= Solution
For a slender <elastic beam> undergoing small bending, assume plane cross-sections remain plane and normal to the <neutral axis>. Fibres a signed distance $z$ from that axis have longitudinal <strain> $\varepsilon_{xx}=-z/R$, where $R$ is the local <radius of curvature>. <Hooke's law> gives <stress> $\sigma_{xx}=-Ez/R$, where $E$ is <Young's modulus>. Zero resultant axial <force> places the <neutral axis> through the section's <centroid> for a homogeneous beam.
The <bending moment> is the moment of this <stress> distribution. Its magnitude is
$$
\boxed{M=\frac{EI_A}{R},\qquad I_A=\int_{\rm section}z^2\,dA.}
$$
The <second moment of area> $I_A$ has units $\mathrm{m^4}$, and the <bending stiffness> $EI_A$ has units $\mathrm{N\,m^2}$. A rectangular section of width $b$ and depth $h$ has $I_A=bh^3/12$, so placing more material far from the <neutral axis> strongly resists bending. This is the purpose of an I-section.
For small deflection $y(x)$, the signed <curvature> is approximately $y''$. A consistent variational sign convention follows from the <elastic energy> and a transverse load $q(x)$:
$$
\Pi[y]=\frac12\int EI_A(y'')^2\,dx-\int qy\,dx,
\qquad \boxed{(EI_Ay'')''=q.}
$$
For constant <bending stiffness> this is the <Euler-Bernoulli beam equation> $EI_Ay''''=q$. A clamped end fixes $y=y'=0$; a simply supported end has $y=0$ and zero <bending moment>; an unloaded free end has zero moment and shear, $y''=y'''=0$ for uniform $EI_A$. Endpoint loads replace the corresponding zero-force <boundary conditions>.
For a cantilever of length $L$ carrying a tip <force> $F$, choose positive deflection along $F$. The internal bending gives $EI_Ay''=F(L-x)$, with $y(0)=y'(0)=0$. Two integrations give
$$
y(x)=\frac{Fx^2(3L-x)}{6EI_A},
\qquad \boxed{y(L)=\frac{FL^3}{3EI_A}.}
$$
This approximation requires small slopes, elastic strains and a length large compared with section dimensions. Shear deformation, rotary inertia, plastic yielding and large <rotations> require extensions. Adding inertia gives <flexural waves> with $\omega^2=(EI_A/\rho_m A_s)k^4$ for a uniform beam, where $\rho_m A_s$ is <mass> per unit length.
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