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Cantilever Beam Deflection Formulas

Reference equations for a prismatic cantilever (Euler–Bernoulli, small deflection) with: applied end/offset moment, partial/trapezoidal line load, and point load. Variables: \(L\) (length), \(x\) (measured from the free end), \(E\) (modulus), \(I\) (second moment), \(c\) (extreme-fiber distance). The fixed support is at \(x=L\).

How to use these formulas

  • Pick the load case below (moment, distributed load, or point load).
  • Use the listed \(V(x)\) and \(M(x)\) expressions to find internal actions.
  • Use boundary conditions at the fixed end (\(x=L\)) to determine constants.
  • Keep units consistent: \(E\) in Pa, \(I\) in m\(^4\), loads in N, lengths in m \(\Rightarrow y\) in m.
Coordinate
\(x=0\) at free end, \(x=L\) at fixed end
Deflection
\(y>0\) upward, \(y<0\) downward
Slope
\(\theta>0\) counter-clockwise rotation of tangent
Moment sign
Positive \(M\) follows the red curved arrow in the diagram

Sign convention used on this page

  • Coordinates: \(x\) is measured from the free end (\(x=0\)) toward the fixed end (\(x=L\)).
  • Deflection: \(y>0\) is upward, \(y<0\) is downward.
  • Slope: \(\theta>0\) rotates the tangent counter-clockwise (centerline inclines up to the right); \(\theta<0\) clockwise.
  • Loads: If \(P>0\) is drawn downward, then it produces \(y<0\) deflection under this convention.
  • Moments: \(M_0\) is positive in the direction shown in the diagram (red curved arrow). Reaction moment sign follows this same convention.
  • Free end: The free end carries no reaction force or reaction moment.
Cantilever beam sign convention showing positive/negative deflection y, slope theta, shear P and moment M directions
Sign convention for deflection \(y\), slope \(\theta\), and bending moment \(M\) used throughout this page.

Singularity (Macaulay) functions

\(\langle x-a\rangle^{n}=0\) for \(x<a\), and \(\langle x-a\rangle^{n}=(x-a)^{n}\) for \(x\ge a\). In particular, \(\langle x-a\rangle^{0}=0\) for \(x<a\) and \(\langle x-a\rangle^{0}=1\) for \(x\ge a\) (a step).

Cantilever with Applied Moment

Cantilever beam fixed at x=L with applied moment M0 at position a measured from the free end
Applied moment \(M_0\) at the free end (set \(a=0\)) or at position \(a\). Fixed support is at \(x=L\).

Reactions & Shear

Reaction at fixed end

$$R_2 = 0$$

Shear at distance \(x\)

$$V(x) = 0$$

Moments & Stress

Reaction moment at fixed end

$$M_2 = M_0$$

Bending moment at distance \(x\)

$$M(x) = M_0\,\langle x-a\rangle^{0}$$

Bending stress at distance \(x\)

$$\sigma(x) = \dfrac{M(x)\,c}{I}$$

Slope & Deflection

Slope at the free end and fixed end

$$\theta(0) = \dfrac{-M_0 (L-a)}{EI}, \qquad \theta(L) = 0$$

Slope at distance \(x\)

$$\theta(x) = \theta(0) + \frac{M_0}{EI}\langle x-a\rangle$$

Deflection at free end and fixed end

$$y(0) = \dfrac{M_0 \,(L^2-a^2)}{2EI}, \qquad y(L)=0$$

Deflection at distance \(x\)

$$y(x) = y(0) + \theta(0)\,x + \frac{M_0}{2EI}\langle x-a\rangle^{2}$$

Cantilever with Distributed Load (Partial / Trapezoidal)

Cantilever beam fixed at x=L with partial trapezoidal distributed load from x=a to x=a+b (intensity wa to wb)
Line load from \(x=a\) to \(x=a+b\): intensity varies linearly from \(w_a\) to \(w_b\). Fixed support is at \(x=L\).

Resultant & Reactions

Resultant of the trapezoidal load

$$W=\tfrac{b}{2}\,(w_a+w_b)$$

Reaction force at fixed end

$$R_2 = W$$

Reaction moment at fixed end

$$M_2= -\Big[w_a\,b\,\big(L-(a+\tfrac{b}{2})\big) +\tfrac{(w_b-w_a)\,b}{2}\,\big(L-(a+\tfrac{2b}{3})\big)\Big]$$

Note: the sign of \(M_2\) follows the page’s moment sign convention (see above).

