Restrained thermal expansion - Revision history

Jan: Add 5 watermarked SVG illustrations from the Expert visual plan

2026-11-02T16:29:48Z

Add 5 watermarked SVG illustrations from the Expert visual plan

← Older revision		Revision as of 17:29, 2 November 2026
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	A member that is free to expand simply becomes longer and carries no extra stress. With Milke's correlation, a 15 m steel beam grows by about 34 mm when heated from 20 °C to 200 °C, and by about 73 mm at 400 °C.<ref name="MILKE-212" /> A length of 15 m matches the long floor beams on the east side of WTC 7.<ref name="N1A-53">National Institute of Standards and Technology (2008), p. 53.</ref> If the ends cannot move apart, the elongation turns into a compressive force instead.<ref name="N1A-21" /><ref name="GM-321">Gross and McAllister (2005), p. 321.</ref>		A member that is free to expand simply becomes longer and carries no extra stress. With Milke's correlation, a 15 m steel beam grows by about 34 mm when heated from 20 °C to 200 °C, and by about 73 mm at 400 °C.<ref name="MILKE-212" /> A length of 15 m matches the long floor beams on the east side of WTC 7.<ref name="N1A-53">National Institute of Standards and Technology (2008), p. 53.</ref> If the ends cannot move apart, the elongation turns into a compressive force instead.<ref name="N1A-21" /><ref name="GM-321">Gross and McAllister (2005), p. 321.</ref>
			[[File:fse-restrained-thermal-expansion-v1.svg\|thumb\|upright=1.5\|alt=One heated beam three times: free and longer, held by a spring with less extension and a force, held by walls with no extension and a large force.\|The same heated beam with free, partly restrained and fully restrained ends: restraint turns elongation into force. Based on National Institute of Standards and Technology (2008), p. 21, and Gross and McAllister (2005), p. 40.]]

	The degree of restraint depends on the geometry, the connection method and the framing system around the member.<ref name="MILKE-210" /> In real buildings, the restraint usually comes from cooler parts of the same structure. In the twin towers of the World Trade Center, heated exterior columns were restrained by adjacent cooler columns.<ref name="GM-297">Gross and McAllister (2005), p. 297.</ref> Hot core columns were held back by the hat truss that linked them to the exterior walls, so their elongation moved loads from the exterior walls to the core.<ref name="GM-297" /> Members restrained in this way gained compressive load and could cause local or global redistribution of loads.<ref name="GM-321" />		The degree of restraint depends on the geometry, the connection method and the framing system around the member.<ref name="MILKE-210" /> In real buildings, the restraint usually comes from cooler parts of the same structure. In the twin towers of the World Trade Center, heated exterior columns were restrained by adjacent cooler columns.<ref name="GM-297">Gross and McAllister (2005), p. 297.</ref> Hot core columns were held back by the hat truss that linked them to the exterior walls, so their elongation moved loads from the exterior walls to the core.<ref name="GM-297" /> Members restrained in this way gained compressive load and could cause local or global redistribution of loads.<ref name="GM-321" />
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	The walk-off mechanism shows how geometry decides the outcome. In the north-east corner of WTC 7, the floor beams framed into a girder from one side only.<ref name="N1A-54">National Institute of Standards and Technology (2008), p. 54.</ref><ref name="McA-527">McAllister et al. (2008), p. 527.</ref> When these beams expanded, they pushed the girder sideways, sheared the bolts of its seated connection, and kept pushing until the girder came off its bearing seat.<ref name="McA-527" />		The walk-off mechanism shows how geometry decides the outcome. In the north-east corner of WTC 7, the floor beams framed into a girder from one side only.<ref name="N1A-54">National Institute of Standards and Technology (2008), p. 54.</ref><ref name="McA-527">McAllister et al. (2008), p. 527.</ref> When these beams expanded, they pushed the girder sideways, sheared the bolts of its seated connection, and kept pushing until the girder came off its bearing seat.<ref name="McA-527" />
			[[File:fse-restrained-thermal-expansion-v3.svg\|thumb\|upright=1.5\|alt=Plan of floor beams pushing a girder sideways from one side, and a section of the girder sliding off an 11 inch seat after 5.5 inches.\|Walk-off of the girder at Column 79 of WTC 7: floor beams framing in from one side expand, shear the seat bolts and push the girder off its 11 in. (279 mm) bearing seat after 5.5 in. (140 mm). Based on McAllister et al. (2008), p. 527.]]

