Beam Deflection Calculator (Propped Cantilever)
Find the deflection and prop reaction of a propped cantilever beam, fixed at one end and simply supported at the other, under a midspan point load or a uniformly distributed load.
🏗️ What is a Propped Cantilever Beam?
A propped cantilever beam is fixed rigidly at one end and simply supported by a roller or pin, the "prop", at the other end. This mix of a moment-restrained fixed support and a simple support with no moment restraint makes it statically indeterminate to the first degree, meaning statics alone cannot solve the reactions, and an extra compatibility condition (zero deflection at the propped end) is needed.
Propped cantilevers appear whenever an existing cantilever is later given additional support, or whenever a design intentionally mixes support types to reduce deflection or bending moment. A balcony slab later shored with a column at its outer edge, a temporary construction stage before a permanent second support is cast, and continuous beams analyzed span-by-span with an adjacent span's stiffness idealized as a prop are all common real-world examples.
A common misconception is that the prop simply carries half the load, similar to a symmetric two-support beam. It does not. Because the fixed end resists both force and moment while the prop resists only force, the load splits unevenly, the prop takes 5/16 of a midspan point load or 3/8 of a uniformly distributed load, with the fixed end carrying the larger remaining share plus a restraining moment.
This calculator computes the deflection and prop reaction of a propped cantilever beam under either a midspan point load or a uniformly distributed load, using the standard indeterminate beam formulas, and shows the full deflected shape along the span as an interactive chart.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Steel Propped Cantilever Under a Midspan Point Load
A 5 m steel beam (E = 200 GPa, I = 80,000,000 mm⁴), fixed at one end and propped at the other, carrying a 60 kN point load at midspan
Example 2 — Same Steel Propped Cantilever Under a Uniformly Distributed Load
The same 5 m propped cantilever carrying a 30 kN/m distributed load across the full span
Example 3 — Timber Propped Cantilever Under a Midspan Point Load
A 3 m timber beam (E = 11 GPa, I = 60,000,000 mm⁴), fixed at one end and propped at the other, carrying a 15 kN point load at midspan
❓ Frequently Asked Questions
🔗 Related Calculators
What is a propped cantilever beam?
A propped cantilever is a beam that is rigidly fixed (moment-restrained) at one end and simply supported by a roller or pin at the other end. This combination of one fixed end and one simple support makes it statically indeterminate to the first degree, requiring compatibility equations in addition to statics to solve.
What is the formula for propped cantilever deflection under a midspan point load?
For a propped cantilever with a point load P at midspan (fixed at x=0, propped at x=L), the deflection at the load point is delta = 7PL^3/(768EI). The prop reaction is R_B = 5P/16, and the fixed end carries the remaining 11P/16 plus a restraining moment.
What is the formula for propped cantilever deflection under a uniformly distributed load?
For a propped cantilever under a uniformly distributed load w, the prop reaction is R_B = 3wL/8. Maximum deflection occurs at x_max = [(15 minus square root of 33) divided by 16] times L, approximately 0.5785L from the fixed end, with delta_max = w times x_max squared times (L minus x_max) times (3L minus 2x_max), all divided by 48EI.
Why is the prop reaction only 5/16 of the point load, not half?
Because one end is rigidly fixed and resists both force and moment, it naturally attracts more of the load than a simple support would. Solving the compatibility equation for zero deflection at the propped end gives R_B = 5P/16 for a midspan point load, leaving 11P/16 for the fixed end reaction.
Where does maximum deflection occur under a uniformly distributed load on a propped cantilever?
Unlike a simply supported or fixed-fixed beam, maximum deflection under a uniformly distributed load does not occur at midspan. It occurs at x_max is approximately 0.5785L measured from the fixed end, the point along the span where the beam's slope momentarily equals zero.
What is a typical deflection limit for a propped cantilever?
A common serviceability limit is span/360 (L/360), the same limit typically used for simply supported and fixed-fixed spans. For a 5 m span, L/360 is about 13.9 mm.
What units should I use for E and I in this calculator?
Enter the modulus of elasticity E in gigapascals (GPa) and the second moment of area I in millimeters to the fourth power (mm^4), matching the units used throughout this beam deflection series. The calculator converts to SI base units (Pa and m^4) internally, then reports both the deflection and the prop reaction in consistent units.
How is a propped cantilever different from a simply supported or fixed-fixed beam?
A simply supported beam has no moment restraint at either end. A fixed-fixed beam is moment-restrained at both ends. A propped cantilever mixes the two, moment-restrained at one end and simply supported at the other, giving it a unique deflected shape and reaction distribution between the two extremes.
Why is this called a statically indeterminate beam?
A propped cantilever has four unknown reaction components (two forces and one moment at the fixed end, one force at the prop) but only three independent equilibrium equations from statics. The extra unknown requires an additional compatibility condition, typically zero deflection at the propped support, to solve, which is why it is called indeterminate to the first degree.
Does increasing the modulus of elasticity always reduce propped cantilever deflection?
Yes, deflection is inversely proportional to E in both loading modes, so a stiffer material (higher E) under the same load and geometry deflects less. Steel (E is about 200 GPa) deflects roughly 18 times less than an equivalent timber beam (E is about 11 GPa) for identical span and section.
What assumptions does this propped cantilever deflection formula make?
This calculator uses classical Euler-Bernoulli beam theory, which assumes linear-elastic material behavior, small deflections relative to the span, a prismatic (constant cross-section) beam, a rigid fixed support at one end, and a simple roller or pin support at the other with no moment restraint.