Tensile membrane action - Revision history

Jan: Add 4 watermarked SVG illustrations from the Expert visual plan

2026-11-02T16:29:54Z

Add 4 watermarked SVG illustrations from the Expert visual plan

← Older revision		Revision as of 17:29, 2 November 2026
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	At midspan, where <math>x = L/2</math>, the first term in the numerator becomes the familiar maximum bending moment of a simply supported beam. Midspan is also where the worked example below applies the formula. The expression then reduces to:		At midspan, where <math>x = L/2</math>, the first term in the numerator becomes the familiar maximum bending moment of a simply supported beam. Midspan is also where the worked example below applies the formula. The expression then reduces to:
	<math display="block">T = \frac{q L^2/8 - M}{d}</math>		<math display="block">T = \frac{q L^2/8 - M}{d}</math>
			[[File:fse-tensile-membrane-action-v1.svg\|thumb\|upright=1.5\|alt=Two sketches of a floor strip: above, a slightly bent beam; below, a deeply sagging strip with tension arrows along it and inward pulls at both supports.\|A floor strip before and after it loses its bending capacity: in the catenary stage the load is carried by tension T, which pulls inward on the supports. Based on Zarghamee, Kitane, et al. (2005), pp. 310–311.]]

	{\| class="wikitable"		{\| class="wikitable"
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	The sag has an even stronger effect. For two assumed sags with no residual moment, one fortieth of the span requires a tension of 5.0 <math>qL</math>, and one tenth only 1.25 <math>qL</math>. A floor can thus reduce the pull on its supports by sagging further, as long as it does not fail. This one-way calculation is a simplification. NIST used it to approximate the tensile forces in the floor system of the towers.<ref name="GM-296" /> It treats the floor as a strip spanning in one direction and leaves out any action of the slab in the second direction.		The sag has an even stronger effect. For two assumed sags with no residual moment, one fortieth of the span requires a tension of 5.0 <math>qL</math>, and one tenth only 1.25 <math>qL</math>. A floor can thus reduce the pull on its supports by sagging further, as long as it does not fail. This one-way calculation is a simplification. NIST used it to approximate the tensile forces in the floor system of the towers.<ref name="GM-296" /> It treats the floor as a strip spanning in one direction and leaves out any action of the slab in the second direction.
			[[File:fse-tensile-membrane-action-v2.svg\|thumb\|upright=1.5\|alt=Line chart: floor tension falls as the sag grows, with curves for no, a quarter and half residual moment and a marked point at a sag of L/17.\|Midspan tension in a sagging floor strip as a multiple of the total load qL, against the sag-to-span ratio, for residual moments of 0, 0.25 and 0.5 times qL²/8. The marked point is the World Trade Center truss model. Based on Zarghamee, Kitane, et al. (2005), p. 311, and Zarghamee, Bolourchi, et al. (2005), p. xxxiii.]]

	== Development during a fire ==		== Development during a fire ==
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	The same truss model shows the later stages in detail. In the model, assumed steel temperatures were raised from 20 °C to 700 °C in 30 minutes and by a further 200 °C in the next 10 minutes.<ref name="ZB-xxxviii">Zarghamee, Bolourchi, et al. (2005), p. xxxviii.</ref> The shear studs on the spandrel failed below 275 °C, and the top chords yielded above 300 °C because steel and concrete expand by different amounts.<ref name="ZB-xxxviii" /> At 565 °C, four compression diagonals buckled under high axial compressive force, and the floor sagged into a catenary shape.<ref name="ZB-xxxix" />		The same truss model shows the later stages in detail. In the model, assumed steel temperatures were raised from 20 °C to 700 °C in 30 minutes and by a further 200 °C in the next 10 minutes.<ref name="ZB-xxxviii">Zarghamee, Bolourchi, et al. (2005), p. xxxviii.</ref> The shear studs on the spandrel failed below 275 °C, and the top chords yielded above 300 °C because steel and concrete expand by different amounts.<ref name="ZB-xxxviii" /> At 565 °C, four compression diagonals buckled under high axial compressive force, and the floor sagged into a catenary shape.<ref name="ZB-xxxix" />
			[[File:fse-tensile-membrane-action-v4.svg\|thumb\|upright=1.5\|alt=Timeline from 0 to 750 °C with failure events of a floor truss model, catenary action at 565 °C and walk-off at 730 °C.\|Failure events in NIST's model of a long-span floor truss under assumed temperatures: the catenary stage starts at 565 °C, connection failures follow up to walk-off at 730 °C. Based on Zarghamee, Bolourchi, et al. (2005), pp. xxxviii–xxxix.]]

