An MRT rail may look like one continuous line of steel, but it begins as separate manufactured lengths that must be joined into a geometry smooth enough for wheels to treat the weld almost as if it were never there.
The mathematics of welded rail sits between metallurgy, geometry, thermal expansion, residual stress, track support and repeated wheel loading.
Traditional jointed rail contains physical gaps and mechanical joints.
Modern MRT track instead uses long welded rail strings so wheels encounter a much more continuous running surface.
LTA’s current 2025 civil design criteria state that main-line and depot rails outside turnout limits are to be welded into continuous strings using flash-butt welding, with other approved rail-welding methods used for particular local applications. The same standard requires excessive weld material to be removed and ground so the rail head matches the adjoining running surface.
This creates a useful mathematical question:
How do we turn two finite rails into one structural and geometric object without creating a new defect at the place where they meet?
This pillar owns the welded-rail continuity problem: weld geometry, heat-affected regions, longitudinal rail stress, thermal expansion, neutral temperature, buckling resistance, weld inspection and the interaction between continuously welded rail and its supports. Rail fastener stiffness remains with Rail Fasteners, Sleepers and Track Support. Running-surface restoration remains with Rail Grinding and Wheel Profiling.
The RFE — Why Weld Rails Into Long Continuous Strings?
The weak objective is:
remove the gaps
The stronger Reason for Existence is:
create a sufficiently continuous rail geometry and structural load path that repeated train passages remain smooth, stable and maintainable while thermal expansion and longitudinal force are carried safely through the rail-support system.
Prompt 1 — What Must Match Across a Rail Weld?
A successful weld must make several fields continuous enough:
- rail-head elevation;
- lateral alignment;
- running-surface curvature;
- cross-sectional profile;
- material continuity;
- load transfer.
Let the left rail profile be yL(x) and right profile be yR(x).
At the weld location x=0, a simple geometric objective is:
yL(0) ≈ yR(0)
and ideally the slopes also match:
dyL/dx ≈ dyR/dx
A small height mismatch Δh can behave like a local roughness input to the wheel.
At train speed v, a short geometric irregularity of wavelength λ produces a forcing frequency scale:
f ≈ v/λ
This links weld finishing directly to wheel–rail vibration.
Prompt 2 — Why Does Welding Create Heat and Residual Stress?
Welding raises a local region of steel to very high temperature.
The hot region expands.
Then it cools and contracts while being constrained by surrounding rail.
Thermal strain is:
εthermal = αΔT
If expansion or contraction is constrained, stress develops.
σ ≈ E α ΔT
This is a simplified fully restrained relation, not a complete weld-stress model.
Real welds contain changing temperature fields, plasticity, phase transformation and local material-property variation.
The key idea is that the weld is not only a shape problem.
It is also a thermal-history problem.
Prompt 3 — What Does Continuously Welded Rail Do When the Weather Changes?
A free rail of length L would change length by:
ΔL = αLΔT
But continuously welded rail is restrained longitudinally by fasteners, sleepers, slab track, ballast and structures.
So much of the thermal expansion becomes longitudinal force instead of visible gap movement.
A simplified axial force is:
Nthermal ≈ EAαΔT
where A is rail cross-sectional area.
When rail temperature rises above its stress-free reference state, compressive force increases.
When rail temperature falls, tensile force increases.
Continuously welded rail removes most visible expansion gaps by moving the temperature problem inside the steel as stress.
Prompt 4 — What Is the Rail Neutral Temperature?
Engineers use a reference temperature at which a continuous rail is approximately free of longitudinal thermal stress under defined conditions.
Call it TN.
Then thermal stress relative to that reference is approximately:
σthermal ≈ Eα(Trail−TN)
If the effective neutral temperature drifts because rail is cut, repaired, destressed or displaced, the same weather produces a different longitudinal stress state.
This is why rail welding, rail replacement and track-support condition interact.
Actual Singapore stress-management procedures and target temperatures are deliberately not reproduced here.
Prompt 5 — How Can Compression Become a Buckling Problem?
A straight rail under increasing compressive force tends to remain straight only while lateral resistance and geometric stability are sufficient.
The classic Euler buckling scale for an ideal isolated member is:
Pcr = π²EI/(KL)²
A railway track is not a simple free column, so this equation is only an analogy.
Real track lateral stability depends on:
- rail longitudinal force;
- initial alignment;
- fastener and sleeper restraint;
- ballast or slab-track resistance;
- curve geometry;
- temperature history;
- maintenance disturbance.
The useful stability condition is conceptual:
compressive rail force < track lateral resistance capacity under current geometry and condition
The track-support pillar owns that restraint layer.
Prompt 6 — Why Must the Weld Be Ground After Joining?
Welding can leave excess material above the desired rail profile.
LTA’s current civil criteria explicitly require excessive weld material to be removed and the weld ground to match the rail head on either side.
Let target rail profile be y*(x).
e(x)=yweld(x)−y*(x)
A finishing process tries to reduce:
RMSweld = √[(1/L)∫e(x)²dx]
but local peak error can matter even when RMS is small.
One sharp mismatch can produce stronger impact than a smoother error spread over a longer distance.
This is why profile, slope and curvature all matter.
