What is Clapeyron's three-moment equation?

The three-moment equation is a relation between the bending moments at three consecutive supports of an indeterminate continuous beam. It was published in 1857 by the French engineer Clapeyron. It is derived from the geometric continuity condition (continuity of the rotation angle over the support) between two adjacent spans, and gives the basic equation for solving the unknown interior support moments. For a symmetric three-span continuous beam with equal spans and a uniform distributed load, the interior support moment has the simple closed-form value M = -wL squared over 10.

Why does a negative bending moment appear over the interior supports?

In a continuous beam, the supports hold the beam against its tendency to bend, producing a hogging moment with tension on the top fibre and compression on the bottom. When both adjacent spans try to sag under load, the beam over the support tries to take a concave-downwards shape, but continuity with the neighbouring spans constrains that rotation, so the support pushes up while a negative bending moment develops with the top of the beam in tension. This reduces the positive moment at the mid-span. The fact that the maximum moment is smaller than for three simple beams placed side by side is the main advantage of a continuous beam.

What is the difference between three simple beams and one continuous beam?

Three simply supported beams placed side by side have a maximum moment of wL squared / 8 = 0.125 wL squared in each span. By contrast, for an equal-span three-span continuous beam, the maximum positive moment in the outer spans is about 0.08 wL squared, and the negative moment at the interior support is wL squared / 10 = 0.10 wL squared. The maximum absolute value drops from 0.125 wL squared to 0.10 wL squared, a 20 percent reduction. The maximum deflection is also roughly halved, from about 0.0130 wL to the fourth / EI for a simple beam to about 0.0069 wL to the fourth / EI for the continuous beam. This is why continuous beams are preferred for bridges and floor slabs.

Why do the maximum positive moment and deflection occur in the outer spans?

In a symmetric equal-span continuous beam, the middle span is restrained at both ends by interior supports, so its rotations are held on both sides and it cannot sag as freely. The outer spans, by contrast, have a free rotation at the simple support at the outside and a restrained rotation only at the interior support, so they behave more like a cantilever-supported beam and develop larger deflections and positive moments. The maximum positive moment is about 0.08 wL squared in the outer spans against only 0.025 wL squared at mid-span of the middle span, almost three times larger. In bridge design the outer spans are often shortened to about 0.8 L to balance the moments between spans.

Continuous Three-Span Beam Simulator — Free Online Calculator

Parameters

Span length L

Distributed load w

kN/m

Flexural rigidity EI

kN·m²

Evaluation position x / L

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All three spans share the same length L, and the distributed load w acts over the entire beam. The "evaluation position x/L" marks a point within the outer span A–B.

Results

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Interior support moment M_B = M_C = −wL²/10

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Maximum positive moment M_+ = 0.08wL²

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Outer-span maximum deflection δ_max

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Interior support reaction R_B = 1.1wL

Beam Model, SFD and BMD

Top = beam model (four supports A, B, C, D and the distributed load w) / Middle = shear force diagram (SFD) / Bottom = bending moment diagram (BMD), peaking negative at the interior supports and positive in the outer spans.

Theory & Key Formulas

A three-span continuous beam has three spans (A–B, B–C, C–D) and four supports A, B, C, D, and is an indeterminate structure. Taking the interior support moments M_B and M_C as unknowns, the system is solved analytically by Clapeyron's three-moment equation.

Three-moment equation for two adjacent spans of equal length L, equal EI, and a uniform load w:

$$M_{i-1} L + 2 M_i (2L) + M_{i+1} L = -\frac{w L^3}{4} - \frac{w L^3}{4}$$

At the end supports M_A = M_D = 0, and by symmetry M_B = M_C, which gives the interior support moment:

$$M_B = M_C = -\frac{w L^2}{10}$$

The reactions then follow directly from equilibrium:

$$R_A = R_D = 0.4\,w L, \qquad R_B = R_C = 1.1\,w L$$

In the outer span A–B, the maximum positive moment occurs at x ≈ 0.4 L, with the maximum deflection near mid-span:

$$M_{+,\text{max}} = 0.08\,w L^2, \qquad \delta_{\max} \approx 0.0069 \frac{w L^4}{E I}$$

Compared with the 0.125 wL² maximum moment of three side-by-side simply supported beams, the continuous configuration reduces the maximum moment by 20 percent. This is the main reason continuous beams are favoured for bridges and floor slabs.

