Vertical Stress Due to Embankment Loading
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An embankment of $3\text{ m}$ high is to be constructed as shown in the figure below. If the unit weight of compacted soil is $19\text{ kN/m}^3$, calculate the vertical stress due to the embankment loading at (A), (B), and (C) points.
This question includes visual content: A trapezoidal embankment cross-section is shown atop a horizontal ground line. The embankment has a height of $3\text{ m}$ and a top width of $6\text{ m}$. It has $1:1$ slopes on both sides. A unit weight $\gamma_{\text{soil}} = 19\text{ kN/m}^3$ is noted. Below the base of the embankment, three points A, B, and C are located on a horizontal line at a depth of $3\text{ m}$ below the base. Point A is located $1.5\text{ m}$ right of the center of the left slope. Point B is vertically aligned with the edge where the left slope meets the base. Point C is $4.5\text{ m}$ to the left of Point B. Horizontal dimension lines show: $4.5\text{ m}$ from C to B, $1.5\text{ m}$ from B to the start of the flat top section, and $6\text{ m}$ for the total width of the flat top section.
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In this problem, we need to calculate the vertical stress at three different points below an embankment that is three meters high. The soil used in the embankment has a unit weight of nineteen kilonewtons per cubic meter.
Vertical Stress under Embankment Loading
To solve this, we use Westergaard's or Boussinesq's solutions for trapezoidal embankments. The formula for vertical stress under a corner or center can be simplified using influence factors based on the geometry.
Let's define our geometric parameters 'm' and 'n' to find the influence factors. For point A, which is under the center of the embankment, we can treat it as two symmetrical parts.
Analysis for Point A (Center)
With a top half-width of three meters and a side slope of three meters, and depth of three meters, we find that m equals one and n equals one.
Using the influence factor chart for an embankment, an 'm' of one and 'n' of one gives us a factor 'I' of approximately zero point four six.
The vertical stress at A is the surface pressure 'q' times this total factor. Fifty-seven times zero point nine one two is fifty-two kilonewtons per square meter.
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