Assignment – Mohr’s Circle for CIVL4230 Advanced Soil Mechanics

This assignment covers the calculation of principal stresses, direction of principal stresses, stresses on the horizontal plane, stress due to strip load, maximum shear stress, principal strain increments, angle of dilation, and orientation of lines of zero extension in soil mechanics for CIVL4230 Advanced Soil Mechanics course.

Running head: ASSIGNMENT – MOHR’S CIRCLE
CIVL4230 Advanced soil mechanics
Assignment – Mohr’s Circle
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ASSIGNMENT – MOHR’S CIRCLE 2
Question 1
(a) The magnitude of the principal stresses
From the initial stresses in figure 1, the Mohr Circle is as shown below
From the graph,
Max Principal Stress = 44.14 kPa
Min Principal Stress =15.86 kPa
Max Shear Stress, τmax = 14.142kPa
(b) The direction of the principal stresses is at angle of 22.5 as shown in the diagram⁰

ASSIGNMENT – MOHR’S CIRCLE 3
(c) The stresses on the horizontal plane H-H.
If the stresses shown in the figure are the principal stresses, then the stresses in the H-H plane
before rotation are:
σx= 35.0kPa
σy= 25.0kPa
τxy = 8.66kPa
Using the equations for stress at a point:
σavg = (σx+σy)/2 = (σ1+ σ2)/2 = 30.000
R =τmax =σ1−σ2/2= √ [(σx−σavg) 2 + τxy2] = 14.142
σ1= σavg + R =44.142
σ2 = σavg − R= 15.858
θp = tan −1 [τxy/ (σx−σavg)] 2 = 22.5
θτmax = θp−π/4 = -22.5
Also:

ASSIGNMENT – MOHR’S CIRCLE 4
σx = σavg + R + cos(2⋅θp) = 40.0
σy = σavg – R + cos (2⋅θp) = 20.0
τxy = R + sin (2⋅θp) = 10.0
Question 2
In the Fig. 2, a load q per unit area acting on a strip of infinite length and of constant width 2a.
The vertical stress at any arbitrary point P due to a line load can be expressed as
Applying the principle of superposition, the total stress σz, at point P due to a strip load
distributed over a width 2a may be written as
Wh
ere b = a
A non-dimensional value of σz/q are always given in form of graphs and expressed more
conveniently as
where β = α , α and δ are angles as shown in figure 2.Thus the principal stresses σ1 and σ3 at
any point P may be obtained from the equations
σ1 = q/π (α + Sin α)
σ3 = q/π (α - Sin α)
The direction of the principal stresses is positive as the load is acting downwards
Question 3
a)
Increment shear stress, Δτxy = q/π {Sinα Sin (α + 2δ)}
Given that:
q= 200 kPa
Width = 10m
Water depth = 2.0m
Bulk unit weight of the soil = 19kN/m3
Distance below = 5m
Area a= B x W

ASSIGNMENT – MOHR’S CIRCLE 5
Area acted upon by the applied load = 10 *5 = 50m2
Area by the soil = 10* 1.5 = 15m2
Area acted upon by water surcharge = 10*2 = 20m2
Stress = F/A
Stress due to soil, σx = 19kN/m3 x 1.5 m = 28.5kPa
Stress due to load, σy = 200kPa
Stress due to water, τxy = 9.98kN/m3 x 2m = 19.96kPa
Max shear stress τmax = √ ( σx−σy
2 )2
+ τxy2
= 88.04kPa
b) The magnitude and direction of the final maximum shear stress
From the Principal stress calculated, Mohr’s Circle is developed as shown
Stress due to soil, σx = 19kN/m3 x 1.5 m = 28.5kPa
Stress due to load, σy = 200kPa
Stress due to water, τxy = 9.98kN/m3 x 2m = 19.96kPa

ASSIGNMENT – MOHR’S CIRCLE 6
The coordinates of the point of tangency are the stresses on the plane of maximum obliquity αm
given as illustrated by the figure below.

