A SIMPLIFIED PROCEDURE FOR STRESS ANALYSIS IN THE

AID OF STRAIN GAGES ELASTO-PLASTIC RANGE WITH THE

by B.N. (Rao

For hydroburst tests, the stresses in the inelastic range are determined from the post-yield strain gages. Keil and J3enningl have described the analytical method in detail for the calculation of stresses from the measured strain values. They have also presented a practical procedure together with the application of a nomograph. Motivated by the work of the above mentioned authors, a simplified procedure, which can be easily programmed on a digital computer, for the stress analysis in the elasto-plastic range with the aid of strain gages is proposed here. The analytical method for elasto-plastic stress analysis used is summarized below.

EXPERIMENTAL STRESS ANALYSIS

Based on a biaxial stress field, an equivalent strain (e,) is determined by substituting the measured principal strains

and e 2 ) into eq (11, derived from the distortion energy principle. The stress (a,) corresponding to the strain ( G ) was obtained from the uniaxial stress-strain curves for the material. In the plastic region, the secant modulus of

elasticity (S) was taken equal to a.. Accordingly, the

‘combined’ Poisson’s ratio for the total deformation lies between Poisson’s ratio v and the ‘plastic’ Poisson’s ratio v = 0.5. The equations are as follows:

E.

where

N* = { 1 - v,(l- V,)}(l- vi)-*

Dr. B.N. Rao is associated with Vikram Sarabhai Space Centre, Structural Engineering Group, Trivandmm, India.

and

N, = { - 1 + V8(4 - u , ) } (1 - JP)-Z

The combined Poisson’s ratio (a secant Poisson’s ratio) v8, is evaluated by the equation

(2) y = - - 1 1 (-- .)- S 8 2 2 E

The three unknowns E., v, and S are to be determined from the two eqs (1) and (2) and the uniaxial stress-strain curve.

A constitutive relatianship that gives the stress as an explicit function of the strain is more useful in this analysis. This relationship is

a = E E { 1 + (L)”}-l‘” (3) e.

where a is the stress, E is the Young’s modulus, e is the strain, E . is the plastic strain, and n is a parameter defining the shape of the nonlinear stress-strain relationship. Note that this single-valued expression represents essentially the inverse of the Ramberg-Osgood equation. However, in eq (31, the stress approaches the plastic stress asymptotically with increasing strain. The constants E~ and n in eq (3) can be determined approximately by

(4) n = In (2) / In (-) E €0

0.

The values of e. and a. in eq (4) are obtained from a point on the curve at the location as indicated in Fig. 1. The accuracy of the representation of the stress-strain curve defined in eq (3) may be improved further by modifying the values of em and n through least-square curve fit.

The value of the secant modulus ( S ) for a specified value of e, can be obtained directly from eq (3) as

38 May/June 1991

500 r

300 c Y K I- YI

Fig. 1-Stress-strain curve of a material - - - eq(3)iE = 105,000 MNlm’ , = 0.00442, n = 4.50321;

0 = data[Ref. 71

Using eqs (2) and (5), one can get the combined Poisson’s ratio, u,, in terms of E. as

(6) 1 1 c, I - 1 1 1

2 2 €0

u, = - - (- - u ) (1 + (-) }

Since, in eqs (1) and (6), the value of u, and E. are inter- dependent, an iteration process is necessary to evaluate proper U, and c, which will satisfy both equations. Once the value of c. is established, the secant modulus (S) can be evaluated directly from eq (5).

With the secant modulus (S) and the combined Poisson’s ratio ( u,), the principal stress u1 and u1 can be calculated by using Hooke’s equations for the biaxial stress conditions

To be valid, eqs (7) and (8) presuppose isotropy, so that coaxiality is guaranteed and, thus, principal stresses and principal strains behave in the same directions. It should be noted that the secant modulus (S) in the proportional limit is equal to the Young’s modulus ( E ) of the material and the combined Poisson’s ratio, u,, is equal to the Poisson’s ratio (v). So the above described procedure can be used for the calculation of the principal stresses in the proportional limit

from the measured principal strains. For this case, the first iteration itself gives the solution.

In order to demonstrate the above-described procedure for the determination of the principal stresses from the measured principal strains, the example in Ref. 1 is con- sidered here. In that example, a strain-gage rosette is applied to the surface of a member made of material (Designation Ti 99, No. 3.7065, Annealed condition) with yield strength, 450 MN/m2, Young’s modulus, E = 105,000 MN/m2, and the Poisson’s ratio, u = 0.3. Loading is applied to the mem- ber until it reaches the elasto-plastic region. From rosette measurements, the principal strains are found to be E, =

Initially, the stress-strain diagram of the material is represented by the relationship as suggested in eq (3). The material constants, c. and n are evaluated. The data generated from the empirical relationship are plotted with the actual stress-strain data in Fig. 1. The iteration process to evaluate proper u, and c, is as follows: initially the combined Poisson’s ratio, u,, is assumed as u and sub- stituted in eq (1) to obtain e. for the specified values of cI and cl . The value of c. is substituted in eq (6) to obtain u,. Using this value of u, in eq (l), E. will be determined. This iteration process has to be continued until these two un-. knowns u, and c, satisfy both eqs (1) and (6) with the desired degree of accuracy. Three to four iterations may be required to get proper values of u, and c,. For this example, the values of u, and c. for each iteration obtained are presented below

0.003, c2 = 0.001.

Iteration ”a c.

0.3 0.003152 0.3086 0.003 18 1 0.3089 0.003182 0.3089 0.003182

The equivalent strain, c, = 0.003182 and the combined Poisson’s ratio, u, = 0.3089. The secant modulus, S = 100323 MN/m2 is obtained from eq (5). The equivalent stress, a, = 319.23 MN/m’. The principal normal stresses at the point of measurement obtained by substituting the values of u, and S in eqs (7) and (8) are ul = 366.48 MN/m2 and ul = 213.48 MN/m2 respectively. The stress analysis results presented in Ref. 1 obtained through the nomograph are found to be less than 3 percent of the results obtained through an iteration process. Since the formulation of the problem is general, the calculation of principal stresses from the measured principal strains in elastic as well as elastic-plastic ranges can be determined. The iteration procedure explained above can be easily programmed on a digital computer.

REFERENCE 1. Keil, S. and Benning, O., “On the Evaluation of Elusto-plastic strains Measured with Strain Gages, ” EXPERIMENTAL MECHANICS, 19 (8), 265-270 (1979).

Experimental Techniquem 99