Real-time calculation and visualization of deflection, bending moment, and shear force for simply supported and cantilever beams under concentrated and distributed loads.
Parameters
Beam type
Young's modulus E
GPa
Material stiffness (steel: 200 GPa, Al: 70 GPa)
Cross-sectionSecondMoment I
Bending stiffness from the cross-section —
Beam length L
m
Load intensity q
kN
Distributed load (kN/m)
Load position a
m
Load application point on the beam
EI (bending stiffness) —
Results
MaxDeflection δ_max
—
mm
MaxMoment M_max
—
kN·m
MaxShear force V_max
—
kN
Bending Stiffness EI
—
N·m²
Beam Model — Click to set load position, drag to adjust magnitude
Click: set load position | Vertical drag: change load magnitude
What exactly is "beam deflection," and why is it so important to calculate?
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Basically, it's how much a beam bends under a load. Think of a diving board sagging when someone stands on the end. We need to calculate it to ensure structures like bridges or building floors don't bend too much, which could lead to failure or discomfort. In this simulator, you can see the deflection curve instantly by adjusting the Beam Type and Load Strength q on the left.
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Wait, really? So the bending moment and shear force are different from the deflection? What do those diagrams tell us?
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Exactly! The deflection is the "what you see"—the bent shape. The bending moment and shear force are the internal "stresses" causing that bend. For instance, the bending moment diagram shows where the beam is under maximum tension or compression—a critical spot for cracks to start. Try switching from a cantilever to a simply supported beam above; you'll see the moment diagram flip completely, revealing a different critical point.
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I see the parameter "EI" – Young's modulus E and Second moment of area I. What happens if I make the beam material stiffer or change its cross-section shape?
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Great question! E is the material stiffness (steel vs. rubber), and I depends on the shape's geometry (an I-beam vs. a solid rectangle). Their product EI is the beam's flexural rigidity—its resistance to bending. In practice, to reduce deflection in a floor joist, you'd increase I by making it taller. Slide the E and I sliders down in the simulator and watch the deflection curve shoot up dramatically!
Physical Model & Key Equations
The fundamental governing equation for beam bending, known as the Euler-Bernoulli beam theory, relates the load distribution to the fourth derivative of deflection. This is the equation solved in the background of this simulator.
$$EI\frac{d^4w}{dx^4}= q(x)$$
Where: E = Young's modulus (material stiffness, in Pa) I = Second moment of area (cross-section shape property, in m⁴) w(x) = Deflection at position x (in m) q(x) = Distributed load applied to the beam (in N/m)
For a simply supported beam with a uniform load q, the solution to the above equation gives the deflection curve. This is a classic result you can validate with the simulator.
$$w(x)=\frac{qx(L^3-2Lx^2+x^3)}{24EI}$$
Where: L = Total length of the beam (in m) x = Position along the beam from one support (in m)
The maximum deflection occurs at the center, x = L/2. Notice how deflection scales directly with load q and inversely with EI.
Frequently Asked Questions
The unit for deflection is mm, for bending moment is N･mm, for shear force is N, and for stress is MPa. These depend on the units of the input values (length in mm, load in N, Young's modulus in MPa), so please ensure a consistent unit system when entering data.
Yes, they can be set simultaneously. You can add a concentrated load at any position on the beam and also superimpose a distributed load for calculation. The results are displayed as the linear superposition of the effects of each load.
For a rectangular cross-section, enter the width b and height h in mm, and the second moment of area I will be automatically calculated. For a circular cross-section, enter the diameter d. For an arbitrary cross-section, you can directly input the value of I.
Consider increasing the cross-sectional dimensions, changing to a material with a higher Young's modulus, adding support points to shorten the span length, or reducing the load. You can modify each parameter in the simulator and check the effects in real time.
Real-World Applications
Bridge Design: Engineers use this exact analysis to determine the required depth and material for bridge girders. They must limit deflection under the weight of traffic (a distributed load) and heavy trucks (point loads) to prevent excessive sway and ensure long-term fatigue life.
Building Floor Systems: The floor you're standing on is analyzed as a beam. Calculating deflection ensures it doesn't feel "bouncy" under furniture and people (uniform load) and can support heavy items like a water tank (a point load). The bending moment diagram identifies where to place reinforcing steel.
Automotive Chassis: The vehicle's frame is a complex beam system. Simulating deflection and stress from engine weight and road forces helps optimize the design for safety and handling, ensuring the chassis is stiff enough to protect the passenger cabin.
Industrial Machine Frames: Precision machines like CNC mills or printing presses require extremely rigid frames. Even tiny deflections under the motor's forces (modeled as point loads) can cause misalignment and ruin product quality, making this analysis critical.
Common Misconceptions and Points to Note
Let's go over a few points where people often stumble when starting to use this type of tool. First, you might tend to think "Young's modulus E and the second moment of area I are independent parameters", but in reality, changing the material can affect not only E but also I. For example, when switching from a steel (E=210GPa) square pipe to an aluminum (E=70GPa) one, to maintain strength you'd likely increase the wall thickness or the cross-sectional size, right? This also increases I, so the deflection won't simply triple just because E becomes one-third. Using this tool to try things out—like reducing E to 1/3 and then perhaps doubling I—gets you closer to reality.
Next, understand the limits of the "point load" model where the entire force acts at a single point. In the real world, force is almost never applied at a true "point". For instance, when a machine part is bolted down, the load is distributed around the bolt hole. The result you get from setting a concentrated load P=1000N in this simulator implicitly indicates that high stress concentrations will occur in its vicinity in reality. When considering a safety factor, you need to keep this "idealization" in mind.
Finally, remember that small deflection does not automatically mean "OK". Stiffness is certainly important, but in some cases, like with an automobile suspension arm, a certain degree of flexibility is necessary for shock absorption. Also, for the fixed end of a cantilever beam, the bending stress $\sigma = \frac{My}{I}$ (where M is the bending moment and y is the distance to the edge of the cross-section) that occurs there is a more direct criterion for failure than the deflection itself—it must not exceed the material's yield strength. Get into the habit of reviewing results from both the deflection and stress perspectives.