4 Mohr's stress circle transformation
model
4.1 Plane stress transformation equatlons
analysed
You may have found using the transformation equations a little tedious,
especially in SAQ 27. Fortunately there is an easy way to analyse plane
stress at a point. Firstly, though, we have to perform a little more algebraic
manipulation.
We can rearrange the transformation equations as follows:
a,-f(an+a,,)=f(on-a,y)ws28+r,sin20
r, = f (an - a,,) sin 20 - r,, cos 29
We can square both sides and add the equations. This allows us to get one
equation featuring the two variables a, and to,using the trigonometric
relation cos' 20 + sin2 20 = 1:
Coo -f(o,, + ay,)]' + ~ g =
2 t(axx- a,y)2 + r,:
Now compare this equation with that of a circle for two variables x and y,
whose centre is at a, b and whose radius is r (Figure 44):
Figure 44
[X
- aI2 + [ y - bI2 = r 2
You can see that the equation relating our two stress variables (a,, r,) is
that of a circle, known as Mohr's circle, where the equivalent parameters
are:
a=f(oxx+~,)r)
b=O
(effectively we have [re - 01')
and
r = &(a,
- a,)'
+ rf,
So if we plotted a, against re for a range of values of O we should expect
to see a circular graph. Moreover our graph of stress plots will always
have its centre on the horizontal axis (r, = 0). because the equivalent b
parameter value is 0. This gives us a quick graphical way of determining
stresses in other directions (i.e. on other elements at the same point).
Plot the graph of a, against re for the results obtained in SAQ 27. Use the
axes as shown in Figure 45.
If we know the stress components on one element face, therefore, we can
determine the stress components on any plane (or element face) at the
same point inclined at an angle to the original element.
R
liiiiiiiiil
scale 10 mm : 10 MN m-'
Figure 45
4.2 Drawing Mohr's stress circle
Figure 46 shows a stress element with a known pair of stresses on it.
Stresses on the other hces have been omitted for clarity. Also shown is
another stress element at the same point, inclined at angle B to the first
element. If we have sufficient data to draw the circle graph the stresses on
the other element Face can easily be shown another point on the cirde,
Figure 47. Note that although plane A'B' is located at angle 8 anticlockwise from plane AB, th;co;rcsponding stress plot is located at 28
anticlockwise from the AB stress dot. (The
28 relationship was explained
,
in Section 3.2.) This complication of doubling the angleon-the oirole graph
always applies with Mohr's circle. The strsss plots of planes 0 apart on
ekmcnts appear as 28 apart on the circle (in the same direction, clockwise
or anticlockwise).
.
Notice that if we have planes on elements at 90° to one another, the
corresponding stress plots will be at 180" to one another on the circle. This
gives us a very quick way of drawing the circle graph, because such stress
plots will be at the ends ofa diameter, and we already h o w that the centre
of the cirde will always be on the I , = 0 axis ( W o n 4.1).
. - .
. .
The circle 1s drawn, not by plottlng many (U,,?,] pomrs, Dur ~y a
geometrical construction after finding the catre and radius by plotting the
end-points of a diameter. Figure 48 shows sn element in a state of plane
...
.
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Figure 48
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For the Mohr's circle of Figure 53 draw the element aligned with the x and
y axes, and indicate the strasses.
N m-'
8dullon
Stresn coordinates of X are (- 100,100) MN m-'.
Stress on faces AB and CD is U, = -100 MN m-' and c = 100 MN m-'.
Stress coordinates of Y are (M, - 100) MN m-2.
Stress on faces BC and DA is U, = 50 MN m-' and T = - 100 MN m-'.
Figure 54 shows this state of plane stress.
SA0 SO
Draw the state6 of streas on the element, aligned with the X and y axes,
from each of the MOWScircles in Figure 55.
Figure 55
We have seen in Section 4.1 how the equation repreacnting the stress field
at a point is that of a circle giving a graph of all the normal and shear atress
~oordinates,and have practised drawing a variety of such circle graph
With reference to the equation, you should be able to quickly compute
sufficient data to draw the circle without having to measure anything to
scale.
In Figure 48, what are the coordinates of the antre of the circle, and what
is the radius?
SA0 32
From Figure 48, what are the values of maximum shear stnss. maximum
normal stress and minimum normal stress at the point?
L 1... L I. , .L,,..l,. JI*.
Figure 53
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