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Compare it to detect a real > content change; `generated` moves on every render and is not a change signal. ![Steel Thermal Expansion and Temperature-Dependent Properties](/images/uploads/33e15f44-df90-4c6a-b133-1da93cccf80a/wiki-hero-a-structural-steel-beam-and-bolted-connection-exposed-to-intense-heat-showing-un-1920x823.jpg) Calculated Values # Steel Thermal Expansion and Temperature-Dependent Properties Learn how temperature changes affect steel expansion, strength, stiffness, and fire performance. ![Portrait of Anders Bergström, steel industry reporter](/images/uploads/a7d8e5bc-7cdf-409b-913b-211e63e8c441/anders-bergstr-m-1920x1920.jpg) **[Anders Bergström](/news/author/anders-bergstrom "Anders Bergström")** Calculated Values 60+ min read Updated Aug 14, 2026 Evidence-reviewed On this pageOn this page - [What Thermal Expansion Means in Steel](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#what-thermal-expansion-means-in-steel "What Thermal Expansion Means in Steel") - [The Thermal-Expansion Coefficient of Carbon and Low-Alloy Steel](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#the-thermal-expansion-coefficient-of-carbon-and-low-alloy-steel "The Thermal-Expansion Coefficient of Carbon and Low-Alloy Steel") - [Temperature-Dependent Physical Properties Beyond Expansion](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#temperature-dependent-physical-properties-beyond-expansion "Temperature-Dependent Physical Properties Beyond Expansion") - [Steel Strength at Elevated Temperature](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#steel-strength-at-elevated-temperature "Steel Strength at Elevated Temperature") - [Deformation, Creep, and Stress Relaxation During Heating](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#deformation-creep-and-stress-relaxation-during-heating "Deformation, Creep, and Stress Relaxation During Heating") - [Phase Transformation and Nonlinear Thermal Expansion](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#phase-transformation-and-nonlinear-thermal-expansion "Phase Transformation and Nonlinear Thermal Expansion") - [Stainless Steels Compared with Carbon and Low-Alloy Steels](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#stainless-steels-compared-with-carbon-and-low-alloy-steels "Stainless Steels Compared with Carbon and Low-Alloy Steels") - [Thermal Restraint, Thermal Stress, and Structural Compatibility](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#thermal-restraint-thermal-stress-and-structural-compatibility "Thermal Restraint, Thermal Stress, and Structural Compatibility") - [Fire Design of Steel Members and Connections Under EN 1993-1-2](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#fire-design-of-steel-members-and-connections-under-en-1993-1-2 "Fire Design of Steel Members and Connections Under EN 1993-1-2") - [How Elevated-Temperature Properties Are Measured](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#how-elevated-temperature-properties-are-measured "How Elevated-Temperature Properties Are Measured") - [Calculating Thermal Expansion: Worked Frameworks Without False Precision](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#calculating-thermal-expansion-worked-frameworks-without-false-precision "Calculating Thermal Expansion: Worked Frameworks Without False Precision") - [Common Errors in Interpreting Steel Thermal-Property Data](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#common-errors-in-interpreting-steel-thermal-property-data "Common Errors in Interpreting Steel Thermal-Property Data") - [Selecting Data for Engineering and Reference Use](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#selecting-data-for-engineering-and-reference-use "Selecting Data for Engineering and Reference Use") - [Reference Tables and Terminology for Steel at Elevated Temperature](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#reference-tables-and-terminology-for-steel-at-elevated-temperature "Reference Tables and Terminology for Steel at Elevated Temperature") ## What Thermal Expansion Means in Steel ### Linear expansion, strain, and coefficient notation Thermal expansion is a temperature-dependent change in a material’s dimensions. When a steel bar is heated, its atoms vibrate more strongly and the average spacing between them generally increases. The bar therefore becomes longer, wider, or thicker. Cooling usually produces the reverse change, although the path may not be perfectly reversible when heating causes phase transformation, plastic deformation, creep, or other metallurgical changes. For a member whose temperature is reasonably uniform, the usual first approximation is: ΔL=α⁢L⁢ΔT OpenStax uses 12 × 10⁻⁶ °C⁻¹ as a representative coefficient for steel. Strong evidence \[1\] \[1\] [**Thermal Expansion**](https://openstax.org/books/university-physics-volume-2/pages/1-3-thermal-expansion). OpenStax. University Physics Volume 2, 2022.OpenStax *University Physics Volume 2* gives this linear relation in its 2022 treatment of thermal expansion and uses 12×10−6∘C−1 as a representative coefficient for steel. #### Symbols in the linear-expansion equation ΔL Change in length. L Original length. ΔT Temperature change. α Coefficient of linear thermal expansion. εth Dimensionless thermal strain. Here, ΔL is the change in length, measured in metres, millimetres, or another length unit; L is the original length in the same unit; ΔT is the temperature change; and α is the coefficient of linear thermal expansion. The coefficient has units of inverse temperature, commonly ∘C−1 or K−1. A temperature interval of 1 °C has the same size as an interval of 1 K, so the numerical value of α is the same in those two unit systems. The equation can also be written as thermal strain: εth=ΔLL=α⁢ΔT Thermal strain is dimensionless. If α=12×10−6∘C−1 and a steel member warms by 100 °C, the calculated free thermal strain is 0.0012, or 0.12%. A 10 m member would lengthen by approximately 12 mm under that simplified assumption: ΔL=(12×10−6)⁢(10000mm)⁢(100)=12mm #### Interpret the result correctly The equation predicts free dimensional change under its stated assumptions. It does not by itself predict thermal stress, restraint force, buckling, or failure. That result is a calculation of dimensional change, not a prediction of stress or failure. Secant coefficient **Secant coefficient** An average thermal-expansion coefficient calculated over a specified temperature interval rather than the local slope at one temperature. The symbol α is often treated as though every steel has one fixed value. That is not correct. Its value depends on chemical composition, metallurgical condition, temperature interval, and the way the value is reported. A mean or secant coefficient over 20 to 600 °C is not necessarily the same as the instantaneous or tangent coefficient at 600 °C. Carbon steel, low-alloy steel, ferritic stainless steel, austenitic stainless steel, and quenched-and-tempered grades can show different expansion behaviour. Representative and code-based thermal-expansion coefficients discussed in the article. | Source context | Temperature range | Coefficient | |---|---|---| | Representative steel value | Ambient temperature | 12 × 10⁻⁶ °C⁻¹ | | BS 5950 Part 8 recommendation | Ambient temperature | 12 × 10⁻⁶ °C⁻¹ | | BS 5950 Part 8 recommendation | 200 to 600 °C | 14 × 10⁻⁶ °C⁻¹ | \[2\] \[2\] [**Fire Design of Steel Structures**](https://core.ac.uk/download/pdf/9554664.pdf). University of Sheffield. Reference on BS 5950 Part 8 fire design, 2006.The University of Sheffield’s 2006 reference on fire design reports that BS 5950 Part 8 recommends 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C. Those figures demonstrate why a single number should not be carried automatically through every temperature range. They are code-based design values for a defined application, not universal constants for all steels. The simple equation also assumes a uniform temperature and a sufficiently small change that a linear approximation is acceptable. Over a broad temperature range, expansion should be calculated by integration: ΔL=L0∫T0T1α⁢(T)dT where α⁢(T) is the temperature-dependent coefficient. If different parts of a section have different temperatures, each part has its own thermal strain. A flange exposed to a fire may expand more quickly than a protected web, creating curvature and internal force even before the average member temperature becomes high. Phase transformation creates another limit. In carbon and low-alloy steels, heating can alter the crystal structure, and the associated transformation strain may add to or offset ordinary lattice expansion over part of the temperature range. Cooling can produce still different dimensional changes depending on the cooling rate and the resulting microstructure. Thermal expansion is therefore separate from temperature-dependent strength, but the two may occur together. A heated steel member can lengthen while its elastic modulus and yield strength decline. ASM references on carbon and low-alloy steels identify the coefficient of linear thermal expansion alongside thermal conductivity and heat capacity as important physical properties. These properties affect one another in practical fire or heating calculations: thermal conductivity controls how quickly heat spreads, heat capacity affects the energy required for a temperature rise, and expansion determines the dimensional response. ![Schematic comparing a freely expanding steel bar with a restrained bar under heating.](/images/uploads/47a41f18-ca24-44a4-854d-b224d441c5c5/wiki-inline-a-steel-bar-comparing-free-thermal-expansion-with-fully-restrained-heating-1920x1094.jpg)[](/images/uploads/47a41f18-ca24-44a4-854d-b224d441c5c5/wiki-inline-a-steel-bar-comparing-free-thermal-expansion-with-fully-restrained-heating-1920x1094.avif "Enlarge image — Schematic comparing a freely expanding steel bar with a restrained bar under heating.")Thermal strain becomes stress only when movement is opposed. ### Free expansion versus restrained expansion A steel member undergoes free expansion when its supports and connections allow the calculated thermal strain to occur. Consider a straight bar lying on low-friction supports, with no axial load and a uniform temperature increase. It can lengthen by ΔL without developing an axial restraint stress. The bar has changed length, but no mechanical force is required to permit that change. #### Caution: restraint is essential A freely expanding member may develop little axial stress. Stress estimates require compatibility, restraint stiffness, temperature distribution, material degradation, and connection behaviour. This distinction is central. Thermal strain does not automatically equal stress. If the same bar is fixed between rigid supports, its total axial length may be unable to change. In a simplified fully restrained case, the mechanical strain needed to cancel the thermal strain is approximately: εmech=−εth=−α⁢ΔT While the steel remains elastic, the corresponding stress is approximately: σth=−E⁢α⁢ΔT where E is Young’s modulus and the negative sign indicates compression during restrained heating. The associated axial force is: N=σth⁢A with A as the member’s cross-sectional area. These expressions are useful teaching approximations, not complete fire-design equations. They assume uniform temperature, complete restraint, elastic behaviour, constant properties, and no slip or movement at the connections. Real restraint is usually partial. A column may be held by floor slabs, beams, bracing, or connections that themselves deform. A pipe may have guides, anchors, and bends that distribute movement. A beam may expand axially while its ends rotate, or it may develop lateral displacement as thermal curvature increases. Connection stiffness and friction change the force path. Consequently, the force is not simply “the force from temperature”; it is the response of the steel member and its surrounding structural system. Heating history matters too. As temperature rises, E falls, so the elastic stress predicted using room-temperature properties becomes increasingly unreliable. At elevated temperature, steel may yield under a stress that would have been elastic at ambient conditions. Under sustained load, creep and stress relaxation can reduce restraint force while increasing deformation. On cooling, permanent plastic strain may remain, and a member that was initially free to expand may not return to its original length. \[3\] \[3\] [**EN 1993-1-2: Resistance of Members and Connections Exposed to Fire**](https://eurocodes.jrc.ec.europa.eu/publications/en-1993-1-2-resistance-members-connections-fire). European Commission Joint Research Centre. Eurocodes, 2005. \[4\] \[4\] [**ANSI/AISC N690-18**](https://www.aisc.org/globalassets/aisc/publications/standards/n690-18w.pdf). American Institute of Steel Construction. AISC Standard, 2018. ANSI/AISC N690-18 specifically addresses combined thermal and mechanical effects, including members that may be restrained against thermal expansion. EN 1993-1-2 addresses the fire design and elevated-temperature resistance of steel members and connections. Their inclusion of restraint and member resistance reflects a practical fact: the same temperature rise can produce little axial force in one arrangement and substantial force in another. ### Why temperature change is not the same as temperature stress Temperature change is a thermal input. Temperature stress is a mechanical response produced when thermal movement is prevented, opposed, or made incompatible with movement elsewhere. A free steel plate can become hotter and expand without thermal stress. A cooler plate welded to it can prevent uniform movement and create stress from the resulting temperature difference. This is why ΔT alone cannot define a steel member’s condition. The analysis may need the temperature distribution through the section and along the member, the rate of heating, thermal gradients, section factor, heat transfer, restraint stiffness, connection behaviour, applied load, creep, relaxation, and phase changes. Mechanical degradation must also be treated separately: loss of yield strength, loss of Young’s modulus, altered Poisson’s ratio, and changes in stress–strain behaviour do not follow directly from the expansion equation. \[5\] \[5\] [**Elevated Temperature Deformation of Structural Steel**](https://www.nist.gov/publications/elevated-temperature-deformation-structural-steel). National Institute of Standards and Technology. NIST IR 88-3899, 1989. Research reflects this separation. NIST IR 88-3899, published in 1989, reports deformation behaviour of structural steel at elevated temperature. ASTM STP 124, published in 1994, reports elevated-temperature thermal expansion and mechanical behaviour for [stainless steels](/categories/stainless-steels "stainless steels"). Other constructional-steel studies measure yield strength, tensile strength, elongation, elastic modulus, and expansion coefficient as separate temperature-dependent properties. The NTIS record for elevated-temperature properties of ASTM A723 and D6AC steels likewise describes measurements of specific heat, thermal expansion, stress relaxation, tensile properties, Young’s modulus, and Poisson’s ratio. #### From temperature rise to thermal force 1. **1. Calculate thermal strain** Estimate free strain from α and ΔT, or integrate α(T). 2. **2. Check compatibility** Determine whether supports and connections permit the movement. 3. **3. Establish the temperature field** Account for gradients, heating rate, section factor, and heat transfer. 