GATE ME Strength of Materials Formula Sheet 2027

Strength of materials is the subject GATE ME's syllabus calls Mechanics of Materials. Its questions are short once the right formula is chosen, and slow when it is guessed. This sheet gives the formulas the subject runs on, a worked one-line example for each group, and the trap that costs marks.

Key takeaways

• Stress transformation underlies pressure-vessel seams, rosettes and the failure theories of Machine Design, so learn it first.
• A thin-walled vessel question rarely stops at hoop stress; the marks are in the strain or the inclined plane that follows.
• Find the reactions first, then walk along the beam; an applied couple makes the bending moment jump.
• Learn the standard deflections as stiffnesses, because vibration questions reuse them.
• Thermal stress comes only from the restrained part of the free expansion.
• The 2027 syllabus rewords several lines, such as thin-walled pressure vessels and mechanical properties, toughness and hardness.

The terms this sheet uses

Stress $\sigma$ is internal force per unit area, and strain $\varepsilon$ is change in length per unit length. A principal plane carries no shear, and the normal stress on it is a principal stress. $E$, $G$ and $K$ are the Young's, shear and bulk moduli, and $\nu$ is Poisson's ratio.

$I$ is the second moment of area about the bending axis; $J$ is the polar moment for torsion. Tension is positive throughout. Keep MPa with millimetres, since $1\ \text{MPa} = 1\ \text{N/mm}^2$.

Elastic constants and Hooke's law

Example: $E = 210$ GPa and $\nu = 0.25$ give $\displaystyle G = \frac{210}{2(1.25)} = 84$ GPa and $\displaystyle K = \frac{210}{3(0.5)} = 140$ GPa. With $\sigma_x = 100$ MPa, $\sigma_y = 50$ MPa and $E = 200$ GPa, $\nu = 0.3$: $\displaystyle \varepsilon_x = \frac{100 - 15}{200\,000} = 4.25 \times 10^{-4}$.

Stress transformation and Mohr's circle

The normal stress on a plane at angle $\theta$ is the mean stress plus a part that turns with $2\theta$:

Mohr's circle has its centre at the mean stress and its radius equal to the maximum in-plane shear. Example: $\sigma_x = 80$, $\sigma_y = 20$, $\tau_{xy} = 40$ MPa give centre 50 and radius $\sqrt{30^2 + 40^2} = 50$, so $\sigma_1 = 100$ MPa, $\sigma_2 = 0$ and $\theta_p = 26.57^\circ$.

Trap: in plane stress with $\sigma_3 = 0$, the absolute maximum shear is half the largest of $\lvert\sigma_1 - \sigma_2\rvert$, $\lvert\sigma_1\rvert$ and $\lvert\sigma_2\rvert$. When both principal stresses share a sign, it exceeds the circle's radius.

For plane strain, use the same equations with $\varepsilon$ for $\sigma$ and $\gamma/2$ for $\tau$. Plotting $\gamma$ instead of $\gamma/2$ doubles the radius.

Thin-walled pressure vessels

Example: $p = 1.5$ MPa, $d = 600$ mm, $t = 6$ mm give $\sigma_h = 75$ MPa and $\sigma_l = 37.5$ MPa. With $E = 200$ GPa and $\nu = 0.3$, the hoop strain is $\displaystyle \frac{75 - 11.25}{200\,000} = 3.19 \times 10^{-4}$ and the volumetric strain $7.13 \times 10^{-4}$. A helical seam is a transformation from these two directions.

Shear force and bending moment

Example: a 6 m simply supported beam under $w = 10$ kN/m has reactions of 30 kN and $\displaystyle M_{\max} = \frac{10 \times 36}{8} = 45$ kN m. This group appeared in 2023, 2024, 2025 and 2026, more steadily than any other in the subject.

Remember: an applied couple makes the bending moment diagram jump by its value and leaves the shear force unchanged. Contraflexure is where $M$ changes sign, not where $V$ is zero.

