Fundamentals of Engineering Exam (FE) Engineering Mechanics and Materials Syllabus

State the principle of work and energy for a particle.

$$U_{1\to2}=\Delta T=\tfrac{1}{2}m v_2^{2}-\tfrac{1}{2}m v_1^{2}$$ The total work done by all forces equals the change in kinetic energy. Work of a force: $U=\int \vec{F}\cdot d\vec{r}$.

State the principle of impulse and momentum for a particle.

$$\int_{t_1}^{t_2}\sum \vec{F}\,dt=m\vec{v}_2-m\vec{v}_1$$ The linear impulse of the net force equals the change in linear momentum. If $\sum\vec{F}=0$, momentum is conserved.

Define the coefficient of restitution $e$ for a direct central impact.

$$e=\frac{v_{B}'-v_{A}'}{v_A-v_B}$$ the ratio of relative separation velocity to relative approach velocity along the line of impact. $e=1$ is perfectly elastic; $e=0$ is perfectly plastic (bodies move together).

What is the mass moment of inertia, and give it for a thin uniform rod of mass $m$ and length $L$ about its center.

The mass moment of inertia $I$ measures resistance to angular acceleration, $I=\int r^{2}\,dm$. For a slender rod about its centroidal transverse axis: $$I_G=\frac{1}{12}m L^{2}$$ and about one end $I=\frac{1}{3}m L^{2}$.

State the equation of rotational motion for a rigid body about its mass center $G$.

$$\sum M_G=I_G\,\alpha$$ The net moment about the mass center equals the centroidal mass moment of inertia times the angular acceleration. Combined with $\sum\vec{F}=m\vec{a}_G$, these govern general plane motion.

What is the kinetic energy of a rigid body in general plane motion?

$$T=\tfrac{1}{2}m v_G^{2}+\tfrac{1}{2}I_G\,\omega^{2}$$ the sum of translational kinetic energy of the mass center and rotational kinetic energy about the mass center.

Define the instantaneous center of zero velocity (IC) for a rigid body in plane motion.

The IC is the point in (or extended from) the body that momentarily has zero velocity. All points move as if the body rotates purely about the IC, so $v=\omega\, r_{IC}$, where $r_{IC}$ is the distance from the point to the IC.

Give the natural (angular) frequency of an undamped mass-spring system and its relation to period.

$$\omega_n=\sqrt{\frac{k}{m}}\quad\text{(rad/s)},\qquad f=\frac{\omega_n}{2\pi},\qquad T=\frac{1}{f}=2\pi\sqrt{\frac{m}{k}}$$

Give the natural frequency of a simple pendulum of length $L$ undergoing small oscillations.

$$\omega_n=\sqrt{\frac{g}{L}},\qquad T=2\pi\sqrt{\frac{L}{g}}$$ independent of mass and (for small angles) of amplitude.

Define the damping ratio $\zeta$ and classify the response for $\zeta<1$, $\zeta=1$, and $\zeta>1$.

$$\zeta=\frac{c}{c_c}=\frac{c}{2\sqrt{km}}$$ $\zeta<1$: underdamped (oscillatory decay); $\zeta=1$: critically damped (fastest non-oscillatory return); $\zeta>1$: overdamped (slow non-oscillatory return).

Define normal (axial) stress and normal strain.

Normal stress: $$\sigma=\frac{P}{A}$$ (force perpendicular to area $A$). Normal strain: $$\varepsilon=\frac{\delta}{L}$$ (change in length per unit original length). Stress units are Pa or psi; strain is dimensionless.

State Hooke's law in one dimension and define Young's modulus.

$$\sigma=E\varepsilon$$ where $E$ (Young's modulus) is the slope of the linear-elastic stress-strain curve, a measure of stiffness. Valid up to the proportional limit.

Define Poisson's ratio and give the relation between $E$, $G$, and $\nu$ for isotropic materials.

Poisson's ratio: $$\nu=-\frac{\varepsilon_{lateral}}{\varepsilon_{axial}}$$ Shear modulus relation: $$G=\frac{E}{2(1+\nu)}$$ Typical metals have $\nu\approx0.3$.

Give the formula for axial deformation of a prismatic bar under load.

$$\delta=\frac{P L}{A E}$$ where $P$ is axial force, $L$ length, $A$ cross-sectional area, $E$ Young's modulus. For varying load/area, integrate: $\delta=\int \frac{P(x)}{A(x)E}\,dx$.

State the torsion formula for shear stress in a circular shaft.

$$\tau=\frac{T\rho}{J}$$ where $T$ is applied torque, $\rho$ the radial distance from the axis (max at outer radius), and $J$ the polar moment of inertia. Maximum stress: $\tau_{max}=\frac{T r}{J}$.

Give the angle of twist for a circular shaft of length $L$ under constant torque.

$$\phi=\frac{T L}{J G}$$ where $T$ is torque, $L$ length, $J$ polar moment of inertia, and $G$ the shear modulus.

