Bearing Capacity of Soil - Complete Loksewa Guide (7th Level, Foundation Engineering)

A one-stop, exam-focused, easy-to-remember guide to Chapter 11: Bearing Capacity of Soil, built for Nepal's 7th Level Loksewa (Overseer / Sub-Engineer, Civil) syllabus under Foundation Engineering and Earth Pressure.

Why this chapter matters

Bearing capacity is one of the highest-weightage topics in the Foundation Engineering section of the Loksewa syllabus. Almost every past paper has at least one 10-mark question from this chapter (usually Terzaghi's theory) plus one or two 5-mark questions (definitions, factors, or failure modes), and 2 to 4 MCQs. If you can master this single chapter well, you lock in guaranteed marks.

Quick Syllabus Map

  • Types of bearing capacity (ultimate, net ultimate, safe, allowable)
  • Factors affecting bearing capacity
  • Modes of foundation (shear) failure
  • Terzaghi's General Bearing Capacity Theory (assumptions, equation, failure zones, factors)
  • Bearing capacity of cohesionless and cohesive soils (special cases)
  • Effect of shape, depth, and load eccentricity/inclination (Meyerhof's extensions - commonly missed but exam-relevant)
  • Effect of water table
  • In-situ determination: Plate Load Test and SPT correlation (commonly asked alongside this chapter)
  • Settlement: types, causes, allowable limits, and its link to bearing capacity

Exam Weightage at a Glance

Marks Typical Question Approx. Frequency
1 mark (MCQ) Definitions, "which failure occurs in loose sand", factors, water table effect 2 to 4 questions almost every exam
5 marks Define bearing capacity / types / factors / differentiate ultimate vs allowable / effect of water table Appears nearly every exam
10 marks Terzaghi's theory with assumptions and equation / modes of failure with sketches / derive the equation Appears in most exams, sometimes twice

Golden rule: If you only have time to prepare one thing from this chapter, prepare Terzaghi's theory (assumptions + equation + sketch of the three zones). It is asked the most and it anchors almost every other question in the chapter.

1. What is Bearing Capacity?

Definition: Bearing capacity is the ability of soil to safely support the load transmitted by a foundation without shear failure or excessive settlement.

One-line version for exam: Bearing capacity is the maximum pressure that soil can safely carry.

Why it matters

If applied pressure exceeds the soil's bearing capacity:

  • Foundation settles excessively
  • Shear failure occurs
  • Building tilts
  • Structural cracks develop
  • Complete collapse may occur

2. Types of Bearing Capacity

(A) Ultimate Bearing Capacity (qu)

The maximum pressure the soil can withstand just before shear failure occurs.

(B) Net Ultimate Bearing Capacity (qnu)

qnu = qu - gamma * Df

where Df = depth of foundation, gamma = unit weight of soil.

This is the ultimate capacity in excess of the original overburden pressure removed during excavation - i.e., the extra pressure the soil can take beyond what it already carried before the foundation trench was dug.

(C) Safe Bearing Capacity (qs)

qs = qu / FOS

Typical FOS = 2.5 to 3, depending on the design code and soil investigation confidence.

(D) Allowable Bearing Capacity (qa)

The maximum pressure that satisfies BOTH:

  • Shear failure criteria, and
  • Settlement criteria

This is the value actually used in foundation design - not qu, not qs alone.

Memory Trick

Ultimate  --->  divide by FOS  --->  Safe
Safe + settlement check  --->  Allowable  (design value)

Think of it as a funnel: you start with the soil's absolute maximum strength (ultimate), you tone it down with a safety factor (safe), then you check settlement on top of that before you finally accept a number for design (allowable).

3. Factors Affecting Bearing Capacity

This is a favourite 5-mark question. Organize your answer under four headings - examiners like structured answers.

Soil Properties - Cohesion (c) - Angle of internal friction (phi) - Unit weight (gamma) - Density / relative density - Compressibility

Foundation Properties - Width (B) - wider footing generally increases bearing capacity in cohesionless soil - Shape (strip, square, circular, rectangular) - Depth (Df) - deeper foundation increases bearing capacity (more surcharge/confinement) - Inclination of the base

Groundwater - A higher water table reduces effective stress and decreases bearing capacity (see Section 8 below for the full mechanism and diagram).

