Integrated Mechanical Testing Laboratory
College of Engineering and Applied Science
Arizona State University

STRESS AND STRAIN CONCENTRATION

Objective

The purpose of this experiment is to demonstrate the existence of stress and strain concentration in the vicinity of a geometric discontinuity in a cantilever beam , and obtain an approximate measure of the elastic (theoretical or geometric) stress concentration factor, Kt.

Theory

The discontinuity here is a simple circular hole , drilled through the depth of the beam on its centerline.

 

 

The sketch shows the stress distribution at two sections of a cantilever beam, and illustrates the presence of stress concentration.

At section A, the stress is uniform across the width of the beam, and calculable from the following relationship:

where :

s= stress, psi (N/m2)
M = bending moment , in-lbs (mN)
I = moment of inertia of beam cross section, in4 (m4)
P = load, lbs (N)
c = half-thickness of beam, in(m)

                    (1)

 

At section B, the nominal stress, based upon the net area of the section, is:

                   (2)

If the location of the hole is selected so that

                   (3)

the nominal stress at section B is the same as that at section A.. The maximum stress at section B, however, is much greater, due to the stress concentration effect. As shown in the sketch, the maximum stress exists at the edge of the hole, on the transverse diameter, and the stress decreases rapidly with the distance from the hole. By definition the stress concentration factor, Kt, is the ratio of the maximum stress at the hole to the nominal stress at the same point. That is,

                   (4)

Since the nominal stresses and the peak stress at the edge of the hole, are all uniaxial, the strain and stress are proportional. Thus, the stress concentration factor is equal to the ratio of the maximum to nominal strains at section B. Therefore,

                   (5)

General

Procedure

In this experiment , the beam will be loaded with the Flexor until a predetermined nominal axial strain level of 2,000m e is reached at Section B (as shown in the sketch). The nominal strain at Section B will be measured , not at that point on the beam, but instead at Section A where the measurement can be made more conveniently and accurately. It is important not to exceed a nominal strain of 2,000m e , since

The actual strain at the edge of the hole is much higher than the nominal, and excessive strain could produce local yielding.

The actual strains in the region of the stress concentration will be measured with three very small strain gages placed in Section B at varying distances from the edge of the hole, with one of the gages directly adjacent to the edge. The strains indicated by the three gages will be plotted at the locations of the respective gage centerlines. A smooth curve can be drawn through the resulting three data points to show the strain distribution in the vicinity of the hole. Since the centerline of the closest gage to the hole cannot physically coincide with the edge of the hole, it is necessary to extrapolate the data to the edge in order to obtain an approximate value for the maximum strain.

The ratio of the maximum to the nominal strain at Section B is the strain concentration due to the disruptive presence of the hole. If the proportional limit of the beam has not been exceeded during the experiment , the stresses are proportional to the strains, and the same ratio represents the stress concentration factor, Kt.

Dr.  Kingsbury, Lab Manager, Integrated Mechanical Testing Laboratory.
Copyright 2003 IMTL, Fulton School of Engineering, Arizona State University. All rights reserved.
Revised: May 25, 2005.