State the flexure (bending stress) formula for a beam.

$$\sigma=\frac{M c}{I}$$ where $M$ is the bending moment, $c$ the distance from the neutral axis to the outer fiber, and $I$ the centroidal moment of inertia. Stress is zero at the neutral axis and maximum at the extreme fibers.

Give the transverse shear stress formula in a beam.

$$\tau=\frac{V Q}{I b}$$ where $V$ is the shear force, $Q$ the first moment of the area above (or below) the point about the neutral axis, $I$ the moment of inertia, and $b$ the section width at that point.

State the differential relationships among distributed load $w$, shear $V$, and moment $M$ in a beam.

$$\frac{dV}{dx}=-w(x),\qquad \frac{dM}{dx}=V(x)$$ Slope of the shear diagram equals negative load; slope of the moment diagram equals shear. The moment is maximum where $V=0$.

Write the general 2D stress-transformation equation for normal stress on a plane rotated by angle $\theta$.

$$\sigma_{x'}=\frac{\sigma_x+\sigma_y}{2}+\frac{\sigma_x-\sigma_y}{2}\cos 2\theta+\tau_{xy}\sin 2\theta$$ The shear transforms as $\tau_{x'y'}=-\frac{\sigma_x-\sigma_y}{2}\sin 2\theta+\tau_{xy}\cos 2\theta$.

Give the in-plane principal stresses for a 2D stress state.

$$\sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^{2}+\tau_{xy}^{2}}$$ On principal planes the shear stress is zero.

Give the maximum in-plane shear stress for a 2D stress state and its relation to the principal stresses.

$$\tau_{max}=\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^{2}+\tau_{xy}^{2}}=\frac{\sigma_1-\sigma_2}{2}$$ It occurs on planes $45^\circ$ from the principal planes.

What does Mohr's circle represent, and what are its center and radius?

Mohr's circle is a graphical representation of 2D stress transformation, plotting $\sigma$ vs. $\tau$. Center: $\sigma_{avg}=\frac{\sigma_x+\sigma_y}{2}$ on the $\sigma$-axis. Radius: $R=\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^{2}+\tau_{xy}^{2}}=\tau_{max}$. Each $\theta$ rotation = $2\theta$ on the circle.

Give the elastic-curve differential equation governing beam deflection.

$$EI\frac{d^{2}y}{dx^{2}}=M(x)$$ Integrating once gives the slope $EI\,y'=\int M\,dx$, and twice gives deflection $y$, with integration constants set by boundary conditions.

State Euler's critical buckling load for a column and define the effective-length factor.

$$P_{cr}=\frac{\pi^{2}EI}{(KL)^{2}}$$ where $KL$ is the effective length. $K=1$ pinned-pinned, $K=0.5$ fixed-fixed, $K=0.7$ fixed-pinned, $K=2$ fixed-free (cantilever). Buckling uses the smallest $I$.

What are the three principal types of primary (atomic) bonding, and which is non-directional?

Ionic (electron transfer, electrostatic attraction), covalent (electron sharing, strongly directional), and metallic (electron sea, non-directional). Metallic and ionic bonds are non-directional; covalent bonds are directional. Secondary bonding (van der Waals, hydrogen) is much weaker.

Compare the FCC, BCC, and HCP crystal structures by atoms per unit cell and atomic packing factor.

FCC: 4 atoms/cell, APF $=0.74$; BCC: 2 atoms/cell, APF $=0.68$; HCP: 6 atoms/cell, APF $=0.74$. FCC and HCP are close-packed (max density); BCC is less dense.

Define yield strength, ultimate tensile strength, and ductility from a tensile test.

Yield strength: stress at onset of plastic deformation (often the $0.2\%$ offset). Ultimate tensile strength (UTS): maximum engineering stress on the curve. Ductility: extent of plastic deformation before fracture, e.g. $\%\,EL=\frac{L_f-L_0}{L_0}\times100$.

Distinguish toughness, resilience, and hardness as mechanical properties.

Toughness: total energy absorbed to fracture (area under the full stress-strain curve). Resilience: elastic energy stored to the yield point, $U_r=\frac{\sigma_y^{2}}{2E}$. Hardness: resistance to localized plastic indentation (e.g., Rockwell, Brinell).

What is fatigue, and what is the endurance (fatigue) limit?

Fatigue is failure under cyclic loading at stresses below the static strength, due to crack initiation and growth. The endurance limit is a stress amplitude (seen in ferrous alloys) below which the material can endure essentially infinite cycles; many non-ferrous metals (e.g., aluminum) have no true endurance limit.

Name the four primary classes of engineering materials and give a distinguishing property of each.

Metals: high stiffness/strength, ductile, electrically/thermally conductive. Ceramics: hard, brittle, high melting point, good insulators. Polymers: low density, flexible, low stiffness, often insulating. Composites: engineered combinations (e.g., fiber + matrix) giving tailored high strength-to-weight ratios.