Beam Deflection Calculator (Fixed-Fixed)
Find the maximum midspan deflection of a fixed-fixed beam, both ends moment-restrained, under a central point load or a uniformly distributed load.
🏗️ What is Fixed-Fixed Beam Deflection?
Fixed-fixed beam deflection is the amount a beam bends when both ends are rigidly connected to their supports so that they resist bending moment as well as shear, unlike a simply supported beam whose ends can freely rotate. This moment restraint at both ends makes a fixed-fixed beam significantly stiffer than an equivalent simply supported beam for the same span, load, and cross-section.
Fixed-fixed conditions occur wherever a beam is cast monolithically with its supports or rigidly bolted and welded so no rotation can occur at the connection. Reinforced concrete beams poured integrally with stiff columns, steel beams with fully welded moment connections, and short-span bridge girders anchored into massive abutments all approximate fixed-fixed behavior. Precise fixity is rarely perfect in practice, but the fixed-fixed idealization gives a useful lower bound on deflection and a different bending moment distribution than the simply supported case.
A common misconception is that fixing both ends only affects the support reactions and has no real impact on deflection. In fact, moment restraint at the supports reshapes the entire bending moment diagram, introducing negative (hogging) moments at the ends and reducing the positive (sagging) moment at midspan. This redistribution is exactly why a fixed-fixed beam under a central point load deflects only one-fourth as much as the same beam simply supported.
This calculator computes the maximum midspan deflection of a fixed-fixed beam under either a central point load or a uniformly distributed load, using the standard Euler-Bernoulli beam formulas, and shows the symmetric deflected shape along the full span as an interactive chart.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Steel Fixed-Fixed Beam Under a Central Point Load
A 5 m steel beam (E = 200 GPa, I = 40,000,000 mm⁴), fixed at both ends, carrying a 40 kN point load at midspan
Example 2 — Same Steel Beam Under a Uniformly Distributed Load
The same 5 m fixed-fixed steel beam carrying a 25 kN/m distributed load across the full span
Example 3 — Timber Fixed-Fixed Beam Under a Central Point Load
A 4 m timber beam (E = 11 GPa, I = 60,000,000 mm⁴), fixed at both ends, carrying a 15 kN point load at midspan
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for fixed-fixed beam deflection under a central point load?
For a beam fixed (moment-restrained) at both ends with a point load P at midspan, maximum deflection is delta_max = PL^3/(192EI), occurring at the center of the span (x = L/2). P is the load, L the span length, E the modulus of elasticity, and I the second moment of area.
What is the formula for fixed-fixed beam deflection under a uniformly distributed load?
For a fixed-fixed beam under a uniformly distributed load w (force per unit length), maximum deflection is delta_max = wL^4/(384EI), also occurring at midspan. This is one-fifth the deflection of an equivalent simply supported beam under the same distributed load.
Why does a fixed-fixed beam deflect less than a simply supported beam?
Fixing both ends against rotation resists the bending moment that would otherwise develop freely at the supports, redistributing moment into the span and reducing curvature. Under a central point load, a fixed-fixed beam deflects 4 times less than an equivalent simply supported beam (denominator 192EI versus 48EI).
What units should I use for E and I in this calculator?
Enter the modulus of elasticity E in gigapascals (GPa) and the second moment of area I in millimeters to the fourth power (mm^4). These are standard units for structural steel and timber sections; the calculator converts them to SI base units (Pa and m^4) internally before computing the midspan deflection in millimeters.
What is a typical deflection limit for a fixed-fixed beam?
A common serviceability limit is span/360 (L/360), the same limit typically used for simply supported spans, since both ends of a fixed-fixed beam are fully restrained. For a 5 m span, L/360 is about 13.9 mm.
Where does maximum deflection occur on a fixed-fixed beam?
For both a central point load and a uniformly distributed load on a fixed-fixed beam, maximum deflection occurs at the exact midspan (x = L/2), by symmetry of the loading and support conditions.
How does span length affect fixed-fixed beam deflection?
Deflection is extremely sensitive to span length. Under a point load, deflection scales with L^3 (doubling the span increases deflection 8x). Under a uniformly distributed load, deflection scales with L^4 (doubling the span increases deflection 16x), assuming all other properties stay fixed.
What is the difference between a fixed-fixed beam and a simply supported beam?
A simply supported beam rests on a pin and roller support, both allowing free rotation and no moment restraint. A fixed-fixed beam is rigidly connected at both ends so they resist both shear and bending moment, cutting deflection and shifting peak bending moment to the supports rather than midspan.
Does increasing the modulus of elasticity always reduce fixed-fixed beam deflection?
Yes, deflection is inversely proportional to E, so a stiffer material (higher E) under the same load and geometry deflects less. Steel (E is about 200 GPa) deflects roughly 18 times less than an equivalent timber beam (E is about 11 GPa) for identical span and section.
What assumptions does this fixed-fixed deflection formula make?
This calculator uses classical Euler-Bernoulli beam theory, which assumes linear-elastic material behavior, small deflections relative to the span, a prismatic (constant cross-section) beam, and rigid supports at both ends that prevent both displacement and rotation.
Can this calculator be used for simply supported or cantilever beams?
No, this calculator specifically applies fixed-fixed beam formulas, moment-restrained at both ends. Simply supported and cantilever beams use different boundary conditions and different deflection formulas, and would give incorrect results here.