Study notes

Qwestrum Engineering360 · Civil Engineering · Railway Engineering

Points and Crossings

Relate the crossing number N to the crossing angle (1 in N), compute the switch and lead geometry from it, and remember a higher N gives a flatter, higher-speed turnout.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

Turnout number N = cot α — higher N, gentler divergence
Components: switch, stock rail, tongue, frog, check rail
Design speed through turnout limited by N and cant

Topic details

Introduction

Points and crossings allow trains to move from one track to another. The turnout comprises a set of switches (movable tongue rails against fixed stock rails) that guide the wheels, and a crossing (frog) where one rail crosses another, with check rails guiding the wheel through the gap.

Scope in B.Tech and GATE syllabus

The crossing angle is expressed as 1 in N, where N is the crossing number; a larger N means a smaller angle, a longer, gentler turnout and a higher permissible speed on the diverging track. Indian Railways uses standard turnouts such as 1 in 8.5, 1 in 12 and 1 in 16.

Why this topic matters in practice

The key geometric quantities — switch angle, lead length and heel divergence — follow from N and the gauge, and setting them out correctly is essential for smooth wheel passage without derailment at the crossing nose.

Key relations & formulas

Switch angle \alpha: tan \alpha = \frac{1}{(2 N)}

(N = turnout number)

Lead length L = (\frac{G}{2}) cot(\frac{\alpha}{2}) + stock rail length

(approximate layout)

Formulas (Indian textbook notation)

$Crossing angle: 1 in N defines divergence$

Notation and sign conventions

Relation 1 —

Switch angle \alpha: tan \alpha = 1 /

Switch angle \alpha: tan \alpha = \frac{1}{(2 N)}

(N = turnout number)

Write this relation with symbols exactly as in Railway Engineering — Satish Chandra & MM Agarwal before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

Lead length L =

Lead length L = (\frac{G}{2}) cot(\frac{\alpha}{2}) + stock rail length

(approximate layout)

Relation 3 —

Crossing angle: 1 in N defines divergence

Formulas (Indian textbook notation)

$Crossing angle: 1 in N defines divergence$

Fundamentals and definitions

The crossing (frog) is where the running edges of two rails intersect at the crossing angle; the crossing number N is the cotangent of that angle, so a 1-in-12 crossing has a flatter angle than a 1-in-8.5. The number is measured by the right-angle method (from the theoretical nose) in Indian practice.

Governing relations in practice

The switch assembly uses tapered tongue rails that fit against the stock rails; the switch angle and the throw of the switch determine how sharply the wheels are diverted. A smaller switch angle gives a smoother, higher-speed entry.

Design and analysis considerations

The lead is the distance from the toe of the switch to the theoretical nose of the crossing; it, together with the radius of the lead curve, governs the speed on the turnout. Check rails opposite the crossing hold the wheel flange to prevent it taking the wrong path at the gap in the running rail.

Advanced theory and extensions

Because the turnout radius is small, speed on the diverging route is restricted; higher-number crossings (1 in 16, 1 in 20) with longer leads permit higher diverging speeds, used at important junctions.

Assumptions and validity limits

State assumptions explicitly before using any relation for points and crossings — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Railway Engineering viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Railway Engineering papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to points and crossings.
4. Use equation 1:

Switch angle \alpha: tan \alpha = 1 /

.
5. Use equation 2:

Lead length L =

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Points and Crossings appears in Indian Railways and metro systems. In Indian civil curricula this topic is tested because it connects theory to track, signalling, and maintenance.

GATE and semester exams often combine points and crossings with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use points and crossings?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Thinking a higher crossing number means a sharper turnout (it is flatter).
• Confusing the switch (movable tongue rail) with the crossing (frog).
• Omitting the check rail’s role at the crossing nose.
• Mixing up the different methods of measuring crossing angle.

Quick revision checklist

Before attempting points and crossings problems, confirm you can:

1. Turnout number N = cot α — higher N, gentler divergence
2. Components: switch, stock rail, tongue, frog, check rail
3. Design speed through turnout limited by N and cant

Revise the solved examples in Railway Engineering — Satish Chandra & MM Agarwal and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Crossing angle for a 1 in 12 turnout

Problem

For a 1 in 12 crossing, determine the crossing angle using the right-angle (cotangent) method.

Solution

For the right-angle method, N = cot α, so α = cot⁻¹(12) = arctan(1/12) = arctan(0.0833) = 4.76° ≈ 4° 46′. A 1 in 8.5 crossing would give a sharper angle of arctan(1/8.5) = 6.71°, confirming that a higher crossing number yields a flatter angle and a higher permissible speed on the diverging track.

Conceptual check — Points and Crossings