Stress is internal force intensity (normal) or (shear). Strain is relative deformation or. The engineering stress–strain curve (RK Bansal / GATE SOM) defines elastic limit, yield, UTS, and fracture; true stress–strain accounts for instantaneous area.

Define normal stress and shear stress. Give SI units.

Normal stress acts perpendicular to a section; shear stress (or ) acts tangent to the section. Both have SI unit or (often in SOM).

What is the difference between engineering stress and true stress?

Engineering stress uses original area:. True stress uses instantaneous area:. After necking, true stress keeps rising while engineering stress falls.

Define longitudinal and lateral strain. How are they related by Poisson’s ratio?

Longitudinal strain; lateral strain. Poisson’s ratio (negative sign because lateral strain is opposite in sense for uniaxial tension).

State Hooke’s law in uniaxial form and the range of validity.

within the elastic (proportional) limit. Beyond that, stress–strain is nonlinear; unloading may leave permanent set.

What is volumetric strain? Express it for a cube under hydrostatic pressure.

Volumetric strain. Under hydrostatic pressure,, where is the bulk modulus.

Define modulus of elasticity, modulus of rigidity, and bulk modulus.

(direct stress/strain); (shear); (volumetric). All have units of stress.

What is factor of safety? How is allowable stress obtained?

Factor of safety (FoS). Allowable (working) stress (brittle) or (ductile design).

A circular steel pin of diameter carries an axial tensile force of. Assuming uniform stress on the cross-section, find the tensile stress in MPa.

**Given:**,. **Find:** Normal stress on the cross-section. **Assumptions:** Uniform axial stress; Saint-Venant region (away from load points); engineering stress on original area. **Solution:** Cross-section area Axial stress Since, (tension). **Result:** (tensile). **Exam trap:** Using diameter instead of radius in, or mixing with N without converting.

A brass rod of length and shortens by under compression. Find the compressive strain and the corresponding stress.

**Given:**,,. **Find:** Strain and stress. **Assumptions:** Linear elastic (Hooke); uniform compression; small deformation. **Solution:** Engineering strain (compressive). Hooke’s law (compressive). **Result:**, (compression). **Exam trap:** Leaving in mm while is in m → strain wrong by. Always convert to the same length unit first.

For an aluminium alloy, and Poisson’s ratio. A tensile specimen develops longitudinal strain. Find the lateral strain and the shear modulus.

**Given:**,, (tension). **Find:** Lateral strain magnitude/sign and. **Assumptions:** Isotropic linear elastic solid; uniaxial tension. **Solution:** Poisson: (negative = lateral contraction). Shear modulus **Result:**,. **Exam trap:** Forgetting the minus sign on lateral strain, or using (wrong formula).

A material has and bulk modulus. Determine Poisson’s ratio.

**Given:**,. **Find:**. **Assumptions:** Isotropic Hookean solid; applies. **Solution:** **Result:**. **Exam trap:** Using or mixing GPa with MPa inconsistently. Check for stable isotropic solids.

Mild steel has yield stress and. Find the modulus of resilience. Also find the allowable stress if FoS based on yield.

**Given:**,, FoS. **Find:** Modulus of resilience and. **Assumptions:** Linear elastic to yield; FoS on yield for ductile design. **Solution:** **Result:**,. **Exam trap:** Using UTS instead of yield for FoS on a ductile steel design, or writing (missing the 2).

At a point in plane stress, the principal stresses are (tension) and (compression). Find the maximum in-plane shear stress and the absolute maximum shear stress.

**Given:**,, (plane stress). **Find:** and. **Assumptions:** Plane stress; Mohr’s circle / principal stress ordering. **Solution:** Ordered principals:,,. In-plane (using only): Absolute max shear uses the algebraically largest and smallest principals: (same here). If both in-plane principals were tensile, you must also check against out-of-plane. **Result:** (in-plane and absolute). **Exam trap:** Ignoring the zero out-of-plane principal when both in-plane stresses have the same sign.

A steel rod is rigidly fixed between two walls. It is heated by. Take and. Find the thermal stress developed. State its nature.

**Given:** Fixed ends,,,. **Find:** Thermal stress and whether tension/compression. **Assumptions:** Fully constrained; uniform temperature; linear elastic; supports rigid. **Solution:** Free thermal expansion would be. Walls cancel this with mechanical compression. Equivalent: Heating + fixed ends → **compressive** stress. **Result:** (compressive). **Exam trap:** Reporting tension on heating, or using without. No stress if free to expand.