	The connections then carried the tension. At about 600 °C, before the gusset plate and the exterior seat bolts failed, the tension between the truss and the exterior columns was about 12 kip (53 kN). It passed through the gusset plate and the strap anchor.<ref name="ZB-81">Zarghamee, Bolourchi, et al. (2005), p. 81.</ref> The interior seat bolts had sheared at 566 °C, and the gusset plate fractured and the exterior seat bolts sheared at around 680 °C.<ref name="ZB-xxxix" /> At about 700 °C, a tension of about 10 kip (44 kN) passed through the strap anchor alone, and at 730 °C the truss walked off its exterior seat.<ref name="ZB-81" /><ref name="ZB-xxxix" /> NIST notes that the actual failure modes and sequence in the towers may have been different, because the model used assumed rather than actual temperature histories.<ref name="ZB-81" />		The connections then carried the tension. At about 600 °C, before the gusset plate and the exterior seat bolts failed, the tension between the truss and the exterior columns was about 12 kip (53 kN). It passed through the gusset plate and the strap anchor.<ref name="ZB-81">Zarghamee, Bolourchi, et al. (2005), p. 81.</ref> The interior seat bolts had sheared at 566 °C, and the gusset plate fractured and the exterior seat bolts sheared at around 680 °C.<ref name="ZB-xxxix" /> At about 700 °C, a tension of about 10 kip (44 kN) passed through the strap anchor alone, and at 730 °C the truss walked off its exterior seat.<ref name="ZB-81" /><ref name="ZB-xxxix" /> NIST notes that the actual failure modes and sequence in the towers may have been different, because the model used assumed rather than actual temperature histories.<ref name="ZB-81" />
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	== Demands on supports and connections ==		== Demands on supports and connections ==
	The tension in a sagging floor becomes an inward pull on whatever supports it. NIST found that substantial sagging of the tower floors produced pull-in forces at the column connections and led to inward bowing of the exterior wall.<ref name="GM-296" /> A floor with tensile forces at its connections does not restrain the exterior wall from bowing inward.<ref name="GM-296" /> In the global analyses, floors with a large sag likewise did not restrain the exterior columns against buckling.<ref name="ZK-310" />		The tension in a sagging floor becomes an inward pull on whatever supports it. NIST found that substantial sagging of the tower floors produced pull-in forces at the column connections and led to inward bowing of the exterior wall.<ref name="GM-296" /> A floor with tensile forces at its connections does not restrain the exterior wall from bowing inward.<ref name="GM-296" /> In the global analyses, floors with a large sag likewise did not restrain the exterior columns against buckling.<ref name="ZK-310" />
			[[File:fse-tensile-membrane-action-v3.svg\|thumb\|upright=1.5\|alt=Section through a tall building edge: several heated floors sag between core and exterior wall and pull the wall columns inward into a bow.\|Sagging floors pull inward on the exterior columns and no longer brace them, so the wall bows inward. Based on Gross and McAllister (2005), p. 296, and Zarghamee, Kitane, et al. (2005), p. 310.]]

	Sagging can also overload the supports of the floor itself. NIST noted that floor sagging between supports may cause tensile failure of the truss seats, or tensile forces that pull the columns inward.<ref name="GM-89" /> NIST also summarised an earlier study of the towers by the University of Maryland and the Israel Institute of Technology. That study postulated, without supporting calculations, that the floors would sag and fail at their connections to the columns.<ref name="GM-324">Gross and McAllister (2005), p. 324.</ref> [[Progressive collapse]] would then follow as the floors below also failed.<ref name="GM-324" />		Sagging can also overload the supports of the floor itself. NIST noted that floor sagging between supports may cause tensile failure of the truss seats, or tensile forces that pull the columns inward.<ref name="GM-89" /> NIST also summarised an earlier study of the towers by the University of Maryland and the Israel Institute of Technology. That study postulated, without supporting calculations, that the floors would sag and fail at their connections to the columns.<ref name="GM-324">Gross and McAllister (2005), p. 324.</ref> [[Progressive collapse]] would then follow as the floors below also failed.<ref name="GM-324" />

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

2026-11-01T23:28:05Z

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

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Mark: New FSE wiki article: Tensile membrane action (written by Claude Opus 5.5)

2026-11-01T22:27:37Z

New FSE wiki article: Tensile membrane action (written by Claude Opus 5.5)

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'''Tensile membrane action''' is the carrying of load by tension in a floor or structural member that has deflected far beyond its normal range, once its bending capacity has been exceeded.<ref name="ZK-310">Zarghamee, Kitane, et al. (2005), p. 310.</ref><ref name="MILKE-235">Milke (2002a), p. 4-235.</ref> In the one-way form that NIST studied in the World Trade Center towers, a sagging floor acts as a tensile structure, a behaviour called catenary action.<ref name="ZK-310" /> Milke notes that in frames with many members, membrane action may let a heated steel member keep its integrity at temperatures above those usually associated with failure.<ref name="MILKE-235" />

