Calculation
θ = F · L² / (2 · E · I)
An end force deflects a cantilever and rotates its tangent at the free end. The angle grows with the square of free length.
Euler-Bernoulli-Biegelinie bei Endkraft · θ = FL²/(2EI)
Calculate Cantilever Tip Rotation Under an End Load
An end force deflects a cantilever and rotates its tangent at the free end. The angle grows with the square of free length.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Tip rotation θ — Magnitude of the angle between the original beam axis and tangent at the free end; relevant to attached-device alignment, not the same as deflection in millimetres.
- Perpendicular end force F — Static force perpendicular to the original beam axis at the free tip, taken from the load case or weight.
- Free length L — Distance from the fixed edge to the end force along the undeformed cantilever, from a drawing.
- Young modulus E — Material stiffness at the considered temperature from a datasheet; structural steel is typically around 210 GPa.
- Second moment of area I — Geometric bending stiffness of the section about the loaded axis. Obtain from profile tables, section drawings or a section calculator; 100 cm⁴ equals 10⁻⁶ m⁴.
Example
With F = 1000 N, L = 2 m, E = 210 GPa and I = 100 cm⁴ = 10⁻⁶ m⁴, θ = 1000·2²/(2·210·10⁹·10⁻⁶) = 0.009524 rad = 0.546°.
Assumptions and limits
Straight prismatic Euler-Bernoulli beam, rigid clamp, linear elasticity, small angles and one static end force. Shear deformation, self-weight, other loads and nonlinear large deflections are excluded.
Understand Cantilever tip rotation under an end load
A force at a cantilever's free end moves it down and tilts its end face. That tip rotation often matters for attached guides and brackets.
What does this quantity describe?
A cantilever is clamped at one end and free at the other. For a straight beam of constant bending stiffness E·I, end force F causes moment F(L−x) at position x. Beam curvature is approximately moment divided by E·I. Integrating from the clamp, where slope is zero, to the tip gives θ = FL²/(2EI). θ is the tip-tangent angle, E Young modulus and I the second moment of area about the bending axis.
Formula and variables
θ = F · L² / (2 · E · I)
Moment at distance x from clamp: M(x) = F(L−x)Tip rotation: θ = FL²/(2EI)
| Symbol / input | Meaning |
|---|---|
| Tip rotation θ | Magnitude of the angle between the original beam axis and tangent at the free end; relevant to attached-device alignment, not the same as deflection in millimetres. |
| Perpendicular end force F | Static force perpendicular to the original beam axis at the free tip, taken from the load case or weight. |
| Free length L | Distance from the fixed edge to the end force along the undeformed cantilever, from a drawing. |
| Young modulus E | Material stiffness at the considered temperature from a datasheet; structural steel is typically around 210 GPa. |
| Second moment of area I | Geometric bending stiffness of the section about the loaded axis. Obtain from profile tables, section drawings or a section calculator; 100 cm⁴ equals 10⁻⁶ m⁴. |
Choose the inputs correctly
F in newtons is the static perpendicular tip force. L in metres runs from clamp face to the force. E in pascals comes from material data; steel is around 210 GPa. I in m⁴ comes from section geometry or a profile table and must match bending direction. Use values for the same beam and load case.
How to use the calculator
Enter force, free length, material stiffness and the correct I. The angle describes tip-tangent orientation. Compare it with allowable alignment error for an attached component; check deflection and strength separately.
Worked example
For F = 1000 N, L = 2 m, E = 210 GPa and I = 100 cm⁴ = 10⁻⁶ m⁴, θ = 1000·4/(2·210·10⁹·10⁻⁶) = 0.009524 rad = 0.546°. Doubling L multiplies θ by four.
Understand the result and units
Length enters squared; doubling E·I halves the angle. Tip rotation is an angle, not vertical tip movement. Two designs can have similar deflections yet different tip alignment.
The calculation uses newtons, metres, pascals and m⁴ and yields radians internally. The shared unit register converts kN, mm, GPa and cm⁴ before calculation and radians to degrees for display.
Typical applications
Alignment of fixtures on cantilevers, sensor brackets, small booms and machine frames under a concentrated tip force.
Assumptions, limits and common mistakes
Straight constant-section beam with linear elasticity, rigid clamp and small rotations. Shear deformation, flexible mounting, self-weight, extra forces, temperature and large deflections are excluded.
Common mistake: I must be for the bending axis. Do not mistake angle θ for tip deflection. L is free cantilever length, excluding the embedded portion.
Frequently asked questions
What is “Cantilever tip rotation under an end load” used for?
Alignment of fixtures on cantilevers, sensor brackets, small booms and machine frames under a concentrated tip force.
Where do the input values come from?
F in newtons is the static perpendicular tip force. L in metres runs from clamp face to the force. E in pascals comes from material data; steel is around 210 GPa. I in m⁴ comes from section geometry or a profile table and must match bending direction. Use values for the same beam and load case.
What does the result not cover?
Straight constant-section beam with linear elasticity, rigid clamp and small rotations. Shear deformation, flexible mounting, self-weight, extra forces, temperature and large deflections are excluded.