Beam Deflection Calculator: Multi-Load Superposition + Span/250

Beam Deflection Calculator

Superposition · span/250 check · required-I reverse mode · BMD/SFD

Beam & Section

Steel 2,00,000 · RCC ≈ 5000√fck (M25 → 25,000)
e.g. ISMB300 ≈ 8.6×10⁷ · rect = bd³/12

Loads (superposed)

Reverse: required I for the deflection limit

Uses the loads above and the selected span limit — deflection scales as 1/I.

Result

Set the beam, section and loads, then calculate to see the maximum deflection, its location, the serviceability badge and the BMD/SFD diagrams

Elastic (Euler–Bernoulli) deflections with standard textbook formulas, superposed exactly and scanned along the span for the true combined maximum. Self-weight is not added automatically — include it as a UDL if required. For RCC, long-term deflection (creep/shrinkage) and cracked-section effects are beyond this elastic check; fixed-fixed point loads are supported at midspan only. BMD/SFD are formula-based illustrations, not FEA.

Beam Deflection Calculator: True Superposition, Span/250 Check and Required-I

Most free beam calculators handle only one load at a time, a single point load at midspan or a single UDL and tell you to use paid software the moment you have more than one load acting together. This tool actually superposes multiple point loads and UDLs on the same beam, in any position and scans the entire combined deflection curve to find the true maximum and exactly where it happens, not just the sum of each load’s individual peak, since those peaks rarely occur at the same point along the span. It checks the result against the IS 456 span/250 or span/360 serviceability limit with a clear Pass or Fail badge, and includes a reverse mode that tells you the minimum moment of inertia needed if your current section fails.

How to Use

Step 1: Set up your beam and section

  • Select the Support Type: Simply Supported, Cantilever (fixed at left) or Fixed-Fixed.
  • Enter the Span L (mm), the full length of your beam.
  • Enter E (N/mm²), the modulus of elasticity. A hint below the field reminds you that steel is typically 2,00,000 N/mm², while RCC is approximately 5000√fck (for M25, that works out to roughly 25,000 N/mm²).
  • Enter I (mm⁴), the moment of inertia of your section. A hint shows a reference value for a common steel section (ISMB300 ≈ 8.6×10⁷ mm⁴) and the formula for a plain rectangular section (bd³/12) if you need to calculate it yourself.
  • Select your Deflection Limit: Span/250 (the IS 456 total deflection limit) or Span/360 (the tighter limit used after partitions are in place).

Step 2: Add your loads

  • Tap Add Load to create a load card. Each card lets you pick the Load Type, Point (kN) or UDL (kN/m).
  • For a point load, enter its magnitude in kN, and its Position a (mm), measured from the left support. For a fixed-fixed beam, the point load position is automatically locked to the exact centre, since the formula used only supports a midspan point load for that support type.
  • For a UDL, just enter the magnitude in kN/m. Since a UDL covers the full span by default, there’s no position field for it.
  • Repeat Step 2 for every load acting on your beam, up to 4 loads total. All of them get superposed together in the same calculation.

Step 3: Calculate

  • Tap Calculate Deflection. The result panel shows your beam’s true combined maximum deflection, exactly where along the span it occurs, the allowable deflection for your selected limit, a clear PASS or FAIL badge and the utilisation percentage (how much of your allowable deflection you’re actually using). If you added more than one load, a per-load breakdown table also appears below, showing each individual load’s own peak deflection and location, with a note reminding you that the combined maximum above is the real answer , not simply the sum of these individual peaks.
  • Below the numbers, the tool draws a combined Bending Moment Diagram (BMD) and Shear Force Diagram (SFD), superposed across all your loads together.

Step 4: Use reverse mode, if your beam fails

  • If your result shows FAIL, open “Reverse: required I for the deflection limit.”
  • Tap Find Required I. Since deflection scales directly with 1/I, the tool works out exactly what moment of inertia your section would need, using the same loads and span limit you already set up, to bring the deflection right down to the allowable limit.

Step 5: Export

  • Use Print / PDF for a clean printable copy, or Copy Result to paste the figures into your report.

Key Features

  • True multi-load superposition, up to 4 point loads and UDLs combined on the same beam, in any position
  • Numeric scan across the full span to find the actual combined maximum deflection and its exact location, not just an added-up estimate
  • Supports Simply Supported, Cantilever and Fixed-Fixed beams
  • IS 456 span/250 or span/360 serviceability check with a clear Pass/Fail badge and utilisation percentage
  • Per-load breakdown table showing each individual load’s own peak, alongside the true combined result
  • Reverse mode for the minimum required moment of inertia when your beam fails the deflection check
  • Combined BMD and SFD diagrams, superposed across every load you’ve added
  • Export as PDF or copy the result.

