How MRT Viaducts and Elevated Railway Structures Work Using Mathematics: How Concrete Carries Trains Above a Moving City

A railway viaduct looks static.

The loads are not.

Each train enters a span.

Wheel loads move along the deck.

The deck bends.

Bearings and piers transmit force.

Foundations carry those reactions into the ground.

wheel load
→ rail and track support
→ viaduct deck
→ bearings
→ pier / column
→ foundation
→ ground

The structure also expands and contracts with temperature, experiences wind, resists braking and traction forces, and vibrates under repeated train passages.

LTA’s December 2025 civil design criteria contain a dedicated chapter for above-ground structures including above-ground trainways. The Jurong Region Line is a current large-scale example: LTA describes it as a 24-kilometre elevated MRT line with 24 stations. The new NS3A station project on the existing North-South Line also includes a new viaduct that will temporarily carry diverted train services while the existing infrastructure is modified.

This article owns the elevated structural load path. Track support above the deck belongs to the fastener/sleeper pillar. Train ride response belongs to Bogies, Suspension and Ride Comfort. Network expansion belongs to its own planning pillar.

The RFE — Why Does the Viaduct Exist?

The weak answer is:

hold the tracks above the road

The stronger definition is:

carry a repeatedly moving railway load through a durable above-ground structure while preserving the track geometry, clearances, vibration performance and city space that allow trains and everything below them to coexist.

Prompt 1 — How Does a Train Become a Moving Structural Load?

Suppose axle load P moves along beam coordinate x at train speed v.

xload(t)=x0+vt

The bending moment at a section changes as the load moves.

For a simple simply supported span of length L with one point load P at distance a from the left support and b=L−a from the right:

RA = Pb/L
RB = Pa/L

The largest bending effect occurs when the train’s axle pattern occupies particular positions on the span.

A real train is a sequence of axle loads, not one point load:

Load train = Σi Pi δ[x−xi(t)]

The viaduct design searches many load positions to find critical force effects.

Prompt 2 — How Does the Deck Bend?

Beam curvature is related to bending moment:

EI d²y/dx² = M(x)

E is elastic modulus and I is second moment of area.

Greater EI means less curvature under the same bending moment.

For a simple centre point load on a simply supported beam:

δmax = PL³/(48EI)

Deflection grows strongly with span length L.

Doubling span length increases this simplified deflection scale eightfold if everything else remains unchanged.

This is why long spans need deeper, stiffer or more structurally efficient forms.

Prompt 3 — Why Use Prestressed Concrete?

Concrete is strong in compression but weaker in tension.

Prestressing introduces deliberate compressive stress before service loads act.

At a section:

σ = P/A ± Pe y/I ± M y/I

where P is prestress force, e eccentricity, A area, y distance from centroid and M service bending moment.

The prestress is arranged so that service-load tensile stress and cracking are controlled.

LTA’s current civil criteria explicitly include prestressed-concrete requirements in the above-ground structures chapter.

Prompt 4 — Why Do Bearings and Expansion Joints Matter?

A viaduct changes length with temperature.

ΔL = αLΔT

For a fictional 100 m structure, thermal expansion coefficient α=10×10⁻⁶/°C and temperature change 30°C:

ΔL = 10×10⁻⁶×100×30
   = 0.03 m
   = 30 mm

If movement is completely restrained, large thermal forces can develop.

Bearings allow intended translations and rotations while still transmitting loads.

The detailed arrangement depends on structure type and is not reproduced here.

Prompt 5 — How Does the Viaduct Vibrate Under Trains?

A simple mode follows:

m q¨ + c q˙ + k q = F(t)

The natural frequency is:

fn=(1/2π)√(k/m)

Train axle spacing creates repeated forcing frequencies depending on speed.

If axle spacing s is encountered at speed v:

fpass ≈ v/s

Design avoids unacceptable resonance and vibration responses.

LTA’s current above-ground criteria explicitly include vibration as a design requirement.

The viaduct is not static concrete. It is a dynamic filter between a moving axle field and the ground below.

Prompt 6 — How Do Piers and Foundations Carry the Load?

Deck reactions enter bearings and piers.

A pier can carry:

• vertical axial force;
• horizontal braking or traction effects;
• wind;
• bending moment from eccentric loading;
• construction-stage forces.

For axial force N and bending moment M:

σ = N/A ± My/I

Foundations then transfer the reactions to soil or rock.

Settlement matters because track above the structure depends on geometry.

A few millimetres of differential structural movement can matter more to the train than a larger uniform settlement that preserves alignment.

Prompt 7 — How Is an Elevated Railway Built Without Stopping the City?

Construction is a staging problem.

foundation
→ pier
→ pier head
→ precast / cast deck segment
→ span completion
→ track and systems
→ testing

Segmental construction reduces the need for large continuous falsework beneath the whole alignment.

