I.
Objective
The purpose of this lab was to examine the curvature and “plastic hinge” formation in several different specimens.
We used samples that differed in material, cross-sectional area, length, and how much they had been annealed beforehand.
Several specimens were exposed to a downward load onto a circular die, and we recorded data on their final loaded
curvature, as well as the unloaded curvature due to springback. Additionally, we also studied the formation of a “plastic
hinge” when an encastered specimen was exposed to a positive moment. As a result of performing these experiments, we
as engineers were able to deduce experimental results from these tests, and compare the experimental data to theoretical
results derived from formulas.
II.
Experimental Methods
2.1
Manual Stretch Forming
The first experiments were performed by the lab technician. He began by stretch-forming specimens of 6061-T6
Aluminum over cylindrical dies. Samples of Aluminum with thicknesses of 1.6 mm and .8mm were formed over
cylindrical dies with curvature radii,  , of 380 mm and 300 mm. Each specimen was connected to a strain gage, which
measured the “microstrain” of the specimen as it was formed over the die. Microstrain is simply the actual strain
experienced by the specimen multiplied by one million. After each specimen was “fully loaded,” meaning that it had been
fully formed over the cylindrical die, we recorded the surface strain in both the loaded and the unloaded configurations.
We performed a similar test on .8 mm 6160-T6 Aluminum over dies that had curvature radii of 300mm, 150mm,
100mm, and 80mm respectively. We recorded the surface strain experienced by each specimen.
2.2
Plastic Hinge
We performed this test in order to simulate a loading moment on an encastered beam. We used a specimen of
6061-T6 Aluminum with the following dimensions: width = 46.5mm, thickness = 1.535 mm and length = 305 mm. We
clamped down one end of the specimen to a table, leaving about 205 mm of the specimen unclamped and hanging over
the edge. This is the part of the beam that is not encastered. We used a ruler that was mounted on a stool and calibrated it
to be even with the unloaded, undeformed surface of the specimen, so that it would be easier to tell when the specimen
began to plastically deform. After the beam was secured, we used a force measurement gauge to apply a force at the free
end of the beam. While one student began applying an upward load orthogonal to the surface of the beam, another student
used a tape measure to measure the length of the moment arm. This length begins at the encastered edge of the beam, and
extends to the point on the beam where the force is being applied. We recorded the load required to bring the specimen
into a semi-plastic state, as well as the load required to create a plastic hinge in the specimen. The formation of a plastic
hinge occurred when the specimen continued to experience further strain while the applied load stayed constant. After
recording the loads at first yield and collapse, we also recorded the length of the specimen that remained elastic. Simply,
we used a ruler to measure the distance of the beam that remained straight in the unloaded configuration.
2.3
Matched Die Forming
In order to examine specimens using the Matched-Die Forming technique, we used a large mechanical tensile
testing machine to press individual specimens between two matching male and female dies. This test was performed on 5
different specimens:
Material
HR Steel 1020
6061-T6 Aluminum
6061-T6 Aluminum
6061-T0 Aluminum
6061-T0 Aluminum
Width (mm)
51.3
25.3
25
25.8
25.8
Thickness (mm)
1.5
.8
1.6
.8
1.6
Figure T-1: Table of Dimensions for Each Specimen
1
Length (mm)
153.4
152.8
152.6
152.4
152.6
Each specimen went through two tests. The first test used dies with  = 80mm, and the second test used dies with  =
40mm in radius. During the first test, each specimen was exposed to a maximum load of 4 kN as it was bent around the 80
mm die. During the second test, loads were variable (an average load of about 3.7 kN). After each test was performed, we
recorded the unloaded radius of curvature, unloaded , of each specimen.
III.
Results
3.1
Manual Stretch Forming
Below is a table of the recorded microstrains obtained from the strain gage of each specimen of 6061-T6
Aluminum:
Thickness
 loaded B10
( die =380 mm)
 loaded B10
( die =300 mm)
 loaded B10
( die =150 mm)
 loaded B10
( die =100 mm)
 loaded B10
( die =80 mm)
1.6 mm
.8 mm
2100
1050
2660
1330
5320
N/A
7980
N/A
9975
N/A
6
6
6
6
6
Figure T-2: Recorded Microstrains for 6061-T6 Aluminum over variable die radii
As indicated by Figure T-2, we did not record microstrains for the .8mm specimen over 150mm, 100mm and 80mm dies.