Shear & Moment (Singularity Form)

Helper: load gradient

$$k=\dfrac{w_b-w_a}{b}$$

Shear at distance \(x\)

$$V(x) = - w_a\langle x-a\rangle^{1} + w_a\langle x-(a+b)\rangle^{1} - \tfrac{k}{2}\langle x-a\rangle^{2} + \tfrac{k}{2}\langle x-(a+b)\rangle^{2}$$

Moment at distance \(x\)

$$M(x) = - \tfrac{w_a}{2}\langle x-a\rangle^{2} + \tfrac{w_a}{2}\langle x-(a+b)\rangle^{2} - \tfrac{k}{6}\langle x-a\rangle^{3} + \tfrac{k}{6}\langle x-(a+b)\rangle^{3}$$

Bending stress at distance \(x\)

$$\sigma(x)=\dfrac{M(x)\,c}{I}$$

Slope & Deflection

Slope at distance \(x\)

$$\theta(x) = \theta(0) - \frac{w_a}{6EI}\big(\langle x-a\rangle^{3}-\langle x-(a+b)\rangle^{3}\big) - \frac{k}{24EI}\big(\langle x-a\rangle^{4}-\langle x-(a+b)\rangle^{4}\big)$$

Deflection at distance \(x\)

$$y(x) = y(0) + \theta(0)\,x - \frac{w_a}{24EI}\big(\langle x-a\rangle^{4}-\langle x-(a+b)\rangle^{4}\big) - \frac{k}{120EI}\big(\langle x-a\rangle^{5}-\langle x-(a+b)\rangle^{5}\big)$$

Boundary Conditions for \(\theta(0)\) and \(y(0)\)

Fixed-end conditions (at \(x=L\))

$$\theta(L)=0,\qquad y(L)=0$$

Free-end values (for \(L\ge a+b\))

$$\theta(0)=\frac{w_a}{6EI}\big((L-a)^3-(L-a-b)^3\big) +\frac{k}{24EI}\big((L-a)^4-(L-a-b)^4\big)$$
$$y(0)=\frac{w_a}{24EI}\big((L-a)^4-(L-a-b)^4\big) +\frac{k}{120EI}\big((L-a)^5-(L-a-b)^5\big)-\theta(0)\,L$$

Cantilever with Point Load

Cantilever beam fixed at x=L with a point load P at distance a from the free end
Point load \(P\) at \(x=a\). For a tip load, set \(a=0\). Fixed support is at \(x=L\).

Reminder: In this section the diagram shows \(P\) acting downward. With \(y>0\) upward, this produces negative deflection.

Reactions & Shear

Reaction force at fixed end

$$R_2 = P$$

Shear at distance \(x\)

$$V(x) = -P\langle x-a\rangle^{0}$$

Moments & Stress

Reaction moment at fixed end

$$M_2 = -P(L-a)$$

Moment at distance \(x\)

$$M(x) = -P\langle x-a\rangle$$

Bending stress at distance \(x\)

$$\sigma(x) = \dfrac{M(x)\,c}{I}$$

Slope & Deflection

Slope at free and fixed ends

$$\theta(L)=0$$
$$\theta(0)=\dfrac{P(L-a)^{2}}{2EI}$$

Slope at distance \(x\)

$$\theta(x) = \theta(0) - \dfrac{P}{2EI}\langle x-a\rangle^{2}$$

Deflection at free and fixed ends

$$y(L)=0$$
$$y(0) = -\dfrac{P}{6EI}\left(2L^{3} - 3L^{2}a + a^{3}\right)$$

Deflection at distance \(x\)

$$y(x) = y(0) + \theta(0)\,x - \dfrac{P}{6EI}\langle x-a\rangle^{3}$$
Notes & assumptions: Euler–Bernoulli beam theory; prismatic member; small deflection; constant \(E\) and \(I\).

References

  • Oberg, Jones, Horton, Ryffel. Machinery’s Handbook, Industrial Press.
  • Young & Budynas. Roark’s Formulas for Stress and Strain, McGraw-Hill.
  • Beer & Johnston. Mechanics of Materials, McGraw-Hill.

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