GEOENG2 Mohr's Circle
-
Upload
jbenedictang -
Category
Documents
-
view
235 -
download
1
Transcript of GEOENG2 Mohr's Circle
-
8/13/2019 GEOENG2 Mohr's Circle
1/28
Mohrs Circle Analyses
Geotechnical Engineering 2
(GEOENG2)
Jonathan Rivera Dungca
-
8/13/2019 GEOENG2 Mohr's Circle
2/28
Stress Acting on Other Planes
Some problems may require computation of the
stresses on other planes
We can obtain these stresses using a graphical
representation called a Mohrs circle, which
was developed by the German engineer Otto
Mohr (1835-1918)
-
8/13/2019 GEOENG2 Mohr's Circle
3/28
Mohrs Circle
It describes the two
dimensional stresses
at a point in a material
It considers the
stresses acting on
each side of a twodimensional element
and plots them on a s
vs. tdiagram
-
8/13/2019 GEOENG2 Mohr's Circle
4/28
Mohrs Circle
Each point on the
circle represents
the normal and
shear stresses
acting on one
side of anelement oriented
at a certain angle
-
8/13/2019 GEOENG2 Mohr's Circle
5/28
Principal Stresses
If the soil element is
rotated to a certain
angle, the shear
stresses will be zero
The planes on each
side of this element
are known aspr incipal planes
-
8/13/2019 GEOENG2 Mohr's Circle
6/28
Principal Stresses
The stresses acting
on them are known
as pr incipal
stresses
The major p r inc ipal
stress, 1, is also
the greatest normalstress that acts on
any plane
-
8/13/2019 GEOENG2 Mohr's Circle
7/28
Principal Stresses
The minor pr inc ipalstress, 3, is the smallest
normal stress that acts on
any plane
These two stresses act at
right angles to each other
In a 3D analysis, there
would also be an
intermediate princ ipal
stress, 2which are at right
angles to both s1 ands3
-
8/13/2019 GEOENG2 Mohr's Circle
8/28
Magnitudes ofs1and s3
2
2
3 22 zxzxzx
t
ssss
s
2
2
122 zx
zxzxt
sssss
-
8/13/2019 GEOENG2 Mohr's Circle
9/28
Angle between szand s1
31
311 2cos21
sssss zz
where:s1= major principal stress
s3= minor principal stress
sx= horizontal stress
sz= vertical stress
tzx= shear stress acting on ahorizontal plane
z= angle between
szand
s1
-
8/13/2019 GEOENG2 Mohr's Circle
10/28
Principal Stresses Beneath a
Circular Loaded Area
-
8/13/2019 GEOENG2 Mohr's Circle
11/28
Mohrs Circle that represents the
combined effects of both the
Geostatic and the Induced Stress
To develop such a circ le, compu te the
geostat ic xand zand the induced x, z,
and zx, then add them by superposi t ion
and use the combined values to develop
the Mohrs Circle. Do not attempt tocombine 1and 3values by
superposi t ion
-
8/13/2019 GEOENG2 Mohr's Circle
12/28
Stresses on Other Planes
After constructing the Mohrs circle, we can obtain
the normal and shear stresses that act on any
plane through that point
ssss
s 2cos22
3131
ss
t 2sin2
31
-
8/13/2019 GEOENG2 Mohr's Circle
13/28
Greatest Shear Stress, tmax
It occurs on the plane Gand H in the figure. These
planes are oriented at 450
angles from the principal
planes, and the shearstress acting on them is
equal to the radius of the
Mohrs Circle
2
31max
sst
-
8/13/2019 GEOENG2 Mohr's Circle
14/28
-
8/13/2019 GEOENG2 Mohr's Circle
15/28
Procedure to compute the stresses
on a given plane1.Draw a soil element that is
aligned with the x and z
axes.
2.Compute sx, sz, txzand tzxand mark these stresses on
the soil element.
3.Plot the points sx
, txz
and
sz, tzxon an s, t diagram
then use these points to
draw the Mohrs circle. The
center of the circle is at2zx ss
s
-
8/13/2019 GEOENG2 Mohr's Circle
16/28
Procedure to compute the stresses
on a given plane4. Compute s1and s3
5. Compute the angle zbetween sz and s1
6. Compare the positions ofthe points on the Mohrs
circle that represents szand s
1
to determine if z
extends clockwise or
counter-clockwise from sz.
-
8/13/2019 GEOENG2 Mohr's Circle
17/28
Procedure to compute the stresses
on a given plane7. Draw another soil element
that is rotated at an angle
z from the first soil
element. The sides of thiselement are the principal
planes. Mark the stresses
s1 and s3 on this soil
element. Since the sidesof this element are the
principal planes, the shear
stresses are zero.
-
8/13/2019 GEOENG2 Mohr's Circle
18/28
Procedure to compute the stresses
on a given plane8. Compute for the value of
tmax(if required). It acts on
the planes oriented 450
from the principal planes.9. Draw a third soil element
with one of its sides
oriented in the direction of
the plane on which thestresses are to be
computed. Mark the
stresses as s
and t on this
element
-
8/13/2019 GEOENG2 Mohr's Circle
19/28
Procedure to compute the stresses
on a given plane10.Determine the angle
betweensand s1, then
locate the point on the
Mohrs circle thatrepresents the plane. This
point is located at an angle
2from the point that
represents s1. Be sure tofollow the sign convention:
angles measured
clockwise from s1are
positive.
-
8/13/2019 GEOENG2 Mohr's Circle
20/28
Procedure to compute the stresses
on a given plane
11.Compute for sand t.
-
8/13/2019 GEOENG2 Mohr's Circle
21/28
Example
The horizontal and vertical stresses at a certain
point in a soil are as follows:sx= 2100 lb/ft
2, sz= 3000 lb/ft2, tzx= -300lb/ft
2
a. Determine the magnitudes and directions of themajor and minor principal stresses.
b. Determine the magnitude and directions of themaximum shear stress.
c. Determine the normal and shear stresses actingon a plane inclined at 350clockwise from the x-axis
-
8/13/2019 GEOENG2 Mohr's Circle
22/28
Example
-
8/13/2019 GEOENG2 Mohr's Circle
23/28
Example
Solution (a):
2
2
2
2
2
1
/3091
)300(2
30002100
2
30002100
22
f tlb
xz
zxzx
t
sssss
2
2
2
2
2
3
/2009
)300(2
30002100
2
30002100
22
f tlb
xz
zxzx
tssss
s
-
8/13/2019 GEOENG2 Mohr's Circle
24/28
Example
Solution (a) Continued :
0
1
31
311
17
20093091
20093091)3000(2cos
2
1
2cos
2
1
ss
sss
zz
-
8/13/2019 GEOENG2 Mohr's Circle
25/28
Example
-
8/13/2019 GEOENG2 Mohr's Circle
26/28
Example
Solution (b):
231max /541
2
20093091
2f tlb
sst
-
8/13/2019 GEOENG2 Mohr's Circle
27/28
Example
Solution (c):
2
3131
/2419
)3517(2cos
2
20093091
2
20093091
2cos22
f tlb
sssss
2
31
/525
)3517(2sin2
20093091
2sin
2
f tlb
sst
-
8/13/2019 GEOENG2 Mohr's Circle
28/28
Example