The tension does not disappear, however. It has to be anchored at the supports, where it pulls inward on columns and walls.<ref name="GM-296">Gross and McAllister (2005), p. 296.</ref> In the twin towers, pull-in forces from sagging floors bowed the exterior walls inward before the towers collapsed.<ref name="ZK-310" /> In a full-scale fire test at Cardington in England, on the other hand, a floor sagged 0.64 m over a 9 m span without any reported structural failure.<ref name="McA-337">McAllister et al. (2008), p. 337.</ref> Membrane action can therefore keep a floor up at large deflections, but only if the connections and the surrounding structure can carry the tension.<ref name="MILKE-235" /><ref name="GM-296" />

== From bending to tension ==
At normal temperatures, a floor carries its load by bending between its supports. In a fire, the steel loses stiffness and strength, and the floor starts to sag.<ref name="GM-89">Gross and McAllister (2005), p. 89.</ref> In its analysis of the World Trade Center floors, NIST listed three causes of sagging between supports. These were the loss of stiffness and strength of the steel trusses and the concrete slab at high temperature, and the yielding or buckling of the truss web members needed for bending. Failure of the connections between slab and trusses, which removes composite action, was the third.<ref name="GM-89" />

When the bending capacity is exhausted, the floor can still carry load as a hanging cable. NIST defines catenary action as the combined action that results when the bending capacity of the truss is exceeded and additional load is carried by the floor acting as a tensile structure.<ref name="ZK-310" /> Part of the load may still be carried by bending, through the residual moment capacity of the deformed floor.<ref name="ZK-310" /> For a floor strip under a uniform load, NIST approximated the tension from this combination of bending and catenary behaviour as follows:<ref name="ZK-311">Zarghamee, Kitane, et al. (2005), p. 311.</ref><ref name="GM-296" />
<math display="block">T = \frac{0.5\, q\, x\, (L - x) - M}{d}</math>
At midspan, where <math>x = L/2</math>, the first term in the numerator becomes the familiar maximum bending moment of a simply supported beam. Midspan is also where the worked example below applies the formula. The expression then reduces to:
<math display="block">T = \frac{q L^2/8 - M}{d}</math>

{| class="wikitable"
! Symbol !! Description !! Unit
|-
| <math>T</math> || horizontal tension in the floor strip || kN
|-
| <math>q</math> || uniform load per unit length of the strip || kN/m
|-
| <math>L</math> || span between supports || m
|-
| <math>x</math> || distance from the support || m
|-
| <math>d</math> || sag of the floor at distance <math>x</math> || m
|-
| <math>M</math> || residual moment capacity of the deformed floor || kN·m
|}

The formula shows the two features that matter most. The tension equals the part of the bending demand that the floor can no longer resist, divided by the sag. A floor that sags less must therefore develop a larger tension to carry the same load. The supports provide vertical reactions of half the total load each, as in ordinary bending, plus the horizontal pull of the tension.<ref name="ZK-311" />

== Worked example ==
The long-span floor trusses of the World Trade Center towers spanned nominally 60 ft (18.3 m) between the exterior walls and the core.<ref name="ZB-xxxiii">Zarghamee, Bolourchi, et al. (2005), p. xxxiii.</ref> In NIST's model of a single long-span truss under assumed fire temperatures, the floor sagged by about 42 in. (1.07 m) after its web diagonals had buckled.<ref name="ZK-311" /> The sag was thus about one seventeenth of the span.

If the floor keeps no residual moment capacity, the midspan formula gives a tension of 18.3 / (8 × 1.07) = 2.14 times the total load <math>qL</math> on the span.<ref name="ZK-311" /> Each support then has to resist a horizontal pull about twice as large as the whole load that the floor carries. If the floor keeps a residual moment of a quarter of <math>qL^2/8</math>, the tension falls to 1.61 <math>qL</math>, and with half to 1.07 <math>qL</math>.

The sag has an even stronger effect. With no residual moment, a sag of one fortieth of the span requires a tension of 5.0 <math>qL</math>, and a sag of one tenth only 1.25 <math>qL</math>. A floor can thus reduce the pull on its supports by sagging further, as long as it does not fail. This one-way calculation is a simplification. NIST used it to approximate the tensile forces in the floor system of the towers.<ref name="GM-296" /> It treats the floor as a strip spanning in one direction and leaves out any action of the slab in the second direction.

