Temperature Gradient Stress Calculator (Basic)

Calculator Inputs

Use presets or enter custom material properties. For partial restraint, set k between 0 and 1.

All fields accept decimals.

Temperature case

Pick uniform for ΔT through the whole member.

Material preset

Preset fills E and α, you can still edit.

Restraint factor k (0–1)

1 = fully restrained, 0 = free movement.

Elastic modulus E (GPa)

Steel ≈ 200, concrete ≈ 30.

Thermal expansion α (microstrain/°C)

Steel ≈ 12, concrete ≈ 10.

Thickness (mm) (optional)

Shown for context in this basic model.

Initial temperature (°C)

Reference condition before temperature change.

Final temperature (°C)

Service condition after temperature change.

Top temperature (°C)

Temperature at one face of the section.

Bottom temperature (°C)

Temperature at the opposite face.

Reset

Formula Used

This basic calculator uses linear-elastic thermal stress relationships with a restraint factor k. Use it for quick checks and comparisons, not final design.

Uniform temperature change (axial)

Assumes the member cannot expand or contract axially.

σ = −E · α · ΔT · k

Where ΔT = T_final − T_initial.

Linear temperature gradient (bending prevented)

Assumes curvature is prevented and temperature varies linearly.

σ(z) = −E · α · (T(z) − T̄) · k

Extreme stress magnitude ≈ |σ|max = E · α · (|ΔT|/2) · k.

Sign convention: Negative = compression, Positive = tension. If heating is restrained, stress is compressive; if cooling is restrained, stress is tensile.

How to Use This Calculator

Select the temperature case: uniform change or linear gradient.
Pick a material preset or enter custom E and α values.
Set restraint factor k based on joint and boundary conditions.
Enter temperatures, then calculate to view stresses above the form.
Use CSV or PDF export for quick documentation and sharing.

Example Data Table

Case	Material	E (GPa)	α (με/°C)	k	Temperatures (°C)	ΔT (°C)	Result (MPa)
Uniform	Steel	200	12	1.00	20 → 60	40	-96.000
Uniform	Concrete	30	10	0.80	15 → 45	30	-7.200
Gradient	Steel	200	12	1.00	Top 60 / Bot 20	40	σ_top -48.000 / σ_bot +48.000
Gradient	Concrete	30	10	0.70	Top 45 / Bot 15	30	σ_top -3.150 / σ_bot +3.150
Uniform	Aluminum	69	23	0.90	25 → 55	30	-42.849

Example results assume linear elasticity, simple restraint modeling, and typical material properties. Always confirm project-specific values and code requirements.

Professional Notes and Practical Context

1) Why temperature gradients matter on site

A temperature difference across a section creates restrained curvature and self-equilibrating stress. Common triggers include early-morning sunlight on one face, hot asphalt adjacent to a slab edge, steam curing, and localized heating from cutting or welding.

2) Typical material properties used for quick checks

For rapid screening, structural steel is often taken as E ≈ 200 GPa with α ≈ 12 με/°C. Normal-weight concrete is frequently approximated as E ≈ 25–35 GPa with α ≈ 8–12 με/°C. Aluminum members may use E ≈ 69 GPa with α ≈ 23 με/°C.

3) Understanding the restraint factor k

The restraint factor (0 to 1) represents how much free thermal movement is prevented. Anchored embeds, stiff connections, or locked bearings push k toward 1. Sliding connections, flexible supports, or gaps reduce k. If you are unsure, run a range such as k = 0.4, 0.7, and 1.0 to bracket outcomes.

4) Uniform change vs. linear gradient

Uniform temperature change mainly drives axial stress when expansion or contraction is restrained. A linear gradient drives opposite stresses at the faces: the hotter face tends to be compressive and the cooler face tensile. This calculator uses a basic “bending prevented” assumption for gradients, suitable for conservative comparisons.

5) Interpreting the sign and magnitude

The stress sign follows a simple convention: negative indicates compression and positive indicates tension. For uniform heating under restraint, compression is expected; cooling produces tension. For gradients, the face temperature relative to the mean controls which side goes into tension, which is often critical for cracking risk.

6) Example magnitude you can sanity-check

With steel (E 200 GPa, α 12 με/°C), a restrained uniform change of 40 °C gives roughly 96 MPa stress at k = 1. For a 40 °C gradient, the extreme face stress is about half that, roughly 48 MPa in compression on the hot face and 48 MPa in tension on the cool face, assuming a linear profile.

7) Where this basic model is most useful

Use it during method statements, temporary works checks, weld sequencing reviews, and lift planning where temperature differences are plausible. Compare connection details or curing controls to reduce ΔT.

8) Limits and good practice

This tool does not model creep, cracking, nonlinear gradients, composite sections, or time-dependent concrete behavior. Treat outputs as first-pass indicators. Confirm critical cases with project standards, code guidance, and detailed analysis when restraint is high or serviceability demands are strict.

FAQs

1) What does the restraint factor k represent?

It scales stress between free movement (k = 0) and full restraint (k = 1). Use higher k for fixed anchors and stiff connections, lower k for sliding or flexible supports.

2) Why is stress negative for restrained heating?

Heating tries to expand the member. If expansion is prevented, compressive stress develops to counteract that expansion. Cooling does the opposite and tends to create tensile stress.

3) How is the gradient case different from uniform change?

A gradient creates opposite stresses at opposite faces because temperatures differ through the thickness. Uniform change mainly creates one axial stress state when the full member temperature shifts together.

4) Does thickness change the gradient stress result here?

In this basic linear-gradient, bending-prevented approach, thickness is informational only. The extremes depend on ΔT across the section, not on thickness. Detailed models can include thickness effects.

5) What units should I use for E and α?

Enter E in GPa and α in microstrain per degree Celsius (με/°C). The calculator converts internally and returns stress in MPa, which aligns with typical construction material reporting.

6) What is a reasonable ΔT range to check?

For sun/shade or curing, start with 10–30 °C and extend to 40–60 °C for extreme exposure or localized heating. Always use measured or specified temperatures when available.

7) Can I use this for final design verification?

Use it for screening and documentation, not final verification. Final checks may require code-based temperature profiles, restraint modeling, cracking criteria, and time-dependent behavior, especially for concrete.