MGJ 27 OCT 96

PURPOSE

The purpose of this exercise is to the study the effects of geometric discontinuities on the stress states in structures and to use photo elasticity to determine the stress concentration factor in a simple structure.

EQUIPMENT

Un-notched beam of birefringent material (an epoxy).

Notched bend beam of the same birefringent material as the un-notched beam.

Four-point flexure loading fixture with load pan and suitable masses for loading

Circular polariscope with monochromatic light source

PROCEDURE

Part 1. Beam under Pure Bending to Determine the Stress-Optical Coefficient of the Material

i) Install the un-notched beam (see Fig. 1) in the four-point flexure loading fixture

ii) Attach the load pan (Note: The combined pan/fixture mass/weight is ~1.204 kg=2.66 lbf)

iii) Apply 2 weights (~9.049 kg = 20 lbf each) one at a time to the load pan.

iv) With the polarizer and analyzer crossed (dark field), focus the camera, and record the image using the thermal printer

v) Determine the maximum fringe orders at the top and bottom of the beam including estimates of fractional fringes orders by counting the fringes.

vi) The stress-optical coefficient can be calculated using the following relation:

(1)

where f is the stress-optical coefficient, is the fringe order, t is the model thickness, and are the plane-stress principal stresses.

Part 2. Notched Beam under Pure Bending to Determine the Stress Concentration Factor

i) Install the notched beam (see Fig. 2) in the four-point flexure loading fixture

ii) Attach the load pan (Note: The combined pan/fixture mass/weight is ~1.204 kg=2.66 lbf)

iii) Apply 2 weights (~2.262 kg = 5 lbf each) one at a time to the load pan. (Note: Do not apply more than 2 of the weights at any one time).

iv) With the polarizer and analyzer crossed (dark field), focus the camera, and record the image using the thermal printer.

v) Determine the maximum fringe orders at the top and bottom of the beam and at the edge of the notch including estimates of fractional fringes orders.

vi) The stress distributions within the beam can be calculated using the relation:

(2)

where f is the stress-optical coefficient determined previously, is the fringe order, t is the model thickness, and are the plane-stress principal stresses.

* REFERENCES

Manual on Experimental Stress Analysis, J.F. Doyle & J.W. Philips, eds, Society for Experimental Mechanics, 1989

Experimental Stress Analysis, J.W. Dally and W.F. Riley, McGraw-Hill, Inc., 1990

Handbook on Experimental Mechanics, A.S. Kobayashi, ed., Prentice Hall, Inc.,1992

Formulas for Stress and Strains, R.J. Roark and W.C. Young, McGraw-Hill, Inc., 1975

Stress Concentration Factors, R.E. Peterson, John Wiley and Sons, Inc., 1974

RESULTS

When loads are applied to a solid body, such as part of a structure or a machine component, stresses which vary from point to point, are set up in the body. At certain points, stress concentrations (sometimes called stress raisers) occur and are potential weak points in the body. Frequently, an alteration in the shape of the body will lead to a reduction in the stresses at such points and to a more even distribution over the whole body. An optimum body is that of uniform load-carrying capability.

The mathematical theory of elasticity provides many valuable solutions involving the stress distributions in bodies of simple geometries and loadings. A common use of these solutions is the determination of stress concentration factors () resulting from discontinuities or other localized disturbances in the stress field of the body. In more complicated problems, commercially available two- and three-dimensional computer programs for finite element and boundary element analyses (FEA and BEM, respectively) can be used to locate and quantify the stress concentrations.

These theoretical and numerical results are exact solutions to problems which may or may not model the actual situations (usually due to assumptions about loads, load applications and boundary conditions). This uncertainty in modeling often requires experimental verification by spot checking the analytical or numerical results. A frequently cited example involves a threaded joint which seldom produces uniform contact at the threads. Contact analyses based on the idealized boundary condition of uniform contact will grossly underestimate the actual maximum stress concentration at the root of the overloaded thread. The uncertainty in the contact condition requires a stress analysis of the actual threaded joint experimentally despite the proliferation of FEA and BEM programs. Experimental stress analysis is also necessary to study nonlinear structure problems involving dynamic loading and/or plastic/viscoplastic deformations. Available FEA programs cannot provide detailed stress analysis of three-dimensional dynamic structures and the constitutive relations for plastic/viscoplastic materials are still under development.

One such experimental procedure often applied to empirically determine stress states is photoelasticity. Photoelasticity is a relatively simple, whole-field method of elastic stress analysis which is well suited for visually identifying locations of stress concentrations. In comparison with other methods of experimental stress analysis, such as a strain gage technique which is a point measurement method, photoelasticity is inexpensive to operate and provides results with minimum effort.

Photoelasticity consists of examining a model similar to the structure of interest using polarized light. The model is fabricated from transparent polymers possessing special optical properties. When the model is viewed under the type (but not necessarily magnitude) of loading similar to the structure of interest, the model exhibits patterns of fringes from which the magnitudes and directions of stresses at all points in the model can be calculated. The principle of similitude can be used to deduce the stresses which exist in the actual structure.

