16-Civ-A4 Geotechnical Materials and Analysis · December 2019
Question 3 of 6: Vertical stress increase under an L-shaped raft
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format: National Examinations (PEO/Engineers Canada) — 16-Civ-A4 Geotechnical Materials and Analysis, December 2019. Six questions, 100 marks, closed book, 3 hours; all questions are to be answered. A formula sheet, an m–n influence chart and a Newmark chart are provided at the back of the paper.
Reference texts: Das & Sobhan, Principles of Geotechnical Engineering, 9th ed. (Cengage); Craig’s Soil Mechanics (Knappett & Craig), 8th ed.; Holtz, Kovacs & Sheahan, An Introduction to Geotechnical Engineering, 2nd ed.
Question 3: Vertical stress increase under an L-shaped raft (20 marks)
Given. An L-shaped raft formed by a 5 m × 4 m rectangle with a 3 m × 2 m rectangular notch removed at the lower-left, carrying a uniform contact pressure. Point A is the re-entrant (inner) corner; point B is the outer corner of the notch, off the loaded area.
Given data
Uniform load $q$
100 kPa
Depth of interest $z$
3 m below base
Full rectangle
5 m × 4 m
Unloaded notch (lower-left)
3 m × 2 m
Find. $\Delta\sigma_z$ at $z=3\ \text{m}$ beneath point B (by the corner influence-factor / m–n chart) and beneath point A (by Newmark’s influence chart), then compare the two methods.
Figure 2 — plan of the L-shaped raft (full 5×4 m less a 3×2 m corner notch). A is the re-entrant corner; B is the outer corner of the notch.
Approach. Both methods give the vertical stress under a corner of a uniformly loaded rectangle, $\Delta\sigma_z = q\,I$, with $I=I(m,n)$ and $m=B/z,\ n=L/z$ (interchangeable). Any point is handled by superposing rectangles that share that point as a common corner — adding loaded rectangles and subtracting the unloaded notch. Newmark’s chart is the graphical form of the same integral, with $\Delta\sigma_z = 0.005\,N\,q$ for $N$ counted blocks.
(i) Point B by superposition of corner rectangles. B is the outer corner of the notch. The loaded L-area, referred to B, is the full 5 m × 4 m rectangle (corner at B) minus the 3 m × 2 m notch (also a corner at B):
$$\Delta\sigma_{z,B}=q\big[\,I(m_1,n_1)-I(m_2,n_2)\,\big].$$
Evaluate the influence factors. With $z=3$ m: full rectangle $m_1=5/3=1.67,\ n_1=4/3=1.33\Rightarrow I=0.214$; notch $m_2=3/3=1.00,\ n_2=2/3=0.67\Rightarrow I=0.145$ (Boussinesq corner integral, matching the provided m–n chart).
$$\Delta\sigma_{z,B}=100\,(0.214-0.145)=\boxed{6.9\ \text{kPa}}$$
Point B lies outside the loaded footprint, so its stress increase is small — the load has to “reach” sideways to B.
(ii) Point A by Newmark’s method. A is the re-entrant corner, with three of its four surrounding quadrants loaded. Split the L into three rectangles that all share the corner A: a 3 m × 2 m block, and two 2 m × 2 m blocks. The Newmark influence value is $I_N=0.005$ per block, so drawing the plan to the depth scale ($z=3$ m) and counting blocks $N$ gives $\Delta\sigma_z=0.005\,N\,q$.
Sum the three corner contributions. $I(3/3,2/3)=0.145$ and $I(2/3,2/3)=0.121$ (twice):
$$\Delta\sigma_{z,A}=q\big[\,0.145+2(0.121)\,\big]=100(0.387)=\boxed{38.7\ \text{kPa}}$$
Equivalently, the number of Newmark blocks the loaded area covers is $N=\Delta\sigma_z/(0.005\,q)=38.7/0.5\approx 77$ blocks — the count a student would obtain by overlaying the scaled plan on the chart.
(iii) Compare the methods. The m–n (chart or closed-form) method is fast, accurate and ideal for rectangular areas evaluated at a corner, but every point that is not a corner needs a superposition of several rectangles and it cannot handle an irregular plan directly. Newmark’s chart handles any shape and any interior point by simply tracing the scaled loaded area and counting blocks, but it is graphical, slower, and its accuracy depends on careful scaling ($=z$) and honest block counting near the boundary.
