What is the formula for deflection of a beam?

The allowable deflection depends on the support and loading conditions. The most common cases are a cantilevered beam with a point load at its free end, and a simply supported beam subjected to a uniformly distributed load. For the same span ratio, the deflection of the cantilevered beam is significantly larger. In the formulas for calculating deflection, the bending stiffness (the product of the modulus of elasticity and the cross-sectional moment of inertia) is used to divide terms that depend on the load and span. The calculation formulas listed above show the exact equations for each support and loading configuration considered by this calculator.

What is the difference between a cantilever beam and a simply supported beam?

A simply supported beam is held at each end by a support that allows free rotation. It can be thought of as a plate resting on two supports. A cantilevered beam has one end fixed and the other end free. This can be thought of like a balcony or diving board. For the same span and load, the deflection of the cantilevered beam is much greater because only one end is fixed.

How does the moment of inertia affect deflection?

Since the moment of inertia is in the denominator of all deflection formulas, a higher value will result in less deflection. Since the moment of inertia increases with the cube of the cross-section height, increasing the height of the beam is more effective for reducing deflection than increasing the width.

What type of load causes more deflection, point or surface?

For the same total load, a point load in the middle of the span will normally cause more deflection than a uniformly distributed load over the length of the beam because the deformation is concentrated at one single point. The calculator allows these two cases to be directly compared by switching between different types of loading.

Beam Deflection Calculator

Beam and load details

Span length (L)

Point load (P)

Modulus of elasticity (E)

Area moment of inertia (I)

Estimate bending stress

Add the distance from the neutral axis to the extreme fiber to get the maximum bending stress.

Maximum deflection (mm)

Maximum bending moment (kN.m)

Maximum deflection (mm)

Maximum bending moment: kN·m
Span-to-deflection ratio (L/delta)
Maximum slope (rad)

Governing formula: δ = P·L³ / (48·E·I), a simply-supported beam with a central point load.

Building codes often cap deflection at about L/360 for floors or L/240 for roofs. This beam's span-to-deflection ratio is 8533, so a larger number is stiffer.

Deflected shape

The beam deflection calculator allows you to calculate how much a loaded beam will deflect. After selecting the support conditions and load type, and entering the span length, load, material strength, and cross-section information, it displays the maximum deflection along with the maximum bending moment and angle of twist. This version supports simply supported beams, cantilevered beams, and two-point supported beams subjected to a point load or uniformly distributed load, and allows you to estimate the bending stress.

What is deflection?

Deflection is the vertical distance that a point on a beam moves from its original position after a load has been applied to it. Loads acting on floors, roofs and bridges are supported by beams, and when these loads act they cause the beams to bend. Excessive deflection can lead to serviceability problems. Even if a beam does not break, floors may feel bouncy, surface materials may crack or doors may stick. For this reason engineers check for deflection and strength separately.

There are three factors that counteract the deflection: The harder the material is, the greater its modulus of elasticity and the less it will deflect. The deeper or thicker the cross section is, the greater its moment of inertia and the less it will deflect. The shorter the span is, the less it will deflect since this increases in proportion to the third or fourth power of length.

Here's how to use this calculator:

First select the type of support for the beam. A simply supported beam is supported at both ends by a pin, a cantilevered beam has one fixed end and one free end, while a double overhanging beam is fixed at both ends. Next choose the type of load: either a single point load or a uniformly distributed load across the span.

Enter the span length, load, material modulus of elasticity and moment of inertia. The maximum deflection will be shown immediately as well as the maximum bending moment. For simply supported beams and cantilevers the maximum rotation angle is also shown. Activate the option for the bending stress and enter the distance from the neutral axis to the outer edge to determine the maximum bending stress.

Formula for deflection of a beam:

For each type of support and load combination there is a closed formula for the maximum deflection. In this case P is the point load, w is the uniformly distributed load per unit length, L is the span, E is the modulus of elasticity and I is the moment of inertia.

Beam and load	Maximum deflection
Simply supported, central point load	$\dfrac{PL^3}{48EI}$
Simply supported, uniform load	$\dfrac{5wL^4}{384EI}$
Cantilever, point load at free end	$\dfrac{PL^3}{3EI}$
Cantilever, uniform load	$\dfrac{wL^4}{8EI}$
Fixed-fixed, central point load	$\dfrac{PL^3}{192EI}$
Fixed-fixed, uniform load	$\dfrac{wL^4}{384EI}$

The two most common cases that occur in everyday life are worth remembering individually.

\delta_{\text{cantilever}} = \frac{PL^3}{3EI} \qquad \delta_{\text{simple}} = \frac{5wL^4}{384EI}

The term EI in each denominator is the stiffness of the cross-section and represents the overall ability of the beam to resist bending. As L is raised to the third or fourth power, doubling the span increases deflection by a factor of eight to sixteen. This is why much deeper beams are required for long spans.

Example calculation

Let's consider a simple bench. The distance between the legs is 1.5 m. The wooden boards are made of Eastern White Pine and have a thickness of 4 cm and a width of 30 cm, so the moment of inertia about the bending axis will be calculated as follows:

I = \frac{b\,h^3}{12} = \frac{30 \times 4^3}{12} = 160\ \text{cm}^4 = 1.6 \times 10^{-6}\ \text{m}^4

The modulus of elasticity for Eastern White Pine is about 6800 MPa. A child weighing 400 N sitting at the center will produce a point load at the center so this can be considered as simply supported beam.