4. **4. Apply temperature-dependent properties** Use appropriate modulus, strength, creep, relaxation, and phase data. 5. **5. Determine structural response** Estimate force, deformation, curvature, instability, or connection demand. The practical rule is straightforward: calculate thermal strain first, then determine whether the structure permits that strain. Only after considering compatibility, restraint, material degradation, and heating history can thermal force or stress be estimated. A coefficient such as 12×10−6∘C−1 is a starting parameter—not a complete description of how steel responds to heat. ## The Thermal-Expansion Coefficient of Carbon and Low-Alloy Steel The thermal-expansion coefficient of carbon and low-alloy steel is often presented as though it were a single material constant. It is not. A quoted value depends on the steel grade, its microstructure, the temperature interval, the heating or cooling path, and whether the number describes a local slope or an average expansion over a range. The surrounding test conditions matter too: specimen orientation, applied stress, heating rate, phase condition, and measurement method can all affect the result. For a freely expanding bar, the basic relation is ΔL=α⁢L⁢ΔT where ΔL is the change in length, L is the original length, ΔT is the temperature change, and α is the linear coefficient of thermal expansion. OpenStax *University Physics Volume 2* gives 12×10−6∘C−1 as a representative value for steel and uses this equation to introduce linear thermal expansion (OpenStax, 2022). That number is useful for an initial estimate. It should not be treated as a universal value for every carbon steel, low-alloy steel, or temperature range. Thermal expansion is also separate from strength loss. A steel member may lengthen because its temperature rises, lose yield strength because its material properties change, and develop stress because supports restrain that lengthening. Those effects interact in a real structure, but they are not the same property. ### Representative ambient-temperature values At ordinary temperatures, 12×10−6∘C−1 is a widely used engineering value for carbon and low-alloy structural steel. In practical terms, a 1 m bar subjected to a uniform 100 °C rise would expand by approximately 12×10−6×1×100=0.0012m, or 1.2 mm, if it were free to move and if the coefficient remained constant across that interval. The University of Sheffield’s 2006 summary of BS 5950 Part 8 gives the same 12×10−6∘C−1 value at ambient temperature. This agreement between a textbook example and a design-code recommendation does not prove that all steel has exactly that coefficient. It shows that the value is a suitable representative assumption for a defined engineering context. The exact coefficient can vary among grades such as [S275](/materials/gost/s275 " — composition, equivalents and standards"), S355, ASTM A36, ASTM A572, and alloyed pressure-vessel or quenched-and-tempered steels. The differences at ambient temperature may be modest compared with changes caused by temperature range and phase transformation, but they can still matter when calculating fit-up tolerances, thermal movements, or restrained forces in long members. Chemical composition affects the crystal structure and the relative amounts of ferrite, pearlite, bainite, martensite, and other constituents. Prior heat treatment affects the starting microstructure. A nominal grade designation therefore does not, by itself, provide every detail needed to reproduce a measured expansion curve. The quoted unit also deserves attention. A coefficient written as 12×10−6∘C−1 is equivalent in numerical size to 12×10−6K−1, because a temperature interval of 1 °C has the same magnitude as an interval of 1 K. The Celsius and kelvin scales differ in their zero points, but not in the size of their increments. This equivalence applies to ΔT and to coefficients expressed per degree of temperature interval. It does not mean that a temperature of 20 °C is interchangeable with 20 K. The coefficient is dimensionless in physical form—strain per temperature interval—but engineering tables commonly write it with units of ∘C−1 or K−1. Confusing a temperature value with a temperature difference can produce a serious calculation error. ### Temperature-range-dependent coefficients Steel’s expansion is not perfectly linear over all temperatures. The coefficient generally changes as temperature rises, and the curve may change more sharply near metallurgical transformations. For carbon and low-alloy steels, the transition from body-centred cubic ferrite to face-centred cubic austenite during heating produces a change in lattice behaviour and can be accompanied by contraction or a reduced net expansion. The exact temperature and shape of the response depend on carbon content, alloying elements, prior microstructure, heating rate, and thermal history. The University of Sheffield summary reports that BS 5950 Part 8 recommends 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C. These are code-based values for specified design use, not a claim that every steel specimen follows a straight line with one value below 200 °C and another value above it. The change from 12 to 14 indicates the practical importance of selecting a coefficient for the temperature interval being analysed. Suppose a 10 m member is heated uniformly from 20 to 600 °C. Using a constant 12×10−6∘C−1 would predict about 69.6 mm of expansion. A higher coefficient over part of the range gives a larger result. The difference may affect movement joints, bearing clearances, connection forces, and the forces calculated for restrained members. Two definitions must be separated. The instantaneous, or differential, coefficient at temperature T is commonly written α⁢(T)=1L⁢dLdT. It describes the local slope of the expansion curve. An average or integrated mean coefficient over T1 to T2 is instead α¯T1,T2=L⁢(T2)−L⁢(T1)L⁢(T1)⁢(T2−T1). An average coefficient may be perfectly suitable for calculating total movement across a broad range, while being unsuitable for a calculation that needs the local strain increment at a particular temperature. If α varies with temperature, the total free strain is more accurately obtained from εth=∫T1T2α⁢(T)dT. That distinction becomes important near phase changes. A table labelled “coefficient at 600 °C” may report a differential value, a mean value from room temperature to 600 °C, or a code simplification. Those interpretations are not interchangeable. Published elevated-temperature work reflects this complexity. NIST IR 88-3899, published on January 1, 1989, examined the deformation behaviour of structural steel at elevated temperature. The study belongs to a broader body of work in which thermal expansion is considered alongside stress–strain response, elastic modulus, creep, and relaxation. *Experimental Studies on the Properties of Constructional Steel at Elevated Temperatures* likewise reports measurements of thermal-expansion coefficient together with yield strength, tensile strength, modulus of elasticity, and elongation. These measurements should not be reduced to a single room-temperature number. ### Why tabulated values must be read with their source conditions A tabulated coefficient is meaningful only with its source conditions attached. The table should identify the steel or alloy, temperature range, heating or cooling direction, specimen condition, measurement basis, and whether the value is instantaneous or averaged. A coefficient measured on a particular heat-treated specimen is evidence about that specimen and test programme; it is not automatically a universal material constant for every grade carrying a similar designation. The source’s purpose also controls how the value should be used. OpenStax presents 12×10−6∘C−1 as a representative teaching value. The University of Sheffield summary reports values recommended by BS 5950 Part 8 for structural design. These sources answer different questions from a laboratory paper that measures a continuous expansion curve. A design calculation should follow the applicable standard rather than combine a textbook coefficient with an unrelated test curve without checking definitions. Other references show why this checking matters. ASTM STP 124, published in 1994, reports elevated-temperature properties of stainless steels, including thermal expansion and mechanical behaviour; stainless grades cannot simply be substituted for carbon or low-alloy steel. EN 1993-1-2 addresses fire design and elevated-temperature resistance of steel members and connections, while ANSI/AISC N690-18 addresses combined thermal and mechanical effects, including members restrained against thermal expansion. Neither standard treats a coefficient alone as a complete fire or thermal-response model. In a restrained member, the free thermal strain is only the starting point. Temperature gradients can bend or distort a section rather than produce uniform elongation. Heat transfer, section factor, exposure conditions, connection stiffness, creep, stress relaxation, and phase transformation can alter the resulting force and deformation. A uniform temperature rise with no restraint calls for a free-expansion calculation. A steep gradient or rigid connection requires a structural and thermal analysis using the applicable grade data and design standard. The sound practice is therefore simple: quote the coefficient with its units, temperature interval, definition, grade, and source. Use 12×10−6∘C−1 as a representative ambient-temperature value when that assumption is justified, and recognise the BS 5950 Part 8 value of 14×10−6∘C−1 from 200 to 600 °C as a code recommendation for its stated context. Do not present either number as the fixed thermal-expansion coefficient of all steel. ## Temperature-Dependent Physical Properties Beyond Expansion Thermal expansion describes a change in dimensions, but it does not describe how quickly a steel member heats, how much energy it absorbs, or how evenly temperature spreads through its cross-section. Those questions depend mainly on thermal conductivity, specific heat, density, surface heat transfer, and the member’s geometry. The ASM reference on carbon and low-alloy steels treats thermal conductivity, coefficient of linear thermal expansion, and heat capacity as separate physical properties for this reason. They interact, but one cannot be substituted for another. A nominal value such as 12×10−6°⁢C−1 does not predict a member’s temperature history. OpenStax University Physics Volume 2 (2022) presents the linear relation ΔL=α⁢L⁢ΔT and uses that coefficient as a representative value for steel. The University of Sheffield’s 2006 review reports that BS 5950 Part 8 recommends 12×10−6°⁢C−1 at ambient temperature and 14×10−6°⁢C−1 from 200 to 600 °C. Those are calculation assumptions over stated ranges, not universal constants for every grade, microstructure, or heating path. The same caution applies to thermal conductivity and heat capacity. ![Cutaway steel I-section showing heat flowing from the exposed flange toward the cooler interior.](/images/uploads/cb681061-2a01-4815-ac47-ed58cc074d01/wiki-inline-heat-moving-from-the-exposed-surface-of-a-steel-beam-through-its-flange-and-web-1520x1920.jpg)[](/images/uploads/cb681061-2a01-4815-ac47-ed58cc074d01/wiki-inline-heat-moving-from-the-exposed-surface-of-a-steel-beam-through-its-flange-and-web-2027x2560.avif "Enlarge image — Cutaway steel I-section showing heat flowing from the exposed flange toward the cooler interior.")Conductivity controls how quickly heat spreads through a steel section. ### Thermal conductivity Thermal conductivity, k, measures a material’s ability to conduct heat through itself. In a simple one-dimensional description, Fourier’s law is q⁢′⁢′=−k⁢dTdx, where q⁢′⁢′ is heat flux and dT/dx is the temperature gradient. A high value of k permits a given heat flux with a smaller gradient; a low value requires a steeper gradient. The minus sign indicates that heat flows toward lower temperature. For a steel beam exposed to fire, heat enters through the surface by convection and radiation, then travels inward by conduction. A flange exposed directly to hot gases can heat much sooner than the web’s centreline or a shaded connection behind it. Conductivity therefore affects the delay between surface heating and core heating. It also controls whether large differences develop between an exposed flange and a protected or cooler face. The value of k is not fixed across all steels or temperatures. Carbon steel, low-alloy steel, stainless steel, and high-alloy grades can have different conductivity at the same temperature. Composition, prior heat treatment, porosity, oxidation, and phase state also matter. As temperature rises, conductivity commonly changes significantly, so a fire calculation that inserts one room-temperature value into the entire heating period can misrepresent the heat flow. Stainless steels, for example, are not interchangeable with carbon steels merely because both are called steel; ASTM STP 124 (1994) reports elevated-temperature properties of stainless steels, including thermal expansion and mechanical behavior, in a separate body of technical work. Conductivity does not directly give the member temperature. The surface temperature must first be determined from the surrounding gas temperature, convection coefficient, radiation, emissivity, contact conditions, and any protection system. The resulting heat flux then produces an internal temperature field governed by conduction. A thick plate and a thin angle exposed to the same furnace temperature will not follow the same temperature-time curve. Their exposed surface area relative to heated volume differs, and so does the path heat must travel to reach the interior. ### Specific heat and heat capacity Specific heat capacity, cp, is the energy required to raise the temperature of a unit mass by one degree. For a mass m undergoing a temperature increase ΔT, the approximate sensible heating energy is Q=mcp⁢ΔT. For a volume V, the corresponding quantity is ρ⁢Vcp⁢ΔT, where ρ is density. This explains the distinction between specific heat and total heat capacity. Specific heat belongs to the material; heat capacity belongs to the particular mass or member. A larger cp means that more energy is needed to produce the same temperature rise, all else equal. It does not mean that the steel will resist mechanical degradation. A member may absorb considerable energy while its yield strength and elastic modulus decline with temperature. NIST IR 88-3899, published in 1989, examined deformation behavior of structural steel at elevated temperature, while other elevated-temperature studies measured yield strength, tensile strength, modulus of elasticity, elongation, and thermal expansion. These mechanical properties must be evaluated separately from the energy balance. Specific heat also varies with temperature and can change sharply near metallurgical transformations. Carbon and low-alloy steels may undergo transformation from ferrite-pearlite structures toward austenite during heating, with associated changes in heat capacity and latent heat. The energy supplied during such a transformation does not produce temperature rise in the same simple way as ordinary sensible heating. Cooling can introduce a different transformation path, so heating and cooling histories matter. The NTIS record for *Elevated Temperature Properties of Steels* describes measurements of specific heat, thermal expansion, stress relaxation, tensile properties, Young’s modulus, and Poisson’s ratio for ASTM A723 and D6AC steels. The range of measured properties is instructive: a temperature model needs more than a coefficient of expansion because steel’s thermal and mechanical response changes together, but not identically. For design, the selected data set should match the grade and the governing standard. EN 1993-1-2 addresses the fire design and elevated-temperature resistance of steel members and connections, and its recommended material properties are not automatically interchangeable with values from a furnace experiment on another grade. The issue is not academic. A small error in heat capacity can alter the predicted temperature-time curve, which then affects calculated strength, stiffness, thermal strain, creep, and connection response. ### Thermal diffusivity and temperature gradients Thermal diffusivity **Thermal diffusivity** The ratio k/(ρcp), describing how quickly a temperature disturbance spreads through a material. Thermal diffusivity combines conductivity with density and specific heat: a=kρ⁢cp. It describes how quickly a temperature disturbance spreads through a material. Conductivity supplies the heat-transfer mechanism, while volumetric heat capacity, ρ⁢cp, measures the material’s resistance to temperature change. A steel with high conductivity but also high heat capacity does not necessarily heat internally faster than one with lower values. The ratio matters. This property explains why a steel section can have a hot surface and a much cooler interior during transient heating. In simplified form, the characteristic diffusion time scales approximately with t∼L2a, where L is the relevant distance from the heated surface. Doubling that distance can increase the characteristic time by roughly four times. A thick box section, massive base plate, or heavily welded connection can therefore maintain substantial internal temperature differences long after its outer surface has become hot. Temperature gradients also arise from unequal exposure. Fire may reach three sides of a column while the fourth side is adjacent to a wall. A beam may have a heated lower flange, a cooler upper flange beneath a slab, and a web with a separate intermediate temperature. Solar heating, welding, induction heating, and localized furnace exposure create similar nonuniform fields outside building fires. Those gradients produce differential thermal expansion. If one part of a cross-section tries to expand more than another, the member bends or develops self-equilibrating stresses. If axial movement is restrained by supports, slabs, diaphragms, fire protection, or adjacent members, additional forces develop. ANSI/AISC N690-18 (2018) specifically addresses combined thermal and mechanical effects, including members restrained against thermal expansion. The resulting stress cannot be calculated from α alone; restraint stiffness, temperature distribution, elastic-plastic behavior, creep, and relaxation are also relevant. #### Local temperature is not member temperature A surface or embedded sensor may miss cooler cores, hot corners, welds, shielded faces, and gradients that control curvature, local buckling, or phase transformation. A thermometer or thermocouple records a local temperature, not “the temperature of the steel” in every engineering sense. A surface thermocouple may report the hottest region while the core remains cooler. A single embedded sensor may miss a hot corner, a weld, or a shielded face. Even a reported average temperature can conceal gradients that control curvature, local buckling, connection forces, or phase transformation. Consequently, thermal analysis should identify where temperatures are evaluated and how the distribution evolves with time. Section factor, exposed perimeter, insulation thickness, surface heat transfer, grade, temperature-dependent k, cp, density, and phase changes all enter that calculation. Expansion is one response among several. The temperature field comes first. ## Steel Strength at Elevated Temperature Steel does not follow one universal strength-versus-temperature curve. The result depends on grade, starting microstructure, section thickness, heating rate, load history, temperature range, and whether the specimen is tested during heating or after being held at a chosen temperature. A carbon steel, a low-alloy pressure-vessel steel, and an austenitic stainless steel may respond differently at the same furnace temperature. The selected design standard also matters because standards simplify test data into recommended reduction factors, limiting temperatures, and calculation procedures. Temperature-dependent steel properties answer different physical or mechanical questions. | Property | Primary question answered | Not a substitute for | |---|---|---| | Thermal expansion coefficient | How much would free length change? | Yield strength or modulus | | Thermal conductivity | How does heat move through the steel? | Specific heat | | Specific heat | How much energy raises temperature? | Thermal conductivity | | Yield strength | When does plastic deformation