Beam Deflection Calculator - Supported Beams

Simply supported, point load at midspan

Simply supported, point load at midspan

δ

maximum deflection

P

point load

L

span between supports

E

Young's modulus

I

moment of inertia (second moment of area)

Source: Uses the formula in American Wood Council Design Aid No. 6, Figure 7 [1].

Simply supported, point load off center

Simply supported, point load off center

The largest deflection is not under the load but toward the middle of the longer segment.

Source: Uses the formula in American Wood Council Design Aid No. 6, Figure 8, written with the shorter segment [1].

Simply supported, uniform load

Simply supported, uniform load over the whole span

δ

maximum deflection

w

load per unit length (the peak, for a triangular load)

L

span, or the cantilever length

E

Young's modulus

I

moment of inertia (second moment of area)

Source: Uses the formula in American Wood Council Design Aid No. 6, Figure 1 [1].

Simply supported, triangular load

Simply supported, triangular load over the whole span

The coefficient is rounded: the solver gives 0.0065222, so the number shown comes from the solver, not the rounded line.

Source: Based on American Wood Council Design Aid No. 6, Figure 5, with its total load W = w × L ÷ 2 [1].

Simply supported, moment at an end

Simply supported, moment at an end

δ

maximum deflection

M₀

applied moment

Source: Based on American Wood Council Design Aid No. 6, Figure 20: an overhang load applies the same end moment [1].

Cantilever, point load at the tip

Cantilever, point load at the tip

Source: Uses the formula in American Wood Council Design Aid No. 6, Figure 13 [1].

Cantilever, point load short of the tip

Cantilever, point load short of the tip

δ

tip deflection

a

distance from the fixed end to the load

δ is at the tip; a is measured from the fixed end.

Source: Uses the formula in American Wood Council Design Aid No. 6, Figure 14, whose b is the a here [1].

Cantilever, uniform load

Cantilever, uniform load over the whole length

Source: Uses the formula in American Wood Council Design Aid No. 6, Figure 12 [1].

Cantilever, triangular load

Cantilever, triangular load rising to the tip

Zero at the fixed end, w at the free end.

Source: Standard formula; no single publisher.

Cantilever, moment at the tip

Cantilever, moment at the tip

Source: Standard formula; no single publisher.

Fixed both ends, point load at midspan

Fixed both ends, point load at midspan

Source: Uses the formula in American Wood Council Design Aid No. 6, Figure 24 [1].

Fixed both ends, uniform load

Fixed both ends, uniform load over the whole span

Source: Uses the formula in American Wood Council Design Aid No. 6, Figure 23 [1].

Any other beam and load

Any other beam and load

Overhanging beams, partial loads, off-center loads on a fixed beam and moments away from an end have no single closed form. Every case, textbook ones included, is solved this way.

Source: Standard method; no single publisher. It reproduces American Wood Council Design Aid No. 6, Figures 20 and 25, for an overhang and an off-center load on a fixed beam [1].

Bending stress

With a section depth

σ

bending stress at the outer fiber, in MPa

M

the largest bending moment

h

the section depth

I

moment of inertia (second moment of area)

For a section symmetric about its neutral axis; otherwise enter twice the distance to the outer fiber. The moment, shear and reactions come from the same solution as the deflection.

Source: Standard formula; no single publisher.

Deflection limit

L

the span, or twice a cantilever's length

δ

the largest deflection, up or down

L/180

roof member with no ceiling (IBC)

L/240

roof member over a nonplaster ceiling (IBC)

L/360

floor member under live load, or roof over a plaster ceiling (IBC)

L/480, L/600, L/1000

tighter limits set by your design standard

The check passes when δ is at most L ÷ n for the limit you pick. For a cantilever, L is twice its length. The ratio is rounded down, so a beam just over a limit never reads as passing. The IBC limits are for live load.

Source: The three code limits and the cantilever rule are those of the International Building Code's Table 1604.3, live load, and its footnote i (2012 edition read) [2]. The tighter limits name no code; no publisher.

Material moduli

With a material chosen

Steel

200 GPa

Aluminum

70 GPa

Concrete

30 GPa, about M35 concrete by IS 456; an M20 concrete is about 22 GPa

Wood (pine)

10 GPa, clear wood at 12% moisture; pines run 8.2 to 13.7 GPa

Representative values. A typed modulus is used only with Custom.

Source: Steel's 200 kN/mm² and concrete's E = 5,000 √fck are IS 456's [3]. Aluminum is within 3% of MIL-HDBK-5J's 6061-T6 and 7075-T6 [4], and pine is mid-range in the USDA Wood Handbook's Table 5-3a [5].