What is a propped cantilever beam?

A propped cantilever is a beam fixed at one end and supported by a pin or roller at the other. Because it has one more support than a simply supported beam, the number of unknown reactions (or reaction moments) exceeds the equations of static equilibrium by one, making it a one-degree statically indeterminate beam that requires an additional compatibility condition. Typical examples include the base of a tower crane jib, continuous bridge supports, and a cantilever canopy stiffened with an outrigger column.

Why are deflections and reactions smaller than a simply supported beam?

Because the right-hand pin takes part of the load, the reaction moment at the left fixed end is reduced. Under a uniformly distributed load w, the simply supported beam has a maximum moment of wL^2/8, while the propped cantilever has a fixed-end moment of magnitude wL^2/8 but a maximum positive moment of only 9wL^2/128 (about wL^2/14.2). The maximum deflection drops from 5wL^4/(384EI) to wL^4/(185EI), about 2.6 times smaller.

Where do the reactions 5wL/8 and 3wL/8 come from?

They follow from the compatibility condition that the deflection at the right-hand pin must be zero. If the pin were removed, the free-end deflection of the remaining cantilever under w would be wL^4/(8EI), and the upward deflection caused by a force R_B alone would be R_B*L^3/(3EI). Setting them equal gives R_B = 3wL/8. Vertical equilibrium then yields R_A = wL - R_B = 5wL/8, and the moment equilibrium gives a unique fixed-end moment M_A = -wL^2/8.

What about a central point load?

For a propped cantilever with a point load P applied at midspan, the fixed-end moment is M_A = -3PL/16, the fixed-end reaction is R_A = 11P/16 and the pin-end reaction is R_B = 5P/16. When UDL and point loads act simultaneously, linear superposition still holds for this statically indeterminate beam, so each set of formulas can simply be added. This simulator displays the superposition of w and P in real time.

Propped Cantilever Beam Simulator — Reactions, Moments, Deflection

Parameters

Beam length L

Uniformly distributed load w

kN/m

Central point load P

Flexural rigidity EI

kN·m²

Left end is fully fixed; right end is a pin (roller). The UDL w (full length) and the central point load P are superposed.

Results

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Fixed-end moment M_A = -wL²/8

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Fixed-end reaction R_A = 5wL/8

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Pin-end reaction R_B = 3wL/8

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Max deflection δ_max ≈ wL⁴/(185·EI)

Beam Model, SFD and BMD

Top = beam diagram (left fixed hatching, right pin △, UDL arrows and central P) / Middle = SFD (shear force diagram) / Bottom = BMD (bending moment diagram)

Theory & Key Formulas

For a beam fixed at the left and pinned at the right (one-degree statically indeterminate), the redundant reaction is found from the zero-deflection condition at the right support. For a UDL $w$ over the full length:

$$M_A = -\frac{wL^2}{8},\quad R_A = \frac{5wL}{8},\quad R_B = \frac{3wL}{8}$$

The maximum positive moment occurs at $x = 5L/8$ from the fixed end ($3L/8$ from the pin):

$$M_{+,\max} = \frac{9\,wL^2}{128}$$

The maximum deflection occurs at $x \approx 0.4215\,L$:

$$\delta_{\max} = \frac{wL^4}{185\,EI}$$

For a central point load $P$ at midspan:

$$M_A = -\frac{3PL}{16},\quad R_A = \frac{11P}{16},\quad R_B = \frac{5P}{16}$$

$w$ and $P$ can be combined by linear superposition. Compared with a simply supported beam, the fixed end reduces the maximum deflection by about 2.6 times and the maximum positive moment by about 1.8 times.

What is the Propped Cantilever Beam Simulator

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Is a "propped cantilever" really different from a regular cantilever? The names sound so similar.

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A propped cantilever is a regular cantilever with one extra support (a "prop") added at the free end. The left end is fully fixed and the right end is a pin (or roller). There are four unknown reactions (M_A, R_A, R_B and a horizontal reaction) but only three equilibrium equations, so we add one compatibility condition. That makes it a one-degree statically indeterminate beam. In the simulator the hatching on the left and the triangle on the right show those supports.

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Wait, indeterminate? Why can't equilibrium alone solve it once we add a support?

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Equilibrium gives only three equations (sum of vertical forces, sum of horizontal forces, sum of moments), and we have four unknown reactions. So we are one equation short. The fix is to add "the deflection at the right support equals zero". If you imagine removing the right support, the cantilever's free-end deflection under w is $wL^4/(8EI)$. Adding only R_B upward gives a deflection $R_B L^3/(3EI)$. Setting them equal gives $R_B = 3wL/8$ immediately. This is called the unit load method or compatibility condition.

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I see! With w=15 kN/m and L=8 m, M_A becomes -120 kN·m in the simulator. Is that a large number?

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For the same conditions, a simply supported beam has a maximum moment of $wL^2/8 = 120$ kN·m at midspan. The fixed-end moment of the propped cantilever has the same magnitude, -120 kN·m. The interesting thing is that the maximum positive moment drops to $9wL^2/128 \approx 67.5$ kN·m. Look at the red curve in the BMD: it dives down at the left and rises into a hump at $5L/8 = 5$ m from the fixed end (i.e. $3L/8$ from the pin). The design effectively trades a large negative moment at the fixed end for a smaller positive peak in the middle.

