Soil Mechanics Questions and Answers – Stress Distribution – Triangular Loadings

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This set of Soil Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “Stress Distribution – Triangular Loadings”.

1. The uniformly varying load is __________ in a beam.
a) rate of loading increases linearly from zero
b) rate of loading increases non-linearly from zero
c) equal load at every point
d) equal load only at supports
View Answer

Answer: a
Explanation: The uniformly varying load is the rate of loading which increases linearly from zero at one end of the support.
The uniformly varying load is rate of loading which increases linearly from zero

2. The triangular load is also known as ___________
a) uniformly distributed load
b) uniformly varying load
c) point load
d) equivalent uniformly distributed load
View Answer

Answer: b
Explanation: The triangular load is also known as uniformly varying load. The uniformly varying load is the rate of loading which increases linearly from zero at one end of the support.
The uniformly varying load is rate of loading which increases linearly from zero

3. For any position of point P subtending angle α with AB, the vertical stress is given by___________
Find the vertical stress for any position of point P subtending angle α with AB
a) \(σ_z=\frac{q}{aπ} [xα(x-α)] \)
b) \(σ_z=\frac{q}{aπ} \)
c) \(σ_z=\frac{q}{aπ}\left[xα-\frac{az}{(x-α)^2+z^2}(x-α)\right] \)
d) \(σ_z= \left[xα-\frac{az}{(x-α)^2+z^2}(x-α)\right] \)
View Answer

Answer: c
Explanation: For any position of point P subtending angle α with AB, the vertical stress is given by,
\(σ_z=\frac{q}{aπ}\left[xα-\frac{az}{(x-α)^2+z^2}(x-α)\right] \)
Where, σz= vertical stress
x=distance from the support A
z=depth from the ground surface
q=load applied.
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4. For point P under the support A, the vertical stress is given by __________
Find the vertical stress for any position of point P subtending angle α with AB
a) \(σ_z=\frac{q}{aπ} [xα(x-α)] \)
b) \(σ_z=\frac{q}{aπ} \)
c) \(σ_z=\frac{q}{π} \left[\frac{az}{a^2+z^2}\right] \)
d) \(σ_z=\left[xα-\frac{az}{(x-α)^2+z^2}(x-α)\right] \)
View Answer

Answer: c
Explanation: For point P under the support A, the vertical stress is given by,
\(σ_z=\frac{q}{π} \frac{sin2α_A}{2} \)
∴ \(σ_z=\frac{q}{π} \left[\frac{az}{a^2+z^2}\right]. \)

5. For point P under the support B, the vertical stress is given by __________
Find the vertical stress for any position of point P subtending angle α with AB
a) \(σ_z=\frac{q}{aπ} [xα(x-α)] \)
b) \(σ_z=\frac{q}{π}α_B \)
c) \(σ_z=\frac{q}{π} \left[\frac{az}{a^2+z^2}\right] \)
d) \(σ_z=\left[xα_B-\frac{az}{(x-α_B )^2+z^2}(x-α_B)\right] \)
View Answer

Answer: b
Explanation: For point P under the support B, the vertical stress is given by,
\(σ_z=\frac{q}{π}α_B \)
Where α_B=angle subtended from point P below the support B to the line AB
q=load applied.

6. For a linearly variable infinite load, for a point P, the vertical stress σz is _________
Find the vertical stress σz for a linearly variable infinite load, for a point P
a) \(σ_z=\frac{q}{aπ} \left[xα+z\right] \)
b) \(σ_z=\frac{q}{aπ} \)
c) \(σ_z=\frac{q}{π}\left[\frac{az}{a^2+z^2}\right] \)
d) \(σ_z=\left[xα-\frac{az}{(x-α)^2+z^2}(x-α)\right] \)
View Answer

Answer: a
Explanation: let the load intensity be q at a horizontal distance a and the load intensity at distance x is qx=q*x/a.
∴ for any point P the load is given by, \(σ_z=\frac{q}{aπ} \left[xα+z\right]. \)

7. For a symmetrically distributed triangular load, under the centre of the triangular load, the vertical stress at any point at a depth z is given by ___________
Find vertical stress at any point at depth z for symmetrically distributed triangular load
a) \(σ_z=\frac{q}{aπ} \left[α_1+α_2 \right] \)
b) \(σ_z=\frac{q}{π} \left[α_1+α_2 \right]\)
c) \(σ_z=\frac{q}{π} \left[\frac{az}{a^2+z^2}\right]\)
d) \(σ_z=\frac{q}{π} \left[α_1-α_2 \right]\)
View Answer