Loading - Vertical (centric) loads - Inclined loads - Eccentric loads - Dynamic / cyclic loads

Mnemonic: "SFGL" - Soil properties, Foundation properties, Groundwater, Loading. Four buckets, easy to reproduce under exam pressure.

4. Modes of Shear Failure

There are three classical modes, first classified by Vesic based on Terzaghi's observations.

(A) General Shear Failure

Occurs in: dense sand, stiff clay (shallow footing, low compressibility soil).

Characteristics: - Well-defined, continuous failure surface from footing edge to ground surface - Sudden, brittle failure - Significant ground heave next to the footing - High bearing capacity; failure load is clearly identifiable on the load-settlement curve

(B) Local Shear Failure

Occurs in: medium dense soils (transitional behaviour).

Characteristics: - Failure surface starts to form below the footing but does not reach the ground surface clearly - Moderate settlement before failure - Slight/limited ground heave - Failure load is not sharply defined - judgment is needed to pick qu

(C) Punching Shear Failure

Occurs in: loose sand, soft clay, or any soil at large depth of embedment.

Characteristics: - Footing punches straight down, compressing soil directly below it - No visible failure surface at the ground surface, little to no heave - Large settlement occurs gradually with increasing load; no clear "collapse" point

Load-Settlement Behaviour (the graph most students forget to draw)

The single best way to remember the difference between these three failures is the shape of the load-settlement curve. General shear gives a sharp peak; punching gives no peak at all; local shear sits in between.

Load settlement curves for general, local, and punching shear failure

Comparison Table

General Local Punching
Dense soil Medium dense soil Loose / soft soil
Sudden failure Gradual failure Progressive penetration
Clear, continuous failure surface Partial failure surface No distinct failure surface
Ground heave visible Slight heave No heave
Small settlement at failure Moderate settlement Large settlement, no clear failure point
Sharp peak on load-settlement curve Gentle, less-defined peak No peak - curve keeps sinking

Memory Trick - "DMC-GLP": Dense = General, Medium = Local, loose (Cohesionless soft) = Punching. Or simply remember by density: Dense - Medium - Loose maps to General - Local - Punching, in that order, like a ladder going down in both soil strength and failure sharpness.

5. Terzaghi's General Bearing Capacity Theory

Developed by: Karl Terzaghi (1943) - the "father of soil mechanics." This is the single most important topic in the chapter.

Applicable to: - Strip footing (long, continuous footing - length much greater than width) - Vertical, centric loading - Homogeneous soil - Shallow foundation (Df less than or approximately equal to B)

Assumptions (very commonly asked - memorise all seven)

  1. Soil is homogeneous.
  2. Soil is isotropic.
  3. Foundation is shallow (Df <= B).
  4. Base of footing is rough (full friction/adhesion develops between footing and soil).
  5. Load is vertical and concentric (no eccentricity, no inclination).
  6. Ground surface is horizontal near the footing.
  7. Failure occurs by general shear failure only (soil above the footing level is treated only as a surcharge, its shear strength is ignored).

Memory Trick - "HISS-RGF": Homogeneous, Isotropic, Shallow, Soil above footing acts only as surcharge, Rough base, Ground horizontal, Failure is general shear. Seven letters, seven assumptions.

The Three Failure Zones (this sketch alone can earn you half the marks of a 10-mark question)

Terzaghi visualised the soil beneath a strip footing failing in three connected zones:

  • Zone I - Elastic wedge (active Rankine zone): A wedge of soil directly under the footing that moves down as a single rigid block with the footing, pushing the soil around it outward.
  • Zone II - Radial shear zone: Curved (logarithmic spiral) shear surfaces radiating outward from the edges of the footing.
  • Zone III - Passive Rankine zone: Soil pushed up and outward at the ground surface, causing the characteristic heave seen in general shear failure.

Terzaghi's three failure zones beneath a strip footing

How to reproduce this sketch fast in the exam: Draw the footing, then a small triangle directly below it (Zone I), then two curved "fan" shapes on either side (Zone II), then two straight-sided triangles further out at the ground surface (Zone III). Label B and Df. That is a full mark-scoring sketch.

