Question

The partial differentiation of velocity of water in soil element with respect to depth z in terms of coefficient of volume change is given by ____________

(a) \(\frac{∂v}{∂z}=m_v \frac{∂^2 \overline{u}}{∂t^2}\)

(b) \(\frac{∂v}{∂z}=m_v  \frac{\overline{u}}{∂t^2}\)

(c) \(\frac{∂v}{∂z}=\frac{∂^2 \overline{u}}{∂t^2}\)

(d) \(\frac{∂v}{∂z}=m_v  \frac{∂\overline{u}}{∂t}\)

I had been asked this question by my school teacher while I was bunking the class.

Origin of the question is Terzaghi’s Theory of One Dimensional Consolidation topic in division One Dimensional Consolidation of Geotechnical Engineering I

Answer 1

The correct answer is (d) \(\frac{∂v}{∂z}=m_v  \frac{∂\overline{u}}{∂t}\)

Easy explanation: From the volume of water squeezed out, ∆q=\(\frac{∂v}{∂z}\) dxdydz and change of volume per unit time, \(\frac{∂∆V}{∂z}=-m_v dxdydz \frac{∂∆σ’}{∂t}\)

we get, \(\frac{∂v}{∂z}=-m_v \frac{∂∆σ’}{∂t}\)   —————–(1)

since, ∆σ=∆σ’+u, partally differentiating it with respect to time t,

\(\frac{∂∆σ’}{∂t}=-\frac{∂\overline{u}}{∂t}\)   ————————(2)

∴ from (1) and (2),

\(\frac{∂v}{∂z}=m_v  \frac{∂\overline{u}}{∂t}.\)