begin? | Tensile strength | | Young’s modulus | How much elastic strain develops? | Yield strength | | Creep | How does strain grow under sustained stress? | Stress relaxation | The measured properties are not interchangeable. *Experimental Studies on the Properties of Constructional Steel at Elevated Temperatures* reports measurements of yield strength, tensile strength, modulus of elasticity, elongation, and thermal-expansion coefficient. The NTIS record for *Elevated Temperature Properties of Steels* identifies a related set of measurements for ASTM A723 and D6AC steels: specific heat, thermal expansion, stress relaxation, tensile properties, Young’s modulus, and Poisson’s ratio. These groups show why a single “strength at temperature” value is inadequate. A design may depend on the onset of permanent deformation, the maximum tensile resistance, elastic deflection, post-yield ductility, lateral strain, or time-dependent stress loss. Temperature also produces effects that must be kept separate. OpenStax University Physics Volume 2 gives the linear expansion relation “ΔL = αLΔT” and uses 12 × 10⁻⁶ °C⁻¹ as a representative coefficient for steel. A University of Sheffield reference reports that BS 5950 Part 8 recommends 12 × 10⁻⁶ °C⁻¹ at ambient temperature and 14 × 10⁻⁶ °C⁻¹ from 200 to 600 °C. Those figures describe free thermal expansion under the stated assumption; they do not specify the loss of strength or stiffness. When expansion is restrained, thermal strain can generate stress, and that stress can alter the mechanical response before any externally applied load is considered. ### Yield strength and tensile strength Yield strength marks a transition from predominantly recoverable elastic strain to permanent plastic strain. At elevated temperature, this transition generally occurs at a lower applied stress than it does at room temperature, but the reduction is not a universal percentage for all steels. A constructional steel with a ferrite-pearlite structure, a quenched-and-tempered low-alloy steel, and a stainless steel can have different temperature-dependent curves because their alloying elements and phase stability differ. Tensile strength is the maximum engineering stress reached in a tensile test. It is not the same property as yield strength. A specimen can lose much of its yield resistance while retaining a different proportion of its ultimate tensile resistance. The separation matters in a heated member: local yielding may begin well before the member reaches its maximum tensile capacity, and a calculation based only on tensile strength can miss excessive deformation or instability. Heating history changes the measured result. A test performed while the specimen is being heated may capture transient behavior, whereas a specimen heated to a target temperature, held there, and then loaded may experience recovery, stress relaxation, or microstructural change before the tensile test begins. NIST IR 88-3899, *Elevated Temperature Deformation of Structural Steel*, was published on January 1, 1989 and addresses deformation behavior at elevated temperature. Its subject is broader than a simple table of strength-reduction factors: deformation depends on temperature, stress, and time. ASTM STP 124 provides elevated-temperature stainless-steel data covering thermal expansion and mechanical behaviour. Strong evidence \[6\] \[6\] [**ASTM STP 124**](https://store.astm.org/stp124-eb.html). ASTM International. ASTM Special Technical Publication 124, 1994.Phase transformation can further interrupt a smooth strength curve. In plain-carbon and low-alloy steels, heating through transformation ranges changes the crystal structure and can alter strength, ductility, and the shape of the stress–strain response. On cooling, the final properties may not return to their original values, particularly when the heating and cooling path produces a different microstructure. Stainless steels add another set of possibilities, including grade-dependent phase stability and transformation behavior. ASTM STP 124, published in 1994, reports elevated-temperature properties of stainless steels, including thermal expansion and mechanical behavior. #### Information required with elevated-temperature strength - **Grade** Identify the exact steel designation and product condition. - **Temperature** State whether the value is transient, steady-state, or after a temperature hold. - **Loading** Report stress, strain rate, and duration at temperature. - **Property definition** Distinguish yield strength, tensile strength, modulus, and elongation. - **Test history** Record heating, cooling, prior heat treatment, and phase condition. For design, “strength at 600 °C,” for example, must therefore be read as a test or code-defined property under specified conditions, not as a permanent material constant. EN 1993-1-2 addresses the fire design and elevated-temperature resistance of steel members and connections, but its calculation rules should not be transferred casually to a different grade, loading history, or standard framework. The temperature in a table is also not automatically the temperature throughout a real member. Heat transfer, section factor, protection, moisture, thermal gradients, and connection details determine the temperature field. ### Elastic modulus and stiffness loss The elastic modulus, also called Young’s modulus, is the slope of the initial elastic portion of a stress–strain curve. It controls the relation between stress and recoverable strain. A lower modulus means that a given stress produces more elastic deformation, even if the material has not yet reached its reduced yield strength. This is a different phenomenon from yield-strength loss. Yield strength answers, “When does permanent plastic deformation begin?” Modulus answers, “How much elastic strain develops before that point?” Heating can reduce both, but not necessarily at the same rate. A member may therefore become noticeably more flexible while still carrying stress below its current yield limit. Conversely, a reduction in yield strength can cause plastic deformation under a stress that would previously have been safely elastic, even where the initial elastic slope has not changed by the same amount. The distinction affects buckling calculations. Member stiffness depends not only on the material modulus but also on geometry, restraint, imperfections, and the temperature distribution along the member. As the modulus falls, Euler-type buckling resistance falls for a given effective length; thermal bowing or restraint can add moments and second-order effects. A column with a modest average temperature can still have a severe stability problem if one flange or one segment becomes much hotter than the rest. The NTIS record for *Elevated Temperature Properties of Steels* specifically includes Young’s modulus and Poisson’s ratio for ASTM A723 and D6AC steels. Poisson’s ratio describes the lateral strain associated with axial strain and can matter in multiaxial stress states, plates, shells, and finite-element models. It should not be silently treated as fixed when a source or material model provides temperature-dependent data. The same caution applies to modulus: a room-temperature elastic constant is not a substitute for a temperature-dependent value. ### Elongation and stress–strain response Elongation records how much extension a tensile specimen undergoes, usually over a specified gauge length, and is commonly reported as a percentage after fracture. It provides information about ductility, but it does not replace the complete stress–strain curve. Two steels can have similar elongation at fracture while differing in yield point, strain hardening, tensile strength, necking behavior, or energy absorbed before failure. At elevated temperature, the stress–strain curve may show a lower initial slope, an earlier yield transition, altered strain hardening, and greater total strain before fracture. The curve may also depend strongly on strain rate and the duration of the test. At sufficiently high temperature, creep and stress relaxation become important: a specimen held at constant strain can lose stress, while a specimen held under constant stress can continue to deform. These are time-dependent responses, not merely temperature corrections to a room-temperature curve. The article does not support a single generic elevated-temperature strength curve for all steels. Limited evidence The available research identifies the relevant measured properties but does not support inventing one generic curve or assigning unverified values at selected temperatures. A defensible curve must state the steel designation, specimen condition, heating procedure, strain rate, temperature measurement method, and whether the test was transient or steady-state. Without those details, a plotted line can imply precision that the experiment did not establish. Material resistance is only one part of elevated-temperature performance. EN 1993-1-2 requires the response of members and connections to be considered in fire design, while ANSI/AISC N690-18 addresses combined thermal and mechanical effects, including members restrained against thermal expansion. In a restrained frame, free expansion is converted partly into axial force and bending; later, loss of yield strength, modulus reduction, creep, relaxation, and geometric instability can change the force path. Section factor and heat transfer govern how quickly steel heats, and gradients can cause curvature even when the average temperature seems moderate. Elevated-temperature resistance is therefore a structural problem as well as a materials problem. ## Deformation, Creep, and Stress Relaxation During Heating Heating a steel member does more than increase its length. The member may expand, develop thermal stress when that expansion is restrained, lose stiffness and strength, and continue deforming while the temperature remains high. These effects interact. A calculation based only on thermal strain and an instantaneous elastic modulus can therefore miss deformation that develops over time or after the material structure has changed. For an unconstrained bar under a uniform temperature increase, the basic relation is ΔL = αLΔT. OpenStax University Physics Volume 2 (2022) uses 12 × 10⁻⁶ °C⁻¹ as a representative coefficient for steel. That value is useful for explaining the mechanism, but it is not a universal constant. The University of Sheffield’s discussion of BS 5950 Part 8 reports 12 × 10⁻⁶ °C⁻¹ at ambient temperature and 14 × 10⁻⁶ °C⁻¹ from 200 to 600 °C. The selected grade, temperature interval, heating history, and design standard all affect the adopted value. Thermal expansion is only one component of total deformation. Mechanical strain may include recoverable elastic strain, permanent plastic strain, time-dependent creep strain, and transformation-related strain. A temperature gradient can also bend or distort a member even when the average temperature would predict a modest overall elongation. The section factor and heat-transfer conditions control how quickly different parts of a section heat, while connections and surrounding construction determine how much movement is actually possible. ### Findings from NIST IR 88-3899 NIST IR 88-3899, *Elevated Temperature Deformation of Structural Steel*, was published by the National Institute of Standards and Technology on January 1, 1989. It is the principal source for the elevated-temperature deformation behavior of structural steel in this discussion. Its importance is not that it supplies one deformation curve for every grade and fire exposure. Rather, it treats deformation as a temperature- and load-dependent response that must be measured and interpreted under defined conditions. The report’s subject is structural steel deformation at elevated temperature, where the usual room-temperature assumption of an almost immediate elastic response becomes inadequate. At increasing temperature, the elastic modulus decreases, so a given stress produces more instantaneous elastic strain. Yield behavior also changes, reducing the stress required for permanent deformation. If the load remains present while the steel is hot, strain can continue to increase even without an increase in applied stress. That additional strain is not obtained by multiplying the current stress by a single temperature-dependent modulus. This distinction matters for members carrying gravity load during a fire or other prolonged heating event. A beam may first expand and develop compressive force against its restraints. As its stiffness falls, the same restraint force can produce greater elastic deformation. If the stress reaches the temperature-dependent yield range, plastic deformation can accumulate. Continued exposure then allows time-dependent deformation to add to the movement. The response is a sequence, not a single point on an elastic stress–strain diagram. NIST IR 88-3899 should also be read with attention to material identity and test history. “Structural steel” covers grades with different compositions, processing histories, and transformation behavior. Carbon and low-alloy steels have temperature-dependent thermal conductivity, heat capacity, and coefficient of linear thermal expansion, as identified in ASM reference material. Stainless steels require separate treatment; ASTM STP 124 (1994) reports elevated-temperature thermal expansion and mechanical properties for stainless steels rather than treating them as interchangeable with carbon structural steel. A further complication is phase transformation. Heating can alter the crystal structure and the associated mechanical response within a temperature range that depends on composition and prior condition. Transformation may change the apparent expansion behavior and can produce strain that is not represented by αLΔT alone. Cooling history matters as well. Steel that has experienced high temperature may not return to its original strength, stiffness, or dimensional state when it cools. ### Time-dependent deformation Creep is deformation that develops with time while a material is subjected to sustained stress, especially at elevated temperature. The defining loading condition is sustained stress or sustained load; the measured quantity is the continuing increase in strain. In a heated steel member, the stress may come from gravity, restraint, imposed displacement, or a combination of these sources. Creep does not require the applied load to increase. A column carrying a steady load can shorten progressively as temperature rises and remains high. A beam can sag further during a period in which its external load is unchanged. The rate depends on temperature, stress, grade, microstructure, prior heating, and the current deformation state. Because the supplied NIST material establishes elevated-temperature deformation behavior without providing a single applicable rate for all steels, no universal creep rate should be assigned to structural steel. The word “sustained” does not mean that the stress must remain numerically constant in a real structure. As deformation changes geometry, contact conditions, restraint forces, and load paths, the stress may change too. A heated restrained beam may develop axial compression during expansion, then lose resistance as its stiffness and yield strength decline. Creep can relieve part of that force while increasing the member’s displacement. In a frame, that movement can redistribute load into columns, connections, slabs, fire protection, or adjacent members. #### Why heating history matters 1. **Heating rate** Controls time available for conduction, recovery, transformation, creep, and oxidation. 2. **Temperature hold** Allows thermal equilibration, stress relaxation, and microstructural change. 3. **Peak temperature** May trigger phase transformation or substantial strength and stiffness loss. 4. **Cooling rate** Influences the phases formed and the residual dimensional state. 