	The seat at Column 79 was 11 in. (279 mm) wide, so the girder lost its support once it had moved at least 5.5 in. (140 mm).<ref name="McA-527" /> The girder had no shear studs that could have held it laterally to the slab.<ref name="McA-527" />		The seat at Column 79 was 11 in. (279 mm) wide, so the girder lost its support once it had moved at least 5.5 in. (140 mm).<ref name="McA-527" /> The girder had no shear studs that could have held it laterally to the slab.<ref name="McA-527" />
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	An analysis of this floor area in the NIST investigation shows how early such failures begin. In a finite element model of the north-east floor framing, the beams were heated to 600 °C and the girder to 500 °C, while the slab stayed cold.<ref name="McA-350">McAllister et al. (2008), p. 350.</ref><ref name="McA-352">McAllister et al. (2008), p. 352.</ref> The first shear stud failed at a beam temperature of only 103 °C.<ref name="McA-353">McAllister et al. (2008), p. 353.</ref> Both seat bolts of the girder at Column 79 had failed at 154 °C, and all but three shear studs at 300 °C.<ref name="McA-353" /> The northmost floor beam began to buckle sideways at 436 °C.<ref name="McA-353" /> NIST carried these failure modes into its larger 16-storey and 47-storey models of the building.<ref name="McA-353" />		An analysis of this floor area in the NIST investigation shows how early such failures begin. In a finite element model of the north-east floor framing, the beams were heated to 600 °C and the girder to 500 °C, while the slab stayed cold.<ref name="McA-350">McAllister et al. (2008), p. 350.</ref><ref name="McA-352">McAllister et al. (2008), p. 352.</ref> The first shear stud failed at a beam temperature of only 103 °C.<ref name="McA-353">McAllister et al. (2008), p. 353.</ref> Both seat bolts of the girder at Column 79 had failed at 154 °C, and all but three shear studs at 300 °C.<ref name="McA-353" /> The northmost floor beam began to buckle sideways at 436 °C.<ref name="McA-353" /> NIST carried these failure modes into its larger 16-storey and 47-storey models of the building.<ref name="McA-353" />
			[[File:fse-restrained-thermal-expansion-v4.svg\|thumb\|upright=1.5\|alt=Dot plot of eight failure events in a heated floor model, starting at 103 °C, with reference lines at 538 °C and 593 °C.\|Beam temperature at each failure event in NIST's finite element model of the north-east floor framing of WTC 7, compared with the ASTM E119 average steel temperatures of 538 °C (columns) and 593 °C (floor beams). Based on McAllister et al. (2008), p. 353, and National Institute of Standards and Technology (2008), p. 21.]]

	At these temperatures, [[steel in fire]] has lost little of its strength. Milke notes that the yield strength of ASTM A36 steel is still about 60 % of its room-temperature value at 538 °C.<ref name="MILKE-212" /> In WTC 7, the critical columns did not exceed about 300 °C, while expansion damaged the floor framing mainly at or below 400 °C.<ref name="N1A-21" /> The failures that started the [[progressive collapse]] of the building occurred mainly at temperatures well below those at which structural steel loses significant strength.<ref name="N1A-53" /><ref name="N1A-66" />		At these temperatures, [[steel in fire]] has lost little of its strength. Milke notes that the yield strength of ASTM A36 steel is still about 60 % of its room-temperature value at 538 °C.<ref name="MILKE-212" /> In WTC 7, the critical columns did not exceed about 300 °C, while expansion damaged the floor framing mainly at or below 400 °C.<ref name="N1A-21" /> The failures that started the [[progressive collapse]] of the building occurred mainly at temperatures well below those at which structural steel loses significant strength.<ref name="N1A-53" /><ref name="N1A-66" />
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	A moderate thrust can also help the slab. Its effect on a reinforced concrete slab is the same as that of a prestressing force acting along the line of the thrust, so it can be treated as fictitious reinforcement.<ref name="FB-248" /> The thrust only helps if its line of action lies below the resultant of the compression zone. Fire tests showed the line of action near the bottom of the member, particularly when the thrust was small.<ref name="FB-248" /><ref name="FB-249">Fleischmann and Buchanan (2002), p. 4-249.</ref> For solid slabs with minimal restraint, Fleischmann and Buchanan place it 25 mm above the bottom after 2 hours of fire exposure, 32 mm after 3 hours and 38 mm after 4 hours.<ref name="FB-249" />		A moderate thrust can also help the slab. Its effect on a reinforced concrete slab is the same as that of a prestressing force acting along the line of the thrust, so it can be treated as fictitious reinforcement.<ref name="FB-248" /> The thrust only helps if its line of action lies below the resultant of the compression zone. Fire tests showed the line of action near the bottom of the member, particularly when the thrust was small.<ref name="FB-248" /><ref name="FB-249">Fleischmann and Buchanan (2002), p. 4-249.</ref> For solid slabs with minimal restraint, Fleischmann and Buchanan place it 25 mm above the bottom after 2 hours of fire exposure, 32 mm after 3 hours and 38 mm after 4 hours.<ref name="FB-249" />
			[[File:fse-restrained-thermal-expansion-v5.svg\|thumb\|upright=1.5\|alt=Section of a concrete slab heated from below, restrained at both ends, with a thrust force acting near the bottom face and a compression zone at the top.\|A concrete slab heated from below and restrained by the surrounding cooler slab: the thermal thrust acts near the bottom and works like prestressing. Based on Fleischmann and Buchanan (2002), pp. 4-247 to 4-249.]]