Prompt 7 — How Are Rail Welds Inspected?
A weld can look acceptable at the surface while containing internal defects.
Inspection therefore combines geometry and material examination.
Public railway engineering commonly uses methods such as ultrasonic testing to detect internal discontinuities.
A simplified ultrasonic travel relation is:
d = vt/2
where v is wave speed in the material and t is echo round-trip time.
The signal amplitude, angle and reflection pattern help trained inspectors interpret whether a discontinuity may exist.
Actual acceptance criteria remain governed by engineering standards and authorised inspection procedures.
Prompt 8 — How Does the Weld Enter the Maintenance Cycle?
One weld may experience millions of wheel passages.
Let stress range from one wheel passage be Δσ.
Repeated stress can accumulate fatigue damage.
D = Σ ni/Ni
where ni is experienced cycles in one stress range and Ni is the reference fatigue life for that range.
Condition evidence may include:
- weld-profile change;
- ultrasonic indications;
- surface cracking;
- impact or vibration signature;
- temperature-stress history;
- local track-geometry movement.
The weld therefore belongs inside Predictive Maintenance and Track Inspection, but this pillar owns the reason the weld is a special local state inside an otherwise continuous rail.
A Fictional Continuously Welded Rail Example
Consider a fictional rail with:
E = 200 GPa α = 12×10⁻⁶ /°C A = 7.5×10⁻³ m² TN = 30°C
Rail temperature rises to 50°C.
ΔT = 20°C σthermal ≈ 200×10⁹ × 12×10⁻⁶ × 20 ≈ 48 MPa compression
Approximate compressive force:
N = σA = 48×10⁶ × 7.5×10⁻³ ≈ 360 kN
Now imagine a local weld profile mismatch of only 0.15 mm spread over 50 mm.
A train at 20 m/s encounters that spatial scale at:
f≈20/0.05=400 Hz
One article now contains two very different mathematical states at the same location:
slow thermal stress over hours + fast wheel-impact excitation over milliseconds
That is why rail welding belongs simultaneously to structural mechanics, vibration and maintenance.
Deletion Tests
- Remove weld geometry: any joined profile is assumed smooth enough for a wheel.
- Remove thermal expansion: continuous rail has no longitudinal temperature force.
- Remove neutral temperature: stress state has no reference condition.
- Remove track restraint: welded rail expands freely like an isolated bar.
- Remove residual stress: welding heat leaves no mechanical history.
- Remove ultrasonic inspection: internal weld quality is inferred only from appearance.
- Remove fatigue: millions of repeated wheel passages do not accumulate damage.
- Remove World Return: post-weld geometry and long-term behaviour are never remeasured.
Rail-Welding Paradoxes
- Removing rail joints creates a smoother railway but increases the importance of thermal longitudinal stress.
- A weld can be metallurgically sound but geometrically noisy.
- A weld can look smooth but contain internal discontinuity.
- The most continuous-looking rail still contains a history of every place where steel was joined.
- A few tenths of a millimetre at one weld can matter because trains encounter it millions of times.
The Rail-Welding Audit
- What rail lengths are being joined?
- What welding process is appropriate for the location?
- How are rail-head height, slope and profile aligned?
- What thermal history does the weld create?
- What residual stress remains after cooling?
- What neutral-temperature state applies to the continuous rail?
- What longitudinal restraint is supplied by fasteners and track support?
- What temperature range changes compression and tension?
- What lateral stability margin exists?
- How is excess weld material removed?
- How is internal weld quality inspected?
- How are welds identified and tracked through maintenance records?
- What repeated wheel-load fatigue does the location accumulate?
- What measured geometry or ultrasonic return would force inspection or repair?
World Return — When the First Train Crosses the New Weld
prepare rail ends → weld → cool → grind to target profile → inspect geometry and material → release to service → trains cross → inspect again → compare long-term weld behaviour
If vibration remains higher at the weld than adjacent rail, finishing geometry may need review.
If ultrasonic condition changes over repeated inspections, the weld becomes a maintenance decision.
If rail alignment changes during hot weather, the weld itself may be healthy while the broader continuous-rail restraint state needs attention.
Continuously welded rail works when two manufactured pieces become one railway object strongly enough for structure, smoothly enough for the wheel, and predictably enough for temperature and maintenance.
Key Equations
yL(0)≈yR(0) Weld height continuity dyL/dx≈dyR/dx Weld slope continuity f≈v/λ Spatial weld geometry to forcing frequency εthermal=αΔT Thermal strain σthermal≈EαΔT Fully restrained thermal stress scale ΔL=αLΔT Free thermal expansion Nthermal≈EAαΔT Continuous-rail thermal force scale σ≈Eα(Trail−TN) Neutral-temperature stress relation Pcr=π²EI/(KL)² Idealised buckling analogy RMSweld=√[(1/L)∫e²dx] Weld-profile error d=vt/2 Ultrasonic echo depth relation D=Σni/Ni Cumulative fatigue damage
Reader-safety note: This article uses public railway-engineering principles only. It does not reproduce Singapore MRT rail neutral-temperature targets, destressing procedures, welding parameters, acceptance thresholds, inspection limits, exact weld locations or emergency repair methods.