Beam Bending Equation Summary Table
The following table summarises the beam bending equations for a series of standard single spans, links to the mathematical proofs and useful calculation tools for each type of span are also included.
The equations for beam bending, reactions, slope and deflection use Macaulay Brackets. Macaulay brackets are represented with square brackets ("[" and "]"), when the value within the brackets is negative, then the bracketed expression is given a value of zero. Macaulay brackets are used to turn individual expressions "on" or "off", depending at what point along the beam you are interested. For more information on Macaulay brackets, please follow this link.
| Span and Load Type | Reactions | Moment Expression | Slope Equation | Deflection Equation | Proof | Tool |
|---|---|---|---|---|---|---|
 | \( R_{A} = Pb/L\) \( R_{B} = P - R_{A}\) | \( M_{x} = - \frac{Pbx}{L} + P[x-a]\) | \( \frac{dv}{dx} = \frac{P}{12EIL}( 2b(L^2 - b^2 - 3x^2)\) \(+6L[x-a]^2)\) | \( v = \frac{P}{6EIL}(bx(L^2 - b^2 -x^2)\) \(+L[x-a]^3)\) | Proof | Tool |
 | \( R_{A} = \frac{\omega L}{2}\) \( R_{B} = \frac{\omega L}{2}\) | \( M_{x} = - R_{A}x + \frac{\omega x^2}{2}\) | \( \frac{dv}{dx} = \frac{\omega}{24EI}(-6Lx^2 + 4x^3 + L^3)\) | \( v = \frac{\omega x}{24EI}(-2Lx^2 + x^3 + L^3)\) | Proof | Tool |
 | \( R_{A} = (c+\frac{b}{2}) \frac{\omega b}{L}\) \( R_{B} = (a+\frac{b}{2}) \frac{\omega b}{L}\) | \( M_{x}= - R_{A}x + \frac{\omega}{2}[x-a]^2 \) \(- \frac{\omega}{2}[x-a-b]^2\) | \( \frac{dv}{dx} = \frac{\omega}{24EIL}(-12bx^2(c+\frac{b}{2}) +4L[x-a]^3\) \(-4L[x-a-b]^3+4bL^2(c+\frac{b}{2})-(L-a)^4+c^4 )\) | \( v = \frac{\omega}{24EIL}(- 4bx^3(c+\frac{b}{2}) + L[x-a]^4 \) \(- L[x-a-b]^4 + 4bxL^2(c+\frac{b}{2}) \) \(- x(L-a)^4+ xc^4)\) | Proof | Tool |
 | \( R_{A} = P\) \( M_{A} = Pa\) | \( M_{x} = M_{A} - R_{A}x + P[x-a]\) | \( \frac{dv}{dx} = \frac{P}{2EI} (2ax - x^2 + [x-a]^2)\) | \( v = \frac{P}{6EI}(3ax^2 - x^3 + [x-a]^3)\) | Proof | Tool |
 | \( R_{A} = \omega L\) \( M_{A} = \frac{\omega L^2}{2}\) | \( M_{x} = M_{A} - R_{A}x + \frac{\omega x^2}{2}\) | \( \frac{dv}{dx} = \frac{\omega x}{6EI}(3L^2 - 3Lx + x^2) \) | \( v = \frac{\omega x^2}{24EI}(6L^2 - 4xL+ x^2)\) | Proof | Tool |
 | \( R_{A} = \omega b\) \( M_{A} = \omega b(a+\frac{b}{2})\) | \( M_{x} = M_{A} - R_{A}x + \frac{\omega}{2}[x-a]^2 \) \(- \frac{\omega}{2}[x-a-b]^2\) | \( \frac{dv}{dx} = \frac{\omega}{6EI}(3 b x (2a + b - x) + [x-a]^3\) \(- [x-a-b]^3)\) | \( v = \frac{\omega}{24EI}(2bx^2(6a + 3b-2x) + [x-a]^4\) \(- [x-a-b]^4)\) | Proof | Tool |
 | \( R_{A} = \frac{Pb^2}{L^3} (3a+b)\) \( M_{A} = \frac{Pab^2}{L^2}\) | \( M_{x} = M_{A} - R_{A}x + P[x-a]\) | \( \frac{dv}{dx} = \frac{P}{2EIL^3}(b^2x (2aL-3ax-bx)\) \(+L^3[x-a]^2)\) | \( v = \frac{P}{6EIL^3}(b^2x^2 (3aL-3ax-bx)\) \(+L^3[x-a]^3)\) | Proof | Tool |
 | \( R_{A} = \frac{\omega L}{2}\) \( M_{A} = \frac{\omega L^2}{12}\) | \( M_{x} = M_{A} - R_{A}x + \frac{\omega x^2}{2}\) | \( \frac{dv}{dx} = \frac{\omega x}{12EI}(L-2x)(L-x)\) | \( v = \frac{\omega x^2 }{24EI}(L-x)^2\) | Proof | Tool |
 | \( M_{A} = \frac{-\omega}{12L^2} (e^3(3e-4L) \) \(- c^3(3c-4L))\) \( R_{A} = \frac{\omega}{2L^3}(e^3(2L-e) \) \(-c^3(2L-c))\) \(\text{where } e = (b + c) = (L - a)\) | \( M_{x} = M_{A} - R_{A}x + \frac{\omega}{2}[x-a]^2 \) \(- \frac{\omega}{2}[x-a-b]^2\) | \( \frac{dv}{dx} = \frac{1}{EI}(M_{A}x - \frac{R_{A}x^2}{2} + \frac{\omega}{6}[x-a]^3\) \(- \frac{\omega}{6}[x-a-b]^3)\) | \( v = \frac{1}{EI}(\frac{M_{A}x^2}{2} - \frac{R_{A}x^3}{6} + \frac{\omega}{24}[x-a]^4\) \(- \frac{\omega}{24}[x-a-b]^4)\) | Proof | Tool |