Stress Concentration
Introduction:
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 to obtain an approximate measure of the elastic stress concentration factor, K. In this instance, the discontinuity is simply a circular hole, drilled through the depth of the beam on its centerline.
Figure 1 shows the stress distribution at two sections of a cantilever beam, and illustrates the presence of stress concentration.
Figure 1
At section A, the stress is uniform across the width of the beam, and calculated from the following equations:
s _{A} = M_{A} c / I_{A} = 6 PL / (bt^{2})
where:
s = Stress, psi
M = bending moment, inlbs
I = Moment of inertia of beam cross section, in4
P = Load, lb
c = halfthickness of the beam, in
At section B, the nominal stress, based upon the net area of section, is:
s _{B (NOM)} = M_{B} c / I_{B} = 6 PL / [(bd)t^{2}]
If the location of the hole is selected so that l/L = (bd)/b, 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 Figure 1, the maximum stress exists at the edge of the hole, on the transverse diameter, and the stress decreases rapidly with distance from the hole. By definition, the stress concentration factor, K, is the ratio of the maximum stress at the stressraiser to the nominal stress at the same point. That is,
K = s _{B} (max) / s _{B (NOM) }= s _{B} (max) / [6 P l / [(bd)t^{2}]
Since the nominal stress at both sections of the beam, and the peak stress at the edge of the hole, are all uniaxial, the strain and stress are proportional, if the proportional limit of the beam material is not exceeded in the experiment. Thus, the stress concentration factor is equal to the ration of the maximum to nominal strains at section B. Therefore,
K = e _{B} (max) / e _{B (NOM) }= e _{B} (max) / e _{A}
Equipment and Supplies:
Flexor, cantilever flexure frame
High strength aluminum alloy beam, 1/4 x 1 x 121/2 in. with stress concentration. Strain gages preinstalled
Model 1011 Portable Strain Indicator
Procedure:
In this experiment, the beam will be loaded with the Flexor until a predetermined nominal axial strain level of 2,000 m e is reached at section B. (See figure ) 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 2,000 m e . Since the actual strain at the edge of the holes is much higher than the nominal, and excessive strain could produce local yielding.
The actual strains in the region of 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 on the graph sheet at the locations of the respective gage centerlines. A smooth curve will 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.
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 material has not been exceeded during the experiment, the stresses are proportional to the strains, and the same ratio represents the stress concentration factor, K.
Acquisition of Data:
Back the calibrated loading screw out of the way, and insert the beam into the flexor with the gaged end in the clamp, and with the gages on the top surface. Center the free end of the beam between the sides of the Flexor, making certain that the end of the beam is inserted into the clamp as far as it will go, and firmly clamp the beam in place with the knurled clamping screw.
The gages will be connected (via the Flexor cable) to the strain indicator one at a time, first with the beam undeflected, and again with the beam deflected. An initial "reference" reading of the strain indicator measuring dial will be obtained for each gage with the beam undeflected, and a final reading with the beam deflected. The differences in these two sets of readings will give the strains at the respective gage locations.
Connect the strain gage leads from the beam to the binding posts of the Flexor as shown in Figure 2. With the loading screw clear of the beam, connect one of the two common leads in the Flexor cable, to the Sbinding post of the 1011 Strain Indicator, and the other common lead to the D2 binding post. Connect the independent lead from Gage # 1 to the P+ binding post. Complete the bridge circuit with the builtin 120ohm precision resistor by connecting the shorting link between the P and D1 binding posts.
(a)
(b)
Figure 2. Wiring Diagrams for Flexor (a) and Strain Indicator (b)
Set the gage factor adjustment of the strain indicator to the value given on the beam for Gages # 1, #2 and #3. This gage factor setting will also be used for Gage # 4 .
Warning: Do not readjust the gage facto setting during the course of the experiment, even though the gage factor of Gage # 4 differs from that of Gages # 1, #2 and #3.
A simple correction will be made later to account for the difference in gage factors, if it exists. Set the measuring control at 5,000, representing 5,000 m e. The zerobeam deflection reading of the strain indicator for Gage # 1 should be recorded as 5,000 m e.
Warning: Do not adjust the balance control again during the experiment.
Turn the strain indicator off, and disconnect the independent gage # 1 lead from the P+ binding post of the instrument. Connect the independent lead from gage ‚ in its place and record the initial reading for gage # 2. Repeat the same procedure for gages # 3 and # 4. Record the initial reading for all for gages.
After obtaining and recording the initial reading for gage # 4 , and while this gage is connected to the strain indicator, add 2,000 m e to this initial reading, and deflect the beam until the strain indicator reads the final value of strain. Record this value of strain as the final reading . The difference between the initial and final reading will give 2,000 m e . Disconnect the gage # 4 and connect gage # 3 and record the final reading. Repeat the same procedure for gages # 2 and # 1 . Record the difference between the initial and final readings for each gage. Use the following table.
Work Sheet
Gage 
Initial Reading 
Final Reading 
Strain 
1 
5,000 


2 



3 



4 


2,000 
The strain indicated by gage # 4 can be corrected for gage factor by the following relationship:
Corrected e _{4} = 2,000 x (gage factor setting of instrument / gage factor of gage # 4 )
The result of this calculation is the nominal strain at sections A and B of the beam, and should be entered in the worksheet.
The following are the data necessary for the calculation of stress concentration factor:
R = Radius of the hole 
0.125 in. 
X1 = Distance between the center of the hole and gage #1 
0.145 in. 
X2 = Distance between the center of the hole and gage # 2 
0.185 in. 
X3 = Distance between the center of the hole and gage # 3 
0.325 in. 
Correction of e _{4} for gage factor:
Corrected e _{4} = e _{4}^{'} = 2,000 x Gage factor setting of Instrument/gage factor of gage # 4
e _{4}^{' }= 2,000 x ( ) / ( ) = m e
Maximum strain at the edge of the hole:
e _{0 }= A + B + C
In order to calculate this strain we have to find the coefficients A, B and C first.
Computation of coefficients for extrapolation:
C = 5.86 (e_{1}  e_{2})  5.44 (e_{2}  e_{3})
C = __________________
B = 3.49 (e_{1}  e_{2})  1.20 C
B = _________________
A = e 1  0.743 B  0.552 C
A = _________________
Maximum strain at the edge of the hole:
e _{0 }= A + B + C
e _{0 }= ( ) + ( ) + ( )
e _{0 }= ________________
Stress Concentration factor:
K = e _{0 }/_{ }e_{ 4}^{'}
K = e _{0 }/ ( )
K = _________________
Report:
Using the formulas above, calculate the stress concentration factor, K
Plot the strains e _{0}, e _{1},e _{2} and e _{3} versus the corresponding dimensionless distance X/R to visualize the stress distribution in the vicinity of the hole. Note that for e _{0}, R/X = 1 at the edge of the hole.
Last update: April 12, 2000 
Prepared by: Serdar Z. Elgun 