The book's last-minute sheet lists every formula on this page with its condition, and the Mechanics of Materials chapter traces each topic through the six counted papers. Both are in the GATE ME 2027 book, with 942 questions with worked solutions.

Bending, shear stress and shear centre

Example: $b = 100$ mm, $d = 200$ mm and $M = 20$ kN m give $Z = 666\,667\ \text{mm}^3$ and $\sigma = 30$ MPa. A shear force of 40 kN on the same section gives a mean of 2 MPa and a maximum of 3 MPa.

Deflection and energy methods

Strain energy is $\displaystyle \frac{P^2L}{2AE}$ in tension, $\displaystyle \int\frac{M^2}{2EI}\,dx$ in bending and $\displaystyle \frac{T^2L}{2GJ}$ in torsion. Castigliano gives the deflection under a load as $\displaystyle \delta = \frac{\partial U}{\partial P}$. A suddenly applied load doubles the static stress, $\displaystyle \sigma = \frac{2P}{A}$.

Example: a 2 m cantilever with $EI = 2 \times 10^6\ \text{N m}^2$ has stiffness $\displaystyle \frac{3 \times 2 \times 10^6}{8} = 750$ kN/m, so a 10 kN tip load deflects it 13.3 mm.

Torsion of circular shafts

Example: $T = 1$ kN m on a 50 mm shaft gives $\displaystyle \tau_{\max} = \frac{16 \times 10^6}{\pi \times 125\,000} = 40.7$ MPa; at 300 rpm it transmits 31.4 kW. Shafts in series add twists; a bonded core and sleeve share one twist and add torques, in proportion to $GJ$.

Columns and thermal stress

Example: a 2 m pinned column with $E = 200$ GPa and $I = 8 \times 10^{-7}\ \text{m}^4$ has $P_{cr} = 395$ kN. A steel bar with $\alpha = 12 \times 10^{-6}$ per K heated 50 K is stressed to 120 MPa if fully restrained, and to 60 MPa if a 0.3 mm gap is first taken up over 1 m.

Strain gauges, rosettes and mechanical properties

Example: rosette readings of 400, 300 and $-100$ microstrain give $\gamma_{xy} = 300$ microstrain, centre 150 and radius $\sqrt{250^2 + 150^2} = 291.5$, so the principal strains are 441.5 and $-141.5$ microstrain.

In one line: a rosette is Mohr's circle for strain with one extra line of arithmetic.

Using this sheet in the exam

Write $\displaystyle I = \frac{\pi d^4}{64}$ and $\displaystyle J = \frac{\pi d^4}{32}$ on the rough sheet in the first five minutes; the exam-day rules guide says what you may carry. Keep full precision on the GATE virtual calculator and round only the final NAT entry. Most questions here carry two marks, so check the marking scheme, common to every GATE paper, before guessing on an MCQ.

More formula sheets: all of GATE ME · Fluid Mechanics · Heat Transfer · Thermodynamics

Quick revision

1. $E = 2G(1 + \nu) = 3K(1 - 2\nu)$.
2. Principal stresses: centre $\pm$ radius; an angle doubles on Mohr's circle.
3. Hoop $\displaystyle \frac{pd}{2t}$, longitudinal $\displaystyle \frac{pd}{4t}$, sphere $\displaystyle \frac{pd}{4t}$.
4. Simply supported UDL gives $\displaystyle \frac{wL^2}{8}$; an applied couple makes $M$ jump.
5. Cantilever $\displaystyle \frac{PL^3}{3EI}$; simply supported $\displaystyle \frac{PL^3}{48EI}$ and $\displaystyle \frac{5wL^4}{384EI}$.
6. $\displaystyle \tau_{\max} = \frac{16T}{\pi d^3}$; $\displaystyle P = \frac{2\pi NT}{60}$.
7. Euler $L_e$: $L$, $2L$, $\displaystyle \frac{L}{2}$, $\displaystyle \frac{L}{\sqrt{2}}$.
8. Thermal stress only from the restrained expansion, $E\alpha\Delta T$ at most.