoelasticity and its Application in Machine Design”, Trans. A.S.M.E., Nov., 1958, V O ~ . 80, pp.1691-1694.

(5) SAMPSON, R. C., “A Three-Dimensional Photo- elastic Method for Analysis of Differential Con- traction Stresses”, Exp. Mechanics, Oct., 1963, VOI. 3, NO. 10, pp.225-237.

(6) TAYLOR, C. E. and SCHWEIKER, J. W., “A Three- Dimensional Photoelastic Investigation of the Stresses Near a Reinforced Opening in a Reactor Pressure Vessel”, Proc. S.E.S. A., 1959, vol. XVIII, NO. 1, pp. 25-36.

(7) DURELLI, A. J. and PARKS, V. J., “Photoelasticity Methods to Determine Stresses in Propellant- grain Models”, Exp. Mechanics, April, 1962,

(8) DURELLI, A. J., PARKS, V. J. and DEL Rro, C. J., “Experimental Determination of Stresses and Displacement in Thick-Wall Cylinders of Com- plicated Shape”, 1966 SESA Annual Meeting, Pittsburgh, Pennsylvania, Nov., 1966.

V O ~ . 2,No. 4, pp. 102-109.

Once or is known, the stresses 0 0 follow from (9) DURELLI, A. J., PARKS, V. J. and DEL RIO C. J.,

“Stresses, Strains and Displacements Associated oe = or f 2 4 F,. (5A) and the stresses a- from with the Restrained Shrinkage of Cylinders with

~~

oz = or f 2n, F, cos 29, Toroidal Cavities”, Mechanics Division, The Catholic University of America, Washington D.C. May, 1966.

In (6A) the minus sign indicates or > 0,

The analysis may also start with the equilibrium condition in the axial direction,

a,, + aT, + T,, - - - = o a Z ar r

where 2, is defined by the equations (3) and (4). In a simplified form, with the square bracket in

equation (8) equated to zero, this analysis has been used in Ref. 9.

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