	Restraint can also act at a much smaller scale. Some researchers explain spalling, the breaking away of concrete from a heated surface, by restrained thermal expansion close to that surface.<ref name="KH-163">Kodur and Harmathy (2002), p. 1-163.</ref> The restrained layer develops compressive stresses parallel to the surface, which are released by brittle fracture of the concrete.<ref name="KH-163" />		Restraint can also act at a much smaller scale. Some researchers explain spalling, the breaking away of concrete from a heated surface, by restrained thermal expansion close to that surface.<ref name="KH-163">Kodur and Harmathy (2002), p. 1-163.</ref> The restrained layer develops compressive stresses parallel to the surface, which are released by brittle fracture of the concrete.<ref name="KH-163" />
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	At a steel temperature of 100 °C, the free thermal strain is 0.98 × 10⁻³.<ref name="MILKE-212" /> With a modulus of 201 GPa at that temperature, a fully restrained member would carry a stress of about 197 MPa, while its yield strength is still 238 MPa. At 116 °C, the two values meet at about 236 MPa, so the member starts to yield after a temperature rise of less than 100 K. At 200 °C, the elastic stress would already be 432 MPa, almost twice the yield strength of 223 MPa at that temperature.		At a steel temperature of 100 °C, the free thermal strain is 0.98 × 10⁻³.<ref name="MILKE-212" /> With a modulus of 201 GPa at that temperature, a fully restrained member would carry a stress of about 197 MPa, while its yield strength is still 238 MPa. At 116 °C, the two values meet at about 236 MPa, so the member starts to yield after a temperature rise of less than 100 K. At 200 °C, the elastic stress would already be 432 MPa, almost twice the yield strength of 223 MPa at that temperature.
			[[File:fse-restrained-thermal-expansion-v2.svg\|thumb\|upright=1.5\|alt=Line chart: the stress in a fully restrained steel member rises steeply with temperature and crosses the slowly falling yield strength at about 116 °C.\|Elastic stress in a fully restrained ASTM A36 steel member compared with its yield strength, from 20 °C to 400 °C. The lines cross at about 116 °C. Based on Milke (2002a), p. 4-212; E₀ = 205 GPa from Luecke, McColskey, et al. (2005), p. 67.]]

	A real member yields, buckles or breaks its connections well before such stresses develop. The example therefore shows the size of the forces involved rather than the true stress in a building. Real members are also usually only partly restrained, and the degree of restraint depends on the connections and the framing around them.<ref name="MILKE-210" /><ref name="GM-40" /> The free elongation of a 15 m beam at 600 °C is about 115 mm, the same order of magnitude as the 140 mm that unseated the WTC 7 girder.<ref name="McA-527" /> This simple estimate ignores the stiffness, rotation and buckling that NIST's models included.		A real member yields, buckles or breaks its connections well before such stresses develop. The example therefore shows the size of the forces involved rather than the true stress in a building. Real members are also usually only partly restrained, and the degree of restraint depends on the connections and the framing around them.<ref name="MILKE-210" /><ref name="GM-40" /> The free elongation of a 15 m beam at 600 °C is about 115 mm, the same order of magnitude as the 140 mm that unseated the WTC 7 girder.<ref name="McA-527" /> This simple estimate ignores the stiffness, rotation and buckling that NIST's models included.