\delta = \frac{PL^3}{48EI} = \frac{400 \times 1.5^3}{48 \times 6.8\times10^{9} \times 1.6\times10^{-6}} \approx 0.00259\ \text{m} = 2.59\ \text{mm}

Under this load the seat surface bends down by about 2.6 mm. By replacing the material or increasing the height of the plank, the denominator E or I changes, which directly leads to a change in this result.

Elasticity modulus and moment of inertia

The modulus of elasticity is determined solely by the material. The stiffer the material, the less deflection there will be in a beam under equal load. If no test results are available, typical values can serve as a useful starting point.

Material	Modulus of elasticity (GPa)
Structural steel	About 200
Aluminum	About 69
Concrete	15 to 50
Timber	7 to 14

The moment of inertia depends on the shape of the cross-section and which axis the beam is bending about. For a rectangular cross-section, the moment of inertia is proportional to the cube of the height, so that the height has much more influence than the width. This explains why floor beams are often erected vertically rather than horizontally, as this allows for greater height and thus a larger moment of inertia.

How much deflection is allowed?

The deflection is not considered as an absolute value but in relation to the span. In building codes, the deflection of floor beams subjected to service loads is normally limited to one three-hundred-and-sixtieth of the span, or L/360. For roof rafters it's approximately L/240. The calculation tool gives the ratio of span to deflection so that direct comparison can be made. A floor beam with a 4 m span and an allowable limit of L/360 would have a maximum deflection of perhaps about 11 mm. The larger the ratio, the higher the stiffness and serviceability of the beam.

Bending moment and bending resistance

The maximum bending moment indicates how much a load is trying to bend a beam and determines the flexural resistance exerted by the construction material. If you activate the option for flexural resistance and enter the distance from the neutral axis to the outer edge (usually half of the cross-section height), the calculation tool will calculate the maximum flexural resistance based on the bending moment, this distance and the moment of inertia. When a safety factor is considered, this resistance must be well below the allowable value for the material, which represents an additional strength check that should be performed in parallel with the deflection check.

This tool is for general educational purposes and initial sizing only. It does not replace a full structural engineering design by a qualified engineer, nor does it check strength, buckling stability, shear capacity or code compliance. All load bearing components must be designed and checked by professionals.

Frequently asked questions

What is the formula for deflection of a beam?: The allowable deflection depends on the support and loading conditions. The most common cases are a cantilevered beam with a point load at its free end, and a simply supported beam subjected to a uniformly distributed load. For the same span ratio, the deflection of the cantilevered beam is significantly larger. In the formulas for calculating deflection, the bending stiffness (the product of the modulus of elasticity and the cross-sectional moment of inertia) is used to divide terms that depend on the load and span. The calculation formulas listed above show the exact equations for each support and loading configuration considered by this calculator.
What is the difference between a cantilever beam and a simply supported beam?: A simply supported beam is held at each end by a support that allows free rotation. It can be thought of as a plate resting on two supports. A cantilevered beam has one end fixed and the other end free. This can be thought of like a balcony or diving board. For the same span and load, the deflection of the cantilevered beam is much greater because only one end is fixed.
How does the moment of inertia affect deflection?: Since the moment of inertia is in the denominator of all deflection formulas, a higher value will result in less deflection. Since the moment of inertia increases with the cube of the cross-section height, increasing the height of the beam is more effective for reducing deflection than increasing the width.
How much deflection is allowed?: Deflection is not considered as an absolute value but in comparison to the span. Generally floor beams deflections are limited to about L/360 and roof beam deflections are limited to about L/240 of their respective spans. The calculation tools will give you the ratio of span to deflection so these values can be compared.
What type of load causes more deflection, point or surface?: For the same total load, a point load in the middle of the span will normally cause more deflection than a uniformly distributed load over the length of the beam because the deformation is concentrated at one single point. The calculator allows these two cases to be directly compared by switching between different types of loading.

Related calculators

Beam Load CalculatorCalculate the support reactions, maximum shear force, and maximum bending moment of a simply supported beam or cantilever under a point or distributed load, with shear-force and bending-moment diagrams.

Control Authority (Elevator Power) CalculatorFind the change in pitching-moment coefficient from elevator effectiveness and deflection, solve for any variable, and scale it to a real pitching moment.

Flexure CalculatorSolve the flexure formula sigma = M y / I for bending stress, moment, or section size, with rectangular and circular shortcuts, a 3 and 4 point bending-test mode for the modulus of rupture, and a deflection check.

Pitching Moment CalculatorCalculate the aircraft pitching moment from the moment coefficient, dynamic pressure, reference area, and characteristic length, or solve for any one of them. Includes a dynamic-pressure helper, a pitch-stability check, and charts.

Buckling CalculatorFind the critical buckling load of a column from its length, cross-section, material and end conditions, using Euler's formula for slender columns and Johnson's parabolic formula for stocky ones, with slenderness ratio, factor of safety and an eccentric-load check.

3D Printing Cost CalculatorWork out the true cost of a 3D print, including filament, electricity, machine wear, labor, and failed prints, then get a selling price from your margin and marketplace fees.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Gere & Goodno, Mechanics of Materials (beam deflection tables)
Standard maximum-deflection formulas for common beam and load cases.
Wikipedia: Euler-Bernoulli beam theory
The elastic beam theory the deflection formulas are derived from.
The Wood Handbook (USDA Forest Products Laboratory)
Reference modulus-of-elasticity values for wood species.