---
title: "Flexible vs rigid; radius of relative stiffness, k"
subject: "Transportation Engineering II"
chapter: 2
key: "semester_7/transportation_engineering_ii:flexible-rigid-differences-relative-stiffness-k"
source: https://www.bctengineer.com/q/semester_7/transportation_engineering_ii/flexible-rigid-differences-relative-stiffness-k
site: BCT Engineering
---

# Flexible vs rigid; radius of relative stiffness, k

Transportation Engineering II · Chapter 2: Highway Pavement

- Asked in: 2080 Baisakh, 4+4 marks
- Video: [Flexible v/s Rigid Pavement | Highway Engineering | Civil Engineering | by Abhishek Sir Unacademy](https://www.youtube.com/watch?v=NgoFVqYvXfs) — Civil Yodha by Unacademy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/transportation_engineering_ii/flexible-rigid-differences-relative-stiffness-k
- Whole chapter: https://www.bctengineer.com/md/semester_7/transportation_engineering_ii/chapter-2

## Question

List out the differences between flexible and rigid pavement. Explain the terms radius of relative stiffness (l) and modulus of subgrade reaction (k) used in the design of rigid pavements.

## Short answer

Flexible pavements use bituminous layers and granular base with grain-to-grain load transfer; rigid pavements use concrete slabs bearing loads through bending. Radius of relative stiffness l = (E h^3/(12(1-µ^2)k))^¼ defines the slab area resisting load. Modulus of subgrade reaction k = p/∆ is pressure per unit deflection, measured via plate-load test.

## Answer

#### Differences between flexible and rigid pavement
| Basis | Flexible pavement | Rigid pavement |
|---|---|---|
| Material | Bituminous surface over granular layers | Cement concrete slab (PQC) on a sub-base |
| Load transfer | Layer to layer by grain contact; load spreads in a cone | Slab action; bending resists the load, which spreads over a large area |
| Design basis | Limiting strains, CBR/resilient modulus, thickness by IRC:37 | Flexural stress and fatigue, Westergaard analysis, IRC:58 |
| Stresses on subgrade | High, so many layers are needed | Very low because of the wide spread |
| Joints | No joints, only cracks | Joints are essential (contraction, expansion, longitudinal) |
| Temperature effect | Surface softens in heat; little stress | Warping and expansion stresses, so joints and dowels are needed |
| Life and cost | 10-15 years, low initial cost, higher maintenance | 20-40 years, high initial cost, low maintenance |
| Repair and opening | Quick repair; opens to traffic soon | Long curing (28 days), difficult repair |
| Surface | Slightly rough, absorbs sound; self-healing | Smooth and rigid; good in water-logged areas, poor skid resistance when polished |

#### Radius of relative stiffness (l)
The radius of relative stiffness $l$ is a measure of how stiff the slab is compared with the subgrade support. It defines the area of slab that takes part in resisting a load (Westergaard).

$$
l=\left[\frac{E\,h^{3}}{12\,(1-\mu^{2})\,k}\right]^{1/4}
$$

where $E$ is the modulus of elasticity of concrete (kg/cm2), $h$ the slab thickness (cm), $\mu$ Poisson's ratio and $k$ the modulus of subgrade reaction (kg/cm3). The unit of $l$ is cm.

- A thicker slab or higher $E$ gives a larger $l$ (a stiffer slab spreads the load over a wider area); a higher $k$ gives a smaller $l$.
- Typical values for 20-30 cm slabs are about 60-100 cm.
- It is used in all the stress equations (Westergaard, Kelly, Bradbury), in the ratios $L/l$ and $B/l$ for the warping coefficient, and to decide joint spacing and the spread of dowel load transfer (about $1.8\,l$).