The Jurong Region Line has publicly used viaduct launching as part of construction.

The new NS3A project gives an even more coupled example: LTA says a new viaduct will support staged diversion of train services so modifications can be made to existing viaduct infrastructure while the North-South Line remains operational.

That makes construction itself a temporary railway state.

existing viaduct
→ temporary diverted geometry
→ modify old infrastructure
→ final permanent geometry

Prompt 8 — How Does a Viaduct Age?

Repeated train passages create fatigue cycles.

Concrete and steel experience environmental exposure.

Bearings and joints move repeatedly.

A simplified fatigue-damage sum is:

D = Σ ni/Ni

where n_i is experienced cycles at stress range i and N_i is reference cycles to failure for that range.

Structural inspection watches for:

• cracking;
• spalling;
• bearing deterioration;
• joint condition;
• settlement or alignment change;
• drainage and waterproofing issues;
• unexpected vibration.

The permanent structure enters the same predictive-maintenance loop as rolling stock and track.

A Fictional Viaduct Example

Consider a fictional simply supported 30 m span.

A simplified central train load P=1,000 kN is used only for teaching.

RA=RB=P/2=500 kN

Maximum simple bending moment:

Mmax = PL/4
     = 1,000×30/4
     = 7,500 kN·m

Now suppose temperature rises 25°C and effective expansion length is 90 m with α=10×10⁻⁶/°C.

ΔL=10×10⁻⁶×90×25
  =22.5 mm

The structure must carry the moving train load while also permitting or resisting this thermal movement according to its bearing arrangement.

If measured midspan dynamic displacement is 15% higher than the calibrated prediction under comparable train loading, the difference becomes World Return evidence for structural review rather than automatic proof of damage.

Deletion Tests

• Remove moving loads: train position on the span no longer matters.
• Remove bending stiffness: every deck has identical deflection.
• Remove thermal movement: concrete never expands in Singapore heat.
• Remove bearings: deck movement and force transfer need no mechanism.
• Remove vibration: moving axle repetition cannot excite the structure.
• Remove foundations: reactions disappear before reaching the ground.
• Remove construction staging: the viaduct appears fully formed without temporary states.
• Remove inspection: long-life structures are assumed unchanged forever.

Viaduct Paradoxes

• A structure that looks still is continuously moving elastically.
• A longer span can remove piers from the street while making structural stiffness harder.
• Allowing controlled movement through bearings can reduce internal force.
• One new viaduct can temporarily carry trains so an old viaduct can be safely altered.
• The rail may require millimetre geometry precision on top of a structure hundreds of metres long.

The Elevated-Structure Audit

1. What train and track loads act?
2. Where are the critical moving-load positions?
3. What shear and bending moments result?
4. What deck stiffness controls deflection?
5. How is prestress used?
6. What thermal movement occurs?
7. Which bearings carry or release which motions?
8. What natural frequencies and dynamic amplification matter?
9. What pier and foundation reactions result?
10. What construction states govern before the structure is complete?
11. What fatigue and environmental deterioration accumulate?
12. What inspection evidence would show the model is drifting from reality?

World Return — The Train Becomes the Proof Load

design moving-load response
→ build structure
→ install track
→ run trains
→ measure deflection, vibration and geometry
→ compare
→ inspect / maintain / recalibrate

If measured structural movement remains inside prediction, the model gains support.

If one span develops increasing vibration while comparable spans do not, its local structure, bearings or track interface deserve attention.

An MRT viaduct works when a moving railway can repeatedly bend the structure without permanently bending the geometry of the railway.

Key Equations

xload=x0+vt
Moving load position

RA=Pb/L, RB=Pa/L
Simple beam support reactions

EIy''=M(x)
Beam curvature

δmax=PL³/(48EI)
Simple centre-load deflection

σ=P/A±Pey/I±My/I
Prestressed-section stress form

ΔL=αLΔT
Thermal expansion

m q¨+c q˙+kq=F(t)
Structural vibration mode

fn=(1/2π)√(k/m)
Natural frequency

fpass≈v/s
Axle-passage forcing frequency

σ=N/A±My/I
Pier combined stress form

D=Σni/Ni
Cumulative fatigue damage

Reader-safety note: This article uses public structural-engineering concepts. It does not reproduce detailed Singapore MRT structural drawings, bearing layouts, exact load combinations, protection-zone tolerances, inspection thresholds or vulnerable locations. Fictional examples are not design values.

Sources and Further Reading

• LTA — 2025 Civil Design Criteria, Chapter 9 Above-Ground Structures
• LTA — Jurong Region Line elevated railway construction
• LTA — NS3A station and staged new viaduct construction
• LTA — RTS Link segmental viaduct construction example
• eduKateSG — Bogies, Suspension and Ride Comfort
• eduKateSG — How MRT Works | It’s Mathematics