What is an overhanging beam?

An overhanging beam has pin supports near both ends, with one part extending beyond an interior support as a free overhang. This tool covers the case with a pin at A (x=0) and a pin at B (x=L_AB), plus a free end D at x=L_AB+c. Balcony slabs, roof eaves and bridge cantilever segments are typical examples. The key feature is that the main span develops a positive bending moment while the overhang root develops a negative bending moment at the same time.

How are the reactions R_A and R_B obtained?

By solving vertical equilibrium together with moment equilibrium about A. With a UDL w acting over the full length L_AB+c and a tip load P at D, taking ΣM_A=0 gives R_B*L_AB = w*L_AB^2/2 + w*c*(L_AB+c/2) + P*(L_AB+c), so R_B = [w*L_AB^2/2 + w*c*(L_AB+c/2) + P*(L_AB+c)] / L_AB. R_A then follows from vertical equilibrium R_A + R_B = w*(L_AB+c) + P.

Why is the root moment M_B negative?

Because the loads on the overhang (UDL w*c and tip load P) act only to the right of support B. Treating the overhang as a cantilever rooted at B, the downward load creates a clockwise moment that puts the top of the beam in tension and the bottom in compression. This is opposite to the main-span sign convention (bottom tension is positive), so we attach a negative sign and write M_B = -(w*c^2/2 + P*c). In reinforced concrete design it directly drives the choice of top reinforcement at the root.

How is the tip deflection at D computed?

The tip deflection delta_D is approximated as the free-end deflection of a cantilever rooted at B: the UDL contribution is w*c^4/(8EI) and the tip-load contribution is P*c^3/(3EI), and the two are added. A more rigorous evaluation also adds rigid-body rotation due to the rotation angle theta_B of the elastic line at B, but a quick estimate uses only the cantilever part. In design practice the result is compared against deflection limits such as span/250 to span/300.

Overhanging Beam Simulator — Reactions, SFD/BMD & Deflection

Parameters

Main span L_AB

Overhang length c

UDL w

kN/m

Tip load P

Flexural rigidity EI = 50000 kN·m² is fixed. The UDL w acts over the full length (L_AB + c).

Results

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Left reaction R_A

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Right reaction R_B

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Root moment M_B

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Tip deflection delta_D

Beam model / SFD / BMD

Top = beam model (supports A, B with UDL and tip load) / Middle = shear force diagram / Bottom = bending moment diagram (negative at the overhang root)

Theory & Key Formulas

Supports A (x=0) and B (x=L_AB) are pinned and an overhang of length c extends from B to the free end D. The following formulas give reactions and moments under a UDL w acting over the whole length and a tip load P at D.

Right reaction from moment equilibrium about A:

$$R_B = \frac{w\,L_{AB}^{2}/2 + w\,c\,(L_{AB}+c/2) + P\,(L_{AB}+c)}{L_{AB}}$$

Left reaction from vertical equilibrium:

$$R_A = w\,(L_{AB}+c) + P - R_B$$

Negative bending moment at the overhang root B:

$$M_B = -\left(\frac{w\,c^{2}}{2} + P\,c\right)$$

Tip deflection at D (cantilever approximation rooted at B):

$$\delta_D \approx \frac{w\,c^{4}}{8\,EI} + \frac{P\,c^{3}}{3\,EI}$$

The maximum positive moment in the main span occurs where shear vanishes, at x = R_A / w. Increasing the overhang c or the tip load P enlarges the magnitude of M_B (negative), so designers must pay particular attention to the tension on the top fiber at the root.

What is the overhanging beam simulator?

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"Overhanging beam" — I haven't heard that term before. How is it different from a plain simply supported beam?

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Loosely speaking, it has a part that sticks out beyond a support (the "overhang"). Balconies, roof eaves and the cantilever walkways on bridges are typical examples. In this tool, A is the left pin and B is the inner pin, and from B a free overhang of length c extends to D where the loads act. The key is that the main span (A to B) and the overhang (B to D) bend in opposite directions.

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Opposite directions? Looking at the BMD I do see it snap sharply downward at B.

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Right, that drop is the negative moment M_B. In the main span the top is in compression and the bottom in tension, but at the overhang root it is reversed — the top is in tension. In reinforced concrete you need extra top reinforcement at B. With the defaults (L_AB=5, c=2, w=10, P=20) the reading is M_B = -60 kN·m. Check the stat card.

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Yes, it shows -60.0 kN·m. And R_A is 13 while R_B is 77 — the two supports are very different.

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Because all the overhang loads are caught near B. R_B is much larger. If you make the overhang longer or the tip load heavier, R_B grows further and R_A even goes negative — the left support lifts up. That is the lever principle. Play with the sliders and you will see it.