5. **Repeated cycles** Can accumulate plastic strain, residual stress, and metallurgical changes. A temperature history is therefore part of the material state. Two specimens at the same current temperature may deform differently if one was heated rapidly and the other held at an intermediate temperature before reaching that point. The difference can arise from accumulated creep, plastic strain, metallurgical change, and different internal stresses. An analysis that records only the final temperature discards information needed to predict the current deformation. ### Stress relaxation under sustained restraint Stress relaxation **Stress relaxation** A reduction in stress with time while total strain or displacement is held approximately constant. Stress relaxation is related to creep but has the opposite primary control condition. In a relaxation problem, deformation or displacement is held approximately constant while the stress required to maintain that restraint decreases with time. A steel member heated between rigid supports provides the basic example. Thermal expansion is prevented, so compressive stress develops; while the member remains hot and restrained, creep and other inelastic mechanisms can reduce that stress even though the imposed length does not change. The distinction is practical. Under sustained load, creep is observed as increasing deformation. Under sustained restraint, relaxation is observed as decreasing stress. A real structure may experience both at once because restraints are rarely perfectly rigid and loads are rarely perfectly fixed. Some expansion is permitted, some force is transferred to neighboring members, and the resulting displacement and stress evolve together. Relaxation does not mean that restraint becomes irrelevant. If the temperature continues to rise, thermal expansion keeps increasing the incompatibility that the structure must accommodate. If the supports, connections, or surrounding members move, the restraint condition changes. On cooling, the member may contract while residual plastic strain and altered material properties prevent it from returning to its original geometry. Reversal of temperature does not guarantee reversal of deformation. ANSI/AISC N690-18 (2018) addresses combined thermal and mechanical effects, including members restrained against thermal expansion. EN 1993-1-2 addresses the fire design and elevated-temperature resistance of steel members and connections. These standards do not turn thermal expansion into a standalone design check. Their application requires the temperature distribution, mechanical loading, restraint, section behavior, connection response, and temperature-dependent material properties to be considered together. For that reason, a fixed coefficient and an instantaneous modulus are screening inputs, not a complete deformation model. A defensible assessment must ask how fast the steel heats, where temperature gradients occur, whether phase transformation is possible, how long the load is sustained, and how the structure permits or prevents movement. Those questions determine whether the dominant response is free expansion, restrained thermal stress, plastic deformation, creep, stress relaxation, or a changing combination of all five. ## Phase Transformation and Nonlinear Thermal Expansion ### Why α can change with temperature The familiar thermal-expansion equation is useful, but it is not a complete description of steel: ΔL=α⁢L⁢ΔT OpenStax *University Physics Volume 2* (2022) presents this relation with 12×10−6∘C−1 as a representative coefficient for steel. That number is a convenient engineering approximation, not a universal material constant. The coefficient may vary with grade, composition, temperature interval, microstructure, and the way the measurement is made. Even in a single phase, atomic spacing does not increase at a perfectly constant rate. As temperature rises, lattice vibrations become more energetic and the average spacing between atoms changes. The slope of the length–temperature curve can therefore change. A coefficient quoted for an interval is usually an average or secant value: αavg=ΔLL0⁢ΔT A tangent, or instantaneous, coefficient would instead describe the local slope at a particular temperature. Those two values are equal only when the expansion curve is sufficiently close to linear over the interval under consideration. The University of Sheffield’s 2006 reference on fire design reports that BS 5950 Part 8 recommended 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C. The change between those code values makes the central point: even a design standard may assign different coefficients to different temperature ranges. It does not imply that every steel grade follows the same curve, or that the value changes abruptly at exactly 200 °C. Composition affects this response. Carbon steels, low-alloy steels, stainless steels, and precipitation-strengthened grades contain different phases and alloying additions. ASM references on carbon and low-alloy steels treat the coefficient of linear thermal expansion alongside thermal conductivity and heat capacity because these properties depend on the material and its thermal state. ASTM STP 124, published in 1994, reports thermal-expansion and mechanical data for stainless steels, whose phase stability and alloy chemistry can produce behavior unlike that of a plain carbon structural steel. The coefficient also does not describe strength loss, stiffness loss, or thermal stress. The paper *An Introduction to the Mechanical Properties of Structural Steel at Elevated Temperatures* discusses thermal expansion alongside Poisson’s ratio, stress–strain behavior, phase transformation, and elastic modulus. These are related responses to temperature, but they are not interchangeable. A steel member may expand freely while its yield strength and elastic modulus decline. If expansion is restrained, the same thermal strain can generate compression, bending, or connection forces. The coefficient supplies one part of that calculation. ![Illustrated heating and cooling paths showing phase transformation and residual dimensional change in carbon steel.](/images/uploads/80eace1f-2aea-4c8b-ba9b-bc54fde1ea62/wiki-inline-a-carbon-steel-specimen-changing-length-through-ordinary-expansion-and-phase-tra-1920x1094.jpg)[](/images/uploads/80eace1f-2aea-4c8b-ba9b-bc54fde1ea62/wiki-inline-a-carbon-steel-specimen-changing-length-through-ordinary-expansion-and-phase-tra-1920x1094.avif "Enlarge image — Illustrated heating and cooling paths showing phase transformation and residual dimensional change in carbon steel.")Phase transformation can make heating and cooling follow different dimensional paths. ### Transformation-related dimensional effects Steel can undergo metallurgical transformations while it is being heated or cooled. In a carbon or low-alloy steel, ferrite, pearlite, bainite, austenite, and martensite may appear in different combinations, depending on composition, prior processing, heating rate, peak temperature, hold time, and cooling rate. A transformation changes the crystal structure and often the specific volume. The resulting dimensional change is superimposed on ordinary lattice expansion or contraction. During heating, transformation to austenite can alter the length–temperature path. The sign and magnitude of the additional dimensional effect depend on the starting microstructure and chemical composition; it should not be represented by a single transformation temperature for “steel.” During cooling, austenite may transform into ferrite and pearlite, bainite, or martensite. Each route has a different transformation range and volume change. Martensitic transformation, in particular, can produce expansion during a stage in which the specimen would otherwise be contracting because its temperature is falling. That effect is not reversible thermal expansion. If a bar is heated from 20 °C to 700 °C and then cooled, its final length need not return to the original value, even after it reaches the original temperature. The residual difference may result from transformed phases, plastic strain, oxidation, creep, or distortion caused by temperature gradients. Only the elastic portion of ordinary thermal expansion is recovered when the temperature returns, and even that statement assumes that no transformation or irreversible mechanical process has occurred. Conceptual components of total strain during heating and cooling. | Strain contribution | Physical origin | Potentially reversible? | |---|---|---| | Thermal strain | Temperature-dependent lattice spacing | Often, if no irreversible process occurs | | Transformation strain | Change in phase or crystal structure | Not generally fully reversible | | Plastic strain | Permanent mechanical deformation | No | | Creep strain | Time-dependent deformation under stress | Not generally fully reversible | A useful way to express the distinction is to separate total strain into contributions: εtotal=εthermal+εtransformation+εplastic+εcreep This is a conceptual decomposition rather than a guarantee that each term can be measured independently in a practical member. The thermal term is associated with temperature-dependent lattice spacing. The transformation term represents the dimensional change caused by a changing phase fraction and crystal structure. Plastic strain and creep arise from mechanical history, stress, and time at temperature. The transformation range must come from a grade-specific source or test. It depends on carbon content and alloying elements, including chromium, nickel, molybdenum, and manganese, as well as on the initial microstructure. Stainless steels illustrate why generic claims are unsafe: an austenitic grade may remain predominantly austenitic over a temperature range in which a ferritic or carbon steel undergoes a major phase change. ASTM STP 124 therefore reports elevated-temperature properties by stainless-steel type rather than treating stainless steel as one material. The same issue affects mechanical properties. As the paper *An Introduction to the Mechanical Properties of Structural Steel at Elevated Temperatures* explains, phase transformation can alter the stress–strain response and elastic modulus as well as the measured thermal expansion. A specimen may show a change in slope, temporary strain development, or altered unloading behavior near a transformation range. Those observations should not be interpreted as a simple increase or decrease in α alone. ### Heating and cooling are not necessarily symmetric A heating curve and a cooling curve can differ even when they pass through the same nominal temperatures. The material remembers its thermal history. Heating rate controls how much time is available for diffusion and phase rearrangement; holding at a temperature changes phase fractions and grain structure; cooling rate determines whether diffusion-controlled products or martensite form. A furnace cycle and a short-duration fire exposure can therefore produce different final dimensions and mechanical properties at the same peak temperature. NIST IR 88-3899, published in 1989, examined deformation behavior of structural steel at elevated temperature. Its subject is broader than free thermal expansion because deformation also depends on stress, time, and temperature. That distinction matters in restrained members: heating can produce compressive force, while relaxation, creep, yielding, and transformation progressively change the force during the exposure. On cooling, the member may not retrace the heating path. It can retain residual stress or permanent strain. Experimental studies of constructional steel have measured thermal-expansion coefficient together with yield strength, tensile strength, elastic modulus, and elongation. Such measurements show why a single coefficient cannot stand in for a complete elevated-temperature model. The NTIS record *Elevated Temperature Properties of Steels* describes measurements of specific heat, thermal expansion, stress relaxation, tensile properties, Young’s modulus, and Poisson’s ratio for ASTM A723 and D6AC steels. These properties interact during a thermal cycle, but each requires its own temperature- and history-dependent treatment. For structural analysis, the temperature field is as important as the nominal peak temperature. A thick flange and a thin web may heat at different rates. One side of a member may be hotter than the other, producing curvature before any phase transformation occurs. Restraint at supports and connections converts free expansion into force. EN 1993-1-2 addresses fire design and elevated-temperature resistance of steel members and connections, while ANSI/AISC N690-18 addresses combined thermal and mechanical effects, including members restrained against thermal expansion. The applicable standard determines which property curves, reduction factors, thermal actions, and calculation procedures are permitted. A coefficient such as 12×10−6∘C−1, or 14×10−6∘C−1 over a stated range, is therefore a starting approximation. It is not a complete heating-and-cooling law. A reliable assessment must identify the grade and designation, temperature distribution, heating history, cooling history, phase behavior, restraint, creep and relaxation, section factor, heat transfer, connection response, and governing design standard. ## Stainless Steels Compared with Carbon and Low-Alloy Steels Material families require separate consideration when selecting thermal and mechanical data. | Material family | Data concern | Transferability | |---|---|---| | Carbon steel | Grade, microstructure, transformation range | Do not assume one curve for all grades | | Low-alloy steel | Alloy chemistry and heat-treatment history | Use grade-specific data | | Austenitic stainless steel | Phase stability, cold work, alloy chemistry | Do not substitute carbon-steel coefficients | | Ferritic or martensitic stainless steel | Different phase stability and thermal history | Check the exact designation | | Duplex stainless steel | Austenite–ferrite balance | Use matching grade and condition | The phrase “steel thermal expansion” hides important differences between material families. Carbon steel, low-alloy steel, and stainless steel do not share one coefficient, one modulus curve, or one sequence of metallurgical changes as temperature rises. Even within stainless steel, chromium content, nickel content, carbon and nitrogen levels, stabilizing additions, heat treatment, cold work, and prior thermal exposure can change the measured response. For a uniform member that is free to expand, OpenStax University Physics Volume 2 (2022) gives the linear relation ΔL=α⁢L⁢ΔT where L is the original length, ΔT is the temperature change, and α is the linear coefficient of thermal expansion. OpenStax uses 12×10−6∘C−1 as a representative value for steel. That figure is useful for explaining the equation, but it is not a universal constant for all steels or for every temperature interval. The University of Sheffield’s 2006 reference reports that BS 5950 Part 8 recommends 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C. Those are code-based assumptions for a stated design context, not a license to assign the same value to every stainless grade. ### Scope of ASTM STP 124 ASTM STP 124, published by ASTM International in 1994, is the central source for the elevated-temperature stainless-steel data discussed here. Its subject is elevated-temperature properties of stainless steels, including thermal expansion and mechanical behavior. It therefore addresses more than a single expansion coefficient. The relevant evidence includes how stainless materials deform, lose stiffness, retain or lose strength, and respond to heating over the temperature ranges covered by the reported tests. That scope matters because a thermal-expansion table cannot answer all structural questions. A designer may need a temperature-dependent coefficient for calculating free strain, but also a reduced elastic modulus for deformation analysis, yield or tensile strength for resistance, specific heat for the energy required to heat the material, and thermal conductivity for the temperature field through a plate, wall, or connection. Phase stability is another separate question. A material can expand in a predictable way while its crystal structure, precipitation state, or strain-hardening condition changes. ASTM STP 124 should be read as a source of measured or compiled behavior for identified stainless-steel materials and test conditions, not as proof that “stainless steel” is a single engineering material. The reported result depends on the grade, specimen condition, heating rate, temperature range, loading path, and test method. Tensile data from a steadily heated specimen do not automatically describe a restrained beam exposed to a nonuniform fire. The same distinction appears in data for carbon and low-alloy steels. ASM references identify thermal conductivity, coefficient of linear thermal expansion, and heat capacity as relevant physical properties. NIST IR 88-3899, published on January 1, 1989, reports research on the elevated-temperature deformation behavior of structural steel. Other elevated-temperature studies measure yield strength, tensile strength, modulus of elasticity, elongation, and expansion coefficient; a separate NTIS record describes measurements of specific heat, thermal expansion, stress relaxation, tensile properties, Young’s modulus, and Poisson’s ratio for ASTM A723 and D6AC steels. The list shows why one coefficient cannot stand in for the whole material model. ### Composition and grade effects Stainless steel is defined by its alloy chemistry and corrosion-resistant chromium content, but stainless designations also separate materials with different microstructures and thermal histories. Austenitic grades such as UNS S30400 and UNS S31600, ferritic grades such as UNS S40900, martensitic grades such as UNS S41000, precipitation-hardening grades such as UNS S17400, and [duplex](/materials/material-no/1.4460 " — composition, equivalents and standards") grades such as [UNS S31803](/materials/material-no/1.4462 " — composition, equivalents and standards") do not form one temperature-response family. The designation alone is not enough unless it is tied to the applicable material standard and condition. “304 stainless steel,” for example, should be identified through the governing specification, such as ASTM A240/A240M where that standard applies to plate, sheet, and strip, together with the specified product form and condition. A designation such as UNS S30400 identifies a composition system, but it does not by itself specify every product requirement, heat treatment, thickness, surface condition, or test result. The applicable ASTM, ASME, EN, or other material standard must be stated. Nickel generally supports austenitic stability, while chromium promotes ferritic or chromium-rich phases depending on the full composition. Carbon, nitrogen, molybdenum, niobium, titanium, and other additions also affect phase balance and precipitation. These changes influence not only corrosion performance but also expansion, conductivity, heat capacity, strength retention, and the temperature at which transformations or precipitation reactions occur. Austenitic stainless steels often have expansion behavior that differs from the values commonly assigned to structural carbon steel, and their strength response may include effects from cold work and strain-induced transformation in certain compositions. Ferritic and martensitic grades have different phase stability and thermal histories. Duplex grades can experience changes in the balance between austenite and ferrite, while precipitation-hardening grades may lose the condition that supplied their room-temperature strength when exposed to elevated temperature. The practical result is direct: a coefficient copied from a carbon-steel table should not be transferred to a stainless designation without checking the grade and source. Nor should room-temperature yield strength be extended upward with a simple linear adjustment. Temperature-dependent properties must be taken from data that match the material specification and the intended exposure. ### Thermal expansion and mechanical behavior in stainless steels Thermal expansion is a