ASSIGNMENT – MOHR’S CIRCLE 7
From the above Mohr’s Circle, the coordinated at the tangency is (47.25, 57.80)
Also, αm = tan-1 (τ/σ)
=tan-1 (57.80/ 47.25)
Angle of maximum stress obliquity= 50.73⁰
Question 4
Before solving the problems, we can first do a compressive analysis of shear box test with the
various equations involved in calculations.
From the tests, given the thickness (L) of the shear zone, the shear strain (Yzx) and the normal
strain (εz) on the horizontal plane can be estimated by;

ASSIGNMENT – MOHR’S CIRCLE 8
Where D is the horizontal (shear) displacement and h the vertical displacement.
The equations for finding the principal strain increments (dε1, dε3) in the shear box test can be
obtained from Mohr’s circle of incremental strain
From the Mohr’s circle, the principal strain increments dε1 and dε3 are given by:
The angle of dilation, V is obtained from the rate of change of vertical displacement with the
compression is taken as positive with horizontal displacement, D
V = tan–1 (–dεz/dYzx) = tan–1 (–dh/dD)
a) The box is 80 mm square and after placing the sample is 30 mm thick and the sample
compresses by 5 mm
Thus, h=D = 5 mm, Assuming L = 0.85cm =8.5mm
γzx = 5/8.5 = 0.588
εz = 5/8.5 = 0.588
V = tan–1 (–dεz/dYzx)
V = tan–1 (–0.588/0.588)= -45⁰
Sin v = -0.707, Cos v =0.707
Therefore the principal direct strain increments are:
dε1 = 0.588 + {(-0.707 +1)/0.707} *0.588/2 = 0.7098
dε3 = 0.588 + {(-0.707 -1)/0.707} *0.588/2 = - 0.1218
dε1 is +ve direction while dε3 is –ve direction

ASSIGNMENT – MOHR’S CIRCLE 9
b) During this process the top surface of the sample moves horizontally by 4.5 mm and
upwards by 2.5 mm
i) The principal strain increments, dε1 and dε3
D = 4.5mm
h =2.5mm
Assuming L= 8.5mm
γzx = 4.5/8.5 = 0.5294
εz = 2.5/8.5 = 0.2941
V = tan–1 (–0.2941/0.5294)= -29⁰
Sin v = -0.4848, Cos v = 8746
Thus;
dε1 = 0.2941 + {(-0.4848 +1)/0.8746} *0.5294/2 = 0.45
dε3 = 0.2941 + {(-0.4848 -1)/0.8746} *0.5294/2 = - 0.155
dε1 is +ve direction while dε3 is –ve direction
(ii) The maximum shear strain increment,
[(dε1-dε3)2+ (γxy) 2]0.5
[(0.45 + 0.155)2 + 0.52942]0.5
= 0.8039
= 0.8 (+ve direction)
(iii) The angle of dilation, 
V = tan–1 (–dh/dD)
= tan–1 (-2.5 -0)/ (4.5 -0)
= -29.05 = -29⁰
(iv) The orientation of the lines of zero extension
Radius of the strain circle is given as
r + r sin v = dε1
r = 0.45/0.5152 = 0.8734
The orientation of the lines of zero extension line, tan α = tanβ = r Cos v/ r (1-Sin v)
= 0.8734 Cos -29 / [0.8734 (1- Sin -29)]
= 0.589 x 100
= 58.9⁰

ASSIGNMENT – MOHR’S CIRCLE 10
c) At zero extension lines :
dε1 = r –r sin v = 0.8734 – 0.8734 Sin -29 = 1.29
Volumetric strain = -2rsin v = -2 x 0.8734 sin -29 = 0.8469
Shear strain across zero extension lines = 2r Cos v = 2 x 0.8734 Cos -29 = 1.52
Assuming these to be stresses and developing the Mohr’s Circle

ASSIGNMENT – MOHR’S CIRCLE 11
Max Principal Stress 2.605 (+ve direction)
Min Principal Stress -0.4676 (-ve direction)

ASSIGNMENT – MOHR’S CIRCLE 12
References
Viswanadham.B.V.S. (2015).Advanced Geotechnical Engineering
Richard H.G. (2004).Mohr Circles, Stress Paths and Geotechnichs
Sihong Liu. (2011).Simulating a direct shear box test by DEM
Mohr’s Circle Calculator. Retrieved from https://mechanicalc.com/calculators/mohrs-circle/
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Sep. 2018].
Chen, F. (1976). Mohr's circle and its applications to engineering design. New York: American
Society of Mechanical Engineers.
Fredlund, D., Rahardjo, H. and Fredlund, M. (2012). Unsaturated Soil Mechanics in
Engineering Practice. Somerset: Wiley.
Hartog, J. (2012). Advanced Strength of Materials. Dover Publications.
Hryciw, R., Athanasopoulos-Zekkos, A. and Yesiller, N. (2012). Geoongress 2012. Reston, Va.:
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Van Dyck, E. (2012). Effects of principal stress direction and the intermediate principal stress
on the stress-strain-strength behavior of a cross-anisotropic fine sand deposit.