Answer: b
Explanation: Considering two triangular loads of AOC and OCB separately and adding them we get,
\(σ_{z1}=\frac{q}{π} α_1\) for triangular load of AOC
\(σ_{z2}=\frac{q}{π} α_2\) for triangular load of OCB
∴ \(σ_z=σ_{z1}+σ_{z2}\)
∴ \(σ_z=\frac{q}{π} \left[α_1+α_2\right].\)
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8. For a symmetrically distributed triangular load, the shear stress τxz at any point at a depth z is given by ___________
Find vertical stress at any point at depth z for symmetrically distributed triangular load
a) \(τ_{xz}=-\frac{qz}{aπ} \left[α_1-α_2 \right] \)
b) \(τ_{xz}=-\frac{q}{π} \left[α_1+α_2 \right]\)
c) \(τ_{xz}=-\frac{q}{π} \left[\frac{az}{a^2+z^2}\right]\)
d) \(τ_{xz}=-\frac{q}{π} \left[α_1-α_2 \right] \)
View Answer

Answer: a
Explanation: For a symmetrically distributed triangular load, the shear stress τxz at any point at a depth z is given by,
\(τ_{xz}=-\frac{qz}{aπ} \left[α_1-α_2 \right] \)where α1 and α2 are angles subtended by the point P at AO and BO respectively
z=depth.

9. For a triangular and uniformly distributed semi-infinite loads, the shear stress τxz in the plane xz is ___________
Find shear stress τxz in plane xz for triangular & uniformly distributed semi-infinite
a) \(τ_{xz}=-\frac{qz}{aπ} α \)
b) \(τ_{xz}=-\frac{q}{π} α\)
c) \(τ_{xz}=-\frac{q}{π} \left[\frac{az}{a^2+z^2}\right]\)
d) \(τ_{xz}=-\frac{q}{π} z\)
View Answer

Answer: a
Explanation: For a triangular and uniformly distributed semi-infinite loads, the shear stress τxz in the plane xz is given by,
\(τ_{xz}=-\frac{qz}{aπ}\) α where α is the angle subtended by point P on OA,
z=depth
q=load intensity.
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10. For a load intensity of q=20kN/m, find the shear stress τxz at a depth 5m from the given diagram.
Find the shear stress τxz at a depth 5m from the given diagram
a) -5.2 kN/m2
b) -6.2 kN/m2
c) -7.2 kN/m2
d) -8.2 kN/m2
View Answer

Answer: b
Explanation: Given,
Load q=20kN/m
α=30°=0.523rad
a=2.68m
z=5m
∴ \(τ_{xz} =-\frac{qz}{aπ} α\)
\(τ_{xz}=-\frac{20*5}{2.68*π}0.523\)
τxz=-6.2 kN/m2.

11. For a triangular and uniformly distributed semi-infinite loads, the vertical stress σz is given by ___________
Find shear stress τxz in plane xz for triangular & uniformly distributed semi-infinite
a) \(σ_z=\frac{q}{aπ}[xα+z] \)
b) \(σ_z=\frac{q}{aπ}(aβ+xα) \)
c) \(σ_z=\frac{q}{π} \left[\frac{az}{a^2+z^2}\right]\)
d) \(σ_z=[xα-\frac{az}{(x-α)^2+z^2}(x-α)]\)
View Answer

Answer: b
Explanation: For a triangular and uniformly distributed semi-infinite loads, the vertical stress σz is given by,
\(σ_z=\frac{q}{aπ}(aβ+xα) \) where α is the angle subtended by point P on OA,
β= angle at point P between horizontal line and PA
q=load intensity.

12. For a load intensity of q=20kN/m, find the vertical stress σz from the given diagram.
Find vertical stress σz from diagram for load intensity of q=20kN/m
a) 5.62 kN/m2
b) 6.23 kN/m2
c) 13.33 kN/m2
d) 8.32 kN/m2
View Answer

Answer: c
Explanation: Given,
Load q=20kN/m
α=60°=1.047rad
β=60°=1.047rad
a=2.68m
the vertical stress σz is,
\(σ_z=\frac{q}{aπ}(aβ+xα) \)
\(σ_z=\frac{20}{2.68*π}(2.68*1.047+2.68*1.047)\)
∴ σz=13.33 kN/m2.

13. Trapezoidal load is encountered in earth fills.
a) True
b) False
View Answer

Answer: a
Explanation: Trapezoidal loadings are quite commonly encountered in earth fills, embankments, highways, rail road, earth dams, etc. The trapezoidal loadings can be split into a triangular loading minus another triangular loading of smaller magnitude.

14. The maximum shear stress is the difference between major and minor principal stresses.
a) True
b) False
View Answer

Answer: b
Explanation: The maximum shear stress is half the difference between major and minor principal stresses.
\(τ_{max}=\frac{1}{2}(σ_1-σ_2).\)

15. The greatest value of maximum shear stress τmax occurs when angle θ is _________
a) π
b) π/2
c) π/3
d) π/4
View Answer

Answer: b
Explanation: The maximum shear stress is given by,
\(τ_{max}=\frac{1}{2} (σ_1-σ_2 )=\frac{q}{π} sinθ\)
Sinθ is maximum when θ= π/2
∴ Sin π/2=1
∴ \(τ_{max}=\frac{q}{π}.\)

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