free-strain property. Temperature-dependent mechanical degradation is a change in stiffness or strength. Thermally induced stress is the force response produced when that free strain is partly or wholly restrained. These are related, but they are not interchangeable. A freely expanding stainless member can develop little axial stress even while its length changes substantially. The same member, fixed between rigid supports, can generate large restraint forces. If the temperature varies through its thickness or along its length, curvature and secondary stresses may develop. Heating rate, temperature distribution, restraint stiffness, connection slip, creep, and stress relaxation then affect the result. A single value of α cannot capture those effects. For stainless steels, the property check should normally include the temperature-dependent coefficient of expansion, elastic modulus, yield or proof strength, tensile strength, heat capacity, thermal conductivity, and phase stability. The modulus controls elastic deformation and the conversion of imposed strain into stress. Strength controls resistance after yielding. Heat capacity and conductivity govern how quickly different parts of a section heat. Phase stability indicates whether the assumed properties remain applicable after exposure or whether transformation, precipitation, sensitization, or other metallurgical changes alter them. The test condition must also be recorded. Data from an isothermal tensile test, a transient-heating test, a previously cold-worked specimen, and a specimen heated once and cooled before testing do not answer the same question. Elevated-temperature deformation can involve creep and relaxation, particularly when load remains for a long period. A short fire exposure and a long furnace hold may produce different residual properties even at the same peak temperature. Design standards reflect this wider problem. EN 1993-1-2 addresses fire design and elevated-temperature resistance of steel members and connections. ANSI/AISC N690-18 addresses combined thermal and mechanical effects, including members restrained against thermal expansion. Such standards require more than inserting a coefficient into ΔL=α⁢L⁢ΔT; the analysis may need section factor, heat transfer, temperature gradients, material reduction factors, restraint, connection behavior, and load interaction. The defensible statement is therefore not that stainless steel expands by one fixed amount or loses strength along one predictable curve. ASTM STP 124 supplies the appropriate stainless-steel elevated-temperature context, while the material standard, grade, product condition, test method, temperature history, and design standard determine which data can be applied. ## Thermal Restraint, Thermal Stress, and Structural Compatibility Thermal expansion becomes a stress problem when a member cannot freely change length or shape. The free, unrestrained change in length is commonly written as ΔL=α⁢L⁢ΔT where α is the coefficient of linear thermal expansion, L is the original length, and ΔT is the temperature change. OpenStax *University Physics Volume 2* uses 12×10−6∘C−1 as a representative value for steel (2022). That value is useful for illustrating the calculation, but it is not a universal constant for every grade or temperature interval. The University of Sheffield’s summary of BS 5950 Part 8 gives 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C. Those numbers describe free expansion only. They do not describe the stress generated when expansion is opposed, and they do not describe the loss of yield strength, elastic modulus, or stiffness that can accompany heating. Thermal expansion, temperature-dependent mechanical degradation, and thermally induced stress must be treated as related but separate parts of the analysis. ### Fully restrained and partially restrained members A fully restrained steel member is prevented from making the relevant free thermal movement. For a prismatic bar heated uniformly while remaining elastic, the idealized axial stress is σth=−E⁢α⁢ΔT when the ends impose zero axial expansion. The negative sign indicates compression during heating. The corresponding axial force is Nth=σth⁢A, provided the modulus E, expansion coefficient α, and stress state remain appropriate for the temperature under consideration. #### Do not use the fully restrained formula uncritically The expression σth = −EαΔT assumes uniform temperature, complete restraint, elastic behaviour, and suitable temperature-dependent properties. Yielding, creep, relaxation, connection slip, and changing restraint can substantially alter the force. This expression is an upper-bound idealization, not a general fire-design result. At elevated temperature, E decreases, yield strength decreases, creep and stress relaxation may occur, and the expansion coefficient may vary with temperature. A member that initially develops a substantial compressive force can later experience force reduction as the steel relaxes or yields. If heating continues, the member may buckle, shed load to adjacent members, or change from compression-dominated behavior to a mechanism involving large deformation. “Fully restrained” also needs a precise definition. A support may block axial translation while allowing rotation. A beam-to-column connection may restrain rotation but provide little axial restraint. A floor slab, brace, fireproofing system, or adjacent frame may supply restraint through contact or compatibility. Local restraint can act at a flange, web, stiffener, bolt line, weld, or connection plate even when the member’s overall axial movement is not prevented. The restraint is therefore a property of the structural system, not of the isolated steel section. Members, connections, supports, bearing surfaces, anchors, diaphragms, and surrounding structure all contribute. A nominally fixed support may deform, a bolted connection may slip or rotate, and a concrete floor may impose restraint that changes as cracking and bond deterioration develop. Partial restraint is often more realistic. It can be represented conceptually by a spring with stiffness kr, so that the restraining force depends on the difference between the member’s free thermal movement and the movement permitted by the surrounding structure. A very stiff restraint approaches the fully restrained case; a flexible restraint permits more expansion and produces less thermal force. The relationship is not simply a reduced version of the fully restrained stress equation because the member and restraint deform together. Partial restraint changes the force–deformation response. At low temperature, the restraint may produce an elastic force. As the steel approaches yield, the member may accumulate plastic strain while the surrounding structure continues to deform. At higher temperature, creep and relaxation can reduce force without eliminating the imposed displacement. Connection slip, bolt-hole deformation, weld flexibility, bearing, local buckling, and contact separation can introduce further changes. A structural model that assigns either zero movement or zero force to every connection can miss these transitions. Cooling creates a different sequence. A restrained member that expanded during heating may contract as it cools, generating tension if the surrounding structure prevents contraction. Residual plastic strain can leave permanent deformation after the temperature returns to ambient. The final force may depend on the entire heating and cooling history rather than on the final temperature alone. ### Thermal gradients and bending Uniform temperature through a cross-section primarily produces axial expansion. A temperature gradient produces curvature. If the top flange of a beam is hotter than the bottom flange, the hotter region attempts to expand more. Compatibility between the parts of the section then requires bending strain, internal stress, or both. The same effect occurs across a web, around a welded joint, or between exposed and shielded faces. For a simple linear gradient through a section depth h, the free thermal curvature can be approximated by κth≈α⁢(Thot−Tcold)h. This curvature may cause vertical displacement even when the average temperature rise, and therefore the average axial expansion, is modest. A beam heated more strongly on one face can bow toward the cooler side. If its ends or intermediate restraints oppose that bowing, secondary moments and shears develop. Diaphragms and slabs may then attract additional forces, while connections experience rotation demands that were absent in a uniform-temperature calculation. Gradients also occur along the member. One segment may be exposed to fire while another remains near ambient temperature; one portion may be enclosed by insulation while another is adjacent to a ventilation opening. Differential expansion between these segments creates axial force and local bending. Near a connection, heat flow into a cooler adjoining member can produce steep temperature changes and concentrated compatibility stresses. Section geometry matters because heat does not penetrate every part at the same rate. Thin elements may heat quickly, while thick flanges, gusset plates, or connection components remain cooler. Thermal conductivity, heat capacity, and the coefficient of linear thermal expansion are identified by ASM as relevant physical properties of carbon and low-alloy steels. Heat transfer, section factor, surface protection, radiation, convection, and moisture also influence the temperature field. A single section-average temperature cannot represent all of these effects. A gradient can also alter mechanical properties unevenly. The hotter flange may have a lower yield strength and elastic modulus than the cooler flange, so the neutral axis, stiffness distribution, and plastic mechanism can shift during heating. Welding residual stresses and pre-existing imperfections may further influence where yielding or local buckling begins. ### Combined thermal and mechanical effects under ANSI/AISC N690-18 ANSI/AISC N690-18 addresses combined thermal and mechanical effects and restrained members. Strong evidence ANSI/AISC N690-18 provides a relevant framework for analyzing combined thermal and mechanical effects in steel structures, including structural members that may be restrained against thermal expansion. Its significance is analytical: thermal actions are considered together with mechanical actions and structural compatibility, rather than being treated as an isolated temperature increment added to an ambient-load calculation. The applicable load combinations, material models, acceptance criteria, and project requirements must still be taken from the governing edition and its referenced provisions. A member carrying gravity, seismic, pressure, or equipment loads before heating does not begin the thermal event with unused capacity. Initial axial force changes the response to thermal expansion; an already compressed column may approach instability as its stiffness falls, while a tensioned tie may develop additional force during heating before yielding or connection slip occurs. Bending moment and axial compression interact, and a thermal gradient can add curvature to the mechanically deformed shape. The analysis should therefore establish the temperature distribution, restraint conditions, mechanical loads, and relevant temperature-dependent properties. For members restrained against thermal expansion, the model must permit the development and redistribution of axial force. For partially restrained systems, connection and support stiffness should be represented where it affects compatibility. Local restraint and connection-induced forces may control even when the member’s average axial expansion is small. ANSI/AISC N690-18 is not a substitute for a temperature calculation or a material-property model. The engineer must account, as applicable, for heat transfer, thermal gradients, creep, relaxation, phase transformation, yielding, local buckling, connection behavior, and the heating and cooling path. EN 1993-1-2 addresses fire design and elevated-temperature resistance of steel members and connections, but its assumptions and reduction factors are not automatically interchangeable with those in ANSI/AISC N690-18. Stainless steels also require care: ASTM STP 124 (1994) reports elevated-temperature thermal-expansion and mechanical behavior for stainless steels, which can differ from carbon and low-alloy structural steel. The practical conclusion is direct: restraint cannot be assigned from the member label alone. It must be determined from the complete load path and its changing stiffness. A coefficient of expansion estimates free movement; only a compatible thermal–mechanical analysis can establish the forces, moments, deformations, and failure modes that occur when the surrounding structure resists that movement. ## Fire Design of Steel Members and Connections Under EN 1993-1-2 ### The scope of EN 1993-1-2 EN 1993-1-2 covers fire design and elevated-temperature resistance of steel members and connections. Strong evidence EN 1993-1-2 is the Eurocode standard for the fire design of steel structures. It deals with the resistance of steel members and connections at elevated temperature, alongside the thermal actions and material properties needed to assess an accidental fire situation. The European Commission Joint Research Centre identifies EN 1993-1-2 as the Eurocode document covering the fire resistance of steel members and connections ([JRC, 2005](https://eurocodes.jrc.ec.europa.eu/publications/en-1993-1-2-resistance-members-connections-fire "JRC, 2005")). That scope is wider than assigning a thermal-expansion coefficient to a beam. A temperature rise produces free expansion, but a real member may be restrained by slabs, columns, bracing, partitions, or its connections. The resulting forces can be small where movement is free, or substantial where movement is prevented. At the same time, steel loses stiffness and strength as temperature increases. The member’s temperature may vary through its depth, along its length, and around its perimeter. A fire calculation therefore combines heat transfer, temperature distribution, mechanical loading, instability, material degradation, and structural restraint. The familiar relation ΔL=α⁢L⁢ΔT is a useful starting point for unrestrained linear expansion. OpenStax University Physics Volume 2 uses 12×10−6∘C−1 as a representative coefficient for steel ([OpenStax, 2022](https://openstax.org/books/university-physics-volume-2/pages/1-3-thermal-expansion "OpenStax, 2022")). That figure is not a universal constant for every steel grade or temperature interval. A University of Sheffield reference states that BS 5950 Part 8 recommends 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C ([University of Sheffield, 2006](https://core.ac.uk/download/pdf/9554664.pdf "University of Sheffield, 2006")). Those values illustrate why the temperature range and the selected design rules matter. EN 1993-1-2 should not be confused with a general catalogue of measured material properties. Experimental data and design provisions serve different purposes. ASM references for carbon and low-alloy steels identify thermal conductivity, heat capacity, and the coefficient of linear thermal expansion as important physical properties. ASTM STP 124, published in 1994, reports elevated-temperature properties of stainless steels, including thermal expansion and mechanical behaviour ([ASTM International, 1994](https://store.astm.org/stp124-eb.html "ASTM International, 1994")). These sources help describe how materials behave; the Eurocode converts selected behaviour into design models, reduction factors, equations, limits, and procedures. The applicable edition of EN 1993-1-2, its National Annex, the governing fire design standard, and any project-specific national provisions must be checked before calculation. A fire-resistance period alone does not define the problem. The designer must establish the fire scenario, the design thermal action, the load combination, the exposed surfaces, the protection system if present, and the boundary conditions for heat flow. ### Member resistance at elevated temperature Steel member resistance falls because elevated temperature changes several mechanical properties at once. Yield strength decreases, the elastic modulus decreases, stress–strain behaviour becomes less sharply defined, and time-dependent deformation becomes increasingly important. The reduction in modulus is especially significant for buckling calculations: a member can lose stability well before a simple comparison with ambient-temperature yield strength suggests collapse. EN 1993-1-2 provides temperature-dependent design treatment for these effects. In a typical assessment, the member temperature is first obtained from a thermal analysis or an approved simplified temperature model. The resistance calculation then uses the relevant elevated-temperature material parameters and checks the governing failure mode. Depending on the member, that may involve cross-section resistance, axial compression, tension, bending, shear, lateral-torsional buckling, or interaction between axial force and bending moment. The temperature used in the resistance check must represent the steel that carries the force. A protected column may have a lower and more uniform temperature than an unprotected beam exposed on three sides. A hollow section, a solid plate, and an open I-section absorb and lose heat differently. The section factor, commonly expressed through heated perimeter relative to steel volume or area, affects the rate of heating. Protection thickness, moisture, contact quality, shadow effects, emissivity, convection, and the fire exposure on each surface can also alter the result. A single average temperature can be misleading when a strong gradient exists. The flange facing the fire may be considerably hotter than the cooler side, creating curvature and secondary stresses. Temperature differences between connected members can force compatibility actions at joints. For this reason, the thermal model and the mechanical