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Structural Simulator

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

Analyze a beam pin-supported at both A and B with a free overhang extending past B to D under a UDL and a tip point load. Real-time reactions, peak moments and tip deflection are computed and visualized.

Parameters
Left overhang a
m
Main span L_AB
m
Right overhang c
m
UDL w
kN/m
Point load P
kN
Presets

Load applied 0%

Flexural rigidity EI = 50000 kN·m² is fixed. The UDL w acts over the full length (a+L_AB+c). In "Moving load" the point load P travels from left to right.

Results
—
Left reaction R_A
—
Right reaction R_B
—
Max positive moment M⁺
—
Max negative moment M⁻
—
Max shear V_max
—
Max deflection δ_max
Deflected shape / SFD / BMD live

Top = deflected shape that grows as load is applied (support reactions shown as ↑) / Middle = shear force diagram SFD (V=0 point = max positive moment) / Bottom = bending moment diagram BMD (M=0 = point of contraflexure marked ●; negative at the overhang root)

Theory & Key Formulas

A beam with left overhang a, main span L_AB and right overhang c, pin-supported at A and B. Reactions and moments under a UDL w (full length) and a point load P are evaluated by the following equations (moments taken about A).

Right reaction from moment equilibrium about A (x measured from A):

$$R_B = \frac{\sum (\text{load}\times \text{distance from A})}{L_{AB}}$$

Left reaction from vertical equilibrium:

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

Negative bending moment at the right overhang root B (single-overhang case):

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

The maximum positive moment in the main span occurs where the shear V(x)=0.

Tip deflection of the overhang (cantilever approximation rooted at B):

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

The point where the BMD changes sign from positive to negative (M=0) is the point of contraflexure, where the curvature of the deflection curve is zero. Increasing the overhang length or the load enlarges the negative moment and shifts the contraflexure point toward the support.

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 max deflection δ_max of 3.40 mm is surprisingly small.
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That is because EI = 50000 kN·m² is a fairly stiff beam. The tool integrates the curvature $M/EI$ twice and imposes zero deflection at the supports to draw the whole elastic line, so the δ_max shown is the value at the overhang tip. A quick estimate is the cantilever form $\delta \approx wc^4/(8EI) + Pc^3/(3EI)$. Try doubling c to 4 m and the deflection jumps sharply; 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 3.40 mm becomes 1.70 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.

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How to Use

  1. Set main span L_AB and overhang length c (slCVal) in meters. slC is the overhang length, not a load-position input.
  2. The UDL w acts over the full length L_AB+c, and point load P always acts at the free end D.
  3. Read R_A, R_B, root moment M_B, tip deflection δ_D, and the SFD/BMD plots.

Example

With main span 4 m, overhang 1 m, w=10 kN/m, P=0 and fixed EI=50000 kN·m², the tool gives R_A=18.75 kN (display 18.8), R_B=31.25 kN upward (display 31.3), M_B=-5.0 kN·m and δ_D=0.025 mm (display 0.03). The maximum positive moment in the main span is M+=17.6 kN·m at x=1.88 m.

Practical Notes

  1. With P=0, the main-span midspan moment can become negative only when the overhang is about 0.7L_AB or longer.
  2. The negative root moment puts the top fiber in tension, so RC detailing must check top reinforcement anchorage.
  3. P is fixed at the free end D. For arbitrary point-load positions, use the SFD/BMD tool or a full continuous-beam model.