Module 6 · Lesson 6.1

What abstraction discards

A line model has no shear deformation. Here is exactly how much that costs.

Why this matters

A real structure is a three-dimensional solid. Every model of it is an abstraction, and every abstraction throws something away permanently — not approximately, but completely. The discarded behaviour cannot be recovered later by refining the mesh, because it is not in the formulation at all.

The engineering question is therefore never 'is this model accurate?' It is 'what did this model throw away, and does the question I am asking depend on it?' That is answerable, and this lesson answers it for the most common abstraction of all.

By the end of this lesson you should be able to

Name what a line model discards, item by item
Compute the shear component of a deflection exactly
Find the span/depth ratio at which the discarded term stops being ignorable
State Saint-Venant's principle and the licence it gives

What you should already know

Beam bending and shear from Structural Analysis Fundamentals
Module 5's point that behaviour does not survive a handover — it does not survive an abstraction either

The ladder

Five levels, coarsest first. Each row is a decision, not a grade:

Level	Made of	Discards, permanently
Hand calculation	A formula, a load, a span	Continuity, load sharing, everything three-dimensional
2D line model	Nodes and lines in one plane	Out-of-plane behaviour, torsion, any stress that is not a member force
3D line model	Nodes and lines in space	Stress within a section, local behaviour at connections and openings
Shell model	Surface elements	Through-thickness variation beyond the plate assumption, detail below the mesh
Solid model	Volume elements	Nothing in principle — and everything you did not mesh finely enough

The last entry is the honest one. A solid model is not a model without assumptions; it is a model whose assumptions have moved into the mesh.

What a line model actually throws away

The most common abstraction in structural engineering is the line model, and the thing it discards most quietly is shear deformation. Euler–Bernoulli beam theory assumes plane sections remain plane and perpendicular to the neutral axis, which is another way of saying it assumes the shear strain is zero.

For a cantilever with a tip load, the true tip deflection has two terms:

δ = PL³/3EI + PL/GA_v

The first is bending, the second is shear, and a line model on Euler–Bernoulli theory reports only the first. How much is missing depends entirely on how stocky the member is:

Span/depth	Bending, mm	Shear, mm	Shear as a share of the total
2	0.3556	0.0640	15.3 %
4	2.844	0.128	4.3 %
8	22.756	0.256	1.1 %
16	182.044	0.512	0.3 %

(A 300 × 600 concrete member, E = 30 GPa, ν = 0.2, 100 kN tip load.)

Notice what the shear column does: it barely changes, while the bending column grows with the cube of the span. Shear deflection is not becoming small — bending is becoming large.

Setting a tolerance and solving for the crossover gives a number rather than a rule of thumb:

Shear exceeds 10 % of the total below a span/depth ratio of 2.55
Shear exceeds 5 % below 3.70

So for ordinary beams at L/d of 15 or 20, ignoring shear is a rounding error, and for a transfer beam, a deep coupling beam, a short pier or a squat wall, it is not.

Saint-Venant's principle

The licence to simplify locally:

Two statically equivalent load systems produce the same stresses at distances large compared with the region over which they differ.

In practical terms, replacing a real bolted connection with a point load changes the stresses near the connection and not further away. Taking the classical decay estimate, the difference has fallen below 5 % within 0.95 of a member depth — about one depth, and the conclusion is not sensitive to the exact rate assumed.

This is what permits everything else in the course. It is why a point load may stand for a bearing, why a support may be a node, and why a frame model may ignore the connection detail entirely. And it comes with its boundary printed on it: near the disturbance, none of that is true, and if the question is about the connection then the abstraction that made the frame model possible is precisely the wrong one.

Cantilever tip deflection, with shear

δ = bending \frac{P L ^{3}}{3 E I} + shear, discarded by a line model \frac{P L}{G A _{v}}

What it calculates: The true tip deflection of a cantilever, separated into the term a line model contains and the term it does not

P: tip load (kN)
L: span (m)
EI: flexural rigidity (kN·m²)
G: shear modulus, E/2(1+ν) (kN/m²)
A_v: shear area — exactly 5/6 of the gross for a rectangle (m²)

This assumes

Linear elastic material and small displacements
A prismatic member, so both terms use the same section throughout

In plain terms: The first term grows with the cube of the span and the second only linearly, so the shear share falls as 1/L². Shear deflection is not becoming small — bending is becoming large.

Try it

Model abstraction explorer

A line model has no shear deformation. Move the member from stocky to slender and watch how much is missing.

Span / depth

Member depth (m)

A 300 mm wide concrete member, 100 kN at the tip.

Bending deflection: 2.844 mm
Shear deflection: 0.128 mm
True total: 2.972 mm
What a line model reports: 2.844 mm
Omitted by the line model: 4.3 %
Shear passes 10 % below: L/d = 2.55

At this slenderness the omitted shear term is small enough to record as an exclusion and move on. It is still an exclusion rather than an absence.

What this shows: Shear deflection does not become small — bending becomes large. The crossover is a computable span/depth ratio.

Worked example

A transfer beam at span/depth 3

Given

A 300 mm × 600 mm concrete member, E = 30 GPa, ν = 0.2
Span 1.8 m, so L/d = 3
100 kN at the tip, cantilevered

Find

What a line model reports, and what it omits

Practice

For the 300 × 600 member above, at what span/depth ratio does shear deformation account for 10 % of the total tip deflection?

Practice

A cantilever's shear deflection is 0.0640 mm and its bending deflection is 0.3556 mm. What percentage of the total does a line model omit?

Check yourself

A line model of a squat shear wall under-predicts its deflection. What is the right response?

Worked example

Which discarded behaviour matters for this question

Given

A 3 m deep transfer truss is to be modelled as line elements
Three questions are asked of it: the chord forces, the deflection, and the stress at a welded gusset

Find

Which questions the line model can answer

Check yourself

What is the difference between a discretisation error and a modelling error?

Check yourself

Why does the shear share of a deflection fall as 1/L²?

Check yourself

What licence does Saint-Venant's principle give a modeller?

Check yourself

A solid model is sometimes described as having no assumptions. What is wrong with that?

Summary

Every abstraction discards behaviour permanently, not approximately
A line model on Euler–Bernoulli theory has no shear deformation at all
Shear is 15 % of the answer at L/d = 2 and 0.3 % at L/d = 16
The crossover is computable: 10 % below L/d = 2.55 for this section
Refining a mesh cannot recover a term the formulation does not contain
Saint-Venant: the disturbance is below 5 % within about one member depth

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint