Where \(\delta = \dfrac{PL^3}{3EI}\) comes from

Cantilever beam, concentrated load \(P\) at the free end. The goal is the vertical displacement of the tip, starting from the moment–curvature relation \(EI \cdot y'' = M(x)\) .

Axis \(x\) along the beam, origin at the fixed end, \(x = L\) at the free end. Axis \(y\) vertical, positive upward. \(P > 0\) is the magnitude of the load, directed downward.

Cantilever beam with tip point load — geometry and sign convention for deflection derivation

Cut at a generic section \(x\) . Free body of the segment from \(x\) to \(L\) : the only external force is \(P\) downward, applied at distance \((L - x)\) from the section. Moment equilibrium about the cut, \(M(x)\) drawn in the assumed positive sense:

\[M(x) = -P(L - x) \tag{1}\]

Sign: \(P\) downward with positive lever arm \((L - x)\) produces concavity downward — negative under the convention \(y'' > 0\) for upward concavity.

The moment is written on the undeformed geometry (first-order analysis). Consistent with the small-deformation hypothesis already embedded in \(EI \cdot y'' = M(x)\) .

(1) into \(EI \cdot y'' = M(x)\) :

\[EI \cdot y''(x) = -P(L - x) \tag{2}\]

\(EI\) constant along \(x\) : the cross-section does not change. (2) is a second-order, linear, non-homogeneous ODE with a first-degree polynomial forcing term.

Integrating (2) with respect to \(x\) :

\[EI \cdot y'(x) = -P\!\left(Lx - \frac{x^2}{2}\right) + C_1 \tag{3}\]

At the fixed end the section does not rotate:

\[y'(0) = 0 \tag{4}\]

(4) into (3): \(C_1 = 0\) . There remains:

\[EI \cdot y'(x) = -P\!\left(Lx - \frac{x^2}{2}\right) \tag{5}\]

Integrating (5) with respect to \(x\) :

\[EI \cdot y(x) = -P\!\left(\frac{Lx^2}{2} - \frac{x^3}{6}\right) + C_2 \tag{6}\]

At the fixed end the displacement is zero:

\[y(0) = 0 \tag{7}\]

(7) into (6): \(C_2 = 0\) . Full deflection curve:

\[y(x) = -\frac{P}{EI}\left(\frac{Lx^2}{2} - \frac{x^3}{6}\right) \tag{8}\]

A cubic polynomial in \(x\) . The third degree comes from a moment that is linear in \(x\) (first degree), integrated twice.

Deflected shape of a cantilever beam under tip load — result of double integration of EI·y″ = M(x)

\(x = L\) in (8):

\[y(L) = -\frac{P}{EI}\left(\frac{L \cdot L^2}{2} - \frac{L^3}{6}\right) = -\frac{P}{EI}\cdot\frac{L^3}{3} \tag{9}\]

The algebra: \(L \cdot L^2/2 - L^3/6 = L^3(3 - 1)/6 = L^3/3\) . The \(L^3\) factor is \(L\) from the moment arm times \(L^2\) from the double integration. The \(1/3\) factor is the combination \(1/2 - 1/6\) .

The tip deflection is the absolute value of \(y(L)\) :

\[\boxed{\delta = \frac{PL^3}{3EI}} \tag{10}\]

Assumptions, in the order they entered: plane sections, linear elastic material, small deformations (all three from \(EI \cdot y'' = M(x)\) , the starting point), moment on the undeformed geometry (1), \(EI\) constant along \(x\) (2), perfect fixed end — zero rotation (4) and zero displacement (7). If any one fails, (10) does not hold.