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Reference

Beam formulas

Closed-form reactions, maximum moments and maximum deflections for the standard beam cases. Every case below can be loaded interactively in thecalculator.

Simply supported, center point load P

Pin A, roller B; P at x = L/2

Reactions
RA=RB=P2R_A = R_B = \frac{P}{2}
Bending moment
Mmax=PL4 at x=L/2M_{max} = \frac{PL}{4}\ \text{at } x = L/2
Deflection
δmax=PL348 EI\delta_{max} = \frac{PL^3}{48\,EI}

Simply supported, point load P at x = a

Pin A, roller B; P at distance a from A (b = L − a)

Reactions
RA=P bL,RB=P aLR_A = \frac{P\,b}{L}, \quad R_B = \frac{P\,a}{L}
Bending moment
Mmax=PabL at x=aM_{max} = \frac{P a b}{L}\ \text{at } x = a
Deflection
δmax=Pab (L2−a2−b2)3/293 EIL\delta_{max} = \frac{P a b \,(L^2 - a^2 - b^2)^{3/2}}{9\sqrt{3}\,EIL}

Simply supported, UDL w (whole span)

Pin A, roller B, uniform w

Reactions
RA=RB=wL2R_A = R_B = \frac{wL}{2}
Bending moment
Mmax=wL28 at midspanM_{max} = \frac{wL^2}{8}\ \text{at midspan}
Deflection
δmax=5 wL4384 EI\delta_{max} = \frac{5\,wL^4}{384\,EI}

Simply supported, applied moment M₀ at x = a

Concentrated couple

Reactions
RA=−M0L,RB=+M0LR_A = -\frac{M_0}{L}, \quad R_B = +\frac{M_0}{L}
Bending moment
M jumps by M0 at x=a\text{M jumps by } M_0 \text{ at } x = a
Deflection
(piecewise linear, no closed form)\text{(piecewise linear, no closed form)}

Cantilever, tip point load P

Fixed at A; P at free tip

Reactions
RA=P,MA=−P LR_A = P, \quad M_A = -P\,L
Bending moment
∣M∣max=P L at fixed end|M|_{max} = P\,L\ \text{at fixed end}
Deflection
δtip=PL33 EI\delta_{tip} = \frac{P L^3}{3\,EI}

Cantilever, UDL w (whole length)

Fixed at A, uniform w

Reactions
RA=wL,MA=−wL22R_A = wL, \quad M_A = -\frac{wL^2}{2}
Bending moment
∣M∣max=wL22 at fixed end|M|_{max} = \frac{wL^2}{2}\ \text{at fixed end}
Deflection
δtip=wL48 EI\delta_{tip} = \frac{wL^4}{8\,EI}

Cantilever, triangular load (0 at tip → w at wall)

Linearly increasing toward the wall

Reactions
RA=wL2,MA=−wL26R_A = \frac{wL}{2}, \quad M_A = -\frac{wL^2}{6}
Bending moment
∣M∣max=wL26 at wall|M|_{max} = \frac{wL^2}{6}\ \text{at wall}
Deflection
δtip=wL430 EI\delta_{tip} = \frac{wL^4}{30\,EI}

Propped cantilever, UDL w

Fixed at A, roller at B

Reactions
RB=3 wL8,RA=5 wL8,MA=−wL28R_B = \frac{3\,wL}{8}, \quad R_A = \frac{5\,wL}{8}, \quad M_A = -\frac{wL^2}{8}
Bending moment
M+=9 wL2128 at x=5L8M_+ = \frac{9\,wL^2}{128}\ \text{at } x = \tfrac{5L}{8}
Deflection
δmax≈0.00541 wL4EI at x≈0.5785 L\delta_{max} \approx 0.00541\,\frac{wL^4}{EI}\ \text{at } x \approx 0.5785\,L

Fixed–fixed, UDL w

Both ends fixed

Reactions
RA=RB=wL2,MA=MB=−wL212R_A = R_B = \frac{wL}{2}, \quad M_A = M_B = -\frac{wL^2}{12}
Bending moment
M+=wL224 at midspanM_+ = \frac{wL^2}{24}\ \text{at midspan}
Deflection
δmax=wL4384 EI\delta_{max} = \frac{wL^4}{384\,EI}

Fixed–fixed, center point P

Both ends fixed; P at L/2

Reactions
RA=RB=P2,MA=MB=−P L8R_A = R_B = \frac{P}{2}, \quad M_A = M_B = -\frac{P\,L}{8}
Bending moment
M+=P L8 at midspanM_+ = \frac{P\,L}{8}\ \text{at midspan}
Deflection
δmax=P L3192 EI\delta_{max} = \frac{P\,L^3}{192\,EI}

Two equal spans, UDL w on both

Pins at A, B, C; each span L

Reactions
RA=RC=3 wL8,RB=10 wL8R_A = R_C = \frac{3\,wL}{8}, \quad R_B = \frac{10\,wL}{8}
Bending moment
M−=wL28 over B;M+=9 wL2128M_- = \frac{wL^2}{8}\ \text{over B;} \quad M_+ = \frac{9\,wL^2}{128}
Deflection
δ≈0.0052 wL4EI per span\delta \approx 0.0052\,\frac{wL^4}{EI}\ \text{per span}

Overhanging (symmetric), UDL w

Supports at distance a from each end, total length L

Reactions
RA=RB=wL2R_A = R_B = \frac{wL}{2}
Bending moment
Msupports=−w a22M_{\text{supports}} = -\frac{w\,a^2}{2}
Deflection
depends on a/L\text{depends on } a/L

P point load · w distributed load per unit length · L span · E Young's modulus · I second moment of area. Sagging moments positive; downward deflections positive-plotted.