TITLE
Stress Concentration
AIM
The The obje object ctiv ives es of this this expe experi rime ment nt were were to demo demons nstr trat atee and and dete determ rmin inee the the stre stress ss concentrations in the vicinity of a geometric discontinuity of a cantilever beam.
THEORY
In a load loaded ed stru struct ctur ural al memb member er,, stress stress distr distrib ibut utio ions ns vary vary acco accord rdin ing g to the the geom geometr etric ic irreg irregul ulari ariti ties es in the the memb member er.. The The peak peak stres stresss reach reaches es a larg larger er magn magnitu itude de near near these these irregularities. xamples for such sections are holes, grooves, notches, sharp corners, crack, etc. The increase in peak stress in these sections is known as stress concentration. The section that causes the increase in peak stress is known as a stress raiser. The following image is of a cantilevered beam with a hole in it !stress raiser". #
%$ Figure 1: Cantilever Cantilever beam with a circul circular ar hole.
In the above cantilever beam, the stress distributions in focus are section & and section '. &t section &, the stress distribution is uniform since there are no stress raisers or any other types of geometric irregularities. The stress at section & can be calculated using the following relationship, where ( is the applied load, ) is the distance from the load to Section &, b is the width of the beam and t is the thickness of the beam. σ A =
6 PL 2
b t
The following e*uation depicts the nominal stress at Section '. & hole is introduced as a stress raiser in this section. σ B=
6 Pl
( b −d ) t
2
In the above e*uation, l refers to the distance from the load to Section ' and d is the diameter of the hole. The values for l and L have been taken such that the nominal stresses in Section & and Section ' are e*ual. +uring this experiment, a strain gauge was located at Section & as well to measure the nominal strain. +ue to the presence of a stress raiser at Section ', the max
stress reaches a maximum peak stress,
σ B
. The relationship between the maximum stress
and the nominal stress at Section ' can be shown as follows
- t is known as the theoretical stress concentration factor. The value of - t is usually greater than $. owever, - t depends only on the loading geometry, therefore, material properties are not considered when calculating - t.
READINGS
In this experiment, readings for strain were taken at three stages using strain gauges, namely, before loading, after loading and after unloading. The strain gauges were located in the following manner
Figure 2: Placement of strain gauges on the beam
The following table contains the readings of the strain gauges for the three stages of loading.
Table 1: Readings of strain gauges
'efore loading &fter loading &fter unloading
/auge $ !x $012" 00000 600$$3 100003
/auge 3 !x $012" 00000 600078 100005
/auge 4 !x $012" 00000 600$42 600040
/auge 5 !x $012" 00000 600$72 00000
The following table contains the properties of the cantilever beam.
Table 2: Properties of the cantilever beam
Property
'eam ole Strain /auges : +istance from the ole centre
)ength Thickness 9idth +iameter /auge $ /auge 3 /auge 4
Value 800 mm 2.8 mm 8$ mm 38.2 mm $8.4 mm $;.; mm 34.; mm
CALCULATIONS
() ()
r ε = A + B x
ε 1= A + B
2
r + C x
4
( ) ( ) 2
12.8
+ C
15.3
12.8
4
15.3
−6
A + 0.7 B + 0.49 C =112 × 10
ε 2 = A + B
( ) ( ) 2
12.8
+ C
18.8
12.8
4
18.8
−6
A + 4.6 B + 0.21 C =75 × 10
ε 3 = A + B
( ) ( ) 2
12.8
+ C
23.8
12.8
4
23.8
−6
A + 0.29 B + 0.08 C =136 × 10
−6
A =143.37 × 10
−6
B =−12.77 × 10
6
−
C =−45.7 × 10
6
−
ε =143.37 × 10
−
12.77 × 10
−
6
() r
x
2 −
−
45.7 ×10
6
() r
x
4
ε ∨¿ r
=1
x
Theoretical maximum strain ( ε max ,theoretical ) =¿ ε max ,theoretical =143.37 × 10
−6
−6
−12.77 × 10
2
×1
−6
− 45.7 × 10
4
×1
−6
ε max ,theoretical =84.9 × 10
The following graph shows the variation of strain with the *uantity
Figure : !raph of "#periments $train vs. r%#
r x
&ccording to the plotted values, & < 0.0005483 ' < 10.00$52$ C<
0.00$572 −6
ε max , , experimental= 420.9 × 10
DISCUSSION •
The factor - t depends mainly on the geometry of the hole or the notch, and not on the material, except when the material deforms severely. This value is normally obtained from the graphs or either by calculations. There are no - t values available for sharp notches and cracks, but those occurrences provide highest stress concentration.