Terzaghi's General Bearing Capacity Equation (the most important formula in this chapter)

For a strip footing:

qu = c * Nc + gamma * Df * Nq + 0.5 * gamma * B * Ngamma

Where: - c = cohesion of soil - gamma = unit weight of soil - Df = depth of foundation - B = width of footing - Nc, Nq, Ngamma = Terzaghi's bearing capacity factors (depend only on phi, the angle of internal friction)

Term-by-term memory trick - "CDW" (Cohesion, Depth/surcharge, Width): - 1st term (c * Nc) = contribution from cohesion - 2nd term (gamma * Df * Nq) = contribution from surcharge / depth - 3rd term (0.5 * gamma * B * Ngamma) = contribution from self-weight / width of the failure wedge

Bearing Capacity Factors (Nc, Nq, Ngamma)

These depend only on the angle of internal friction, phi. As phi increases, all three factors increase - and they increase very fast (this is why the chart below uses a log scale).

Terzaghi's bearing capacity factors versus phi

Approximate values (Terzaghi's original table - good enough to quote in an exam):

phi (degrees) Nc Nq Ngamma
0 5.7 1.0 0.0
10 9.6 2.7 1.2
20 17.7 7.4 5.0
30 37.2 22.5 19.7
40 95.7 81.3 100.4

Takeaway line for the exam: "Soils with higher angle of internal friction generally have higher bearing capacity, since all three bearing capacity factors increase with phi."

Special Cases (frequently tested as a follow-up to the main derivation question)

Pure Clay (phi = 0, undrained condition):

Since Nq = 1 and Ngamma = 0, the equation simplifies to:

qu = c * Nc + gamma * Df

(with Nc = 5.7 for a strip footing under Terzaghi's theory)

Cohesionless Soil (c = 0, e.g. clean sand):

qu = gamma * Df * Nq + 0.5 * gamma * B * Ngamma

Memory Trick: Clay keeps the "c" term and the surcharge term but drops the width term (Ngamma = 0, since clay's strength does not come from friction/self-weight). Sand drops the cohesion term entirely (c = 0) but keeps both the surcharge and the width term.

Shape Factors - Terzaghi's Modified Equation (often missing from notes, but asked in exams and interviews)

Terzaghi's original equation is only for a strip (continuous) footing. For other shapes, shape factors are applied:

Square footing:

qu = 1.3 * c * Nc + gamma * Df * Nq + 0.4 * gamma * B * Ngamma

Circular footing:

qu = 1.3 * c * Nc + gamma * Df * Nq + 0.3 * gamma * B * Ngamma

Rectangular footing (B = width, L = length):

qu = c * Nc * (1 + 0.3*B/L) + gamma * Df * Nq + 0.5 * gamma * B * Ngamma * (1 - 0.2*B/L)

Memory Trick - shape multipliers on the cohesion term: Strip = 1.0, Rectangle = 1 + 0.3(B/L), Square = 1.3, Circle = 1.3 with a smaller (0.3) width-term multiplier. As the footing becomes more "compact" (square/circular vs strip), the cohesion term's contribution goes up but the self-weight (Ngamma) term's contribution goes down.

Limitations of Terzaghi's Theory (common interview / short-answer question)

  • Valid mainly for shallow, strip footings; not accurate for deep or eccentric/inclined loading without modification.
  • Assumes general shear failure only, so it overestimates capacity for loose sand or soft clay where local/punching failure actually governs.
  • Ignores the shear strength of soil above the footing base (treats it only as surcharge), which under-predicts capacity for some cases.
  • Does not directly account for water table position (must be corrected separately, see Section 8), compressibility of soil, or footing rigidity.

Beyond Terzaghi - other bearing capacity theories (good to know, sometimes asked as "who else proposed...")

  • Meyerhof (1963): Extended the theory to include shape, depth, and inclination factors, and introduced the effective width method for eccentric loads (see Section 7).
  • Hansen (1970) and Vesic (1973): Further generalised the equation with additional factors for depth, inclination, ground slope, and base tilt - widely used in modern practice and codes.
  • Skempton (1951): Proposed Nc values for cohesive soil that vary with the Df/B ratio, rising from 5.14 (surface strip footing) up to about 9 for deep footings - useful specifically for undrained clay.

One-line summary for the exam: "While Terzaghi's theory remains the classical basis for bearing capacity analysis, later theories by Meyerhof, Hansen, and Vesic extended it to cover shape, depth, inclination, and eccentricity effects more realistically."