What is the Continuous Three-Span Beam Simulator

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What is the difference between a "continuous beam" and a simple beam? Just more supports?

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Just more supports, but mechanically that makes a huge difference. Place three simply supported beams side by side and you have three separate beams. A continuous beam, by contrast, is one beam that passes over the interior supports. The rotation of the beam over each interior support is continuous, and that is what makes it "statically indeterminate". Equilibrium alone does not give you the reactions and moments — you also need compatibility of deformations. With the default values (L=5 m, w=20 kN/m) the simulator shows an interior support moment of "-50 kN·m". The negative sign means the top of the beam is in tension; the beam tries to hog up over the support.

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I have heard of the "three-moment equation". Why "three"?

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Because it relates the bending moments at three consecutive supports that bracket two adjacent spans. It is a classical result Clapeyron published in 1857, the standard hand-calculation tool for continuous beams. For an equal-span, uniformly loaded, symmetric three-span continuous beam, the interior support moment comes out to a neat $M_B = M_C = -wL^2/10$. Remembering just that, you can quickly size a bridge or a residential floor beam. The "Interior support moment" card in the simulator shows the actual value.

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Compared with three simple beams side by side, what is the gain from making it continuous?

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A smaller maximum moment, that is the whole story. With simple beams the maximum moment is $wL^2/8 = 0.125\,wL^2$ at each mid-span. Making it continuous, the maximum positive moment (in the outer spans) is $0.08\,wL^2$, and the maximum negative moment (at the interior supports) is $0.10\,wL^2$. The largest absolute value is 0.10 wL² instead of 0.125 wL² — a 20 percent reduction. So the same section can carry 20 percent more load, or the same load can be carried with a section 20 percent smaller. The deflection roughly halves. That is why continuous beams are preferred for bridges and floor slabs.

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Looking at the BMD at the bottom, it jumps to a big negative value at the interior supports. Positive at mid-span, negative at the supports — is this the signature of a continuous beam?

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Exactly, it is the "fingerprint" of a continuous beam. Mid-span, the bottom is in tension and the moment is positive; at the interior support, the top is in tension and the moment is negative. In a reinforced concrete continuous beam, the main reinforcement is placed at the bottom near mid-span and at the top over the interior supports — the bar layout literally follows the shape of the BMD. Look at the SFD too: there is a large jump over the interior support, which is the signature of the 1.1 wL reaction. That is almost three times the 0.4 wL at the end supports — the load concentrates at the interior support.

Frequently Asked Questions

The three-moment equation also applies to continuous beams with unequal spans, but because the coefficients depend on the span-length ratios, the closed-form solution is not as simple as in the symmetric case. The general form is M_{i-1}·L_i + 2·M_i·(L_i + L_{i+1}) + M_{i+1}·L_{i+1} = -w_i·L_i³/4 - w_{i+1}·L_{i+1}³/4. In practice, bridge designers often choose the outer spans to be about 0.8 times the middle span to balance the moments between spans; the optimization of such unequal spans is typically done with continuous-beam analysis programs or general structural-analysis software such as SAP2000 or Midas.

Only the right-hand side of the three-moment equation (the load term, equivalent to the elastic load on the conjugate beam) needs to change to handle point loads, partial distributed loads, applied moments or thermal loads. For example, a point load P at mid-span gives -PL²/8 on the right-hand side. In practice these elementary cases are combined by superposition, or moving-load envelopes are obtained from influence lines. Bridge design routinely requires influence-line analysis for heavy vehicles.

Support settlement can be added to the three-moment equation as an extra term on the right-hand side: a relative settlement Δ between adjacent supports contributes a term 6EI·Δ/L. Because a continuous beam is indeterminate, settlements directly change the moment distribution. For example, if the interior support settles downward, the magnitude of the interior support moment decreases and the positive moment at mid-span increases. Where differential settlement of the foundation is expected, the design must either account for this effect or release it by using movable (roller) supports.