Mark: FSE wiki: corrections after independent fact-check (written by Claude Opus 5.5)

2026-11-01T23:30:59Z

FSE wiki: corrections after independent fact-check (written by Claude Opus 5.5)

← Older revision		Revision as of 00:30, 2 November 2026
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	An analysis of this floor area in the NIST investigation shows how early such failures begin. In a finite element model of the north-east floor framing, the beams were heated to 600 °C and the girder to 500 °C, while the slab stayed cold.<ref name="McA-350">McAllister et al. (2008), p. 350.</ref><ref name="McA-352">McAllister et al. (2008), p. 352.</ref> The first shear stud failed at a beam temperature of only 103 °C.<ref name="McA-353">McAllister et al. (2008), p. 353.</ref> Both seat bolts of the girder at Column 79 had failed at 154 °C, and all but three shear studs at 300 °C.<ref name="McA-353" /> The northmost floor beam began to buckle sideways at 436 °C.<ref name="McA-353" /> NIST carried these failure modes into its larger 16-storey and 47-storey models of the building.<ref name="McA-353" />		An analysis of this floor area in the NIST investigation shows how early such failures begin. In a finite element model of the north-east floor framing, the beams were heated to 600 °C and the girder to 500 °C, while the slab stayed cold.<ref name="McA-350">McAllister et al. (2008), p. 350.</ref><ref name="McA-352">McAllister et al. (2008), p. 352.</ref> The first shear stud failed at a beam temperature of only 103 °C.<ref name="McA-353">McAllister et al. (2008), p. 353.</ref> Both seat bolts of the girder at Column 79 had failed at 154 °C, and all but three shear studs at 300 °C.<ref name="McA-353" /> The northmost floor beam began to buckle sideways at 436 °C.<ref name="McA-353" /> NIST carried these failure modes into its larger 16-storey and 47-storey models of the building.<ref name="McA-353" />

	At these temperatures, [[steel in fire]] has lost little of its strength. Milke notes that the yield strength of ASTM A36 steel is still about 60 % of its room-temperature value at 538 °C.<ref name="MILKE-212" /> In WTC 7, the critical columns ~~stayed below~~ 300 °C, while expansion damaged the floor framing mainly at or below 400 °C.<ref name="N1A-21" /> The failures that started the [[progressive collapse]] of the building occurred mainly at temperatures well below those at which structural steel loses significant strength.<ref name="N1A-53" /><ref name="N1A-66" />		At these temperatures, [[steel in fire]] has lost little of its strength. Milke notes that the yield strength of ASTM A36 steel is still about 60 % of its room-temperature value at 538 °C.<ref name="MILKE-212" /> In WTC 7, the critical columns did not exceed about 300 °C, while expansion damaged the floor framing mainly at or below 400 °C.<ref name="N1A-21" /> The failures that started the [[progressive collapse]] of the building occurred mainly at temperatures well below those at which structural steel loses significant strength.<ref name="N1A-53" /><ref name="N1A-66" />

	As heating continues, the direction of the forces can reverse. In the twin towers, the floors expanded early in the fires, but as their temperature rose they sagged and began to pull inward on the exterior walls.<ref name="N1-144">National Institute of Standards and Technology (2005), p. 144.</ref> In frames with many members, load transfer or [[tensile membrane action\|membrane action]] can then let a heated steel member keep its integrity at temperatures above those usually associated with failure.<ref name="MILKE-235">Milke (2002a), p. 4-235.</ref> At these higher temperatures, creep adds a time-dependent deformation. Milke notes that creep can become significant in structural steel above 460 °C, and that its rate rises about 300 times for A36 steel between 460 °C and 520 °C.<ref name="MILKE-212" />		As heating continues, the direction of the forces can reverse. In the twin towers, the floors expanded early in the fires, but as their temperature rose they sagged and began to pull inward on the exterior walls.<ref name="N1-144">National Institute of Standards and Technology (2005), p. 144.</ref> In frames with many members, load transfer or [[tensile membrane action\|membrane action]] can then let a heated steel member keep its integrity at temperatures above those usually associated with failure.<ref name="MILKE-235">Milke (2002a), p. 4-235.</ref> At these higher temperatures, creep adds a time-dependent deformation. Milke notes that creep can become significant in structural steel above 460 °C, and that its rate rises about 300 times for A36 steel between 460 °C and 520 °C.<ref name="MILKE-212" />

Mark: New FSE wiki article: Restrained thermal expansion (written by Claude Opus 5.5)

2026-11-01T22:20:52Z

New FSE wiki article: Restrained thermal expansion (written by Claude Opus 5.5)

New page

'''Restrained thermal expansion''' is the expansion of a heated structural member that the cooler structure around it partly or fully prevents, so that forces develop in the member and its connections.<ref name="FB-247">Fleischmann and Buchanan (2002), p. 4-247.</ref><ref name="N1A-21">National Institute of Standards and Technology (2008), p. 21.</ref> Fleischmann and Buchanan describe the effect for a floor slab heated from below. The heated part tends to expand, and the cooler surrounding slab resists it with a force called the thermal thrust.<ref name="FB-247" /> In steel frames, expanding beams push on girders and columns, and the resulting forces can buckle beams or break connections.<ref name="N1A-21" />

