About Beam Deflection Calculation
The results are approximate. Before use, verify the calculations against the applicable standards and consult a specialist. The developer is not responsible for the consequences of use without project verification.
The Beam Deflection and Strength Calculator calculates the deflection and strength of a steel beam under a static uniformly distributed or point load. From the cross-section dimensions, it determines the section properties, then accounts for the beam self-weight and calculates the maximum bending moment, shear force, deflection, normal stress, and shear stress.
The calculation applies to straight steel beams with a constant cross-section, small deformations, and linear-elastic material behavior. For a point load, the force is applied at midspan for beams supported at both ends and at the free end for a cantilever.
Guidelines and recommendations
Calculation model
Beam deflection. The calculation is based on classical beam bending theory. It uses a steel modulus of elasticity of E = 210000 MPa and the second moment of area I. The greater the second moment of area, the smaller the deflection for the same span and load.
Section geometry. The entered dimensions are used to calculate the cross-sectional area A in mm2, second moment of area I in mm4, elastic section modulus W in mm3, and first moment of area Q in mm3. The cross-section is treated as an idealized geometric shape without fillet radii or flange slopes.
The elastic section modulus is determined from the distance between the neutral axis and the most distant point of the cross-section:
W = I / ymax
For a cross-section that is not symmetric about the bending axis, its centroid position is calculated first. For an angle section, the calculation assumes bending in the plane corresponding to the orientation shown in the calculator.
Material and steel properties
Modulus of elasticity. A value of E = 210000 MPa is used for all steel grades. This value is used in the beam deflection calculation and represents the stiffness of steel in the elastic range.
Density. To calculate beam mass and self-weight, the calculator uses a steel density of ρ = 7850 kg/m3 and gravitational acceleration g = 9.80665 m/s2.
Yield strength. For steel grades S235, S275, S355, and S420, the calculator uses fy = 235, 275, 355, and 420 MPa, respectively. These values are used for the stress checks. For a specific steel product, the actual fy may depend on product thickness and the applicable product standard.
Load units and self-weight
Distributed load. It can be entered in kg/m or kN/m. A value entered in kg/m is treated as kgf/m and converted to force units using g = 9.80665 m/s2:
1 kgf/m = 0.00980665 kN/m
Point load. It can be entered in kg or kN. A value entered in kilograms is treated as kilogram-force:
1 kgf = 9.80665 N
Self-weight. The cross-sectional area A is first calculated from the entered dimensions, then the beam mass is determined from its volume and steel density. The self-weight is converted into a uniformly distributed load acting over the full beam length:
qself = ρ · A · g
For an external distributed load, the total distributed load is:
q = qext + qself
When an external point load P is applied, the beam self-weight still acts as the distributed load qself. Therefore, the beam's own weight is included in every support and load configuration.
Bending moment and shear force
Internal forces. After determining the total load, the calculator finds the maximum absolute bending moment M and maximum shear force V. The equations depend on the support condition.
- Simply supported:
M = qL2/8 + PL/4, V = qL/2 + P/2.
- Fixed-pinned:
M = qL2/8 + 3PL/16, V = 5qL/8 + 11P/16.
- Fixed-fixed:
M = qL2/12 + PL/8, V = qL/2 + P/2.
- Cantilever:
M = qL2/2 + PL, V = qL + P.
Here, L is the beam span, q is the total uniformly distributed load including self-weight, and P is the external point load. When the external load is distributed, P = 0.
Maximum beam deflection
Simply supported beam. Maximum deflection from the distributed load and a point load at midspan is calculated by superposition:
d = 5qL4/(384EI) + PL3/(48EI)
Fixed-fixed beam. Fixing both ends significantly reduces deflection:
d = qL4/(384EI) + PL3/(192EI)
Cantilever. Maximum deflection occurs at the free end:
d = qL4/(8EI) + PL3/(3EI)
Fixed-pinned beam. The position of maximum deflection depends on the ratio between the distributed load and the point load. With only a distributed load:
d = (39 + 55√33)qL4/(65536EI)
With only a point load applied at midspan:
d = √5 · PL3/(240EI)
When a point load and distributed self-weight act simultaneously, their deflection curves are added together. The maximum value is then determined along the beam length because its position does not necessarily coincide with midspan.