Under uniaxial tension, a material has. Show that the volumetric strain is zero, and state what this idealization means physically.

**Given:** Uniaxial stress;; longitudinal strain. **Find:** Volumetric strain; physical meaning. **Assumptions:** Isotropic Hooke; small strain. **Solution:** Strains:,. means **incompressible** response (volume preserved). Rubber-like elasticity and fully plastic flow of metals are often modelled this way. **Result:** when in uniaxial stress. **Exam trap:** Thinking is typical for steel elastically; is the plastic/incompressible limit.

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Core concept

Stress and Strain Basics

Stress is internal force intensity (normal) or (shear).

31 min15 Interview8 GATEGATEInterview

Level

Advanced

Topic tags

Stress and strain, Som Civil, Civil

Useful for

GATE favourite, Interview ask

Source

Strength of Materials

Learning outcomes

What you'll learn in this topic

1
Normal stress: — tensile (positive) or compressive; SI unit $\mathrm{N/m^2}=\mathrm{Pa}$ (often MPa).
2
Shear stress: on a plane parallel to the force.
3
Normal strain: (dimensionless); shear strain (or in radians) for small angles.
4
Hooke’s region: until proportional limit; $E$ = Young’s modulus.

Key Formulas

Important equations & their meanings

\sigma=\frac{P}{A}

Worked Examples

Step-by-step solved problems

1Axial stress and strain in a steel rod
2Single vs double shear in a pin
3Reading the stress–strain curve

Common Mistakes

Avoid these errors in exams & interviews

Using radius in A=π d²/4, or diameter in A=π r², or forgetting the /4.
Confusing engineering stress with true stress after necking.
Mixing N/mm² and N/m² (factor 106).

Practice Questions

Strengthen your concepts with questions

23+Practice Questions

Interview Questions

Most asked interview problems

15Interview Questions

GATE Questions

Previous year GATE questions

8GATE Questions

Applications in real world

Design checksCore engineering calculations
Plant workTroubleshooting on site
InterviewsGATE / campus rounds
Product teamsDay-to-day engineering use

Visual concept

Problem

Model

Compute

Verify

Model the physics, compute, then verify against limits.

Recommended Book

Strength of MaterialsRK Bansal

Read: Syllabus unit

View on Amazon

University exams

Important Topic

Industry relevance

High

Concept difficulty

Hard

Average time

31 min

Core assumptions (state these in exams)

1. Continuum — material is continuous; stress/strain defined at a point as averages over a small area/volume.
2. Homogeneous and isotropic unless anisotropy is stated (composites, wood).
3. Uniform stress on the section for simple axial/shear formulas (Saint-Venant: valid away from load application points).
4. Small deformations — geometry based on undeformed dimensions (engineering stress/strain).
5. Plane sections remain plane for elementary bar theory.
6. Quasi-static loading — inertia neglected; no wave propagation.
7. Temperature constant unless thermal strain is included separately.

If the bar is tapered or stepped,

\sigma=\frac{P}{A}(x)

varies; integrate for elongation rather than using a single

A

Step-by-step problem approach

1. Identify load type: axial, shear, bearing, or combined.
2. Draw FBD; find internal

P

V

on the critical section.
3. Compute area carefully —

A=\pi \frac{d^2}{4}

(not

\pi r^2

with

d

), hollow, or net area after holes.
4.

\sigma=\frac{P}{A}

\tau=\frac{V}{A}

; convert to MPa.
5. For deformation:

\delta=\frac{PL}{AE}

in the elastic range (Hooke).
6. On stress–strain questions: locate the described point (yield, UTS, fracture).
7. Check units: N and mm² → MPa directly; N and m² → Pa, then ÷

10^6

for MPa.
8. State assumptions (uniform stress, small strain, elastic).

Advanced problem-solving framework

Use this sequence for long-form mastery and repeatable scoring:
1. Identify objective, system boundary, and required output.
2. Write all givens in SI units and classify each as measured, assumed, or estimated.
3. Choose the governing model and relation (Normal stress:

\sigma=\frac{P}{A}

— tensile (positive) or compressive; SI unit

\mathrm{N/m^2}=\mathrm{Pa}

(often MPa).) with one-line justification.
4. Solve symbolically first to catch structural mistakes early.
5. Substitute values with careful unit tracking.
6. Cross-check by sign, order of magnitude, and limiting case.
7. Write a short engineering conclusion tied to safety, performance, reliability, or cost.