6. Effect of Foundation Shape, Depth, and Eccentric/Inclined Loading

This section fills a gap that is very commonly examined but often missing from short notes.

Depth factor

As Df increases, bearing capacity increases because of the additional confining surcharge from the soil above the footing level. This is why deeper foundations are used in weaker soils or for heavier structures.

Eccentric Loading - Meyerhof's Effective Width Method

When a load is applied with an eccentricity "e" (not exactly at the centre of the footing), part of the footing becomes less effective. Meyerhof's approach uses a reduced, "effective" footing dimension for design:

B' = B - 2e   (effective width)

The bearing capacity equation is then applied using B' in place of B, and the ultimate load is:

Qu = qu * B' * L'

Why this matters: Eccentric loads (common under columns with moment, or retaining walls) reduce the effective area available to resist load, so the same footing carries less safe load than it would under pure centric loading.

Inclined Loading

Inclined loads (e.g., from wind, earthquake, or a sloping strut) reduce bearing capacity because part of the load tries to push the foundation sideways rather than straight down. Inclination factors (which reduce Nc, Nq, Ngamma) are applied in Meyerhof's and Hansen's equations to account for this.

Memory Trick: Centric + vertical = full bearing capacity. Any deviation - eccentricity or inclination - always reduces bearing capacity, never increases it.

7. In-Situ Determination of Bearing Capacity (commonly paired with this chapter)

While Terzaghi's theory is analytical, real design also uses field tests. This is frequently examined as a companion topic.

Plate Load Test (PLT)

A steel plate (usually 300 mm to 750 mm) is loaded incrementally at the proposed foundation level, and settlement is recorded at each load step to plot a load-settlement curve, from which the plate's ultimate/allowable bearing capacity is read directly.

Key limitation: The plate is much smaller than an actual footing, so results must be extrapolated using scaling relationships (approximately, bearing capacity in sand scales up with footing width, while settlement in clay scales up with footing width too) - a very common "limitation" question.

SPT (Standard Penetration Test) Correlation

The N-value from a Standard Penetration Test is commonly correlated to allowable bearing capacity of sandy soils using empirical charts/formulas (e.g., Teng's or Peck-Hanson-Thornburn correlations), and is widely used in Nepal for quick estimation on ordinary building foundations where a full lab-based analysis is not feasible.

Memory Trick: PLT = "small-scale direct load test," SPT = "index-based correlation test." Both are indirect substitutes for Terzaghi-style analytical calculation when detailed soil parameters (c, phi) are not available.

8. Effect of Water Table

When the water table rises toward or above the foundation level:

  • Effective stress in the soil decreases.
  • The unit weight acting below the water table must be taken as the submerged (buoyant) unit weight, gamma_sub = gamma_sat - gamma_water, which is much lower than the dry/bulk unit weight.
  • Because both the surcharge term (gamma * Df) and the self-weight term (gamma * B) in Terzaghi's equation depend on unit weight, a rise in water table directly reduces qu.
  • Overall, bearing capacity decreases.

Effect of ground water table on bearing capacity

Correction factors (Rw1, Rw2): Many codes apply reduction factors Rw1 (for water table above the footing base) and Rw2 (for water table below the footing base but within the influence zone, roughly B below the base) directly to the NqDf and NgammaB terms respectively, ranging from 0.5 (water at/above base) to 1.0 (water at depth B or more below the base).

This is why the position of the groundwater table is always checked and reported before foundation design - it is one of the first things a geotechnical report will state.

9. Settlement and Bearing Capacity

Even if soil does not fail in shear, excessive settlement alone can make a foundation unsafe or unserviceable (cracking, tilting, misalignment of machinery, etc). This is why design always checks BOTH shear failure and settlement criteria - this is precisely the definition of allowable bearing capacity from Section 2.

Types of Settlement (frequently examined, often missing from short notes - add this to your revision)

1. Immediate (Elastic) Settlement - Occurs immediately after load application, mainly in cohesionless soils and unsaturated soils, due to elastic/distortion deformation without any change in water content.

2. Consolidation Settlement - Occurs slowly over time in saturated fine-grained (clayey) soils, as excess pore water pressure dissipates and effective stress increases, squeezing water out of the soil skeleton. - Split further into: - Primary consolidation - settlement due to dissipation of excess pore water pressure (governed by Terzaghi's consolidation theory, coefficient of consolidation Cv). - Secondary consolidation (creep) - additional settlement that continues after excess pore pressure has fully dissipated, due to plastic readjustment of soil particles (governed by the secondary compression index, C-alpha).