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The tip deflection of 1.47 mm is surprisingly small.

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That is because EI = 50000 kN·m² is a fairly stiff beam. We use $\delta_D \approx wc^4/(8EI) + Pc^3/(3EI)$, and with c = 2 m that is the number you get. Try doubling c to 4 m and the deflection jumps more than sixteen-fold; the c^4 term makes the overhang length critical. Design codes typically limit deflection to about span/250 or span/300.

FAQ

In structural engineering the sign convention is "positive when the bottom of the beam is in tension". In the main span the bottom is in tension, so the moment is positive; at the overhang root the top is in tension, so the moment is negative. That is why the BMD shows the main span bulging downward (positive) while the overhang portion bulges upward (negative). Reinforced concrete designers must therefore place the main bars on the top face at the overhang root.

Yes. When the tip load P or the overhang c is large and the main span L_AB is short, the left support A tends to lift up — its reaction becomes downward (negative). Concretely, R_A < 0 means support A has to pull the beam down. In real structures this requires anchor bolts or counterweights and is a major design check. Move the sliders here to find combinations that flip R_A negative; it shows up immediately in the stat card.

In the main span the bending moment is M(x) = R_A*x - w*x^2/2, and the shear V(x) = R_A - w*x vanishes at x = R_A/w, which is where the moment reaches its local maximum. With the defaults R_A = 13 kN and w = 10 kN/m, that gives x* = 1.3 m and M+ ≈ 8.45 kN·m. The "M+" label at the top of the BMD card reports this value. Always check that x* falls inside the main span (0 < x* < L_AB) before quoting this maximum.

This tool focuses on quantities that do not depend on section rigidity — reactions, moments, SFD and BMD — plus deflection. Because the tip deflection delta_D is inversely proportional to EI, you can scale the displayed value to any EI. For example, with EI doubled to 100000 kN·m² a reading of 1.47 mm becomes 0.74 mm. Keeping EI fixed avoids overloading beginners with too many sliders while still letting users rescale the deflection mentally.

Real-world applications

Balconies and cantilever slabs: Apartment balconies are textbook overhanging beams. The interior wall or main beam plays the role of support B, and everything beyond is the overhang. A handrail load (tip point P) and live load plus self-weight (UDL w) combine to determine the root moment M_B, which controls the top reinforcement layout. In practice the slab is treated as a continuous beam to account for interaction with the interior span.

Cantilever bridge construction: Long-span bridges are often built by the cantilever method, with segments cast symmetrically out from a pier. At every construction stage the tip carries the weight of the form traveller as a concentrated load, and the negative moment at the pier becomes the critical case. The mental model used in this tool maps directly onto the pier-region check.

Crane and machinery outriggers: The outriggers of a mobile crane act like overhanging beams. The pad at the ground is point D, the body attachment is B and the opposite outrigger plays the role of A. As the load swings around, the reaction split between the supports shifts dramatically, and once R_A becomes negative the machine tips over. Stability calculations are exactly this overhanging-beam model applied to the chassis.

Roof eaves and overhead signs: Eaves and large signs that protrude horizontally from a building carry snow loads, wind loads, and — for signs — sometimes upward wind-induced lift. Upward loading reverses the sign of P, flipping M_B and putting tension on the bottom fiber instead of the top. The practical approach is to simulate both load directions in this tool to ensure safety in every scenario.

Common misconceptions and pitfalls

The most common pitfall is to treat the overhang as a pure cantilever and ignore the main span. Looking at just the overhang it does behave like a cantilever, but the root B is not a true fixed end — the elastic line of the main span has a rotation angle theta_B at B. A rigorous evaluation writes delta_D = (cantilever part) + theta_B * c, and theta_B becomes important when the main span is short and the overhang is long. This tool computes only the cantilever part as a quick estimate; for tighter design checks use a continuous-beam analysis or an FEM model.

Next, do not confuse the sign of a reaction with physical "uplift". R_A > 0 means support A pushes the beam upward; R_A < 0 means the beam tries to lift off A and the support must instead pull the beam downward. The latter cannot be carried by an ordinary pin or roller, so anchor bolts or kentledge are needed. Push the sliders here until R_A turns negative — you can feel directly when the structure becomes dangerous.

Finally, do not equate a negative BMD value with weakness. What matters is which side of the section is in tension. In the positive-moment region of the main span the bottom is in tension, and in the negative-moment region near the overhang the top is in tension. Reinforcement must be on the tension side. The point where the BMD crosses zero is the point of contraflexure, an important location where the top and bottom main reinforcement are switched in detailing.

Overhanging Beam Simulator — Reactions, SFD, BMD & Tip Deflection

What is the overhanging beam simulator?

FAQ

Real-world applications

Common misconceptions and pitfalls

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