Next, solve one "variant version" of the same problem by changing one assumption (loading type, losses, property constancy, boundary condition, or uncertainty level). This builds transfer ability — essential for difficult exams where numbers and wording are changed deliberately.

For viva/interviews, convert your written method into a 45-second explanation format:
"Objective -> model selected -> key assumption -> result -> practical implication."
This makes your answers concise and technically credible.

Exam, interview, and note-making strategy

To make this topic genuinely reusable, maintain notes in four blocks: concept summary, assumptions checklist, solved template, and common error-correction logic. This transforms passive reading into active revision material for class tests, semester exams, GATE-style practice, and interviews.

A practical weekly cycle:
- Day 1: read and annotate the topic.
- Day 3: solve one moderate numerical from memory.
- Day 5: give a 60-second oral explanation.
- Day 7: solve one mixed problem integrating this topic with a prerequisite.
- Day 14: do a timed review to test retention.

For interview readiness, prepare concise answers to:
1. Where is this used in real engineering?
2. Which assumption is most risky if wrong?
3. How do you sanity-check the result quickly?
4. What trade-off does this result influence?

These four questions are asked repeatedly in technical panels, and practicing them creates confidence.

Use this page as a living notebook: append class doubts, lab observations, previous-year tricks, and personal mnemonics. That personalization is what turns a study page into a repeat-visit resource students trust.

Industry scenarios and decision context

Engineering decisions are made under constraints: deadline, budget, material availability, process capability, safety requirements, and maintenance realities. So while solving stress and strain basics, do not treat the answer as "final truth" without context. The numerical output is a decision input, not the decision itself.

Ask these context questions after every solved example:
- If load uncertainty increases, does design margin remain acceptable?
- If manufacturing tolerance drifts, will performance degrade critically?
- If operating temperature/humidity changes, are properties still valid?
- If maintenance is delayed, what failure mode appears first?

Students who practice contextual questioning develop judgment faster and perform better in internships, design tasks, and technical interviews. This context-first style is a major retention driver because learners see immediate real-world value.

Long-form revision worksheet

Use this worksheet when preparing notes:
A) One-paragraph concept explanation in your own words.
B) Symbol and units table for key variables.
C) Validity limits and assumptions list.
D) One standard solved pattern with all steps.
E) One variant problem where an assumption changes.
F) One industry-use explanation with failure consequence.
G) Three common mistakes and their correction rules.

If you can fill all seven blocks without external help, your topic depth is strong enough for repeat use and long retention. If not, revisit the corresponding section and strengthen the missing block.

This structured worksheet approach is intentionally longer than quick revision notes because it is designed for durable mastery. It supports exactly the product goal you mentioned: students should keep coming back because the page is complete enough to build serious notes.

Definition and physical meaning

Stress is the intensity of internal forces that develop within a body to resist external loads. At a cut section, resolve the resultant into a normal component

P

(perpendicular to the section) and a shear (tangential) component

V

Normal (direct) stress

\sigma=\frac{P}{A}

- Tension: fibres elongate; conventionally

\sigma>0

.
- Compression: fibres shorten;

\sigma<0

(or reported as compressive magnitude).

Shear stress

\tau=\frac{V}{A}

where

A

is the area of the plane parallel to

V

Strain measures deformation relative to original size:

\varepsilon=\frac{\delta}{L}=\frac{L-L_0}{L_0}\qquad\text{(normal)}

\gamma=\frac{\delta_s}{L}\approx\theta\quad(\theta\text{ in radians, small)}

Physical parameters

Symbol	Meaning	SI unit
$P$ , $V$	Axial / shear force	$\mathrm{N}$
$A$	Cross-sectional area	$\mathrm{m^2}$
$\sigma$ , $\tau$	Normal / shear stress	$\mathrm{Pa}$ ( $\mathrm{N/m^2}$ )
$\delta$ , $L$	Elongation / length	$\mathrm{m}$
$\varepsilon$ , $\gamma$	Normal / shear strain	— (dimensionless)
$E$	Young’s modulus	$\mathrm{Pa}$
$\nu$	Poisson’s ratio	—

Practical unit:

1\,\mathrm{MPa}=10^6\,\mathrm{Pa}=1\,\mathrm{N/mm^2}

Core assumptions (state these in exams)