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And the deflection is only 4.15 mm — much smaller than a simply supported beam, right?

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The maximum deflection of a simply supported beam is $5wL^4/(384EI)$. With the same EI of 80000 kN·m² it works out to about 10.8 mm. The propped cantilever gives $wL^4/(185EI) = 4.15$ mm, about 2.6 times smaller. Because the right support shares the load, both reactions and deflections are reduced — that is the main benefit of going one-degree indeterminate. Add a point load P and the results change via superposition; try the P slider to feel it.

Frequently Asked Questions

As long as the beam is linearly elastic (Hooke's law), the deformations are small and the support conditions do not depend on the load, linear superposition holds rigorously even for indeterminate beams. For this propped cantilever, the solutions for UDL w and central point load P (M_A, R_A, R_B and deflection) can each be obtained separately and simply added to get the response to the combined load. Once the material yields, superposition no longer holds, so the assumption must always be checked when designing for ultimate limit states.

The propped cantilever is asymmetric (fixed on the left, pinned on the right), so the deflection curve is not symmetric either. The negative moment at the fixed end shifts the peak of the deflection toward the fixed end, landing at $x \approx 0.4215L$ rather than at the midspan. This value is obtained by differentiating the deflection function $w(x)$ and setting it to zero, and it is tabulated in standard beam deflection tables.

For a UDL, the differential relation $dV/dx = -w$ makes the shear decrease linearly from R_A = 5wL/8 at the left to -R_B = -3wL/8 at the right. The zero-crossing is at $x = R_A/w = 5L/8$ from the fixed end, which coincides with the location of the maximum positive moment in the BMD: this is the direct consequence of $dM/dx = V$. When a point load P is added at midspan, the SFD jumps vertically by P at $x = L/2$.

Typical examples include a cantilever canopy stiffened by an outrigger column at its tip, a bridge overhang propped by a temporary bent during erection, the base of a tower-crane jib with a tip pendant cable, and a cantilever balcony strengthened by an external diagonal brace. Because both deflection and stress are significantly smaller than a pure cantilever, this configuration is adopted for reinforcement, weight reduction or improved vibration performance.

Real-World Applications

Stiffening canopies and balconies in architecture: Cantilever canopies and balconies projecting from a building can show conspicuous deflection over time or under changed loading, and an auxiliary column (a "prop") is often installed at the tip to revive them. Going from a pure cantilever to a one-degree indeterminate propped cantilever cuts the tip deflection by about 2.6 times without increasing the fixed-end moment. The reduction in δ_max shown by this simulator illustrates that effect directly.

Continuous bridge supports and temporary erection bents: Continuous-span bridges use intermediate piers to limit midspan stresses. During cantilever erection, a temporary bent is sometimes placed under the projecting end to form a propped cantilever, allowing the overhang to be extended while stresses are controlled. The split between the fixed-end moment $-wL^2/8$ and the positive moment $9wL^2/128$ feeds directly into the construction stress check.

Machine frames and sensor arms: Long sensor arms in measurement equipment or machine tools suffer from high stress at the cantilever root. Adding a tiny roller support at the tip turns the arm into a propped cantilever, raising the first natural frequency and reducing dynamic deflection during acceleration. Increase the EI slider and you can see the deflection δ_max drop inversely proportional to EI.

Tower crane jibs and stay cables: The horizontal jib of a tower crane is held up by counter-jib and pendant cables that effectively pull its tip upward. The geometry and loading are not exactly the simulator's model, but the idea of stiffening a cantilever with a remote support is the same, and this rationalization of support conditions is widely used to carry the self-weight of the jib together with the hoisted load.

Common Misconceptions and Cautions

The most common misconception is to overestimate the fixed-end moment compared with a simply supported beam. The fixed-end moment of a propped cantilever under UDL w is $-wL^2/8$, exactly the same magnitude as the maximum moment $wL^2/8$ of a simply supported beam with the same loading. People often think "fixing one end concentrates all the moment there", but in fact the positive moment is reduced to $9wL^2/128 \approx wL^2/14.2$ in exchange for the fixed end carrying the same magnitude — the total demand does not increase. The peak-to-trough ratio of the red BMD curve in the simulator makes this clear.

The next common error is to assume superposition does not apply to indeterminate beams. Under linear elasticity, small deformations and load-independent supports, superposition holds exactly even for indeterminate beams. Try moving the w slider and the P slider separately, and then compare with both loads applied together: each reaction, moment and deflection equals the sum of the individual solutions. On the other hand, when the material yields or a support lifts off, the problem becomes nonlinear and superposition breaks down.

Finally, take care not to assume that "because the right end is pinned, both deflection and rotation are zero there". A pin (roller) support constrains the vertical displacement only and leaves the rotation (slope) free. The BMD value at the right end x = L is exactly zero because the bending moment at a pin support is necessarily zero. The slope $dw/dx$ at the right end, however, is finite. This is different from a fully clamped end, so the distinction between support conditions must always be checked first.

Propped Cantilever Beam Simulator — One-Degree Indeterminate Beam

What is the Propped Cantilever Beam Simulator

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Cautions

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