A reader recently asked a sharp question in the comments on one of my posts: what’s the best analysis approach for determining thermally induced stresses and forces in reinforced concrete structures, given that standard commercial workflows using software like ETABS seem far from robust and don’t capture the phenomenon effectively? That question deserves a proper answer, because the honest response is: your instinct is correct, and the reason is structural, not a software bug.

1. Why the Rail-Steel Formula Doesn’t Transfer Directly

In a recent post on Union Pacific’s white rail coatings, I used the simplest possible thermally constrained stress relationship: for a fully restrained linear-elastic member,

\[ \sigma = E \alpha \Delta T \]

This works cleanly for rail steel because steel is (to good approximation) linear-elastic, doesn’t crack under tension the way concrete does, and creep at ordinary service temperatures is negligible on any engineering timescale that matters. None of those three assumptions hold for reinforced concrete, which is exactly why a naive application of this formula — or a linear-elastic FE model that implicitly assumes it — overpredicts thermal stress, often dramatically.

2. Three Ways Concrete Refuses to Behave Like Steel

Restraint is almost never total. A rail is fixed at both ends over a long, continuously-fastened length, so full restraint (\( R = 1 \)) is a fair assumption. A concrete wall cast onto an existing foundation, a slab on a flexible support, or a frame member with rotational flexibility at its ends is only partially restrained. Design guidance captures this with a restraint factor \( R \), typically well below 1, derived from the relative stiffness of the new concrete versus whatever it’s cast against or connected to.

Creep relaxes a large fraction of the stress. Under sustained load, concrete creeps — and thermal strain from hydration or seasonal temperature change is about as sustained a load as exists. This creep relaxation isn’t a minor correction; it can relieve a substantial share of the elastically-predicted stress, particularly at early age when creep compliance is highest. A model that doesn’t account for time-dependent creep is, in effect, analyzing a different, stiffer material than the one actually in the structure.

The material cracks, and that changes the answer. Once tensile capacity is locally exceeded, the concrete cracks, stiffness drops at that location, and the remaining restraint force redistributes. A linear-elastic model has no mechanism to represent this — it will keep predicting an ever-increasing stress in a location that has, in reality, already cracked and shed load. This is the deepest reason ETABS-style linear-elastic workflows struggle here: cracking is not a side effect to check afterward, it’s part of the mechanics that determines the stress distribution in the first place.

3. The Industry-Standard Simplified Method: Restraint Factor Design

Rather than reaching straight for nonlinear finite element analysis, the concrete industry’s standard practical approach — codified in ACI 207.2R (Report on Thermal and Volume Change Effects on Cracking of Mass Concrete) and CIRIA C766/C660 (Control of Cracking Caused by Restrained Deformation in Concrete, used extensively in the UK and Europe) — folds all three effects above into a single design equation for the crack-inducing strain:

\[ \varepsilon_{cr} = K \left[ \alpha_c T_1 + \varepsilon_{ca} \right] R – 0.5\, \varepsilon_{ctu} \]

where \( \alpha_c \) is the coefficient of thermal expansion of the concrete, \( T_1 \) is the relevant temperature difference (typically peak hydration temperature minus ambient, or minus the temperature of the restraining element), \( \varepsilon_{ca} \) is autogenous shrinkage strain, \( R \) is the restraint factor for the specific geometry, \( \varepsilon_{ctu} \) is the tensile strain capacity of the concrete, and \( K \) is an allowance for creep relaxation — commonly taken as about 0.65 when \( R \) is derived per CIRIA’s own method, or 1.0 when \( R \) is instead derived using the BS EN 1992-3 approach, since the two restraint-factor conventions already bake creep in differently. If \( \varepsilon_{cr} \) exceeds the tensile strain capacity, cracking is predicted, and the reinforcement design proceeds from there to control crack width rather than to prevent cracking outright, which for most restrained concrete elements isn’t a realistic design target.

ACI 207.2R takes a parallel approach, providing restraint-factor charts and a worked calculation method: for a given wall or slab geometry, the degree of restraint at the surface is estimated from the ratio of member dimension to a characteristic distance from the restrained edge, then combined with the concrete’s tensile strength, elastic modulus, and thermal expansion coefficient to find the temperature change that will cause surface cracking.

The practical workflow, in short:

1. Run a transient heat transfer analysis (or use tabulated adiabatic temperature rise data) to get a realistic time-history of \( T_1 \), rather than assuming a uniform section-wide \( \Delta T \).
2. Estimate the restraint factor \( R \) from code charts or from a separate stiffness analysis of the specific geometry.
3. Apply the creep-adjusted crack-inducing strain equation and compare against tensile strain capacity.
4. If cracking is predicted, proceed to reinforcement design for crack width control rather than treating the crack prediction itself as a design failure.

4. When to Escalate to Dedicated Nonlinear Software

The restraint-factor method is deliberately simplified and empirical, and it can be conservative in some geometries and non-conservative in others — a limitation acknowledged in the literature comparing it against more rigorous approaches. For complex geometries, unusual restraint conditions, or high-consequence structures where the simplified method’s uncertainty isn’t acceptable, the next step up is dedicated nonlinear concrete FE software with age-dependent creep models and smeared or discrete cracking formulations built in, rather than a general-purpose linear-elastic structural package pressed into a job it wasn’t designed for. DIANA FEA is the tool most frequently cited in the concrete-cracking research literature for this purpose. For early-age mass concrete specifically — large pours where heat of hydration, not seasonal temperature swing, is the dominant driver — ConcreteWorks (developed with Texas DOT sponsorship) is a purpose-built, more accessible option that couples the thermal and mechanical analysis specifically for that problem.

Closing Thought

The underlying lesson generalizes well beyond concrete: any time a structural analysis tool assumes linear-elastic behavior, but the real material relieves stress through creep, cracking, or plasticity, that tool will systematically overpredict the driving force and underpredict how the structure actually copes with it. The fix isn’t a better mesh or a finer time step in the same linear-elastic package — it’s recognizing that the physics has changed and reaching for a method built for the material actually in front of you.

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References

1. ACI Committee 207, ACI 207.2R-07: Report on Thermal and Volume Change Effects on Cracking of Mass Concrete, American Concrete Institute, 2007.
2. Bamforth, P.B., CIRIA C766: Control of Cracking Caused by Restrained Deformation in Concrete, Construction Industry Research and Information Association, 2018 (updating CIRIA C660, 2007).
3. Jędrzejewska, A. et al., “Standardized models for cracking due to restraint of imposed strains — the state of the art,” Structural Concrete, 2023.
4. VibrationData blog, “Why Union Pacific Is Painting Its Rails White: Thermal Stress, Sun Kink, and Coating Chemistry,” July 2026 — for the fully-restrained \( \sigma = E\alpha\Delta T \) baseline case.