Beam Deflection Formulas: Simply Supported, Cantilever and Fixed Beams

These tables give the largest deflection of a single-span beam for the load cases that come up most in practice: simply supported beams, cantilevers, beams fixed at both ends and propped cantilevers. Each row also gives where the largest deflection happens, the largest bending moment, the support reactions and the slope at the ends, so one table covers a hand check of a beam.

For the deflection at any point along the span, and plots of deflection, slope, moment and shear, each simply supported and cantilever case links to the free beam calculator, which uses the same formulas.

Notation

L is the span, or the length of a cantilever.
w is a uniform load per unit length. For a triangular load, w is the load at the peak. (The beam calculator calls it q.)
P is a point load and M an applied moment.
a and b locate a load: a from the left support or the fixed end, b = L − a.
E is the modulus of elasticity and I the moment of inertia about the bending axis. EI is the beam's bending stiffness.
δ is deflection and θ is slope, in radians.

Use consistent units. In US units that usually means kips and inches: w in kips per inch (kips per foot divided by 12), L in inches, E in ksi and I in in⁴, which gives δ in inches. The formulas assume a straight, elastic beam whose cross section doesn't change along the span, and small deflections. They leave out shear deformation, which is small except for short, deep beams.

Simply supported beam deflection formulas

A pin at one end and a roller at the other. The largest deflection of a symmetric load is at midspan.

Load case	Largest deflection	Where	Largest moment	Reactions	End slope
Uniform load	5wL⁴ / (384EI)	Midspan	wL² / 8 at midspan	wL / 2 each	wL³ / (24EI)
Point load at midspan	PL³ / (48EI)	Midspan	PL / 4 under the load	P / 2 each	PL² / (16EI)
Point load at any point	Pb(L² − b²)^3/2 / (9√3·LEI) with a ≥ b. Under the load: Pa²b² / (3LEI)	x = √((L² − b²)/3) from the left support, on the longer side of the load	Pab / L under the load	Pb / L left Pa / L right	Pab(L + b) / (6LEI) at the left end
Two equal point loads, each a from a support	Pa(3L² − 4a²) / (24EI)	Midspan	Pa between the loads	P each	Pa(L − a) / (2EI)
Triangular load, zero at one end	0.00652 wL⁴ / EI	0.519L from the zero end	wL² / (9√3) = 0.0642 wL² 0.577L from the zero end	wL / 6 at the zero end wL / 3 at the peak	7wL³ / (360EI) zero end wL³ / (45EI) peak end
Triangular load, peak at midspan	wL⁴ / (120EI)	Midspan	wL² / 12 at midspan	wL / 4 each	5wL³ / (192EI)
Moment at one end	ML² / (9√3·EI) = 0.0642 ML² / EI	0.423L from the moment	M at the loaded end	M / L each, opposite ways	ML / (3EI) loaded end ML / (6EI) other end
Equal moments at both ends	ML² / (8EI)	Midspan	M along the whole span	0	ML / (2EI)

Cantilever beam deflection formulas

Fixed at one end and free at the other. The largest deflection and slope are at the free end, and the largest moment is at the support, where the wall or column has to resist it.

Load case	Deflection at the free end	Largest moment (at the support)	Reaction	Slope at the free end
Uniform load	wL⁴ / (8EI)	wL² / 2	wL	wL³ / (6EI)
Uniform load over length a from the support	wa³(4L − a) / (24EI)	wa² / 2	wa	wa³ / (6EI)
Point load at the free end	PL³ / (3EI)	PL	P	PL² / (2EI)
Point load a from the support	Pa²(3L − a) / (6EI) Under the load: Pa³ / (3EI)	Pa	P	Pa² / (2EI)
Moment at the free end	ML² / (2EI)	M along the whole length	0	ML / (EI)
Triangular load, largest at the support	wL⁴ / (30EI)	wL² / 6	wL / 2	wL³ / (24EI)
Triangular load, largest at the free end	11wL⁴ / (120EI)	wL² / 3	wL / 2	wL³ / (8EI)

Fixed beam deflection formulas

Both ends held against rotation, as in a beam built solidly into walls or welded into stiff columns. The end moments make the beam five times stiffer than a simply supported one under a uniform load, but the supports have to be able to take those moments.

Load case	Largest deflection	Where	Moments	Reactions
Uniform load	wL⁴ / (384EI)	Midspan	wL² / 12 at the supports wL² / 24 at midspan	wL / 2 each
Point load at midspan	PL³ / (192EI)	Midspan	PL / 8 at the supports PL / 8 at midspan	P / 2 each

Propped cantilever deflection formulas

Fixed at one end and pinned at the other. The same formulas cover each span of a beam continuous over two equal spans with the same load on both, because the middle support holds the beam level, as a fixed end would.