These forces appear at modest temperatures, long before steel loses much strength. In World Trade Center Building 7, restrained expansion of long floor beams started the collapse while the steel was mostly at or below about 400 °C.<ref name="N1A-21" /><ref name="N1A-66">National Institute of Standards and Technology (2008), p. 66.</ref> Restraint is not always harmful, however. In a concrete slab, a thrust that acts low in the section works like extra reinforcement and can lengthen the fire endurance.<ref name="FB-248">Fleischmann and Buchanan (2002), p. 4-248.</ref> In [[structural fire engineering]], restraint appears both as a condition of standard fire tests and as an effect to be modelled in analyses of whole structures.<ref name="MILKE-210">Milke (2002a), p. 4-210.</ref><ref name="IFEG-9">National Research Council of Canada et al. (2005), p. 2.6-9.</ref>

== Free and restrained expansion ==
Structural steel and concrete expand when they are heated.<ref name="N1-144">National Institute of Standards and Technology (2005), p. 144.</ref> For an average carbon steel, the coefficient of thermal expansion is 11.4 × 10⁻⁶ per kelvin at room temperature, and it hardly depends on the microstructure of the steel.<ref name="KH-166">Kodur and Harmathy (2002), p. 1-166.</ref> The coefficient grows with temperature, and Milke gives the correlation <math>\alpha_T = (0.004\,T + 12) \times 10^{-6}</math> per °C for structural steel.<ref name="MILKE-212">Milke (2002a), p. 4-212.</ref> Near 700 °C, steel even contracts for a while, because its crystal structure changes from ferrite and pearlite to austenite.<ref name="KH-166" />

A member that is free to expand simply becomes longer and carries no extra stress. With Milke's correlation, a 15 m steel beam grows by about 34 mm when heated from 20 °C to 200 °C, and by about 73 mm at 400 °C.<ref name="MILKE-212" /> A length of 15 m matches the long floor beams on the east side of WTC 7.<ref name="N1A-53">National Institute of Standards and Technology (2008), p. 53.</ref> If the ends cannot move apart, the elongation turns into a compressive force instead.<ref name="N1A-21" /><ref name="GM-321">Gross and McAllister (2005), p. 321.</ref>

The degree of restraint depends on the geometry, the connection method and the framing system around the member.<ref name="MILKE-210" /> In real buildings, the restraint usually comes from cooler parts of the same structure. In the twin towers of the World Trade Center, heated exterior columns were restrained by adjacent cooler columns.<ref name="GM-297">Gross and McAllister (2005), p. 297.</ref> Hot core columns were held back by the hat truss that linked them to the exterior walls, so their elongation moved loads from the exterior walls to the core.<ref name="GM-297" /> Members restrained in this way gained compressive load and could cause local or global redistribution of loads.<ref name="GM-321" />

== Effects on steel-framed buildings ==
In a steel frame, the first parts to suffer are often the connections. NIST found that restrained thermal expansion of beams and girders in WTC 7 caused three types of connection failure. These were bolts sheared by the increased axial forces, seated connections that walked off after their bolts had sheared, and connection welds that failed in shear.<ref name="McA-527" />

The walk-off mechanism shows how geometry decides the outcome. In the north-east corner of WTC 7, the floor beams framed into a girder from one side only.<ref name="N1A-54">National Institute of Standards and Technology (2008), p. 54.</ref><ref name="McA-527">McAllister et al. (2008), p. 527.</ref> When these beams expanded, they pushed the girder sideways, sheared the bolts of its seated connection, and kept pushing until the girder came off its bearing seat.<ref name="McA-527" />

The seat at Column 79 was 11 in. (279 mm) wide, so the girder lost its support once it had moved at least 5.5 in. (140 mm).<ref name="McA-527" /> The girder had no shear studs that could have held it laterally to the slab.<ref name="McA-527" />

Restraint can also hold members in place, however. In NIST's floor models of the twin towers, trusses whose seat bolts and gusset plates failed under horizontal loads rarely walked off their seats, because the exterior columns restrained the expansion of the floor.<ref name="GM-225">Gross and McAllister (2005), p. 225.</ref> Without that restraint, the horizontal loads on those bolts and plates would have been smaller.<ref name="GM-225" />

