16-Civ-A4 Geotechnical Materials and Analysis · December 2017
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Examinations — 16-Civ-A4 Geotechnical Materials and Analysis, December 2017. Closed book, 3 hours, 100 marks. Answer all six questions. Charts (rectangular-area influence chart, Newmark chart) and a formula sheet are provided at the back of the exam.
Reference texts: R.F. Craig & J. Knappett, Craig’s Soil Mechanics (8th ed.); B.M. Das, Principles of Geotechnical Engineering; R.D. Holtz, W.D. Kovacs & T.C. Sheahan, An Introduction to Geotechnical Engineering (2nd ed.); M. Budhu, Soil Mechanics and Foundations.
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
The response is graphical. Along a horizontal plane at depth z, the Boussinesq vertical stress from a surface point load is
$$\sigma_z=\dfrac{3Q}{2\pi z^{2}}\left[\dfrac{1}{1+(r/z)^{2}}\right]^{5/2},$$
where r is the horizontal offset from the load. Two features control every curve: the value on the axis (r = 0) is $\sigma_{z,\max}=3Q/2\pi z^{2}$, which falls as $1/z^{2}$, and the width of the bulb grows in proportion to z. The distribution is therefore a bell (a “pressure bulb” section) that is tall and narrow just under the load and becomes short and wide with depth, while the area under each curve (the total load carried) stays equal to Q.
The horizontal (radial) stress σr obeys a companion Boussinesq expression that contains Poisson’s ratio ν: $\sigma_r=\dfrac{Q}{2\pi}\left[\dfrac{3r^{2}z}{R^{5}}-\dfrac{1-2\nu}{R(R+z)}\right]$ with $R=\sqrt{r^{2}+z^{2}}$. Its shape differs from the vertical bell. For an incompressible soil (ν = 0.5, e.g. saturated clay loaded undrained) the second term vanishes, so σr is zero directly beneath the load, rises to twin peaks at r = z√(2/3) ≈ 0.8z on either side, and then decays. For ν < 0.5 the second term adds a small tension, so the value on the axis is slightly negative. The peak is only about one-fifth of the on-axis vertical stress at the same depth (0.089Q/z² against 0.477Q/z²). Comparing z = 1 m and z = 5 m, the humps move outward in proportion to z (from about 0.8 m to about 4 m off the axis) and their height falls as 1/z², so the 5 m curve is 25 times lower and five times wider.
Reason for the behaviour. A point load is transmitted into the half-space along radiating stress paths. Immediately below the load the entire load acts over a vanishingly small area, giving an extremely high stress concentration. As depth increases the same load is shared over an ever-larger area (the “2:1” or bulb spreading), so the peak stress collapses as $1/z^{2}$ while the load spreads sideways — hence the bells lower and broaden with depth. The horizontal stress behaves differently on the axis because, directly under the load, the soil is squeezed vertically and the horizontal push comes only from lateral bulging. That push is strongest where the stress paths are inclined at about 40° below the load, which is why σr peaks off the axis. This spreading is the physical basis for the pressure-bulb rule of thumb that stresses become negligible below roughly two footing-widths of depth.