If the bar is tapered or stepped,

\sigma=\frac{P}{A}(x)

varies; integrate for elongation rather than using a single

A

Engineering stress–strain curve

A standard tensile test plots engineering stress

\sigma=\frac{P}{A_0}

versus engineering strain

\varepsilon=\frac{\delta}{L_0}

Key points on the curve (mild steel)
1. Proportional limit — end of linear

\sigma

–

\varepsilon

; Hooke’s law holds.
2. Elastic limit — unloading returns to zero permanent set (nearly coincides with proportional limit for many metals).
3. Yield point (upper/lower for mild steel) — large strain at nearly constant stress;

\sigma_y=\frac{P_y}{A_0}

.
4. Ultimate tensile strength (UTS) — maximum engineering stress;

\sigma_u=\frac{P_{\max}}{A_0}

.
5. Fracture / breaking point — specimen separates; engineering stress falls after necking because

A_{0}

is fixed while

P

drops.

Ductile vs brittle
- Ductile (mild steel, Al): large plastic strain, clear yield, necking.
- Brittle (cast iron, concrete in tension): little plasticity; fracture near elastic limit.

True stress and true strain

\sigma_{\mathrm{true}}=\frac{P}{A_{\mathrm{inst}}}\qquad\varepsilon_{\mathrm{true}}=\int_{L_0}^{L}\frac{dL}{L}=\ln\left(\frac{L}{L_0}\right)

Before necking (volume constancy

A L = A_{0} L_{0}

\sigma_{\mathrm{true}}=\sigma_{\mathrm{eng}}(1+\varepsilon_{\mathrm{eng}})

Types of loading and stress states

Axial (tensile/compressive): uniform

\sigma=\frac{P}{A}

on a transverse section (ideal bar).

Shear (single/double): rivet or pin;

\tau=\frac{P}{A}

\frac{P}{2A}

for double shear.

Bearing / crushing: contact pressure

\sigma_b=\frac{P}{d\cdot t}

for a pin in a plate of thickness

t

Bending: normal stress varies linearly over depth (flexure formula — separate module).

Torsion: shear stress varies with radius in circular shafts.

Combined loading: use superposition in the elastic range; principal stresses via Mohr’s circle when both

\sigma

and

\tau

act.

Volumetric strain (small strain, isotropic):

\varepsilon_v=\varepsilon_x+\varepsilon_y+\varepsilon_z

For uniaxial stress with Poisson effect:

\varepsilon_v=\varepsilon(1-2\nu)

Bulk modulus relates hydrostatic pressure to volumetric strain:

K=-\frac{p}{\varepsilon}_v

Poisson’s ratio and lateral strain

When a bar is stretched longitudinally, it contracts laterally:

\nu=-\frac{\varepsilon_{\mathrm{lateral}}}{\varepsilon_{\mathrm{longitudinal}}}

For uniaxial tension

\sigma_x

(others free):

\varepsilon_x=\frac{\sigma_x}{E},\qquad\varepsilon_y=\varepsilon_z=-\nu\frac{\sigma_x}{E}

Limits: theoretically

-1<\nu<0.5

for isotropic stable solids; metals

\nu\approx 0.25

–

0.33

; rubber

\approx 0.5

(nearly incompressible).

Exam note: If

\nu=0.5

and plastic incompressible flow is assumed,

\varepsilon_v=0

Fundamentals: factor of safety and allowable stress (Bansal Ch. 1)

Working (design) stress must stay below a material limit by a factor of safety (FoS):

\sigma_{\mathrm{allow}}=\frac{\sigma_{\mathrm{limit}}}{\mathrm{FoS}},\qquad \mathrm{FoS}=\frac{\sigma_{\mathrm{limit}}}{\sigma_{\mathrm{working}}}.

Which limit?
- Ductile materials: usually yield

\sigma_y

(prevent permanent set).
- Brittle materials: usually ultimate

\sigma_u

(fracture governs).

FoS accounts for load uncertainty, material scatter, stress concentrations, and idealised analysis. Always use the FoS stated in the problem — do not invent a value in GATE numericals unless asked conceptually.

Concept: elasticity, elastic limit, and Hooke’s region

Elasticity — ability to regain original shape on unloading.
Elastic limit — maximum stress with no permanent set.
Proportional limit — end of linear

\sigma

–

\varepsilon

; Hooke’s law

\sigma=E\varepsilon

holds up to this point.

For many metals the proportional and elastic limits nearly coincide; mild steel then shows a distinct yield plateau. Beyond yield, permanent (plastic) strain remains after unloading — do not use

\delta=\frac{PL}{AE}

there without an elastic–plastic model.