3.2
Plastic Hinge
After we had manually applied a steadily increasing load to the encastered specimen, we first encountered yield in
the steel at a load of 35 N. This occurred when the specimen no longer had a perfectly linear shape when unloaded. The
moment necessary to bring the specimen into the plastic regime can be calculated as follows:
` a
M e  F e Arsin 
[1]
` a
In our experiment, we measured the length of the moment arm orthogonal to the applied force ( sin   1), and at first
yield, this length was 195 mm. Using equation [1], we find that M e = 6.835 N*m. We continued to increase the load until
we observed the formation of a plastic hinge. Hinge formation occurred at 44N, with a moment arm of .185 mm. Using
equation [1], except substituting values for F LL and the current moment arm, we can derive M LL = 8.14 N*m. Once we
documented the forces and moment arms required to yield and collapse the material, we measured the extent of the beam
that remained elastic during the process. This is the section of the beam that remained linear and did not bend under the
load. For the measured hinge extent, we observed that 125mm of the 200mm beam remained elastic.
3.3
Matched Die Forming
After performing matched die experiments on 5 specimens, we recorded the values of strain and curvature for
each specimen using a circular chart. Figure T-3 is a table of these recorded values for unloaded radius,  , and curvature,
 , for each specimen when formed over 80mm and 40mm dies. Refer to Figure T-1 for the dimensions of each specimen.
b
c
b
c
b
c
b
c
Material
Thickness
 die  80mm
 die  80mm
 die  40mm
 die  40mm
6061-T6 Al
6061-T6 Al
6061-T0 Al
6061-T0 Al
1020 Steel
1.6 mm
0.8mm
1.6mm
0.8mm
1.5mm
240mm
800mm
100mm
120mm
90mm
4.2m-1
1.25 m-1
10 m-1
8.3 m-1
11.25m-1
60mm
100mm
45mm
50mm
45mm
16.7m-1
10m-1
22.5m-1
20m-1
22.5m-1
Figure T-3: Experimental values of curvature and radius for each specimen
2
IV.
Discussion
4.1
Manual Stretch Forming
For the discussion of this particular experiment, I will consider the .8mm and 1.6mm specimens of 6061-T6
Aluminum. In order to obtain a theoretical value for the surface strain experienced by each specimen, the following
equation can be used:
x2
` a
` a
11 x 2  @ x 2  @ ffffff

[2]
For the purpose of this experiment,  is the radius of the die and x 2 is half of the thickness of the specimen (because the
strain on the surface is half the thickness from the neutral axis of the beam). The negative sign at the beginning of the
formula indicates that a strain at the top surface is tensile, whereas the strain at the bottom surface is tensile (since the
moment is negative). Using this formula and inputting values for die and x 2 , we can construct a new table, Figure T-4,
with theoretical strains derived from formula [2]. Figure T-4 compares theoretical surface strains with surface strains
recorded from the strain gages.
b
c
b
c
b
c
b
c
T(mm)
 theoretical die  380mm
 exp die  380mm
 theoretical die  300mm
 exp die  300mm
1.6
0.8
2105
1052
2100
1050
2667
1333
2660
1330
Figure T-4: Theoretical and Experimental Surface Strains for Aluminum Specimens
As is evident from this table, surface strain values obtained from the strain gages are nearly identical to the surface strains
derived from equation [2].
With these strain values in mind, we can calculate whether or not each specimen has exceeded the elastic regime
using the following relationship:
y
 y  ffffffff
E
[3]
If the surface strain exceeds  y , then the specimen has reached yield stress and is no longer fully elastic. For 6061-T6
Aluminum, which has a Young’s Modulus of 69,000 MPa and a yield stress of 300 MPa, microstrain  y ~ 4350. With this
value in mind, we can conclude that neither of the specimens should have exceeded the elastic regime after being exposed
to a 380mm and a 300mm die. We can calculate the curvature needed to plastically deform a specimen using the
following formula:
y
y
e a hffffff 2 fffffffff
fffff
Eh
[4]
2
Substituting in values for yield stress, Young’s Modulus and height, we derive  e = 5.43 (  e = 183 mm) for 1.6mm
specimen and 10.87 (  e = 91 mm) for .8mm specimen. Therefore, it can be concluded that the radii of each die was too
big to yield each specimen.