model cannot be selected independently of the section geometry and fire exposure. The steel grade remains relevant, although the high-temperature rules may group grades under common design assumptions. Carbon-manganese structural steels and stainless steels do not have identical thermal or mechanical responses. Research also shows that stress relaxation, creep, and phase transformation can affect deformation and load redistribution. NIST IR 88-3899, published on 1 January 1989, examined the deformation behaviour of structural steel at elevated temperature ([NIST, 1989](https://www.nist.gov/publications/elevated-temperature-deformation-structural-steel "NIST, 1989")). A separate study, *An Introduction to the Mechanical Properties of Structural Steel at Elevated Temperatures*, discusses Poisson’s ratio, thermal expansion, phase transformation, stress–strain response, and elastic modulus. Measurements reported in *Experimental Studies on the Properties of Constructional Steel at Elevated Temperatures* include yield strength, tensile strength, modulus of elasticity, elongation, and thermal-expansion coefficient. These results should not be inserted directly into an EN 1993-1-2 calculation without checking their status. A test value for ASTM A723 or D6AC steel, for example, is evidence about a specified material and test method, not automatically a Eurocode design value for a different grade, product form, or loading history. The NTIS record for *Elevated Temperature Properties of Steels* describes measurements of specific heat, thermal expansion, stress relaxation, tensile properties, Young’s modulus, and Poisson’s ratio for ASTM A723 and D6AC steels. Such data can inform engineering judgment, but design requires compatibility with the adopted standard. ### Connections, stability, and thermal input Connections often control the fire response even when the connected members retain sufficient calculated resistance. Bolts, welds, end plates, angles, cleats, haunches, and supporting components may heat at different rates. A connection can attract additional force when thermal expansion is restrained, or it can lose resistance through reduced bolt strength, plate yielding, weld deterioration, or local buckling. The force path must therefore be traced at elevated temperature rather than assumed to remain identical to the ambient-temperature arrangement. EN 1993-1-2 includes provisions for fire design of connections, while connection resistance and component behaviour must be read with the relevant Eurocode documents, including EN 1993-1-8 where applicable. The design may require checks on bolts, welds, plates, angles, component deformation, and the transmission of axial force, shear, and moment. A protected connection also needs a credible protection detail. Stopping protection at the edge of a member does not establish that the joint has the same temperature as the member. Stability is affected by both thermal expansion and reduced stiffness. A restrained beam may develop compressive axial force as it heats; later, sagging, catenary action, or connection deformation may change the force pattern. Columns can experience second-order effects as stiffness falls. Frames can redistribute load into cooler members, but that redistribution is not automatically beneficial because it may overload connections, bracing, or floors. A calculation based only on the initial gravity load and an isolated member can miss these system effects. Thermal input must be defined before temperature can be calculated. The designer should identify the nominal fire curve or other specified fire model, duration, exposed faces, convection and radiation assumptions, and any active or passive protection. The section factor and the thermal properties of steel and protection govern heat transfer. Thermal gradients, restraint, creep, relaxation, and phase changes can then influence the mechanical response. ANSI/AISC N690-18 provides a useful comparison outside the Eurocode system: it addresses combined thermal and mechanical effects, including members restrained against thermal expansion ([AISC, 2018](https://www.aisc.org/globalassets/aisc/publications/standards/n690-18w.pdf "AISC, 2018")). The practical position is clear: thermal expansion is one input, not the fire design. An actual EN 1993-1-2 assessment must verify the adopted edition, National Annex, fire scenario, steel grade, section condition, protection, temperature distribution, load combination, member stability, and connection model. Only then can a temperature-dependent resistance check represent the structure rather than a coefficient in isolation. ## How Elevated-Temperature Properties Are Measured Elevated-temperature steel data do not come from one universal test. A laboratory may measure free expansion, load-bearing capacity, stiffness, heat storage, or time-dependent deformation, and each measurement answers a different question. The constructional-steel experimental study and the NTIS record for *Elevated Temperature Properties of Steels* show the range clearly: reported properties include thermal-expansion coefficient, yield strength, tensile strength, elongation, Young’s modulus, Poisson’s ratio, specific heat, and stress relaxation. These quantities are related, but they are not interchangeable. A steel bar can expand while carrying no load, lose strength while its temperature is held constant, or develop substantial stress when expansion is restrained. Those are three separate responses. Test results must therefore be read with the grade, specimen condition, temperature path, loading arrangement, and applicable standard in view. ![Steel specimen mounted in a dilatometer for measuring thermal expansion.](/images/uploads/d7e45936-0b27-4cce-9383-ca6950e189a7/wiki-inline-a-steel-dilatometer-specimen-being-heated-while-its-length-change-and-temperatur-1920x1288.jpg)[](/images/uploads/d7e45936-0b27-4cce-9383-ca6950e189a7/wiki-inline-a-steel-dilatometer-specimen-being-heated-while-its-length-change-and-temperatur-1920x1288.avif "Enlarge image — Steel specimen mounted in a dilatometer for measuring thermal expansion.")A dilatometer records length change as a steel specimen follows a controlled temperature path. ### Thermal-expansion testing Thermal expansion is commonly measured with a dilatometer. The specimen is heated while its change in length is recorded, usually with a displacement sensor linked to a temperature measurement near the gauge section. For small, uniform temperature changes, the basic relation is ΔL=α⁢L⁢ΔT where L is the initial length, ΔL is the change in length, ΔT is the temperature change, and α is the linear coefficient of thermal expansion. OpenStax *University Physics Volume 2* (2022) uses 12×10−6∘C−1 as a representative value for steel. The University of Sheffield’s 2006 reference reports that BS 5950 Part 8 recommends 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C. Those figures are useful reference values, not a single coefficient applicable to every steel and every interval. A test may report a secant coefficient, calculated from the total expansion between two temperatures, or a tangent coefficient, taken from the local slope of the expansion curve. The two values can differ when the curve is nonlinear. Chemical composition, prior heat treatment, cold work, and phase transformation also affect the result. Carbon and low-alloy steels change their expansion behavior near transformation ranges as ferrite, pearlite, austenite, and other phases develop. Stainless steels can show different curves again; ASTM STP 124, published in 1994, records elevated-temperature thermal-expansion and mechanical data for stainless steels rather than treating “steel” as one material. A dilatometer test also requires careful temperature control. The furnace temperature may not equal the specimen temperature, particularly at a high heating rate or with a thick specimen. A temperature gradient produces an apparent length change that includes uneven thermal distortion. Specimen geometry matters for the same reason: a long, slender sample may equilibrate differently from a short, thick one. Contact friction, sensor alignment, oxidation, and the reference material used for instrument correction can influence the measured displacement. Free-expansion data do not directly give the stress in a restrained member. If expansion is prevented, the thermal strain is converted into mechanical stress according to the restraint and the temperature-dependent stiffness, with plasticity, creep, and relaxation reducing or redistributing that stress. ANSI/AISC N690-18 addresses combined thermal and mechanical effects, including members restrained against thermal expansion. A coefficient alone cannot predict the force in a fixed column, frame, bolt group, or connection. Specific heat is usually measured by calorimetric methods rather than by the dilatometer. It describes the heat required to raise the temperature of a unit mass by one degree. Its value affects the temperature rise produced by a given heat input, while thermal conductivity affects how quickly heat moves through the section. The NTIS record identifies measurements of specific heat alongside thermal expansion and mechanical properties for ASTM A723 and D6AC steels. These thermal properties should not be substituted for one another: expansion describes strain, specific heat describes energy storage, and conductivity describes heat flow. ### Tensile, modulus, and stress–strain testing Elevated-temperature tensile testing places a heated specimen in a calibrated testing machine and records force, extension, and temperature. From the resulting stress–strain curve, investigators may determine yield strength, tensile strength, elongation, and Young’s modulus. The definitions require attention. Yield strength may be obtained from a yield point, a proportional limit, or a specified offset strain. Tensile strength is the maximum engineering stress. Elongation depends on gauge length, specimen shape, fracture location, and whether the reported value is measured after fracture or continuously during the test. Young’s modulus is usually determined from the initial slope of the stress–strain curve, but that slope becomes difficult to measure as temperature rises. Machine compliance, thermal expansion of the grips, seating effects, strain-sensor drift, and small amounts of inelastic strain can distort it. Poisson’s ratio requires simultaneous measurement of axial and transverse strain. At elevated temperature, transverse strain gauges may lose accuracy, so optical or extensometer methods may be used instead. The NTIS work on ASTM A723 and D6AC specifically included Young’s modulus and Poisson’s ratio, while *Experimental Studies on the Properties of Constructional Steel at Elevated Temperatures* reported yield strength, tensile strength, modulus of elasticity, elongation, and thermal-expansion coefficient. A “strength at 600 °C” value is incomplete without the test condition. In a transient test, the specimen is heated while the load is applied or increased, so temperature and strain evolve together. In a steady-state test, the specimen is first heated to a selected temperature and held until thermal equilibrium is approached, then loaded. A third approach holds the load while temperature increases. These procedures can produce different curves because steel has time-dependent deformation and because the material may transform during the temperature hold. Loading rate is equally important. A rapid tensile test can record a higher apparent strength than a slow test because creep and stress relaxation have less time to occur. A long dwell at temperature can lower the subsequent yield strength or alter the shape of the stress–strain curve. The atmosphere matters too. Air can promote oxidation and scale formation; controlled gas or vacuum conditions may suppress some surface reactions. Oxidation reduces the effective cross-section and can affect gripping or strain measurement. The constructional-steel study should therefore be read as a defined experimental program, not as a universal strength curve for all constructional steels. Grade designation, specimen preparation, heating schedule, temperature accuracy, dwell period, strain rate, and property definition all belong with the reported number. EN 1993-1-2 provides a design framework for the fire resistance of steel members and connections, but design values and reduction factors selected under that standard should not be mixed casually with raw data from a laboratory program conducted under different conditions. ### Specimen history, heating rate, and data interpretation The specimen’s history begins before it enters the furnace. ASTM A723 and D6AC, for example, are not interchangeable labels for one generic steel; their composition, processing, strength level, and prior heat treatment affect elevated-temperature response. A rolled constructional-steel coupon, a quenched-and-tempered forging, and a cold-worked stainless-steel specimen can respond differently even at the same nominal temperature. Heating rate controls the time available for conduction, phase change, recovery, creep, and oxidation. A fast ramp can leave the centre of a thick specimen cooler than its surface. A slow ramp can permit transformations before the nominal test temperature is reached. Dwell time then determines whether the specimen approaches thermal equilibrium and whether stress relaxation or microstructural change proceeds during the hold. NIST IR 88-3899, published on January 1, 1989, examined deformation behavior of structural steel at elevated temperature; its relevance lies partly in treating deformation as temperature- and time-dependent rather than as an instantaneous loss of strength. Stress relaxation is measured by imposing a fixed strain, or a closely controlled deformation, and observing the fall in stress with time at a constant temperature. Creep testing reverses the emphasis: a sustained stress is applied and strain accumulates. Neither behavior can be inferred reliably from a short tensile test. Nor can a free-expansion coefficient describe relaxation in a restrained beam. For design interpretation, the test temperature should be connected to the member’s actual temperature distribution. Section factor, heat transfer, thermal gradients, restraint, connection behavior, applied load, and fire exposure duration can all change the structural result. Two programs reporting “yield strength at 500 °C” may have measured different material states and different kinds of yield. Their values should remain separate unless the test methods, steel grades, thermal histories, and definitions are demonstrably comparable. That discipline prevents a representative coefficient, a transient tensile result, and a code reduction factor from being presented as though they were measurements of the same property. ## Calculating Thermal Expansion: Worked Frameworks Without False Precision Thermal expansion is often introduced with a compact equation: ΔL=α⁢L⁢ΔT OpenStax University Physics Volume 2 (2022) presents this linear relation and uses 12×10−6∘C−1 as a representative coefficient for steel. That number is useful for demonstrating the calculation, but it is not a universal property of every steel grade over every temperature range. A calculation should therefore show which coefficient was selected, the temperature interval to which it applies, and whether the result describes a genuinely free member. ### Free-length-change calculation Let: - L0 = initial length at the reference temperature; - T0 = reference temperature; - T1 = final temperature; - ΔT=T1−T0; - α = linear coefficient of thermal expansion; - ΔL = change in length; - L1=L0+ΔL = final length. For a member whose temperature is sufficiently uniform and whose coefficient can be treated as constant, ΔL=α⁢L0⁢(T1−T0). The sign follows directly from the temperature interval. Heating gives ΔT\>0 and, for a positive coefficient, ΔL\>0. Cooling gives ΔT\<0, so the calculated change is negative. Calculated free expansion (mm) Illustrative expansion calculations using the coefficients and dimensions stated in the article; these are not universal material-property measurements. Consider a deliberately labeled illustrative case. Assume a straight steel bar has L0=6.000m, starts at T0=20∘⁢C, reaches T1=220∘⁢C, and is assigned α=12×10−6∘C−1. The coefficient is an assumption for this example, not a claim about the bar’s grade. ΔT=220−20=200∘⁢C ΔL=(12×10−6∘C−1)⁢(6.000m)⁢(200∘⁢C) ΔL=0.0144m=14.4mm. The predicted free length is therefore 6.0144m. Since a temperature difference of one kelvin has the same size as one degree Celsius, α may be expressed per kelvin or per degree Celsius in this difference calculation, provided the units are used consistently. The arithmetic is simple; selecting α is not always simple. A University of Sheffield reference (2006), discussing BS 5950 Part 8, reports 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600∘⁢C. If the illustrative bar were heated from 200∘⁢C to 600∘⁢C, using the latter code-based value would give ΔL=(14×10−6)⁢(6.000)⁢(400)=0.0336m, or 33.6mm. That result is not interchangeable with one calculated using 12×10−6∘C−1, and neither value should automatically be transferred to stainless steel, alloy steel, or a different design standard. The formula predicts free movement only. It does not predict the force in a support, the temperature at which a connection slips, or the distortion caused by a temperature gradient. A beam heated more on one flange than the other may curve even when its average axial length change is modest. A member attached to a cooler frame may also experience restraint before its entire cross-section reaches the same temperature. ### Temperature-dependent or integrated coefficients A constant α is acceptable when the temperature interval is narrow, the selected data source treats the coefficient as constant over that interval, and the accuracy required does not justify a temperature-dependent model. It can also be a reasonable screening assumption when the temperature field and boundary conditions are still