3. Total Settlement

Total Settlement = Immediate Settlement + Primary Consolidation Settlement + Secondary Consolidation Settlement

Memory Trick - "IPS": Immediate (instant, elastic) -> Primary consolidation (pore pressure dissipates, takes time) -> Secondary consolidation (creep, continues after pore pressure is gone). Settlement happens in that chronological order.

Differential Settlement

Uneven settlement between different parts of the same structure (e.g., between two columns) is often more damaging than uniform total settlement, since it causes tilting, cracking in walls/beams, and jamming of doors and windows. Most codes place a tighter limit on differential settlement than on total settlement.

Typical Allowable Settlement Limits (commonly asked as a short question)

  • Isolated footings on sand: total settlement generally limited to about 25 mm to 40 mm.
  • Raft foundations: somewhat higher total settlement is tolerated, since the structure moves more uniformly.
  • Differential settlement between adjacent columns is usually limited to a small fraction of the span (commonly quoted as roughly 1/300 to 1/500 of the distance between columns, depending on the code and structure type), to avoid cracking in framed structures.

(Exact numeric limits vary by code - IS 1904 / relevant Nepal Building Code provisions - so in the exam it is safer to state the concept and the general order of magnitude rather than quoting a single number as absolute.)

10. Engineering Applications

Bearing capacity analysis is required for:

  • Buildings
  • Bridges
  • Retaining walls
  • Towers
  • Chimneys
  • Hydropower structures
  • Industrial foundations

Full Chapter Memory Box (final revision before exam)

  • Bearing Capacity = soil's ability to support foundation loads safely.
  • Ultimate (qu): pressure at shear failure.
  • Net Ultimate (qnu) = qu - gamma*Df.
  • Safe (qs) = qu / FOS.
  • Allowable (qa): governs design; satisfies both shear and settlement criteria.
  • Three failure modes: General (dense, sharp peak), Local (medium, gradual), Punching (loose/soft, no peak).
  • Terzaghi's assumptions: HISS-RGF (7 points).
  • Terzaghi's equation: qu = cNc + gammaDfNq + 0.5gammaBNgamma.
  • Nc, Nq, Ngamma depend only on phi and all increase with phi.
  • Pure clay: qu = cNc + gammaDf. Cohesionless: qu = gammaDfNq + 0.5gammaB*Ngamma.
  • Shape factors: strip (1.0) < rectangle < square/circle (1.3) on the cohesion term.
  • Eccentric load: use effective width B' = B - 2e (Meyerhof).
  • Higher water table -> lower unit weight below GWT -> lower bearing capacity.
  • Settlement types, in order: Immediate -> Primary consolidation -> Secondary consolidation.
  • In-situ tests: Plate Load Test (direct, small-scale) and SPT correlation (index-based) for quick estimation.

Mock Exam Set (weighted exactly like the real Loksewa paper)

Section A - 5 Marks Questions (answer any one style shown, all follow the same structure)

Q1. Define bearing capacity and differentiate between ultimate, safe, and allowable bearing capacity. (5 Marks)

Model Answer: Bearing capacity is the maximum pressure that soil can safely carry without shear failure or excessive settlement.

  • Ultimate bearing capacity (qu) is the pressure at which the soil fails in shear.
  • Safe bearing capacity (qs) is qu divided by a factor of safety (typically 2.5 to 3), giving a safety margin against shear failure alone.
  • Allowable bearing capacity (qa) is the pressure adopted for design, satisfying both the shear-failure criterion and the settlement criterion; it may be lower than qs if settlement, not shear, governs.

In short: Ultimate -> (divide by FOS) -> Safe -> (check settlement) -> Allowable (design value).

Q2. List and explain the factors affecting bearing capacity of soil. (5 Marks)

Model Answer (use SFGL structure): Soil properties (cohesion, friction angle, unit weight, density, compressibility), Foundation properties (width, shape, depth, base inclination), Groundwater position, and Loading type (vertical, inclined, eccentric, dynamic). [Expand each with one line as shown in Section 3.]