#### Modulus of subgrade reaction (k)
The **modulus of subgrade reaction** $k$ is the pressure required to cause unit deflection of the subgrade (Winkler's foundation assumption):

$$
k=\frac{p}{\Delta}\ \ (\text{kg/cm}^3)
$$

It is found by a **plate-load test**: a rigid 75 cm diameter plate is loaded in increments, and $k$ is taken at 0.125 cm deflection (so $k = p_{0.125}/0.125$). The test value is corrected as follows:

- Correction for plate size: $k_{75}= k_{30}/2$ when a 30 cm plate is used.
- Correction for saturation (soaking) of the subgrade soil.
- A sub-base (e.g. lean concrete or granular) raises the effective $k$; IRC:58 gives tables of $k$ over granular or cement-treated sub-bases.
- Roughly, IRC:58 relates $k$ to the soaked CBR of the subgrade: about 2 kg/cm3 for CBR 2 and 5 to 6 kg/cm3 for CBR 10.

Rigid pavement stresses depend only weakly on $k$ (through $l$), so approximate values are acceptable.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How do rigid and flexible pavements differ in the way they carry wheel load?
   A. Rigid: layer-to-layer grain contact in a cone; flexible: slab bending action
   B. Rigid: slab action with bending resisting the load over a large area; flexible: layer-to-layer grain contact spreading in a cone
   C. Both carry load by slab bending but with different thickness
   D. Flexible pavement carries load by dowel bars; rigid by tie bars
   Answer: B. A concrete slab resists load by bending and spreads it widely, while a flexible pavement spreads load through granular layers by grain-to-grain contact. (Key point: Load transfer mechanism)

2. Which pair of statements is correct for rigid pavement compared with flexible pavement?
   A. Higher subgrade stress and no joints
   B. Very low subgrade stress and joints are essential
   C. Very low subgrade stress and no joints
   D. Higher subgrade stress and joints are essential
   Answer: B. The wide spread of load in a rigid slab gives very low subgrade stress, and joints are essential to control temperature stresses. (Key point: Subgrade stress and joints)

3. What does the radius of relative stiffness l measure?
   A. Stiffness of the slab compared with that of the subgrade support
   B. The radius of the tyre contact area
   C. The distance between dowel bars
   D. The temperature difference across the slab
   Answer: A. l expresses how stiff the slab is relative to the subgrade and defines the area of slab taking part in resisting a load. (Key point: Meaning of l)

4. In l = [E h^3 / (12(1-mu^2) k)]^(1/4), what happens to l if the slab thickness is increased and k is increased?
   A. Thicker slab increases l; higher k decreases l
   B. Both increase l
   C. Both decrease l
   D. Thicker slab decreases l; higher k increases l
   Answer: A. h and E are in the numerator so l grows with them, while k is in the denominator so a stiffer subgrade gives a smaller l. (Key point: Effect of h and k on l)

5. How is the modulus of subgrade reaction k defined and found?
   A. Pressure per unit deflection (p/Delta), from a plate-load test at 0.125 cm deflection
   B. Deflection per unit pressure from a CBR test
   C. Ratio of flexural strength to compressive strength of concrete
   D. Load per unit area from a sieve analysis
   Answer: A. k = p/Delta (Winkler assumption), taken from a 75 cm plate load test at 0.125 cm deflection. (Key point: Definition of k)

## Related in Transportation Engineering II

- [Flexible vs rigid design; Westergaard temperature stresses](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aflexible-vs-rigid-design-westergaard-temperature) — Transportation Engineering II, chapter 2
- [Flexible pavement design with 8 CBR values](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aflexible-design-eight-cbr-340-cvd) — Transportation Engineering II, chapter 2
- [Short note: radius of relative stiffness](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Ashort-note-radius-relative-stiffness) — Transportation Engineering II, chapter 2
- [Flexible pavement design: four lane, ADT 8000 trucks](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aflexible-design-adt-8000-four-lane) — Transportation Engineering II, chapter 2
- [Flexible pavement failures; deflection overlay design](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Aflexible-failures-deflection-overlay) — Transportation Engineering II, chapter 4
- [Pavement design factors; IRC rigid design steps](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Apavement-factors-irc-rigid-steps) — Transportation Engineering II, chapter 2, asked 2×

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