Formula / Logic Used

Simply Supported Beam, Point Load at position a

δ(x)=Pbx(L2−b2−x2)6LEIfor x≤a,b=L−a\delta(x) = \frac{Pbx(L^2 – b^2 – x^2)}{6LEI} \quad \text{for } x \leq a, \qquad b = L – a

Simply Supported Beam, Full-Span UDL

δ(x)=wx(L3−2Lx2+x3)24EI\delta(x) = \frac{wx(L^3 – 2Lx^2 + x^3)}{24EI}

Cantilever, Point Load at position a (measured from the fixed end)

δ(x)=Px2(3a−x)6EIfor x≤a\delta(x) = \frac{Px^2(3a – x)}{6EI} \quad \text{for } x \leq a

Cantilever, Full-Span UDL

δ(x)=wx2(6L2−4Lx+x2)24EI\delta(x) = \frac{wx^2(6L^2 – 4Lx + x^2)}{24EI}

Fixed-Fixed Beam, Centre Point Load

δ(x)=Pxr2(3L−4xr)48EI,xr=min⁡(x, L−x)\delta(x) = \frac{Px_r^2(3L – 4x_r)}{48EI}, \qquad x_r = \min(x,\ L-x)

Fixed-Fixed Beam, Full-Span UDL

δ(x)=wx2(L−x)224EI\delta(x) = \frac{wx^2(L-x)^2}{24EI}

Superposition and True Maximum

For any combination of loads, the tool adds every load’s deflection curve together point by point across the span, then scans across 2001 points to find where this combined curve actually peaks:

δtotal(x)=∑iδi(x),δmax=maxx∈[0,L]⁡δtotal(x)\delta_{total}(x) = \sum_i \delta_i(x), \quad \delta_{max} = \max_{x \in [0,L]} \delta_{total}(x)

Serviceability Check

Allowable=Ln,n=250 or 360Allowable = \frac{L}{n}, \qquad n = 250\ \text{or}\ 360

The beam passes if δmax≤Allowable\delta_{max} \leq Allowable

Reverse Mode: Required Moment of Inertia

Since deflection is inversely proportional to I, the required I scales directly off your current result:

Irequired=Icurrent×δmaxAllowableI_{required} = I_{current} \times \frac{\delta_{max}}{Allowable}

Who Should Use This Tool

Structural design engineers checking beam serviceability under real, combined loading, not just a single textbook load case. Also useful for diploma and B.Tech Civil Engineering students learning superposition, span/250 serviceability checks and BMD/SFD concepts for their design coursework.

Frequently Asked Questions (FAQs)

1. How do I calculate deflection when a beam has more than one load acting on it?

Calculate the deflection curve for each load separately using its own formula, then add all the curves together point by point along the span, since deflection is linear for small elastic deformations. This tool does this automatically for up to 4 loads and scans the combined curve to find exactly where the true maximum occurs.

2. Why can’t I just add up the maximum deflection of each individual load?

Each load’s own peak deflection usually happens at a different point along the span, so simply adding those individual peak numbers together overestimates the real combined maximum. This tool scans the actual superposed curve across the full span instead, so the maximum deflection shown is the real value at one real location, not an inflated estimate.

3. What is the difference between the span/250 and span/360 deflection limits?

Span/250 is the IS 456 limit for total deflection considering the full load history of the beam. Span/360 is a tighter limit typically applied to the deflection that happens after partitions or brittle finishes are already in place, since those are more sensitive to movement. You can check your beam against either limit here.

4. My beam fails the deflection check. What do I do next?

Open the reverse mode section and tap Find Required I. Since deflection is inversely proportional to the moment of inertia, the tool calculates exactly how much stiffer your section needs to be, using the same loads and limit, to bring deflection down to an acceptable value.

5. Does this tool account for long-term deflection in RCC beams?

No, this tool calculates elastic, short-term deflection using standard Euler-Bernoulli formulas. Long-term effects like creep and shrinkage in RCC members and reduced stiffness from cracking, need separate factors on top of this elastic result, which this tool’s disclaimer notes clearly.

Related Tools

Read More>>>

Scroll to Top