Controlled engineering arithmetic

Idealized 1D Thermoelastic Stress Check

Evaluate the signed uniaxial linear-thermoelastic relation from controlled modulus, mean CTE, temperature change and imposed total axial strain. This is not a piping-code, pressure-vessel, rail, joint or component acceptance calculation.

Tension positiveSigned strainNo material presetsReference arithmetic only

Controlled-input calculation

Do not use this scalar result as an allowable-stress or fitness-for-service decision. Real restraints have finite stiffness; temperature and properties may be nonuniform; local stress, multiaxiality, plasticity, creep, buckling, fatigue, joints and support movement require an appropriate structural model and governing code.

Young’s modulus E GPa, valid for the stated temperature and material condition Mean linear CTE ᾱ µm/(m·K), signed and valid over the full interval Temperature change ΔT K (a Celsius temperature difference has the same numerical value) Imposed total axial strain εtotal µm/m; enter 0 only if the modeled axial total strain is truly zero Controlled E and ᾱ source / condition Record grade, heat/temper, direction, temperature interval and source revision Boundary and total-strain basis Record how εtotal and the stress-free reference state were established I confirm that E and ᾱ are controlled values for the exact material condition, direction and temperature interval, and that the evaluated state remains linear elastic. I confirm that a uniform, small-strain, uniaxial thermoelastic model is appropriate and that significant gradients, multiaxial stress, local concentration, plasticity, creep, buckling, fatigue and joint/support effects are handled elsewhere.

Enter controlled inputs; no example values are preloaded.

Результат арифметического расчёта

Thermal strain ᾱΔT—

Imposed total strain εtotal—

Elastic/mechanical strain εtotal − ᾱΔT—

Signed uniaxial stress σ—

Relation, signs and dimensions

εthermal = ᾱ ΔT
εmechanical = εtotal − εthermal
σ = E εmechanical

With E in GPa and strains in µm/m, the implemented output is:

σ [MPa] = E [GPa] × (εtotal − ᾱΔT) [µm/m] ÷ 1000

Правило знаков

Axial tension and extension are positive.
Compression and shortening are negative.
ΔT = T − Tref; heating is positive.
For a positive CTE and truly zero total axial strain, heating gives negative (compressive) stress.

Dimensional check

ᾱ: µm/(m·K)
ᾱΔT: µm/m
1 µm/m = 10⁻⁶
1 GPa × 10⁻⁶ = 0.001 MPa

What this does not establish

It does not determine the actual restraint strain or support stiffness.
It does not calculate pipe flexibility, stress intensification, nozzle loads, pressure stress, joint loads or expansion-joint behavior.
It does not compare the result with yield, allowable stress, fatigue, creep or code acceptance limits.
It does not represent a temperature gradient, bonded dissimilar materials, thick sections or a multiaxial stress state.
It is not an ISO formula and does not establish ASME B31.3 compliance.

A full-restraint value is the special arithmetic case εtotal = 0 in this uniaxial model. It is not proof that a real installation is fully fixed or that the calculated elastic stress can physically develop without relaxation, yielding, creep, slip or instability.

Evidence and classification

MIT OpenCourseWare, Mechanics and Materials I, Lecture 15 (1 Nov 2006), pages 1–3

Gives thermal strain αΔT, total strain as elastic plus thermal strain, and the linear isotropic thermoelastic equations. Its constrained-channel example derives σxx = −αΔTE for heating under zero axial strain and zero transverse stress.

NIST Engineering Metrology Toolbox — CTE FAQ

Explains why material identity and the temperature range used to average CTE matter. It does not provide the former generic six-material preset table.

ASTM E228-22

Active test method for measuring linear thermal expansion of solid materials with a push-rod dilatometer. It is a property-measurement source, not a structural acceptance formula.

ASME B31.3-2024 official catalog page

Process piping design includes flexibility and stress-intensification requirements. Exact code calculations and acceptance limits require the licensed governing edition and project design basis; none are reproduced or inferred here.

Controlled 1D Thermoelastic Stress Check

Controlled-input calculation

Результат арифметического расчёта

Relation, signs and dimensions

Правило знаков

Dimensional check

What this does not establish

Evidence and classification

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