An analysis of this floor area in the NIST investigation shows how early such failures begin. In a finite element model of the north-east floor framing, the beams were heated to 600 °C and the girder to 500 °C, while the slab stayed cold.<ref name="McA-350">McAllister et al. (2008), p. 350.</ref><ref name="McA-352">McAllister et al. (2008), p. 352.</ref> The first shear stud failed at a beam temperature of only 103 °C.<ref name="McA-353">McAllister et al. (2008), p. 353.</ref> Both seat bolts of the girder at Column 79 had failed at 154 °C, and all but three shear studs at 300 °C.<ref name="McA-353" /> The northmost floor beam began to buckle sideways at 436 °C.<ref name="McA-353" /> NIST carried these failure modes into its larger 16-storey and 47-storey models of the building.<ref name="McA-353" />

At these temperatures, [[steel in fire]] has lost little of its strength. Milke notes that the yield strength of ASTM A36 steel is still about 60 % of its room-temperature value at 538 °C.<ref name="MILKE-212" /> In WTC 7, the critical columns stayed below 300 °C, while expansion damaged the floor framing mainly at or below 400 °C.<ref name="N1A-21" /> The failures that started the [[progressive collapse]] of the building occurred mainly at temperatures well below those at which structural steel loses significant strength.<ref name="N1A-53" /><ref name="N1A-66" />

As heating continues, the direction of the forces can reverse. In the twin towers, the floors expanded early in the fires, but as their temperature rose they sagged and began to pull inward on the exterior walls.<ref name="N1-144">National Institute of Standards and Technology (2005), p. 144.</ref> In frames with many members, load transfer or [[tensile membrane action|membrane action]] can then let a heated steel member keep its integrity at temperatures above those usually associated with failure.<ref name="MILKE-235">Milke (2002a), p. 4-235.</ref> At these higher temperatures, creep adds a time-dependent deformation. Milke notes that creep can become significant in structural steel above 460 °C, and that its rate rises about 300 times for A36 steel between 460 °C and 520 °C.<ref name="MILKE-212" />

== Effects on concrete slabs ==
Concrete floors show the same effect in the form of thermal thrust. When the underside of an interior slab panel is heated, the hot part tries to expand and the cooler slab around it pushes back.<ref name="FB-247" /> Before 1960, no research had measured the size of this force. In that year the Portland Cement Association started a floor furnace in which the restraint could be varied and monitored.<ref name="FB-247" />

The tests showed a strong link between the allowed expansion and the thrust. If no expansion is allowed, the thrust becomes very high and can crush the concrete.<ref name="FB-248" /> A small amount of allowed expansion lowers the thrust considerably.<ref name="FB-248" /> For conventionally reinforced normal-weight specimens, the maximum thrust was roughly 2,000 kN when only about 3 mm of expansion was allowed over a length of 5.5 m. It dropped to roughly 500 kN when 25 mm was allowed.<ref name="FB-247" /> Lightweight concrete develops considerably less thrust than normal-weight concrete, which is believed to result from its lower modulus of elasticity and lower coefficient of expansion.<ref name="FB-248" />

A moderate thrust can also help the slab. Its effect on a reinforced concrete slab is the same as that of a prestressing force acting along the line of the thrust, so it can be treated as fictitious reinforcement.<ref name="FB-248" /> The thrust only helps if its line of action lies below the resultant of the compression zone. Fire tests showed the line of action near the bottom of the member, particularly when the thrust was small.<ref name="FB-248" /><ref name="FB-249">Fleischmann and Buchanan (2002), p. 4-249.</ref> For solid slabs with minimal restraint, Fleischmann and Buchanan place it 25 mm above the bottom after 2 hours of fire exposure, 32 mm after 3 hours and 38 mm after 4 hours.<ref name="FB-249" />

Restraint can also act at a much smaller scale. Some researchers explain spalling, the breaking away of concrete from a heated surface, by restrained thermal expansion close to that surface.<ref name="KH-163">Kodur and Harmathy (2002), p. 1-163.</ref> The restrained layer develops compressive stresses parallel to the surface, which are released by brittle fracture of the concrete.<ref name="KH-163" />

== Restraint in fire resistance tests ==
The standard fire resistance test ASTM E119 distinguishes restrained and unrestrained assemblies. Both ratings can be derived from a test on a restrained assembly, but tests on unrestrained assemblies set no limiting temperatures for the steel members.<ref name="GM-40">Gross and McAllister (2005), p. 40.</ref> Before 1970, the standard did not define restrained and unrestrained ratings at all.<ref name="GM-40" /> A table in the standard relates real construction to the two conditions. Steel beams welded, riveted or bolted to the framing members, for example, count as restrained, and simply supported end spans and all types of wood construction count as unrestrained.<ref name="MILKE-211">Milke (2002a), p. 4-211.</ref>

