# Soil Mechanics Questions and Answers – Stress Distribution – Newmark’s Influence Chart

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This set of Soil Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “Stress Distribution – Newmark’s Influence Chart”.

1. _________ chart is used to find the vertical stress on Westergaard’s equation.
a) Influence chart
b) Isocurve chart
c) Isobar chart
d) Fenske’s chart

Explanation: The influence chart is used to find the vertical stress on Westergaard’s equation. Whereas, an isobar is a curve or contour connecting all points below the ground surface or ground level of equal vertical pressure. In an isobar, at each and every point the vertical pressure is the same.

2. _________ is more accurate method of determining the vertical stress at any point.
a) Isobar chart
c) Influence chart
d) Fenske’s chart

Explanation: The Influence chart or the influence diagram is a more accurate method of determining the vertical stress at any point under a uniformly loaded area of any shape than the equivalent point load method.

3. The Newmark’s influence chart consists of _________
a) a single circle only
b) a number of circles and radiating lines
c) bar diagram
d) small rectangular unit areas

Explanation: The Newmark’s influence chart consists of a number of circles and radiating lines, is so prepared that the influence of each area unit is same at the centre of the circle.
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4. If a uniformly loaded circular area is divided into 20 sectors, then the influence value if is given by ___________
a) $$\frac{1}{20}\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
b) $$20\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
c) $$20\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{5}{2}\right]$$
d) $$\frac{1}{20}\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{5}{2} \right]$$

Explanation: Let a uniformly loaded circular area is divided into 20 sectors.
σz=vertical stress at depth z
The stress at each unit area will be $$\frac{σ_z}{20}$$
∴ $$\frac{σ_z}{20}=\frac{q}{20}\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
The influence factor is given by $$i_f=\frac{1}{20}\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2} \right]$$.

5. If the influence value $$i_f=\frac{1}{35} \left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$ for a uniformly loaded circular area, then the circular area is divided into _________ sectors.
a) 20
b) 35
c) 7
d) 14

Explanation: let the vertical stress be σz.
The Boussinesq’s vertical pressure σz under a uniformly loaded circular area is given by,
$$σ_z=q\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
If the area is divided into 35 area units, then the stress in each unit is given by,
$$\frac{σ_z}{35}=\frac{q}{35}\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
Therefore for $$i_f=\frac{1}{35} \left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right],$$
The circular area is divided into 35 sectors.

6. If a uniformly loaded circular area is divided into 44 sectors, then the influence value if is given by ___________
a) $$\frac{1}{44} \left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
b) $$44\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
c) $$44\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{5}{2}\right]$$
d) $$\frac{1}{44} \left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{5}{2}\right]$$

Explanation: Let a uniformly loaded circular area is divided into 44 sectors.
σz=vertical stress at depth z
The stress at each unit area will be $$\frac{σ_z}{44}$$
∴ $$\frac{σ_z}{44}=\frac{q}{44}\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
The influence factor is given by $$i_f=\frac{1}{44} \left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$.

7. In Newmark’s influence chart method, the point below which pressure is required should lie within the loaded area.
a) True
b) False

Explanation: In Newmark’s influence chart method, the point below which pressure is required may lie within the loaded area or outside the loaded area as pressure distributes uniformly in all directions in a homogeneous soil mass.

8. The vertical stress under the corner of a uniformly loaded rectangular area of size a, b at depth z and m=a/z, n=b/z is given by ___________
a) $$σ_z=\frac{2q’}{πz}\frac{1}{\left[1+(\frac{x}{z})^2\right]^2}$$
b) $$σ_z=\frac{q}{4π} \left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}\right]$$
c) $$σ_z=\frac{q}{4π} \left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}* \frac{m^2+n^2+2}{m^2+n^2+1}+tan^{-1}⁡\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1} \right]$$
d) $$σ_z=q\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$

Explanation: The vertical stress under the corner of a uniformly loaded rectangular area of size a, b at depth z and m=a/z, n=b/z is given by,
$$σ_z=\frac{q}{4π} \left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}* \frac{m^2+n^2+2}{m^2+n^2+1}+tan^{-1}⁡\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1} \right]$$
m and n are interchangeable terms. The above form of solution is after Newmark(1935).

9. The vertical stress under the corner of a uniformly loaded rectangular area of size 2m*4m at depth 5m and load of 80 kN/m2 is given by ___________
a) 6.22 kN/m2
b) 7.45 kN/m2
c) 8.12 kN/m2
d) 9.23 kN/m2

Explanation: Given,
Size of rectangle=2m*4m
Depth z=5m
Therefore m=2/5=0.4
n=4/5=0.8
∴ the vertical stress at the corner of rectangle is given by,
$$σ_z=\frac{q}{4π}\left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}* \frac{m^2+n^2+2}{m^2+n^2+1}+tan^{-1}⁡\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1} \right]$$
∴ $$σ_z=\frac{80}{4π}\left[\frac{2*0.4*0.8\sqrt{(0.4^2+0.8^2+1)}}{0.4^2+0.8^2+0.4^2 0.8^2+1}* \frac{0.4^2+0.8^2+2}{0.4^2+0.8^2+1}+tan^{-1}\frac{2*0.4*0.8\sqrt{(0.4^2+0.8^2+1)}}{0.4^2+0.8^2-0.4^2 0.8^2+1} \right]$$
∴ σz=7.45 kN/m2.