Three spans are the smallest configuration that contains all three basic elements of a continuous beam — "outer span + interior span + outer span". A two-span beam has no interior spans, only outer spans. For four or more spans, the spans near the centre are sandwiched between other interior spans, so the boundary effect diminishes and the solution approaches that of an infinite continuous beam (M_interior ≈ -wL²/12). The three-span continuous beam, with its memorable solution M = -wL²/10, is the canonical textbook example. In bridge engineering, the three-span continuous bridge (outer span + main span + outer span) is one of the most common configurations.

Real-World Applications

Continuous-girder bridges: The three-span continuous beam is one of the most basic configurations in bridge design. Bridges crossing a river or valley are often built with abutments on each bank and two piers in between, forming a three-span continuous bridge. Compared with three simple spans side by side, the maximum moment is smaller, so the girder depth can be reduced and the riding quality is smoother. Elevated viaducts of high-speed railways, continuous box-girder bridges of expressways, and the approach spans of long-span bridges are all around us in this form.

Floor slabs and continuous beams in buildings: In reinforced concrete buildings, floor slabs and beams that span multiple columns or walls behave naturally as continuous beams. The basic reinforcement layout is bottom main bars under the mid-span for positive moment and top main bars over the interior supports (column positions) for negative moment, mirroring the shape of the BMD. Designers use methods such as Ferguson's approach or applications of the three-moment equation to obtain the moment distribution in each span.

Mechanical engineering and pipe support: Long piping supported by multiple stands, or long shafts in rotating machinery, are also analyzed as continuous beams. The concentration of negative moment and shear at the interior supports directly drives the design of supports and bearings. In plant piping design, the support spacing is routinely optimized using the three-moment equation to keep maximum deflections and stresses within allowable values.

Underground structures and continuous footings: Strip foundations (continuous footings) can also be modelled as continuous beams loaded by the ground reaction as a distributed load. From the balance between subgrade reaction beneath the footing and superstructure loads, the moment and reaction at each support are computed by the three-moment equation, and the reinforcement and thickness are determined. The more rigorous "beam on elastic foundation" model, which accounts for the soil modulus of subgrade reaction, requires numerical analysis, but the continuous-beam approximation is widely used in preliminary design.

Common Misconceptions and Cautions

The most common misconception is to assume that the maximum moment of a continuous beam occurs at mid-span. For an equal-span, uniformly loaded three-span continuous beam, the largest absolute value is the negative moment over the interior supports ($0.10\,wL^2$), not the positive moment at mid-span (about $0.08\,wL^2$ in the outer spans, only $0.025\,wL^2$ in the middle span). Do not carry over the simple-beam intuition that "the positive moment is largest at mid-span"; always draw the whole BMD and check the negative moment over the supports. The negative value on the simulator's "Interior support moment" card is often the governing design value.

The next most common error is to overlook the fact that the interior support carries a reaction of 1.1 wL. The interior support of a continuous beam carries about 2.75 times the 0.4 wL of the end supports. This is an important design load for the piers or columns and foundations under the interior supports, and is more than twice the wL/2 = 0.5 wL of three simple beams side by side. Missing this concentration in geotechnical investigation, foundation design or column sizing for the interior support can lead to inadequate bearing capacity or local deformation and failure. Vary w and L in the simulator and check the magnitude of the interior reaction R_B.

Finally, recognise that the closed-form solution of the continuous beam assumes idealizations: equal spans, uniform stiffness and a uniform distributed load. Real bridges and buildings are far more complex: spans differ, sections vary along the beam, live (moving) loads act on some spans but not others, supports settle, and the structure is exposed to temperature and shrinkage effects. For instance, the loading pattern "live load only on the middle span" relieves the positive moment in the outer spans while increasing it in the middle span above the theoretical value. In real design, all plausible load patterns are evaluated and each member is sized for the worst-case envelope. Use this simulator as an introductory tool to build intuition for the symmetric uniform-load reference case.

Continuous Three-Span Beam Simulator — Three-Moment Equation

What is the Continuous Three-Span Beam Simulator

Frequently Asked Questions

Real-World Applications

Common Misconceptions and Cautions

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