A disadvantage of photoelasticity is the necessity to test a polymer model which may not be able to withstand extreme loading conditions such as high temperature and/or high strain rates. Although photoelasticity is generally applied to elastic analysis, limited studies on photo plasticity and photo viscoelasticty indicate the potential of extending the technique to nonlinear structural analysis. Further details of photoelasticity can be found in listed references.

Show all work and answers on the Worksheet, turning this in as the In-class Lab report.

MGJ 27 OCT 96

NAME______________________________________DATE______________

EQUIPMENT IDENTIFICATION______________________________________

1) The properties of two birefringent polymers often used for photoelastic experiments are in Table 1.

Table 1 Selected Properties of Two Birefringent Polymers Used in Photoelastic Experiments

Homolite 100 (polyester) Epoxy (Araldite, Epon)

Selected Properties (R.T.) Selected Properties (R.T.)

Elastic Modulus, E(GPa)	3.9		Elastic Modulus, E(GPa)	3.3
Proportional Limit	48		Proportional Limit	55
Poisson's ratio,	0.35		Poisson's ratio,	0.37
Stress Optical Coefficient, f (kN/m)*	24		Stress Optical Coefficient, f (kN/m)*	11
Figure of Merit Q=E / f (1/m)	162,500		Figure of Merit Q=E / f (1/m)	300,000

* in green light with wavelength 546 nm

2) For the two beams and loading fixtures, confirm the following information. See Figs. 3 and 4 for nomenclature.

Un-notched beam Notched Beam

Calibration load, Pc =Pweight+Pfixture+Ppan (N)			Test load, Pc =Pweight+Pfixture+Ppan (N)
Outer Span, Lo (mm)			Outer Span, Lo (mm)
Inner Span, Li (mm)			Inner Span, Li (mm)
Height, h (mm)			Height, h (mm)
Thickness, b (mm)			Thickness, b (mm)
Radius of Notch, R (mm)	----		Radius of Notch, R (mm)
Depth of notch, h1 (mm)	----		Depth of notch, h1 (mm)

Note that the calibration and test loads must include the mass of the fixture and pan as well as the masses of the added weights.

3) A unique aspect of the four-point flexure loading arrangement is that the region of interest (the section of the beam within the inner loading span) experiences a pure bending moment as shown in Fig. 4.

For the un-notched beam, determine the following:

Moment of Inertia for the rectangular cross section beam, =_________________mm4

Maximum moment when the calibration load, Pc, was applied, =_____________________N mm

4) At the outer free edge of the beam (y=c=h/2) the stress state is uniaxial and the photo elastic relation can be used to determine the stress optical coefficient directly from the beam bending relation.

The fringe value at the lower, outer edge of the beam recorded at the calibration load, =________________.

Maximum distance to the outer edge of the beam from the neutral axis, c=h/2=___________mm

Maximum uniaxial bending stress at the outer free edge of the beam =__________MPa.

Calculated stress optical coefficient for the material, =________N/mm.

Compare this value to that shown in the table. How do the values compare? Discuss any discrepancies and possible reasons.

5) At the free edge of the notch the stress state is uniaxial and the photoelastic relation can be used to calculate the normal stress using the relation between the fringe order at the free edge, the stress optical coefficient for the material, and the specimen thickness.

The fringe value at the free edge of the notch recorded at the test load, =______.

Calculate normal stress at the free edge of the notch, =________MPa.

6) One way to define a stress concentration factor, kt, is the ratio of the stress at the discontinuity in a body to the stress that would have been at the same point in the body without the discontinuity such that .

The notched beam is symmetric, therefore the neutral axis is the midpoint of the beam. The distance from the neutral axis to the edge of the notch is, =________mm

The moment in the beam at the test load, Pt, is =____________mm.

Stress in an un-notched beam at the same point at the edge of the notch in the notched beam is =_____________MPa.

The stress concentration factor is the ratio of the stresses at the same location for the notched and un-notched beams, =___________.

7) Several authors have compiled stress concentration factors for simple geometries. The most "famous" compilation is Peterson's book of stress concentration factor graphs. A curve fit for a double-notched beam in pure bending (Roarke and Young) is described as follows for 0.25²:²2.0. In this case, =____________ and the stress concentration factor is:

where =___________

=__________

=____________

=____________

such that: =________________

Alternatively, kt can be "picked off" a plot such that kt =________________

8) Compare the kt measured from the photoelastic analysis to that determined from a compiled handbook. Determine the percent differences between the two values. Since many compiled stress concentration factors were determined from photoelastic analyses, discuss possible reasons for differences between the measured and compiled values of kt.

9) Extra effort: Using the fringe orders across the notched beam, assume the stress state is uniaxial, plot the stress across the height of the beam. Compare this stress distribution to that of the un-notched beam at the same load.