---
title: "Stress concentration and Saint-Venant's principle"
subject: "Strength of Materials"
chapter: 3
key: "semester_3/strength_of_materials:stress-concentration-saint-venant"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/stress-concentration-saint-venant
site: BCT Engineering
---

# Stress concentration and Saint-Venant's principle

Strength of Materials · Chapter 3: Simple Stress and Strain

- Asked in: 2081 Bhadra, 2+2+3 marks
- Video: [Axial Loading: St Venant's Principle and Stress Concentrations](https://www.youtube.com/watch?v=2VnQrm7FaPM) — ECUSW
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/stress-concentration-saint-venant
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-3

## Question

Describe stress concentration and stress concentration factor. What information does Saint-Venant's principle provide on this? Why factor of safety is used in design?

## Short answer

Stress concentration is local stress increase near geometric discontinuities, quantified by factor Kt = σmax/σnom. Saint-Venant's principle states this disturbance is local, stresses becoming uniform at about one cross-section dimension away. Factor of safety accounts for load, material, and analysis uncertainties, deterioration, and safety requirements.

## Answer

#### Stress concentration and stress concentration factor
Stress concentration is the localised rise of stress near a sudden change in cross-section or geometry (holes, notches, grooves, keyways, fillets, sharp corners, threads), where the flow of internal force is disturbed. The stress at such a place is much higher than the average stress calculated from $P/A$.

The **stress concentration factor** is

$$
K_t = \frac{\sigma_{max}}{\sigma_{nom}}
$$

where $\sigma_{max}$ is the peak stress at the discontinuity and $\sigma_{nom}$ is the nominal stress based on the net area. $K_t \ge 1$ and depends on geometry (not on the material, as long as it is elastic). For a small circular hole in a wide plate in tension $K_t = 3$.

```text
 Stress flow lines crowd near a hole:
  ─────┐   ┌─────
  ─────┤ ○ ├─────   σ_max at the edge of the hole
  ─────┘   └─────
```

Effects are serious for brittle materials and for fatigue loading, whereas ductile materials under static loading yield locally and redistribute stress.

#### Saint-Venant's principle
It states that the stress and strain produced at points sufficiently far from the place of application of a load (or from a discontinuity) depend only on the **resultant** of the load, not on the exact way it is distributed; the differences between two statically equivalent loadings become negligible at a distance about equal to the largest cross-sectional dimension of the member.

With regard to stress concentration, it tells us that the disturbance is **local**: high stresses exist only in the neighbourhood of the hole, notch or point load, and at a distance of about one width from it the stress becomes uniform ($P/A$). So the formula $\sigma = P/A$ may be used away from these places and the concentration need be considered only locally.

#### Why a factor of safety is used
The factor of safety ($FOS = \dfrac{\text{ultimate or yield stress}}{\text{permissible (working) stress}}$) is used because of:
1. Uncertainty in the actual loads (overloading, impact, dynamic effects).
2. Variation in material properties, defects and non-uniformity.
3. Approximations in analysis and stress concentrations, workmanship and fabrication errors.
4. Deterioration with time (corrosion, fatigue, wear).
5. The need for safety of life and property, and to keep stresses in the elastic range.

Typical values: 1.5 to 2 for steel (yield basis), 3 to 4 for brittle materials (ultimate basis), higher for dynamic loads.

## Check yourself

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

1. What is the stress concentration factor Kt?
   A. σnominal / σmax
   B. Yield stress / working stress
   C. σmax / σnominal at a discontinuity
   D. Strain / stress
   Answer: C. Kt compares the peak stress at a hole, notch or fillet with the nominal stress on the net area. (Key point: Definition)

2. What does Saint-Venant's principle say about stress concentration?
   A. Stress is high everywhere
   B. The disturbance is local; about one width away the stress is uniform P/A
   C. Only the load's point of application matters far away
   D. Stress concentration depends on the material
   Answer: B. Far from the discontinuity only the resultant matters, so P/A can be used there. (Key point: Saint-Venant)

3. When is stress concentration most serious?
   A. Brittle materials and fatigue loading
   B. Ductile materials under static load
   C. Compression only
   D. Elastic range only
   Answer: A. Ductile materials yield locally under static load and redistribute stress. (Key point: Affected materials)

4. What is the factor of safety?
   A. Permissible / ultimate
   B. Load / area
   C. Yield strain / ultimate strain
   D. Ultimate or yield stress / permissible stress, to allow for uncertainties
   Answer: D. FOS covers load uncertainty, material variability, analysis approximations and deterioration. (Key point: Factor of safety)

## Related in Strength of Materials

- [Stress concentration and its effect in brittle material](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astress-concentration-brittle-effect) — Strength of Materials, chapter 3
- [Resultant stress on planes EF and GE, principal stresses](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aresultant-stress-plane-ef-ge) — Strength of Materials, chapter 4
- [Principal stresses, stress invariant, equivalent stress](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Astress-invariant-equivalent-stress) — Strength of Materials, chapter 4
- [Simply supported I-beam: stress at flange-web junction](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Ai-beam-flange-web-junction-stress) — Strength of Materials, chapter 7

## Related in other subjects

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- [Saint-Venant, second-order explicit scheme](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Asaint-venant-second-order-explicit) — Computational Techniques in Civil Engineering, chapter 5, asked 2×
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