Real assemblies rarely match either test condition. NIST noted that a floor assembly like those in the twin towers is neither restrained nor unrestrained, but somewhere in between, and that testing under both conditions bounds its performance under standard exposure.<ref name="GM-40" /> Test furnaces also limit the size of specimens. Milke points out that continuous beams, real floor assemblies and long columns are not tested, so the test is comparative and cannot show actual performance.<ref name="MILKE-210" /> In 2002, furnaces in the United States could test floor spans of less than 18 ft (5.5 m).<ref name="GM-40" />

NIST drew a firm conclusion from WTC 7. The ASTM E119 test does not capture the effect of expanding or sagging floor beams on girders, connections and columns.<ref name="N1A-59">National Institute of Standards and Technology (2008), p. 59.</ref> Because thermal expansion can create an unsafe condition before the steel loses strength, the current rating system, which does not include thermal expansion effects, is not conservative.<ref name="N1A-59" />

== Worked example: a fully restrained steel member ==
The worst case is a member that cannot expand at all, because its free thermal strain then turns completely into stress. The free thermal strain follows from integrating Milke's coefficient of thermal expansion from the initial temperature <math>T_0</math> to the steel temperature <math>T</math>.<ref name="MILKE-212" /> The result is:
<math display="block">\varepsilon_\mathrm{th}(T) = 12 \times 10^{-6}\,(T - T_0) + 0.002 \times 10^{-6}\,(T^2 - T_0^2)</math>
In an elastic member held at both ends, the restrained stress is the product of this strain and the modulus of elasticity at the steel temperature. The member starts to yield when this stress reaches the yield strength at the same temperature. Both properties fall as the steel heats up:
<math display="block">\sigma = E_T\,\varepsilon_\mathrm{th}(T) \ge \sigma_{yT}</math>
For temperatures up to 600 °C, Milke gives the modulus of elasticity and the yield strength of A36 steel as fractions of their values at 20 °C.<ref name="MILKE-212" /> Above 600 °C, other expressions apply.<ref name="MILKE-212" /> The two correlations are:
<math display="block">E_T = \left[1 + \frac{T}{2000 \ln(T/1100)}\right] E_0, \qquad \sigma_{yT} = \left[1 + \frac{T}{900 \ln(T/1750)}\right] \sigma_{y0}</math>

{| class="wikitable"
! Symbol !! Description !! Unit
|-
| <math>\alpha_T</math> || coefficient of thermal expansion of steel at temperature <math>T</math> || 1/°C
|-
| <math>\varepsilon_\mathrm{th}</math> || free thermal strain || –
|-
| <math>T</math> || steel temperature || °C
|-
| <math>T_0</math> || initial steel temperature || °C
|-
| <math>\sigma</math> || stress in the fully restrained member || MPa
|-
| <math>E_T</math> || modulus of elasticity at temperature <math>T</math> || MPa
|-
| <math>E_0</math> || modulus of elasticity at 20 °C || MPa
|-
| <math>\sigma_{yT}</math> || yield strength at temperature <math>T</math> || MPa
|-
| <math>\sigma_{y0}</math> || yield strength at 20 °C || MPa
|}

The example uses steel of grade A36, the grade with a nominal yield strength of 36 ksi (248 MPa) that was used in WTC 7.<ref name="MAC-4">MacNeill et al. (2008), p. 4.</ref> For the modulus at 20 °C, it takes 205 GPa, the value NIST used for the steels of the World Trade Center.<ref name="LUE-67">Luecke, McColskey, et al. (2005), p. 67.</ref> The initial temperature is 20 °C.

At a steel temperature of 100 °C, the free thermal strain is 0.98 × 10⁻³.<ref name="MILKE-212" /> With a modulus of 201 GPa at that temperature, a fully restrained member would carry a stress of about 197 MPa, while its yield strength is still 238 MPa. At 116 °C, the two values meet at about 236 MPa, so the member starts to yield after a temperature rise of less than 100 K. At 200 °C, the elastic stress would already be 432 MPa, almost twice the yield strength of 223 MPa at that temperature.

A real member yields, buckles or breaks its connections well before such stresses develop. The example therefore shows the size of the forces involved rather than the true stress in a building. Real members are also usually only partly restrained, and the degree of restraint depends on the connections and the framing around them.<ref name="MILKE-210" /><ref name="GM-40" /> The free elongation of a 15 m beam at 600 °C is about 115 mm, the same order of magnitude as the 140 mm that unseated the WTC 7 girder.<ref name="McA-527" /> This simple estimate ignores the stiffness, rotation and buckling that NIST's models included.