uncertain; extra decimal places in α would then create false confidence. For a wider range, or where the coefficient changes materially with temperature, write the free strain as εth=∫T0T1α⁢(T)dT and the length change as ΔL=L0∫T0T1α⁢(T)dT. If the temperature range is divided into intervals over which different constant values are assigned, ΔL=L0⁢(α1⁢ΔT1+α2⁢ΔT2+⋯+αn⁢ΔTn). For example, suppose the same 6.000m bar is assigned 12×10−6∘C−1 from 20 to 200∘⁢C, and 14×10−6∘C−1 from 200 to 600∘⁢C. The illustrative expansion becomes ΔL=6.000⁢\[(12×10−6)⁢(180)+(14×10−6)⁢(400)\]. Thus, ΔL=6.000⁢(0.00216+0.00560)=0.04656m, or 46.56mm. This is a piecewise approximation, not a measured property curve. A real calculation should take the values and interpolation rules from the applicable material specification, test data, or design standard. Temperature history can matter as well. Structural steels may undergo metallurgical changes during heating and cooling, and the coefficient need not describe a reversible, single-phase response through the entire cycle. Phase transformation can alter the expansion curve; transformation strain may also contribute to total deformation. ASTM STP 124 (1994) reports elevated-temperature thermal-expansion and mechanical data for stainless steels, while NIST IR 88-3899 (1989) addresses deformation behavior of structural steel at elevated temperature. Those subjects cannot be reduced to one ambient-temperature coefficient. Thermal expansion must also be kept separate from strength loss. The coefficient estimates a free geometric strain. It does not state how yield strength, tensile strength, elastic modulus, Poisson’s ratio, creep, or stress relaxation change. Research on constructional steel at elevated temperatures measures these properties separately, and the NTIS record for *Elevated Temperature Properties of Steels* describes testing of ASTM A723 and D6AC steels for thermal expansion, specific heat, tensile behavior, Young’s modulus, Poisson’s ratio, and stress relaxation. ### Checks on units and assumptions First, verify the initial length. Use the length at the stated reference temperature, not a drawing dimension whose temperature is unknown. For a long member, distinguish total length from the distance between the restraints or connection centrelines. Second, calculate the temperature interval explicitly: ΔT=T1−T0. Do not substitute the final temperature for the temperature change. A bar going from −20∘⁢C to 80∘⁢C has ΔT=100∘⁢C, not 80∘⁢C. Third, check coefficient units. If α is 12×10−6∘C−1, multiplying by a temperature difference in degrees Celsius produces a dimensionless strain. Multiplying that strain by metres produces metres. A value reported in 10−6/K has the same numerical use for a temperature difference in kelvins. Fourth, ask whether the member is free. If it is free to move axially, the calculated ΔL is primarily a displacement prediction. If movement is prevented, the same free thermal strain becomes an imposed strain. In a simplified, fully restrained, elastic member, σth=−E⁢(T)⁢α⁢(T)⁢ΔT, with compression on heating under the stated sign convention. The corresponding force is Nth=σth⁢A, where A is the relevant cross-sectional area. This elementary result is only a starting point: partial restraint, joint flexibility, friction, initial load, temperature gradients, yielding, creep, relaxation, and changing elastic modulus can all reduce or redistribute the force. At elevated temperature, E⁢(T) is not the ambient elastic modulus, and an elastic calculation may cease to describe the member after yielding or transformation. EN 1993-1-2 addresses fire design and elevated-temperature resistance of steel members and connections. ANSI/AISC N690-18 addresses combined thermal and mechanical effects, including members restrained against thermal expansion. The selected standard governs the permitted material curves, thermal model, restraint treatment, and safety format. A coefficient alone is not a design model. ## Common Errors in Interpreting Steel Thermal-Property Data ### Treating one coefficient as a material constant The expression ΔL=α⁢L⁢ΔT, given in *OpenStax University Physics Volume 2* (2022), is a useful first approximation for linear thermal expansion. It is not permission to assign one value of α to every steel at every temperature. OpenStax uses 12×10−6∘C−1 as a representative coefficient for steel. The University of Sheffield’s 2006 reference on fire design reports that BS 5950 Part 8 recommends 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C. Those are defined assumptions for particular calculation ranges, not universal physical constants. The distinction matters because published expansion values may be instantaneous coefficients at a stated temperature or mean coefficients averaged between two temperatures. A mean value from 20 to 600 °C cannot automatically be substituted for an instantaneous value at 600 °C. Chemical composition, prior heat treatment, cold work, grain structure, and phase condition all affect the result. Austenitic stainless steels, for example, generally have different expansion behavior from carbon and low-alloy structural steels. ASTM STP 124, published in 1994, reports elevated-temperature thermal expansion and mechanical behavior for stainless steels rather than treating “stainless steel” as one material. Temperature itself can change the slope. Carbon and low-alloy steels may pass through transformation ranges in which ferrite, pearlite, bainite, or martensite-related structures change toward austenite during heating. That transformation can produce an expansion departure, or even a temporary contraction, that a straight-line calculation misses. On cooling, the reverse transformations and the heating history affect the path again. A specimen heated once at a slow rate is not necessarily described by the same curve as a specimen rapidly heated and then held. The coefficient also describes free thermal strain, not the complete movement of a structure. A freely expanding bar develops approximately εth=∫T0Tα⁢(T)dT. If the temperature is nonuniform, the integral differs from one point to another and creates curvature or secondary forces. A restrained bar cannot freely develop this strain. In a simplified elastic case, restraint produces a stress related to E⁢(T)⁢εth, but that stress is altered by yielding, creep, stress relaxation, connection slip, and geometric deformation. A single α therefore cannot predict member force, connection force, or fire resistance. A quoted value is incomplete unless it identifies the source, steel grade or designation, temperature interval, heating condition, and measurement method. “Steel: 12×10−6∘C−1” lacks all five qualifications. ### Confusing thermal strain with strength retention Thermal expansion answers a kinematic question: how much would an unconstrained material change length? Strength retention answers a mechanical question: how much load can the material carry at a specified temperature, strain rate, and deformation history? They are related through the structural response, but they are not interchangeable. A steel member heated from 20 to 600 °C may develop substantial thermal strain even if it carries no external load. Its yield strength and tensile strength may also fall, while its elastic modulus declines, changing the relationship between stress and elastic strain. These changes do not follow directly from α. Conductivity controls how quickly heat spreads; heat capacity controls the energy required to raise temperature; neither one gives yield strength. The thermal expansion coefficient controls free strain. Young’s modulus, yield strength, tensile strength, elongation, Poisson’s ratio, creep, and stress relaxation describe different parts of the mechanical response. The common assumption that a hot modulus equals the room-temperature modulus is especially damaging. A room-temperature modulus may be used in an initial calculation, but it cannot represent the tangent or elastic stiffness of a heated member without a stated justification. The article *An Introduction to the Mechanical Properties of Structural Steel at Elevated Temperatures* treats modulus, Poisson’s ratio, thermal expansion, phase transformation, and stress–strain behavior as separate temperature-dependent quantities. The NTIS record for *Elevated Temperature Properties of Steels* describes measurements of specific heat, thermal expansion, stress relaxation, tensile properties, Young’s modulus, and Poisson’s ratio for ASTM A723 and D6AC steels. The separation is deliberate: each property requires its own data and interpretation. Tensile data also do not directly equal member resistance. A tensile coupon has a controlled geometry, a known heated length, and a prescribed test history. A column, beam, or connection has a temperature field, section factor, imperfections, residual stress, local buckling risk, axial restraint, and interaction between axial force and bending. EN 1993-1-2 addresses fire design and elevated-temperature resistance of steel members and connections because member resistance requires more than multiplying a room-temperature capacity by a tabulated strength-reduction factor. The same principle applies outside fire design. Heating condition changes the reported result. In a steady-state test, the specimen is heated to a target temperature and then loaded while temperature is approximately constant. In a transient-state test, load may be held while temperature rises. Creep and stress relaxation can become significant during a hold. A restrained member may initially develop compression as it expands, then shed that force through yielding and relaxation. After cooling, permanent strain and residual stress may remain. A tensile curve from a rapidly loaded coupon cannot be treated as a direct prediction of that entire cycle. ANSI/AISC N690-18 explicitly addresses combined thermal and mechanical effects, including members restrained against thermal expansion. That provision reflects a basic structural fact: restraint can create large forces before material failure, while loss of stiffness and strength changes how those forces redistribute. Thermal strain is one input. It is not a strength curve. ### Ignoring grade, phase, and test-method differences “Steel at 500 °C” is not a sufficiently identified test condition. The grade must be stated. ASTM A723 and D6AC steels, carbon structural steels covered by common building specifications, and austenitic stainless grades do not share one elevated-temperature response. Even within a grade, normalized, quenched-and-tempered, welded, cold-worked, and stress-relieved material may show different results. Phase condition is equally important. Heating through a transformation interval changes the microstructure and may alter expansion, yield behavior, ductility, and subsequent cooling properties. A temperature-only table can conceal whether the material remained below transformation, crossed it during heating, or was held long enough for the transformation to progress. NIST IR 88-3899, published on January 1, 1989, reports research on deformation behavior of structural steel at elevated temperature; its relevance is not limited to a single number because deformation depends on stress, temperature, and time. Test method can change the apparent property even when the grade is identical. Thermal expansion measured by dilatometry may be reported as a continuous curve, while a design table may provide an average coefficient over a broad interval. Yield strength depends on the adopted offset or proof-stress definition, specimen orientation, strain rate, and whether the test is steady-state or transient-state. Modulus may mean an initial elastic slope, a secant modulus, or a tangent modulus. Poisson’s ratio may vary with temperature and loading range. Creep data require duration, stress, and temperature; stress-relaxation data require a prescribed strain or displacement history. The reporting minimum is therefore clear: identify the source, exact grade and designation, temperature range, heating rate or heating condition, duration at temperature, atmosphere where relevant, specimen orientation, and test method. State whether the value is measured, fitted, or adopted from a design standard. Then identify the standard governing the calculation. BS 5950 Part 8, EN 1993-1-2, and ANSI/AISC N690-18 do not constitute interchangeable data packages. Finally, thermal analysis must include conductivity, heat capacity, heat transfer, temperature gradients, and section factor before mechanical resistance is assessed. Connections may restrain expansion differently from adjacent members, and restraint can make local temperature differences structurally significant. A coefficient is necessary for many calculations. By itself, it is nowhere near sufficient. ## Selecting Data for Engineering and Reference Use Selecting a thermal-expansion coefficient or elevated-temperature strength value is not a matter of finding a single number labelled “steel.” The usable value depends on the question being asked, the grade and product condition, the temperature interval, the property definition, and the standard governing the calculation. Thermal expansion, mechanical degradation, and thermally induced stress are related but separate subjects. A source that is adequate for explaining one may be unsuitable for the other. #### Source-selection hierarchy - **Design standard** Use the governing standard for design checks and permitted calculation procedures. - **Grade-specific data** Use certified or experimentally documented results for material behaviour. - **Research reports** Use defined test programmes for deformation, strength, creep, relaxation, or phase effects. - **Educational references** Use textbooks for equations and first-order explanations. - **Representative values** Use generic coefficients only when their assumptions and limits are stated. A practical source-selection hierarchy follows that distinction. Use the governing design standard for a design check. Use grade-specific certified data or experimentally documented results for material behaviour. Use textbooks and educational references for basic equations and first-order explanations. Generic representative coefficients can clarify the subject, but they should not be transferred automatically into safety-critical design. ### Match the source to the question For a basic explanation of free thermal expansion, OpenStax *University Physics Volume 2* is an appropriate reference. Its 2022 treatment gives the linear relation ΔL=α⁢L⁢ΔT and uses 12×10−6∘C−1 as a representative coefficient for steel. That equation describes an unconstrained member under a specified temperature change; it does not predict the force in a restrained member, the loss of yield strength, or the deformation produced by a non-uniform temperature field. The distinction becomes clear when comparing educational and engineering references. The University of Sheffield reference, published in 2006, reports that BS 5950 Part 8 recommends 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C. Those figures are useful evidence that the adopted coefficient can vary with temperature and with the source’s design convention. They are not proof that every carbon steel, low-alloy steel, stainless steel, or heat-treated product follows that same curve. For physical-property selection, ASM references on carbon and low-alloy steels are more suitable than a general physics text. They identify thermal conductivity, coefficient of linear thermal expansion, and heat capacity as relevant properties, which matters in transient heating calculations. A thermal model needs heat capacity and conductivity as well as expansion. The expansion coefficient alone cannot establish the temperature distribution through a thick flange, plate, vessel wall, or connection. For stainless steels, ASTM STP 124, published in 1994, reports elevated-temperature properties including thermal expansion and mechanical behaviour. Its role is different from that of OpenStax: it provides documented elevated-temperature information for stainless-steel materials rather than a representative classroom value for steel as a broad category. The selected data still need to match the stainless grade, thermal condition, test direction, temperature range, and property definition. Research reports can answer questions that tables do not. NIST IR 88-3899, *Elevated Temperature Deformation of Structural Steel*, published on January 1, 1989, reports research on the deformation behaviour of structural steel at elevated temperature. It is relevant when the issue is strain development, not merely free expansion. Likewise, *An Introduction to the Mechanical Properties of Structural Steel at Elevated Temperatures* discusses Poisson’s ratio, thermal expansion, phase transformation, stress–strain behaviour, and elastic modulus. That combination is important because a restrained steel member may develop thermal stress while its modulus and yield strength are changing. The paper *Experimental Studies on the Properties of Constructional Steel at Elevated Temperatures* is appropriate when measured property changes are required. It reports measurements of yield strength, tensile strength, modulus of elasticity, elongation, and thermal-expansion coefficient. These properties do not degrade in identical ways or at identical temperatures. A yield-strength curve cannot be substituted for a modulus curve, and neither can be inferred safely from an expansion coefficient. The NTIS record for *Elevated Temperature Properties of Steels* describes measurements of specific heat, thermal expansion, stress relaxation, tensile properties, Young’s modulus, and Poisson’s ratio for ASTM A723 and D6AC steels. That specificity is a strength of the record and also a limitation: results for ASTM A723 and D6AC should not be presented as generic data for all structural steels. Stress relaxation is particularly relevant to sustained restraint and long heating periods, where an initial thermal stress can decline even while temperature remains high. ### Material designation and