Week 11 Cheatsheet — Normal Stress and Strain + Mechanical Properties

How this week breaks down

Three layers, all from one tensile test. Skim this once, then revise from the in-depth note.

Layer	What you compute
Stress & strain	Translate force + geometry into $\sigma =P/A$ and $\epsilon =\Delta L/L_0$.
Mechanical properties	Read $E$, ${\sigma }_y$, UTS, fracture stress and %EL/%AR off the $\sigma$–$\epsilon$ diagram.
Design checks	Apply factor of safety $\Rightarrow {\sigma }_{\text{allow}}$; check lateral deformation via Poisson’s ratio.

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1 — Stress and strain

Definitions

Quantity	Formula	Common units
Axial stress	$\sigma =P/A$	Pa = N/m² ; MPa = N/mm² ; GPa = kN/mm²
Axial strain	$\epsilon =\Delta L/L_0=(L-L_0)/L_0$	dimensionless (or %)

Unit rule. Force is always in newtons. $1\text{MPa}=1{\text{N/mm}}^2=10^6\text{Pa}$. If you mix kN with mm², the stress comes out in GPa — convert first.

Recipe — average normal stress in a member

1. Free-body-diagram the joint or section. Solve $\sum F_x=0$, $\sum F_y=0$ for the axial force $P$.
2. Compute $A=\pi d^2/4$ (or rectangular $A=b\times h$).
3. $\sigma =P/A$. Mind the units (kN $\to$ N; mm² $\to$ m² if you want pascals).

Worked snippets

Setup	Work	Answer
$P=10\text{kN}$, $A=500{\text{mm}}^2$	$\sigma =10000/500$	$20\text{MPa}$
Brass cylinder, $L_0=500\text{mm}$, $\epsilon =0.005$	$\Delta L=0.005\times 500$	$\Delta L=2.5\text{mm}$, $L_f=502.5\text{mm}$

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2 — Elastic deformation and Young’s modulus

Hooke’s law

$\sigma =E\epsilon ⟺E=\frac{\Delta \sigma }{\Delta \epsilon }$

Valid only in the linear elastic region. Outside it, the relationship is no longer proportional.

Reading the $\sigma$–$\epsilon$ diagram

Feature	What it means
Slope of linear part	Young’s modulus $E$
Proportional limit	End of strict linearity
Yield stress ${\sigma }_y$	Transition elastic $\to$ plastic
UTS ${\sigma }_{\text{UTS}}$	Peak stress on the engineering curve
Fracture stress	Stress at break (engineering curve falls after necking)

Engineering vs. true

Variant	Stress	Strain
Engineering	${\sigma }_{\text{eng}}=F/A_0$	${\epsilon }_{\text{eng}}=\Delta L/L_0$
True	${\sigma }_{\text{true}}=F/A$	${\epsilon }_{\text{true}}=\delta /L_0$

After necking, engineering stress falls while true stress rises (instantaneous area shrinks).

0.2% offset yield

Draw a line from $\epsilon =0.002$ on the strain axis, parallel to the elastic slope. Where it crosses the curve = offset yield stress.

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3 — Ductility

$\%EL=\frac{L_f-L_0}{L_0}\times 100,\%AR=\frac{A_f-A_0}{A_0}\times 100$

%EL > 5% generally counts as ductile; brittle materials fracture before significant plastic strain.

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4 — Factor of safety and allowable stress

$SF=\frac{{\sigma }_{\text{failure}}}{{\sigma }_{\text{allowable}}}⟺{\sigma }_{\text{allowable}}=\frac{{\sigma }_{\text{failure}}}{SF}$

Rule-of-thumb $SF$ values (slide 19):

Application	Typical $SF$
Structural members in buildings	$2.0$
Pressurised vessels	$3.5$ – $4.0$
Automobile	$3.0$
Aircraft / spacecraft	$1.2$ – $3.0$

Ductile materials use lower $SF$; brittle materials use higher.

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5 — Poisson’s ratio

${\epsilon }_{\text{long}}=\frac{\Delta L}{L_0},{\epsilon }_{\text{lat}}=\frac{\Delta d}{d_0},\nu =-\frac{{\epsilon }_{\text{lat}}}{{\epsilon }_{\text{long}}},0\le \nu \le 0.5$

Typical values (slide 22):

Material	$\nu$	Material	$\nu$
Aluminium	$0.33$	Cast iron	$0.22$–$0.30$
Brass	$0.34$	Titanium	$0.34$
Lead	$0.43$	Tungsten	$0.28$
Steel	$0.30$	Concrete	$0.10$–$0.20$
Glass	$0.22$	Clay	$0.41$

Use $\Delta d={\epsilon }_{\text{lat}}\cdot d_0$ to find diameter change, then $A_f=\pi d_f^2/4$ for the necked area.

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Common mistakes

1. Mixing kN with mm² — the result is GPa, not MPa. Always convert force to newtons.
2. Substituting $E$ in MPa for steel — should be $\approx 207\text{GPa}$. Off by 1000.
3. 0.5% strain $=0.5$ — it’s $0.005$. Convert percentages first.
4. Forgetting the minus sign on Poisson’s ratio — definition has it; $\nu$ comes out positive for normal materials.
5. Rounding people up — always round down so the allowable stress is respected.
6. Confusing yield, UTS, and fracture stress — they’re three distinct points on the curve.
7. Differentiating engineering vs. true stress after necking — engineering uses $A_0$ always.

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Key formulas at a glance

$$\sigma =\frac{P}{A},\epsilon =\frac{\Delta L}{L_0},\sigma =E\epsilon$$ $$\%EL=\frac{L_f-L_0}{L_0}\times 100,\%AR=\frac{A_f-A_0}{A_0}\times 100$$ $$SF=\frac{{\sigma }_{\text{failure}}}{{\sigma }_{\text{allowable}}},\nu =-\frac{{\epsilon }_{\text{lat}}}{{\epsilon }_{\text{long}}}$$

For the why behind each formula and worked end-to-end examples, see the in-depth note.