4.2
Plastic Hinge
Earlier in the Results Section (3.2), the experimental results for yield load, collapse load and length of the beam
that remained elastic were presented. We can derive theoretical results for these values. In order to calculate the load
required to yield the specimen, we can first derive an expression for the stress as a function of the length and height of the
specimen:
b
c
L @x1 x 2
Mx 2 @P
11  fffffffffffff
 fffffffffffffffffffffffffffffffffffffffffffffff
I
I
3
[5]
Taking x1 to be zero (since this is the location of maximum stress), we can rewrite the equation for stress:
b
c
 y x2
xffffff
2
11 x1  0  @ ffffffffffffffff

@2
y
hfffff
h
[6]
2
Next, by deriving the equation for yielding moment and substituting in  11 and x 2 =(h/2), we can come up with an
expression for yielding moment, and yielding load, in an encastered beam:
b
c
b
c
PL
M e  R@11 x1  0 x 2 dA  R@11 x1  0 x 2 wdx 2  R fffffffffx 22 wdx 2
I
2
ffffffffffffffffffff
bh
y
# Me 
6
2
ffffffffffffffffffff
M
y bh
ffffffffff
e
# Pe 

L
6L
[7]
[8]
[9]
By using the dimensions discussed in Section 2.2, and the resultant moment arm discussed in Section 3.2 (original length
is used in small deflections, but in the case of large-scale deflection, using the current moment arm provides a more
accurate result), as well as  y = 300 MPa, we can plug these values into [9] and calculate P e ~ 28.1 N, which is about 7
Newtons lower than the experimental value. This discrepancy could be chalked up to human error. The beam might have
already left the elastic regime, but its non-linear shape might not have been obvious, or even visible for that matter.
In a similar manner, we can theoretically calculate the collapse load. By the time of collapse, the stress profile of
the beam is a constant -  y above the beam’s neutral axis and  y below. Using this property, we can calculate collapse
moment, M LL , of the beam:
[10]
M  R y x 2 dA
 y bh
# M LL  ffffffffffffffffffff
4
2
ffffffffffffffffffff
y bh
# P LL 
4L
2
[11]
[12]
Using the same process as discussed in the yielding load case, and substituting the values into [12], we can calculate a
collapse load of ~44.4 N, which is very close to the measured experimental value. Since this collapse load does not
depend on judgment of whether or not a line is straight, an experimental value that is very close to the theoretical value is
easily obtained.
4.3
Matched Die Forming
By using equation [8], the elastic bending moments for the 1.6mm and .8mm Aluminum specimens can be easily
1.6mm
calculated. By inputting the dimensions from Figure T-1, and using  y = 300 MPa, we can derive the following: M e
A8mm
= 3.24 N*m, and M e
= .81 N*m. In addition to this, we can also calculate the elastic curvature of each specimen
using [4]. In doing so, we conclude that 1.6mm
= 5.43m-1 and Ae8mm = 10.87m-1. By using these theoretical values for M e
e
and  e , we can construct an Elastic-Perfectly Elastic Idealization plot of each specimen. This plot represents the behavior
of a specimen that goes from perfectly elastic to perfectly plastic after it reaches M e . In other words, the plot has an initial
slope equal to E*I (Young’s Modulus * Moment of Inertia) until it reaches M e , where the slope goes to zero. The plot is
idealized because in reality, there is a transitional period after M e where the specimen exhibits both elastic and plastic
qualities. Elastic-Perfectly Plastic Idealizations for each specimen can be found in Figures A-1 and A-2 attached to the
end of the report.