How the deflection limit is determined
Deflection limit. The calculated beam deflection is compared with a limit expressed as L/n. The larger the coefficient n, the smaller the permitted deflection:
dlim = L / n
If the user does not specify a custom value of n, the calculator applies its built-in practical guideline according to beam span:
- for L ≤ 1000 mm: n = 120;
- for 1000 < L ≤ 3000 mm: n increases linearly from 120 to 150;
- for 3000 < L ≤ 6000 mm: n increases linearly from 150 to 200;
- for 6000 < L ≤ 24000 mm: n increases linearly from 200 to 250;
- for 24000 < L ≤ 36000 mm: n increases linearly from 250 to 300;
- for L > 36000 mm: n = 300.
Within each intermediate range, n does not change abruptly but is interpolated linearly between the stated boundary values. For example, at a span of 4000 mm, n ≈ 166.7 and the deflection limit is about 24 mm. If the user specifies a custom L/n value, it replaces the automatic guideline.
Strength check
Normal stress. Maximum bending stress is calculated from the maximum bending moment and elastic section modulus:
σ = M / W
The strength condition is:
σ ≤ fy / γM0
The calculator uses γM0 = 1.0. Therefore, for steel grade S235, for example, the calculated normal stress is compared with 235 MPa.
Shear stress. It is calculated from the maximum shear force V, first moment of area Q, second moment of area I, and the relevant material width b:
τ = V · Q / (I · b)
The corresponding limit is:
τlim = fy / (√3 · γM0)
With γM0 = 1.0, this corresponds to approximately 135.7 MPa for S235, 158.8 MPa for S275, 205.0 MPa for S355, and 242.5 MPa for S420. Normal stress and shear stress are checked separately against their corresponding limits.
How to interpret the result
Stiffness. The calculated beam deflection is compared with dlim. If d ≤ dlim, the selected beam satisfies the adopted deflection criterion. If the calculated deflection exceeds the limit, a stiffer combination of span and cross-section or a different specified L/n criterion is required.
Strength. Normal and shear stresses are checked separately against their respective limits. When a condition is satisfied, the calculator also shows the reserve as a percentage, calculated from the ratio between the limiting stress and the calculated stress:
reserve = (σlim/σ - 1) · 100%
The same principle is used for shear stress with τ and τlim.
Related European standards
EN 1990 "Eurocode - Basis of structural design". This standard establishes the general principles of structural design, including ultimate limit state and serviceability limit state checks. Permissible deflection depends on the intended use of the structure and the applicable serviceability criteria, so the built-in L/n scale is used by the calculator as a practical guideline.
EN 1991 "Eurocode 1 - Actions on structures". This series of standards defines the principles for determining actions and loads on structures. The load used in the beam calculation should correspond to the relevant loading situation.
EN 1993-1-1 "Eurocode 3 - Design of steel structures - Part 1-1: General rules and rules for buildings". This standard is related to the steel properties used in the strength checks, cross-section resistance, and the factor γM0. The calculator uses γM0 = 1.0.
EN 10025-2 "Hot rolled products of structural steels - Part 2: Technical delivery conditions for non-alloy structural steels". This standard specifies mechanical properties for common structural steels. The nominal fy values used by the calculator for S235, S275, S355, and S420 should be considered together with the actual thickness and type of steel product.
FAQs
Why can a beam pass the strength check but fail the deflection check?
Strength and stiffness are different criteria. Bending stress mainly depends on the ratio M/W, while beam deflection is also highly sensitive to span and the second moment of area I. A steel beam can therefore have sufficient strength while still deflecting more than the adopted limit.
Why does section depth have such a strong effect on beam deflection?
For a rectangular cross-section, the second moment of area is I = bh3/12, so the section depth is raised to the third power. Even a moderate increase in depth can significantly increase stiffness and reduce beam deflection. Width also affects the result, but its influence is linear for this section shape.
Why include self-weight if the external load is much larger?
Self-weight acts continuously over the entire beam length and creates additional bending moment and deflection. For a heavy section or a long span, this contribution can become significant. The calculator determines it automatically from the cross-sectional area, steel density of 7850 kg/m3, and beam length.
How should I choose the deflection limit L/n?
If no separate criterion has been specified, the automatic value used by the Beam Deflection and Strength Calculator can serve as a practical guideline, varying gradually from about L/120 for short beams to L/300 for long spans. If a specific limit such as L/250 or L/300 has been established for the structure, enter the corresponding value of n. The larger n is, the stricter the deflection limit.
Why can calculated section properties differ slightly from steel profile tables?
The calculator derives A, I, and W directly from the entered dimensions of an idealized cross-section. Real rolled profiles may include fillet radii, shape details, and manufacturing dimensions that are reflected in tabulated section properties. Therefore, a small difference from catalogue values is possible even when the main dimensions are the same.