traceability Every value copied into a calculation or database should carry an exact material designation. “Structural steel” is not enough. Record the grade, product standard, heat-treatment condition where relevant, product form, and test orientation. ASTM A723 and D6AC, for example, identify materials in the NTIS record; they are not interchangeable labels. A stainless-steel result from ASTM STP 124 also requires the exact grade or designation used in the reported test. Traceability must include the source title, publisher or institution, publication year, edition, table or figure number, and any stated test method. The temperature interval must be recorded rather than replaced by a single midpoint value. State whether the coefficient is instantaneous, secant, mean over an interval, or an integrated expansion value. “Coefficient of thermal expansion” can refer to different quantities, and confusing them can introduce an avoidable error into ΔL. The same discipline applies to mechanical properties. Identify whether a reported strength is yield strength, proportional-limit stress, proof stress, tensile strength, or an effective value used in a code model. Record whether modulus means Young’s modulus measured during heating, after cooling, or during a specified loading path. Heating history matters: phase transformation, prior plastic strain, cooling rate, and repeated thermal cycles can alter the response. A source record should also state whether the specimen was free to expand or mechanically restrained. An expansion test and a restrained-heating test answer different questions. The latter may include stress redistribution, creep, relaxation, and contact or connection effects. Without that context, a numerically precise value can be misleading. ### When code values supersede generic references For a design check, the governing standard takes precedence over a generic reference unless the design authority explicitly permits another basis. EN 1993-1-2 addresses the fire design and elevated-temperature resistance of steel members and connections within the Eurocode system. Its adopted material relationships, reduction factors, thermal properties, and calculation procedures are therefore the controlling source for a design carried out to that standard, subject to the applicable National Annex and project requirements. ANSI/AISC N690-18 has a different field of application but illustrates the same principle. Published in 2018, it addresses combined thermal and mechanical effects, including structural members that may be restrained against thermal expansion. A calculation under that standard must follow its definitions and load-effects framework rather than importing a coefficient from OpenStax or a value reported for another grade and test programme. Code values are not universal physical constants. They are selected representations intended to support a defined design method. A code may prescribe a coefficient, reduction factor, or stress–strain relationship that differs from a measured value because it must provide a consistent calculation procedure, account for uncertainty, or align with other provisions. The designer must therefore use the edition specified by the project and verify amendments, National Annex provisions, and material scope. Even when a code supplies the coefficient, the calculation still requires temperature distribution, thermal gradients, restraint, heat transfer, section factor, connection behaviour, creep, relaxation, and possible phase changes to be considered where relevant. A uniform temperature assumption may be acceptable for one member and unsafe for another. A free-expansion calculation may describe movement, while the structural problem is actually the force generated by restraint. #### Choosing a reference Use an educational source for the expansion equation, matched research or material data for documented behaviour, and the governing design standard for design decisions. The correct hierarchy is therefore clear: use OpenStax or a comparable educational source for the equation; use ASM, ASTM STP 124, NIST IR 88-3899, the Sheffield reference, the constructional-steel experimental study, or the NTIS record for documented material behaviour matched to its designation; and use EN 1993-1-2, ANSI/AISC N690-18, or the applicable governing standard for design decisions. Each selected value should state the exact grade or designation, temperature interval, property definition, source edition or publication year, and limitations. Without those fields, the number is a citation fragment, not engineering data. ## Reference Tables and Terminology for Steel at Elevated Temperature ### Symbols and definitions Thermal expansion is a geometric response, not a strength value. For a uniform temperature change in a simple, unrestrained member, OpenStax University Physics Volume 2 (2022) gives the linear relation: ΔL=α⁢L⁢ΔT Here, ΔL is the change in length, L is the original length, ΔT is the temperature change, and α is the coefficient of linear thermal expansion. OpenStax uses 12×10−6∘C−1 as a representative value for steel. That number is useful for illustrating the calculation, but it is not a universal constant for every grade, temperature interval, or metallurgical condition. The coefficient may be reported in K−1 or ∘C−1. For temperature differences, one kelvin and one degree Celsius have the same numerical size, so these units are numerically interchangeable. The temperature itself must still be identified as °C or K. A table may define α in two different ways. The **instantaneous**, or differential, coefficient is the local slope: α⁢(T)=1L⁢dLdT It describes expansion over a very small interval near temperature T. The **mean**, or secant, coefficient between T1 and T2 is: α¯T1,T2=L⁢(T2)−L⁢(T1)L⁢(T1)⁢(T2−T1) A mean value over 20–600 °C should not be substituted automatically for an instantaneous value at 600 °C. If expansion changes rapidly near a phase transformation, the difference can matter. For a varying coefficient, the more appropriate expression is: εth=ΔLL0=∫T0Tα⁢(θ)d⁢θ where εth is thermal strain and θ is an integration variable. A table that lists thermal strain directly may already include temperature-dependent expansion and transformation effects; applying α⁢ΔT again would double-count them. BS 5950 Part 8, as reported by the University of Sheffield in 2006, recommends 12×10−6∘C−1 at ambient temperature and 14×10−6∘C−1 from 200 to 600 °C. These are code-based assumptions for the stated application, not proof that all carbon, low-alloy, stainless, or tool steels follow the same curve. ### Property categories Reference tables become easier to interpret when properties are separated by physical role. **Thermal transport properties** describe how heat enters and moves through a component. Thermal conductivity, k, measures heat flow driven by a temperature difference and is commonly expressed in Wm−1K−1. It can vary substantially with grade, temperature, composition, and microstructure. ASM references on carbon and low-alloy steels identify thermal conductivity, coefficient of linear thermal expansion, and heat capacity as relevant physical properties. Heat capacity, C, is the energy required to raise the temperature of a specified body by one kelvin. Specific heat capacity, cp, is heat capacity per unit mass, usually in Jkg−1K−1. “Heat capacity” and “specific heat” are often used loosely in engineering tables, but they are not identical quantities: one depends on the amount of material, while the other is normalized by mass. The subscript p indicates measurement at approximately constant pressure; another dataset may use cv, at constant volume. Thermal diffusivity, a or κ, describes the rate at which a temperature disturbance spreads: a=kρ⁢cp where ρ is density. It is expressed in m2s−1. A steel with high conductivity does not necessarily have high diffusivity if its heat capacity or density is also high. **Mechanical properties** describe resistance to deformation and fracture at a stated temperature and test condition. Yield strength is the stress associated with the specified onset of plastic deformation. The applicable definition may use a yield point, a proof stress such as 0.2% offset, or another standard convention. Tensile strength is the maximum engineering stress reached in a tensile test. Neither value is a direct substitute for the other. Young’s modulus and elastic modulus usually refer to the slope of the elastic portion of a stress–strain curve, E, in pascals. At elevated temperature, the measured slope can depend on loading rate, prior heating, strain range, and whether recoverable and time-dependent strains have been separated. Poisson’s ratio, ν, is the ratio of transverse strain to longitudinal strain in uniaxial elastic loading, with a sign convention commonly reported by magnitude. NIST IR 88-3899, published on January 1, 1989, reports research on the deformation behavior of structural steel at elevated temperature. Other experimental work reports yield strength, tensile strength, modulus of elasticity, elongation, and thermal-expansion coefficient. The NTIS record for *Elevated Temperature Properties of Steels* describes measurements for ASTM A723 and D6AC steels, including specific heat, thermal expansion, stress relaxation, tensile properties, Young’s modulus, and Poisson’s ratio. ASTM STP 124 (1994) reports elevated-temperature properties of stainless steels, including thermal expansion and mechanical behavior. These sources show why a generic “steel at 600 °C” row is inadequate without a grade and test description. **Time-dependent and metallurgical properties** require additional care. Creep is progressive deformation under sustained stress, commonly important at elevated temperature. Stress relaxation is a decrease in stress with time while total strain is held approximately constant. A restrained heated member can therefore lose stress through plasticity, creep, or relaxation even though its temperature remains high. Phase transformation **Phase transformation** A change in crystal structure or phase assemblage that can alter expansion, heat capacity, conductivity, strength, and stiffness. A phase transformation is a change in crystal structure or phase assemblage, such as austenite formation in carbon steel during heating. It can alter expansion, heat capacity, conductivity, strength, and stiffness. Heating rate, cooling rate, prior microstructure, and alloy chemistry affect the temperature range and size of the response. Stainless steels and carbon steels should not be assumed to share the same transformation behavior. ### Reading a temperature-dependent dataset Minimum traceability fields for elevated-temperature steel-property data. | Dataset field | What to record | Why it matters | |---|---|---| | Material | Exact grade, designation, product form, and condition | Properties vary among grades and processing histories | | Temperature | Range, reference temperature, and temperature field | Mean, local, and member temperatures are different quantities | | Test method | Dilatometry, tensile, creep, relaxation, or calorimetry | Each method measures a different response | | Heating history | Heating rate, dwell, cooling rate, and cycles | Time and phase history affect results | | Property definition | Instantaneous, secant, yield, proof, tensile, or modulus value | Similar labels may represent different quantities | | Standard | Edition, National Annex, and governing provisions | Code values are method-specific design representations | Start with the material identity. “Structural steel” is not a sufficient designation. Look for a grade such as ASTM A36, ASTM A572 Grade 50, ASTM A723, D6AC, an EN designation such as S355, or a stainless designation specified by the source. Confirm whether the table concerns the base metal, weld metal, heat-affected zone, or a particular heat treatment. Next, identify what the numbers mean. A column headed “thermal expansion coefficient” may list instantaneous α⁢(T), mean α¯ from a stated reference temperature, or an engineering approximation adopted by a design standard. A column headed “thermal strain” may give total measured strain, free thermal strain, or strain after correction for test-frame compliance. Read the temperature interval and reference temperature before performing a calculation. Mechanical data require the same discipline. “Yield strength at 500 °C” is incomplete unless the table states the test method, heating history, strain rate, specimen orientation, and whether the result is a short-duration tensile value or a design reduction factor. Tensile strength generally falls with temperature, but the curve need not be smooth. Elastic modulus also decreases, while time-dependent deformation can become important before or after the temperature at which a design code changes its reduction factors. Do not confuse material temperature with a uniform member temperature. A thick beam, plate, bolt, or connection may have a hot surface and a cooler core. The thermal gradient creates differential expansion and bending even when the average temperature seems moderate. Restraint changes the problem again: a free member can expand, whereas a restrained member develops thermal stress, with its magnitude controlled by stiffness, connection slip, plasticity, creep, relaxation, and temperature history. EN 1993-1-2 addresses fire design and elevated-temperature resistance of steel members and connections. ANSI/AISC N690-18 addresses combined thermal and mechanical effects, including members restrained against thermal expansion. A practical dataset record should therefore end with a traceable statement: **grade or designation; source and edition; temperature range; heating and cooling history; test condition; property definition; units; and design standard**. If any of these are missing, the value may still support comparison, but it should not be presented as a universal design property. #### Core elevated-temperature terms Thermal strain Expansion strain caused by temperature change, normally ΔL/L for free one-dimensional expansion. Thermal gradient Temperature variation across distance within a member. Restraint Limitation on free thermal movement that generates force or stress. Creep Progressive time-dependent strain under sustained stress. Young’s modulus Elastic modulus describing the initial elastic stiffness. **Glossary** - \*\*α\*\* — coefficient of linear thermal expansion; specify instantaneous or mean. - \*\*ΔL\*\* — change in length. - \*\*L\*\* — original or reference length used in the expansion calculation. - \*\*ΔT\*\* — temperature difference between two stated temperatures. - **Thermal strain** — expansion strain caused by temperature change, normally ΔL/L for one-dimensional free expansion. - \*\*Thermal conductivity, k\*\* — ability to conduct heat, in Wm−1K−1. - \*\*Heat capacity, C\*\* — heat required per kelvin for a specified body. - \*\*Specific heat, cp\*\* — heat required per kelvin per unit mass. - \*\*Thermal diffusivity, a\*\* — rate of temperature equalization, k/(ρ⁢cp). - **Yield strength** — stress marking the specified onset of plastic deformation. - **Tensile strength** — maximum engineering tensile stress in a test. - \*\*Young’s modulus, E\*\* — elastic modulus measured in longitudinal tension or compression. - **Elastic modulus** — slope describing elastic stiffness; confirm the test definition. - \*\*Poisson’s ratio, ν\*\* — transverse-to-longitudinal elastic strain ratio. - **Creep** — time-dependent strain under sustained stress. - **Stress relaxation** — falling stress at approximately fixed total strain. - **Phase transformation** — change in crystal structure or phase assemblage. - **Restraint** — limitation on free thermal movement that generates force or stress. - **Thermal gradient** — temperature variation across distance within a member. ## References 1. \[1\] OpenStax. [Thermal Expansion](https://openstax.org/books/university-physics-volume-2/pages/1-3-thermal-expansion). University Physics Volume 2, 2022. [](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#wiki-cite-ref-1) https://openstax.org/books/university-physics-volume-2/pages/1-3-thermal-expansion 2. \[2\] University of Sheffield. [Fire Design of Steel Structures](https://core.ac.uk/download/pdf/9554664.pdf). Reference on BS 5950 Part 8 fire design, 2006. [](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#wiki-cite-ref-2) https://core.ac.uk/download/pdf/9554664.pdf 3. \[3\] European Commission Joint Research Centre. [EN 1993-1-2: Resistance of Members and Connections Exposed to Fire](https://eurocodes.jrc.ec.europa.eu/publications/en-1993-1-2-resistance-members-connections-fire). Eurocodes, 2005. [](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#wiki-cite-ref-3) https://eurocodes.jrc.ec.europa.eu/publications/en-1993-1-2-resistance-members-connections-fire 4. \[4\] American Institute of Steel Construction. [ANSI/AISC N690-18](https://www.aisc.org/globalassets/aisc/publications/standards/n690-18w.pdf). AISC Standard, 2018. [](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#wiki-cite-ref-4) https://www.aisc.org/globalassets/aisc/publications/standards/n690-18w.pdf 5. \[5\] National Institute of Standards and Technology. [Elevated Temperature Deformation of Structural Steel](https://www.nist.gov/publications/elevated-temperature-deformation-structural-steel). NIST IR 88-3899, 1989. [](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#wiki-cite-ref-5) https://www.nist.gov/publications/elevated-temperature-deformation-structural-steel 6. \[6\] ASTM International. [ASTM STP 124](https://store.astm.org/stp124-eb.html). ASTM Special Technical Publication 124, 1994. [](/wiki/calculated-values/steel-thermal-expansion-and-temperature-dependent-properties#wiki-cite-ref-6) https://store.astm.org/stp124-eb.html **Steel thermal expansion at a glance** Representative coefficient 12 × 10⁻⁶ °C⁻¹ BS 5950 Part 8 ambient value 12 × 10⁻⁶ °C⁻¹ BS 5950 Part 8 value from 200 to 600 °C 14 × 10⁻⁶ °C⁻¹ Basic equation ΔL = αLΔT [Back to Calculated Values](/wiki/calculated-values "Calculated Values — Steel Wiki")