Load case	Largest deflection	Where	Moments	Reactions	Slope at the pin
Uniform load	wL⁴ / (185EI) more exactly 0.00542 wL⁴ / EI	0.422L from the pinned end	wL² / 8 at the fixed end 9wL² / 128 in the span 3L/8 from the pinned end	5wL / 8 fixed end 3wL / 8 pinned end	wL³ / (48EI)
Point load at midspan	PL³ / (48√5·EI) = 0.00932 PL³ / EI Under the load: 7PL³ / (768EI)	0.447L from the pinned end	3PL / 16 at the fixed end 5PL / 32 under the load	11P / 16 fixed end 5P / 16 pinned end	PL² / (32EI)

Deflection at any point along the beam

The tables give the largest value. The deflection at a distance x along the beam, measured from the left support or the fixed end, for the four most common cases:

Simply supported, uniform load	δ(x) = wx(L³ − 2Lx² + x³) / (24EI)
Simply supported, point load at midspan (x ≤ L/2)	δ(x) = Px(3L² − 4x²) / (48EI)
Cantilever, uniform load	δ(x) = wx²(6L² − 4Lx + x²) / (24EI)
Cantilever, point load at the free end	δ(x) = Px²(3L − x) / (6EI)

The beam calculator shows these equations, and the slope, moment and shear equations, for all 17 of its simply supported and cantilever cases, and evaluates them at any point.

How the end supports change deflection

Take the 20 ft W12x26 from the beam deflection example (I = 204 in⁴) under a uniform load of 1.0 kip/ft, and change only the supports:

Supports	Formula	Deflection
Simply supported	5wL⁴ / (384EI)	0.61 in
Propped cantilever	wL⁴ / (185EI)	0.25 in
Fixed at both ends	wL⁴ / (384EI)	0.12 in

Real connections are seldom fully fixed. A bolted steel shear tab or a joist hanger is close to a pin, so design with the simply supported formulas unless the connection is built and designed to carry the end moment.

Worked example: steel cantilever

A W8x18 steel beam cantilevers 6 ft past a column to carry a canopy. Under live load it takes a uniform load of 0.3 kip/ft and a 1.5 kip point load at the tip, from a hanger. Is its deflection acceptable?

Values: L = 6 ft = 72 in, w = 0.3 / 12 = 0.025 kip/in, E = 29,000 ksi, I_x = 61.9 in⁴ for a W8x18.

Step 1: deflection from the point load

δ_P = PL³ / (3EI) = 1.5 × 72³ / (3 × 29,000 × 61.9) = 0.104 in

Step 2: deflection from the uniform load

δ_w = wL⁴ / (8EI) = 0.025 × 72⁴ / (8 × 29,000 × 61.9) = 0.047 in

Step 3: add them and check the limit

δ = 0.104 + 0.047 = 0.151 in

For a cantilever, the International Building Code (IBC) Table 1604.3 limits are taken with the span equal to twice the cantilever's length. With the L/360 live load limit, the allowed tip deflection is 2 × 72 / 360 = 0.40 in. 0.151 in is well under that, so the beam passes for deflection. It still needs its bending and shear checks, and the column has to take the moment at the support: 0.3 × 6² / 2 + 1.5 × 6 = 14.4 kip·ft.

Combining load cases

Deflection is proportional to load, so when several loads act on the same beam you can work out each one's deflection and add them (superposition), as the example does. Add values at the same point: the largest deflections of two cases are often at different places, so adding the two largest values gives an answer on the safe side, not the exact one. For more than a couple of loads, or for continuous beams, the shear and moment diagram calculator adds them for you.

Values of E

Material	Modulus of elasticity E
Structural steel	29,000 ksi (200 GPa)
Aluminum	10,000 ksi (69 GPa)
Douglas Fir-Larch No. 2 lumber	1,600,000 psi (11 GPa)
Concrete, f′_c = 4,000 psi	57,000√f′_c = 3,600 ksi (25 GPa)

Wood E varies by species and grade, so take it from the design values for the lumber you're using. A concrete beam cracks under load, which lowers its stiffness, so concrete deflection uses an effective moment of inertia rather than the full section's.

Common deflection limits

Member	Live load	Total load
Floor beams and joists	L/360	L/240
Roof members supporting a plaster ceiling	L/360	L/240
Roof members supporting a non-plaster ceiling	L/240	L/180
Roof members with no ceiling	L/180	L/120

From IBC Table 1604.3. For a cantilever, L is twice the cantilever's length. Check the code adopted where you build.

Check a beam with the calculators

The beam calculator gives deflection, slope, moment and shear along the beam for 17 simply supported and cantilever cases, with the equations. Free, with no account.
The shear and moment diagram calculator handles several loads at once, overhangs and continuous beams.
The beam designer checks steel and wood beams for deflection, bending and shear together, and suggests sizes that pass.

Frequently asked questions

What is the deflection formula for a simply supported beam?

5wL⁴/(384EI) at midspan for a uniform load, and PL³/(48EI) for a point load at midspan.

What is the deflection formula for a cantilever beam?

wL⁴/(8EI) at the free end for a uniform load, and PL³/(3EI) for a point load at the free end.

What is the deflection of a beam fixed at both ends?

wL⁴/(384EI) for a uniform load, a fifth of a simply supported beam's, and PL³/(192EI) for a point load at midspan, a quarter.

Do I use factored loads for deflection?

No. Deflection is a serviceability check, so it uses the loads as they are, without load factors.

Why is my deflection far too big or too small?

Usually the units are mixed, such as a load in kips per foot with a span in inches. Convert everything to kips and inches first: divide w in kips/ft by 12 and multiply L in feet by 12. L is cubed or raised to the fourth power, so a span left in feet is off by 1,728 or 20,736 times.