10. The influence factor for the vertical stress under the corner of a uniformly loaded rectangular area of size 1m*2m at depth 5m and load of 80 kN/m2 is given by ___________
a) 0.6212
b) 0.7465
c) 0.0328
d) 0.0624

Explanation: Given,
Size of rectangle=1m*2m
Depth z=5m
Therefore m=1/5=0.2
n=2/5=0.4
∴ the vertical stress at the corner of rectangle is given by,
$$σ_z=\frac{q}{4π}\left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}* \frac{m^2+n^2+2}{m^2+n^2+1}+tan^{-1}⁡\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1} \right]$$
∴ σz=kq
The influence factor $$k=\frac{1}{4π}\left[\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1}* \frac{m^2+n^2+2}{m^2+n^2+1}+tan^{-1}⁡\frac{2mn\sqrt{(m^2+n^2+1)}}{m^2+n^2+m^2 n^2+1} \right]$$
$$k=\frac{1}{4π} \left[\frac{2*0.2*0.\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2+0.2^2 0.4^2+1}*\frac{0.2^2+0.4^2+2}{0.2^2+0.4^2+1}+tan^{-1}\frac{20.20.4\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2-0.2^2 0.4^2+1}\right]$$
∴k=0.0328.

11. The Westergaard’s equation is given by ___________
a) $$σ_z=\frac{1}{\left[1+2(\frac{r}{z})^2 \right]^\frac{3}{2}}$$
b) $$σ_z=\frac{1}{2\left[1+2(\frac{r}{z})^2 \right]^\frac{3}{2}}$$
c) $$σ_z=\frac{1}{π\left[1+2(\frac{r}{z})^2 \right]^\frac{3}{2}}\frac{Q}{z^2}$$
d) $$σ_z=\frac{1}{π\left[1+2(\frac{r}{z})^2 \right]^\frac{3}{2}}$$

Explanation: The Westergaard’s equation is given by,
$$σ_z=\frac{1}{π\left[1+2(\frac{r}{z})^2 \right]^\frac{3}{2}}\frac{Q}{z^2}$$
Z= depth of point at which load is found

12. The Westergaard’s influence factor is given by _____________
a) $$K_W=\frac{1}{π\left[1+2(\frac{r}{z})^2 \right]^\frac{3}{2}}$$
b) $$K_W=\frac{Q}{z^2}$$
c) $$K_W=\frac{1}{π\left[1+2(\frac{r}{z})^2 \right]^\frac{5}{2}} \frac{Q}{z^2}$$
d) $$K_W=\frac{1}{π\left[1+2(\frac{r}{z})^2 \right]^3}\frac{Q}{z^2}$$

Explanation: The Westergaard’s equation is given by,
$$σ_z=\frac{1}{\left[1+2(\frac{r}{z})^2 \right]^\frac{3}{2}} \frac{Q}{z^2}$$
∴ $$σ_z=K_W \frac{Q}{z^2}.$$
Where KW is the Westergaard’s influence factor given by,
$$K_W=\frac{1}{π\left[1+2(\frac{r}{z})^2\right]^\frac{3}{2}}.$$

13. For a uniformly loaded rectangular area, the Newmark’s influence factor given by ___________
a) $$K= \left[\frac{20.20.4\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2+0.2^2 0.4^2+1}*\frac{0.2^2+0.4^2+2}{0.2^2+0.4^2+1}+tan^{-1}\frac{20.20.4\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2+0.2^2 0.4^2+1}\right]$$
b) $$K= \frac{1}{4π} \left[\frac{20.20.4\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2+0.2^2 0.4^2+1}*\frac{0.2^2+0.4^2+2}{0.2^2+0.4^2+1}+tan^{-1}\frac{20.20.4\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2+0.2^2 0.4^2+1}\right]$$
c) $$K= \frac{1}{4π}$$
d) $$K= \frac{q}{4π} \left[\frac{20.20.4\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2+0.2^2 0.4^2+1}*\frac{0.2^2+0.4^2+2}{0.2^2+0.4^2+1}+tan^{-1}\frac{20.20.4\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2+0.2^2 0.4^2+1}\right]$$

Explanation: The Newmark’s influence factor for a uniformly loaded rectangular area of size a, b at depth z is given by,
$$K= \frac{1}{4π} \left[\frac{20.20.4\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2+0.2^2 0.4^2+1}*\frac{0.2^2+0.4^2+2}{0.2^2+0.4^2+1}+tan^{-1}\frac{20.20.4\sqrt{(0.2^2+0.4^2+1)}}{0.2^2+0.4^2+0.2^2 0.4^2+1}\right]$$
Where m=a/z
n=b/z.
m and n are interchangeable.

14. Find the vertical pressure at depth 5m for a uniformly loaded circular area of 80 kN/m2 load and radius of 5m.
a) 51.72 kN/m2
b) 54.12 kN/m2
c) 78.325 kN/m2
d) 12.24 kN/m2

Explanation: Given,
depth z=5m
The Boussinesq’s vertical pressure σz under a uniformly loaded circular area is given by,
$$σ_z=q\left[1-\left[\frac{1}{(1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
∴ $$σ_z=80\left[1-\left[\frac{1}{1+(\frac{5}{5})^2}\right]^\frac{3}{2}\right]$$
∴ σz=51.72 kN/m2.

15. Find the influence factor for the vertical pressure at depth 5m for a uniformly loaded circular area of 80 kN/m2 load and radius of 1m.
a) 0.6212
b) 0.0571
c) 0.0328
d) 0.0624

Explanation: Given,
depth z=5m
The Boussinesq’s vertical pressure σz under a uniformly loaded circular area is given by,
$$σ_z=q\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2} \right]$$
∴ σz=kq
∴ $$k=\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^\frac{3}{2}\right]$$
$$K=\left[1-\left[\frac{1}{1+(\frac{1}{5})^2}\right]^\frac{3}{2} \right]$$
k=0.0571.

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