<p>I don't know where else to post this.</p>
<p>I am reading a Geotechnical engineering textbook by B.M. Das, 6th edition.</p>
<p>Problem 3.1 is as follows:</p>
<p>For a given soil, show that
<span class="math-container">$$e=\frac{\gamma_{sat} - \gamma_w}{\gamma_d - \gamma_{sat} + \gamma_w}$$</span></p>
<p><span class="math-container">$e=\frac{V_v}{V_s}$</span> where <span class="math-container">$V_v$</span> is the volume of voids I a specific soil and <span class="math-container">$V_s$</span> is the volume of solid material.</p>
<p><span class="math-container">$\gamma_{sat}$</span> <span class="math-container">$\gamma_d$</span> <span class="math-container">$\gamma_w$</span> are the unit weight of saturated soil (all voids filled with water), dry soil (no water) and water in <span class="math-container">$kN/m^3$</span></p>
<p>My attempt:
<span class="math-container">$W_s,W_w$</span> are the weight of solids and weight of water in the saturated sample. Total volume is <span class="math-container">$V_{tot}=V_s+V_v=1$</span></p>
<p><span class="math-container">$$e=\frac{\gamma_{sat} - \gamma_w}{\gamma_d - \gamma_{sat} + \gamma_w}=\frac{\frac{W_s+W_w}{V_{tot}}-\frac{W_w}{V_v}}{\frac{W_s-W_s-W_w}{V_{tot}}+\frac{W_w}{V_v}}=\frac{V_v(W_s+W_w)-W_w}{W_w - V_vW_w}=\frac{V_vW_s+W_w(V_v-1)}{W_wV_s}=e*\frac{W_s}{W_w}-1$$</span></p>
<p>I can't find a way beyond this and something is wrong. We can move around the variables to get <span class="math-container">$e=\frac{V_vW_s}{V_sW_w}-1=\frac{\gamma_s}{\gamma_w}-1=G_s-1$</span>
,<span class="math-container">$G_s$</span> is defined as the specific gravity and is constant for the solid material regardless of the void ratio (<span class="math-container">$e$</span>). Therefore <span class="math-container">$e=G_s-1$</span> must be wrong because <span class="math-container">$e$</span> is supposed to be variable in a soil with constant <span class="math-container">$G_s$</span></p>

CC BY-SA 4.0
2023-03-25T00:44:36.880Z
This question is on topic here, but it might be a better fit on SE Engineering because it might have more people who could answer this soil mechanics question. If you don't get a satisfactory answer here post the question there & use the tags `soil-mechanics` & `geotechnical-engineering`.
Fred
CC BY-SA 4.0
2023-03-30T16:28:09.357Z
I think there's either an error in the question or an error in your transcription.  It should be saturated minus dry in the numerator, not saturated minus water.  In that case the math works out straightforwardly.
damp_civil
<geophysics><soil>
Geotechnology: Prove a formula, calculated with unit weights, for void ratio
Erik Eriksson
# Geotechnology: Prove a formula, calculated with unit weights, for void ratio

(Asked by: username_0 on 2023-03-24)

I don't know where else to post this.

I am reading a Geotechnical engineering textbook by B.M. Das, 6th edition.

Problem 3.1 is as follows:

For a given soil, show that
$$e=\frac{\gamma\_{sat} - \gamma\_w}{\gamma\_d - \gamma\_{sat} + \gamma\_w}$$

$e=\frac{V\_v}{V\_s}$ where $V\_v$ is the volume of voids I a specific soil and $V\_s$ is the volume of solid material.

$\gamma\_{sat}$ $\gamma\_d$ $\gamma\_w$ are the unit weight of saturated soil (all voids filled with water), dry soil (no water) and water in $kN/m^3$

My attempt:
$W\_s,W\_w$ are the weight of solids and weight of water in the saturated sample. Total volume is $V\_{tot}=V\_s+V\_v=1$

$$e=\frac{\gamma\_{sat} - \gamma\_w}{\gamma\_d - \gamma\_{sat} + \gamma\_w}=\frac{\frac{W\_s+W\_w}{V\_{tot}}-\frac{W\_w}{V\_v}}{\frac{W\_s-W\_s-W\_w}{V\_{tot}}+\frac{W\_w}{V\_v}}=\frac{V\_v(W\_s+W\_w)-W\_w}{W\_w - V\_vW\_w}=\frac{V\_vW\_s+W\_w(V\_v-1)}{W\_wV\_s}=e\*\frac{W\_s}{W\_w}-1$$

I can't find a way beyond this and something is wrong. We can move around the variables to get $e=\frac{V\_vW\_s}{V\_sW\_w}-1=\frac{\gamma\_s}{\gamma\_w}-1=G\_s-1$
,$G\_s$ is defined as the specific gravity and is constant for the solid material regardless of the void ratio ($e$). Therefore $e=G\_s-1$ must be wrong because $e$ is supposed to be variable in a soil with constant $G\_s$

**Tags:** geophysics, soil

> **Comments:**
> **username_1**: This question is on topic here, but it might be a better fit on SE Engineering because it might have more people who could answer this soil mechanics question. If you don't get a satisfactory answer here post the question there & use the tags `soil-mechanics` & `geotechnical-engineering`.  
> **username_2**: I think there's either an error in the question or an error in your transcription.  It should be saturated minus dry in the numerator, not saturated minus water.  In that case the math works out straightforwardly.  

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