These idealizations are superimposed with the experimental Moment vs. Curvature plot. This makes it easy to
compare the idealized case with the experimental case. Both plots are plotted up to a loading curvature of 25m-1, which
corresponds to a 40mm radius (the radius of the second die used). The experimental plots were obtained by recording the
4
unloaded curvature values of each specimen at each die on the abscissa, and then tracing a line with a slope equal to the
initial elastic slope and plotting the point where this line intersects with a vertical line at curvatures 12.5m-1 and 25m-1.
Although these plots were obtained experimentally, theoretical values for plot points can be obtained using the
following:
H
I
L
@c
h
hffffff
2
c
3
M Ebc
fffffffffffffffffff
l
 y x 2 dx 2  Z  y x 2 dx 2  Z  y  2 dx 2M
K 3   y bj
hffffff
@
@c
c
M  @bLJZ
f
g2
hffff
2
i
k [13]
@c 2m
2
Substituting in the following for one power of  y :
Ec   y
[14]
We can obtain an expression for the moment while the specimen is in the elastic-plastic regime:
f
# M  b y
g
2
2
hfffffff cffffff
@
4
3
[15]
Using values from Figure T-1, as well as the curvatures for each die, it is possible to find the theoretical moment at each
loaded curvature and plot the theoretical Moment vs. Curvature plot for both specimens.
On Figure A-1 and Figure A-2, two instances of unloading were documented at the loaded curvatures of each
die. We can formally calculate these unloaded curvatures using:
M loaded
unloaded  loaded    loaded @ ffffffffffffffffffffff
EI
[16]
where loaded is the curvature of the die, M loaded is the moment at each curvature found from [15], E is the Young’s
Modulus and I is the bending moment of inertia. By using the reciprocal of the result found in [16], and using [2] to find
the surface strain in each case ( x 2 = thickness/2) , we end up with:
b
c
b
c
b
c
b
c
Thickness [mm]
unloaded die  80mm
 surf die  80mm
unloaded die  40mm
 surf die  40mm
1.6
0.8
239mm
667mm
.003347
.000600
59.9mm
99mm
.01335
.004040
Figure T-4: Theoretical Unloaded Curvatures and Surface Strains for 6061-T6 Aluminum Specimens
Comparing Figure T-4 with the curvature values in Figure T-1,bit becomes evident
that the theoretical values are very
c
0.8mm
close to the experimental values, with the exception of unloaded die  80mm , which was off by about 133 mm. This
might be due to the material experiencing strain hardening in the experiment.
Using the curvatures obtained in the above section, and following a similar procedure for the 3 other specimens
(1.6mm 1020 steel, .8mm 6061-T0 and 1.6mm 6061-T0 Al), we can predict the unloaded curvatures for each specimen.
We can then compare these results with the unloaded curvatures found experimentally in the hand-drawn plots attached at
the very end of this report.
Material
6061-T6 Al
6061-T6 Al
6061-T0 Al
6061-T0 Al
HRS 1020 Steel
Thickness
[mm]
1.6
.8
1.6
.8
1.5
b
c
b
c
u die  80mm [1/m]
u die  40mm [1/m]
 y [MPa]
E [MPa]
4.2
1.5
10.8
9.2
9.5
16.7
10.1
23.4
21.6
22.0
300
300
60
60
300
69,000
69,000
69,000
69,000
200,000
Figure T-5: Curvature Predictions for 5 samples of variable material and thickness
5
As is evident from the above table, 6061-T6 Aluminum experienced the most springback. Springback can be found be
subtracting the unloaded curvature from the loaded curvature (keep in mind, l is 12.5 for 80mm and 25 for 40mm). The
largest discrepancies between loaded and unloaded curvature values indicate greater springback. Generally speaking,
specimens with a smaller sheet thickness, higher yield strength and lower elastic modulus experienced the most
springback.
V.
Conclusion
This lab focused primarily on finding the unloaded curvatures for several different specimens, as well as the
amount of load required to create a plastic hinge in an encastered beam, through experimentation. Once this experimental
data was recorded, we were able to derive theoretical results using many of the formulas discussed above and contrast
them with the experimental results. It became evident that in many cases, the experimental data was very close to the
theoretical results, but in other cases there were noticeable discrepancies possibly due to human error and judgment flaws
(case in point: yield load on an encastered beam.)
6