Q3. Explain the effect of the ground water table on bearing capacity. (5 Marks)

Model Answer: [Use Section 8 - effective stress reduction, submerged unit weight, Rw1/Rw2 correction factors, and the diagram description: soil below GWT contributes less to both the surcharge and self-weight terms of Terzaghi's equation, so qu falls as the water table rises.]

Section B - 10 Marks Questions

Q4. Explain Terzaghi's General Bearing Capacity Theory with its assumptions and equation. (10 Marks)

Model Answer Outline: 1. Introduce Terzaghi (1943), state applicability (strip footing, shallow, centric vertical load, homogeneous soil). 2. List all seven assumptions (HISS-RGF). 3. Draw and label the three failure zones (Zone I elastic wedge, Zone II radial shear, Zone III passive Rankine) - reproduce the sketch from Section 5. 4. Write the equation: qu = cNc + gammaDfNq + 0.5gammaBNgamma, defining every term. 5. State that Nc, Nq, Ngamma depend only on phi and increase with phi; mention the special cases for pure clay and cohesionless soil. 6. Optionally add one line on limitations to show deeper understanding.

Q5. Explain the different modes of shear failure beneath a shallow foundation with sketches. (10 Marks)

Model Answer Outline: 1. Name the three modes: general, local, punching shear failure, and note they were classified based on soil density/consistency. 2. For each: state the soil type it occurs in, describe the failure surface, ground heave, and settlement behaviour (use the comparison table from Section 4). 3. Draw/describe the load-settlement curve for each (sharp peak for general, gentle peak for local, no peak for punching) - this single graph often carries a big share of the marks since it visually proves understanding. 4. Conclude with the memory trick: Dense - Medium - Loose maps to General - Local - Punching.

Mock MCQ Set (with expanded coverage beyond the original notes)

1. Terzaghi's bearing capacity theory is mainly applicable to: A. Pile foundation B. Strip footing C. Well foundation D. Raft foundation Answer: B

2. Punching shear failure usually occurs in: A. Dense sand B. Stiff clay C. Loose sand D. Rock Answer: C

3. Bearing capacity generally increases with: A. Higher cohesion B. Higher friction angle C. Greater foundation depth D. All of the above Answer: D

4. The foundation should normally be designed using: A. Ultimate bearing capacity B. Net ultimate bearing capacity C. Allowable bearing capacity D. Punching bearing capacity Answer: C

5. A rise in the groundwater table generally: A. Increases bearing capacity B. Decreases bearing capacity C. Has no effect D. Doubles the bearing capacity Answer: B

6. In Terzaghi's equation, the bearing capacity factor Ngamma becomes zero when: A. phi = 0 (pure clay) B. c = 0 C. Df = 0 D. B = 0 Answer: A

7. Which zone in Terzaghi's failure mechanism lies directly beneath the footing? A. Passive Rankine zone B. Radial shear zone C. Elastic wedge (Zone I) D. Surcharge zone Answer: C

8. Meyerhof's "effective width" concept (B' = B - 2e) is used to account for: A. Water table effects B. Eccentric loading C. Shape of footing D. Depth of footing Answer: B

9. Which type of settlement in clay continues even after excess pore water pressure has fully dissipated? A. Immediate settlement B. Primary consolidation settlement C. Secondary consolidation (creep) settlement D. Elastic settlement Answer: C

10. For a general shear failure, the load-settlement curve typically shows: A. No peak at all B. A sharp, well-defined peak C. A gradually increasing curve with no failure point D. A perfectly straight line Answer: B

Interview / Viva Questions (short, conceptual - often asked after the written exam)

  1. Why is allowable bearing capacity used instead of ultimate bearing capacity for design?
  2. Why does dense sand usually fail by general shear while loose sand fails by punching shear?
  3. How does the water table affect bearing capacity, and how is this correction applied?
  4. Why is settlement checked even when shear failure does not occur?
  5. What are the limitations of Terzaghi's bearing capacity theory?
  6. How does Meyerhof's theory differ from Terzaghi's theory?
  7. Why does an eccentric load reduce the safe bearing capacity of a footing?
  8. What is the practical difference between a Plate Load Test and an SPT-based estimate of bearing capacity?

End of Chapter 11 revision guide. Revisit the Memory Box and the four diagrams the night before your exam - together they cover roughly 90 percent of what has historically been asked from this chapter.