Step-by-step problem approach

1. Identify load type: axial, shear, bearing, or combined.
2. Draw FBD; find internal

P

V

on the critical section.
3. Compute area carefully —

A=\pi \frac{d^2}{4}

(not

\pi r^2

with

d

), hollow, or net area after holes.
4.

\sigma=\frac{P}{A}

\tau=\frac{V}{A}

; convert to MPa.
5. For deformation:

\delta=\frac{PL}{AE}

in the elastic range (Hooke).
6. On stress–strain questions: locate the described point (yield, UTS, fracture).
7. Check units: N and mm² → MPa directly; N and m² → Pa, then ÷

10^6

for MPa.
8. State assumptions (uniform stress, small strain, elastic).

Common mistakes in exams

• Using radius in

A=\pi \frac{d^2}{4}

, or diameter in

A=\pi r^2

, or forgetting the

/4

.
• Confusing engineering stress with true stress after necking.
• Mixing N/mm² and N/m² (factor

10^6

).
• Applying

\sigma=\frac{P}{A}

at a stress concentration without a factor

K_{t}

.
• Taking double-shear area as

A

instead of

2A

.
• Reporting strain with units (strain is dimensionless).
• Using ultimate stress as allowable without dividing by FoS.

Detailed conceptual understanding

Stress and Strain Basics should be studied as a complete reasoning chain: definition, governing assumptions, physical interpretation, boundary conditions, and limits of validity. In som civil, strong students do not stop at "what is the formula"; they explain why the model applies, which simplifications are being used, and what error appears when those simplifications break. This is the key difference between memorized learning and engineering understanding.

A high-quality conceptual pass should answer these questions in writing:
1. Which quantity is being predicted or controlled?
2. Which variables dominate sensitivity and why?
3. Which assumptions are explicit, and which are hidden?
4. What real-world effects are neglected in first-pass analysis?
5. Which engineering decision depends on this output?

When revising, rewrite the concept in your own words and attach one real scenario from lab, workshop, project, internship, or industry case. This habit transforms abstract theory into retrievable memory. If a topic cannot be explained without reading the page, it is not yet mastered.

Use this page as a note source: create a "concept map" with cause-effect arrows and keep updating it whenever you solve new problems. Students who maintain evolving concept maps typically retain topics longer and return less to emergency cramming.

Advanced problem-solving framework

\sigma=\frac{P}{A}

— tensile (positive) or compressive; SI unit

\mathrm{N/m^2}=\mathrm{Pa}

Exam, interview, and note-making strategy

These four questions are asked repeatedly in technical panels, and practicing them creates confidence.

Industry scenarios and decision context

Common misconceptions and correction patterns

Most weak performance comes from repeated misconception patterns, not from lack of intelligence. Typical patterns include unit inconsistency, wrong model selection, assumption mismatch, and skipping interpretation after substitution.

Correction pattern to practice:
1. Detect: identify exactly where logic diverged.
2. Diagnose: state why that step is invalid.
3. Repair: rewrite with correct model/assumption.
4. Verify: run a sanity check and compare trends.

Maintain a personal "mistake log" with three columns: mistake, reason, correction rule. Reviewing this log before exams has a larger performance impact than reading new theory repeatedly.

Use the same correction discipline in interviews: acknowledge the slip, state corrected logic, and proceed. This demonstrates professional maturity and keeps the discussion positive even when you initially miss a step.

Ask about this topic

What you'll learn in this topic

Key Formulas

Worked Examples

Common Mistakes

Practice Questions

Interview Questions

GATE Questions

Applications in real world

Visual concept

Recommended Book

What should I learn next?

Core assumptions (state these in exams)

Step-by-step problem approach

Advanced problem-solving framework

Exam, interview, and note-making strategy

Industry scenarios and decision context

Long-form revision worksheet

Definition and physical meaning

Core assumptions (state these in exams)

Engineering stress–strain curve

Types of loading and stress states

Poisson’s ratio and lateral strain

Fundamentals: factor of safety and allowable stress (Bansal Ch. 1)

Concept: elasticity, elastic limit, and Hooke’s region

Step-by-step problem approach

Common mistakes in exams

Detailed conceptual understanding

Advanced problem-solving framework

Exam, interview, and note-making strategy

Industry scenarios and decision context

Common misconceptions and correction patterns

Long-form revision worksheet