Chapter 8 — Retaining Wall Analysis

Chapter VIII covers earth pressures and the structures that resist them. It opens with the classical active and passive pressure theory, then treats rigid walls (gravity and cantilever) and flexible walls (sheet piles and diaphragm walls), including anchored walls and braced excavations.

8.1 Active and passive pressure

8.1.1 Cohesive soil (\(\varphi = 0\), \(c \neq 0\))

For a vertical wall with a horizontal backfill surface, ignoring wall–soil friction, the active pressure (áp lực chủ động) is the minimum lateral pressure the soil exerts on the wall — the condition in which the wall has moved away enough for the soil to mobilise its shear strength along the failure plane:

\[\sigma_a = K_a\,\sigma'_z - 2c\sqrt{K_a}\]

where \(\sigma'_z\) is the effective vertical stress, \(c\) is the cohesion, and \(K_a\) is the coefficient of active earth pressure.

1. The coefficient \(K_a\) is found from:

\[K_a = \tan^2\!\left(45^\circ - \frac{\varphi}{2}\right) = \frac{1 - \sin\varphi}{1 + \sin\varphi}\]

2. The active thrust \(Q_a\) for a wall of height \(H\) is:

\[Q_a = \tfrac{1}{2}H\,\sigma_a = \tfrac{1}{2}K_a\,\gamma\,H^2\]

and its point of application is at \(\tfrac{1}{3}H\) from the base of the wall.

8.1.2 Granular soil

The same framework with \(c = 0\), plus the effect of wall friction and sloping backfill.

Figure. Earth pressures acting on a retaining wall.

8.2 Rigid retaining wall analysis

8.2.1 Definition and classification

Gravity, semi-gravity and cantilever walls.

8.2.2 Analysis of a rigid wall

Stability against overturning, sliding and bearing failure, plus internal structural design.

8.2.3 Some types of low rigid wall

8.3 Flexible retaining wall analysis

8.3.1 Definition and classification

Sheet piles and diaphragm (slurry) walls.

8.3.2 Analysis of a cantilever (fixed-base) wall

8.3.3 Analysis of an anchored wall

8.3.4 Analysis of a braced excavation

Braced excavations (hố đào với tường chống xà) — the case most relevant to urban deep excavation and to the instrumentation in this site's Foundation & Excavation page.

8.3.5 Base stability and wall deformation of an excavation

8.4 Terminology

English	Vietnamese (book)
retaining wall	tường chắn
active earth pressure	áp lực chủ động
passive earth pressure	áp lực bị động
earth pressure at rest	áp lực tĩnh (nghỉ)
coefficient of active earth pressure	hệ số áp lực chủ động
active thrust	lực chủ động
rigid wall	tường chắn cứng
flexible wall	tường chắn mềm
cantilever wall	tường công-xôn
sheet pile	tường cừ
anchored wall	tường chắn có neo
braced excavation	hố đào chống xà

8.5 Key takeaways

• Active pressure is the minimum; passive is the maximum; at rest is between.
• For cohesive soil, \(\sigma_a = K_a\sigma'_z - 2c\sqrt{K_a}\) and \(K_a = \tan^2(45^\circ - \varphi/2)\).
• The active thrust on a wall of height \(H\) is \(Q_a = \tfrac{1}{2}K_a\gamma H^2\), acting at \(\tfrac{1}{3}H\) above the base.
• Rigid walls are checked for overturning, sliding and bearing.
• Flexible walls (sheet pile, diaphragm) and braced excavations are the critical case for urban deep excavation monitoring.