Computational Engineering / From physical structure to analytical model

What abstraction discards

Comparing bending and shear deflection in a cantilever model.

Lesson review: expert verification remains open

This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

A computational engineering reviewer must check algorithms, convergence claims, statistical assumptions and numerical examples beyond the cases explicitly recorded.
A computational structural specialist should validate abstraction explorer equations, section assumptions and local-model limits before design use.
Exact shell-node counts and runtime estimates require the actual mesh/solver; the course numbers remain illustrative.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

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Why this matters

Every structural model uses assumptions about geometry, material behaviour, loading and boundary conditions. Identify which behaviours the formulation represents and which it omits. Mesh refinement can reduce discretisation error, but cannot introduce a deformation mechanism absent from the element formulation.

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

Choosing an idealisation

A hand calculation, a line-element model and a shell or solid model can each answer useful questions. Their capabilities depend on the chosen theory. Hand methods can represent continuity and load sharing; 3D beam elements can recover section stresses under their section assumptions. Shells and solids retain assumptions about material behaviour, loading, boundary conditions and the scale resolved by the mesh.

Bending and shear deformation

Euler–Bernoulli beam theory neglects shear deformation. Timoshenko beam formulations include it, so shear omission is not a property of all line models.

For the linear, prismatic cantilever idealisation used here, with a tip load P:

δ = PL³/(3EI) + PL/(GA_v).

The first term is bending deflection; the second uses an effective shear area. For the rectangular section example, use A_v = 5A/6. This is the adopted shear-correction model, not a universal exact shear area.

For a 300 × 600 mm member with E = 30 GPa, ν = 0.2 and P = 100 kN:

Span/depth	Bending, mm	Shear, mm	Shear share of 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%

With section and load fixed, shear deflection grows linearly with span and bending deflection grows with span cubed. The ratio of shear to bending deflection varies as 1/L². The shear fraction of the total is δ_s/(δ_b + δ_s), which approaches that scaling for slender members.

Under these assumptions, shear exceeds 10% of total deflection below L/d ≈ 2.55 and 5% below L/d ≈ 3.70. Assess the omitted term against the accuracy needed for the question. For very stocky members, also examine whether the beam idealisation represents the local stress and deformation field adequately.

Saint-Venant's principle

Differences between statically equivalent local loading distributions tend to diminish away from the loaded region in suitable elastic bodies. The distance required depends on geometry, material behaviour and the response quantity. There is no universal 5% error boundary at one member depth.

Simplified load or support representations may therefore suit a global analysis while being unsuitable for a local connection or bearing assessment. Check the local region with an appropriate model when it affects the decision.

Cantilever tip deflection, with shear

\delta = \underbrace{\frac{PL^3}{3EI}}_{\text{bending}} + \underbrace{\frac{PL}{GA_v}}_{\text{shear contribution}}

What it calculates: The bending and shear contributions in the stated linear beam idealisation

P: tip load (kN)
L: span (m)
EI: flexural rigidity (kN·m²)
G: shear modulus, E/2(1+ν) (kN/m²)
A_v: effective shear area; 5/6 of gross area is adopted for this rectangular-section example (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

For a fixed prismatic cantilever section and tip load, why does the ratio of shear to bending 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 supports simplification away from local disturbances; the required distance is problem-dependent