== Design and analysis ==
After the WTC 7 investigation, NIST recommended that buildings be evaluated explicitly for maximum credible fires, with any active fire protection assumed to be ineffective.<ref name="N1A-65">National Institute of Standards and Technology (2008), p. 65.</ref> It named five features that call for special attention. These are long-span floors, connections that cannot accommodate thermal effects, framing that pushes girders sideways, shear studs that can fail from differential expansion in composite floors, and girders without shear studs.<ref name="N1A-65" /> NIST stated that the effects of restraint of free thermal expansion were not considered in the structural design of WTC 7. This applied especially to the long spans on the east side, and these effects led to the initiation of the collapse.<ref name="N1A-66" />

The remedies NIST lists include connections and framing systems that better resist thermal expansion.<ref name="N1A-65" /> Thermal insulation can also serve a second purpose. It is normally used to protect the strength of steel, but it can also keep the steel cooler to limit thermal expansion.<ref name="N1A-65" /> NIST also called for research on the effects of fire on the whole structural system, on interactions between members and connections, and on the scaling of fire tests to full-size structures.<ref name="N1A-66" />

Restraint is hard to capture with hand calculations. The International Fire Engineering Guidelines recommend special-purpose computer programs for moment-resisting frames and members affected by axial restraint, citing Buchanan.<ref name="IFEG-9" /> The Australian data sheets for the Fire Safety Verification Method likewise ask that thermally induced deflections and stresses be considered, with input from a structural engineer.<ref name="ABCBD-43">Australian Building Codes Board (2020a), Data Sheet B3, p. 43.</ref> Where the interaction of elements matters, detailed modelling of the whole structure or major parts of it may be required, for example a [[finite element analysis]] with material properties that vary with temperature.<ref name="ABCBD-43" /> The floor models that NIST used to study WTC 7 are an example of this approach.<ref name="McA-350" /><ref name="McA-353" />

== See also ==
* [[Concrete in fire]]
* [[Critical steel temperature]]
* [[Fire resistance]]
* [[Spalling]]
* [[Strength reduction at elevated temperatures]]

== Notes ==
<references />

== References ==
* Australian Building Codes Board. (2020a). ''Data sheets: Fire Safety Verification Method'' (Handbook annex, Print version 1.1).
* Fleischmann, C., & Buchanan, A. (2002). Analytical methods for determining fire resistance of concrete members. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), ''SFPE handbook of fire protection engineering'' (3rd ed., pp. 4-239–4-256). National Fire Protection Association; Society of Fire Protection Engineers.
* Gross, J. L., & McAllister, T. P. (2005). ''Structural fire response and probable collapse sequence of the World Trade Center towers'' (NIST NCSTAR 1-6). National Institute of Standards and Technology.
* Kodur, V. K. R., & Harmathy, T. Z. (2002). Properties of building materials. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), ''SFPE handbook of fire protection engineering'' (3rd ed., pp. 1-155–1-181). National Fire Protection Association; Society of Fire Protection Engineers.
* Luecke, W. E., McColskey, J. D., McCowan, C. N., Banovic, S. W., Fields, R. J., Foecke, T., Siewert, T. A., & Gayle, F. W. (2005). ''Mechanical properties of structural steels'' (NIST NCSTAR 1-3D). National Institute of Standards and Technology.
* MacNeill, R., Kirkpatrick, S., Peterson, B., & Bocchieri, R. (2008). ''Global structural analysis of the response of World Trade Center Building 7 to fires and debris impact damage'' (NIST NCSTAR 1-9A). National Institute of Standards and Technology.
* McAllister, T. P., Gann, R. G., Averill, J. D., Gross, J. L., Grosshandler, W. L., Lawson, J. R., McGrattan, K. B., Pitts, W. M., Prasad, K. R., Sadek, F. H., & Nelson, H. E. (2008). ''Structural fire response and probable collapse sequence of World Trade Center Building 7'' (NIST NCSTAR 1-9, Vols. 1–2). National Institute of Standards and Technology.
* Milke, J. A. (2002a). Analytical methods for determining fire resistance of steel members. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), ''SFPE handbook of fire protection engineering'' (3rd ed., pp. 4-209–4-238). National Fire Protection Association; Society of Fire Protection Engineers.
* National Institute of Standards and Technology. (2005). ''Final report on the collapse of the World Trade Center towers'' (NIST NCSTAR 1).
* National Institute of Standards and Technology. (2008). ''Final report on the collapse of World Trade Center Building 7'' (NIST NCSTAR 1A).
* National Research Council of Canada, International Code Council, Department of Building and Housing, & Australian Building Codes Board. (2005). ''International fire engineering guidelines'' (2005 ed.). Australian Building Codes Board.

[[Category:Structural Fire Engineering]]