== Development during a fire ==
Membrane action is usually the last stage of a longer sequence. Early in a fire, heated members expand, and [[restrained thermal expansion]] increases the loads in the heated elements.<ref name="ZK-311" /> In the twin towers, the floors expanded in the early stages of the fires. As their temperatures rose, they sagged and began to pull inward on the exterior walls instead.<ref name="N1-144">National Institute of Standards and Technology (2005), p. 144.</ref>

NIST's model of a single long-span truss shows this sequence in detail. In the model, assumed steel temperatures were raised from 20 °C to 700 °C in 30 minutes and by a further 200 °C in the next 10 minutes.<ref name="ZB-xxxviii">Zarghamee, Bolourchi, et al. (2005), p. xxxviii.</ref> The shear studs on the spandrel failed below 275 °C, and the top chords yielded above 300 °C because steel and concrete expand by different amounts.<ref name="ZB-xxxviii" /> At 565 °C, four compression diagonals buckled and the floor sagged into a catenary shape.<ref name="ZB-xxxix">Zarghamee, Bolourchi, et al. (2005), p. xxxix.</ref>

The connections then carried the tension until they too began to fail. The interior seat bolts sheared at 566 °C, and the gusset plate fractured and the exterior seat bolts sheared at around 680 °C.<ref name="ZB-xxxix" /> At 730 °C, the truss walked off its exterior seat.<ref name="ZB-xxxix" /> By the time catenary action develops, [[steel in fire]] has lost much of its stiffness and strength, so the reserve of the floor lies mainly in its ability to hang between its supports.<ref name="GM-89" /><ref name="ZK-310" />

== Demands on supports and connections ==
The tension in a sagging floor becomes an inward pull on whatever supports it. NIST found that substantial sagging of the tower floors produced pull-in forces at the column connections and led to inward bowing of the exterior wall.<ref name="GM-296" /> A floor with tensile forces at its connections does not restrain the exterior wall from bowing inward.<ref name="GM-296" /> In the global analyses, floors with a large sag likewise did not restrain the exterior columns against buckling.<ref name="ZK-310" />

Sagging can also overload the supports of the floor itself. NIST noted that floor sagging between supports may cause tensile failure of the truss seats, or tensile forces that pull the columns inward.<ref name="GM-89" /> Other fires show the same weakness at connections. In the 1970 fire at One New York Plaza, steel beams were deflected by several inches and their connecting bolts sheared off. No floors collapsed, but there were connection failures and distortion of the floor beams.<ref name="McA-338">McAllister et al. (2008), p. 338.</ref> In World Trade Center 5, bolts tore out of fire-weakened beam webs at field-bolted connections, and large sections of floor failed.<ref name="McA-338" />

Membrane action is therefore closely tied to the question of [[progressive collapse]]. In a frame, load transfer lets stronger members carry loads that heated members can no longer carry, and membrane action may let heated members keep their integrity.<ref name="MILKE-235" /> If the connections cannot hold the resulting forces, however, the failure of one floor can spread to the columns and walls that support it.<ref name="GM-296" /><ref name="McA-338" />

== Evidence from tests and fires ==
The most detailed full-scale evidence in the NIST reports comes from the Cardington test building of the Building Research Establishment in England. This eight-storey steel braced frame was completed in 1993 to represent a typical office building, with simple shear connections and bays of 9.0 m along its length.<ref name="McA-336">McAllister et al. (2008), p. 336.</ref> Seven fire tests were carried out in it.<ref name="McA-336" />

In Test No. 6, office furniture with a fire load of 46 kg/m² burned in a test area of 135 m².<ref name="McA-337" /> Gas temperatures exceeded 900 °C within 10 minutes and reached just over 1200 °C within 40 minutes, and the heat release rate peaked at 58 MW.<ref name="McA-337" /> No structural failures were reported, but the floor sagged by up to 0.64 m in a span of 9 m, about one fourteenth of the span.<ref name="McA-337" /> In Test No. 3, with compartment temperatures of about 960 °C, the largest floor beam deflection was 428 mm, also in a 9 m span.<ref name="McA-338" />

NIST cites the tests as a comparison with other building fires. It does not analyse how the Cardington floors carried their load at such deflections.<ref name="McA-336" /> Milke's general statement explains the possibility: in frames with many members, load transfer or membrane action may let steel members keep their integrity at temperatures above those normally associated with failure.<ref name="MILKE-235" />

== Modelling and its limits ==
Tensile membrane action cannot be captured by checking members one at a time. Milke notes that a frame analysis is needed to capture load transfer, and that available frame analyses range from algebraic methods to finite element models.<ref name="MILKE-235" /> Predicting membrane action is therefore a task for [[structural fire engineering]] analysis of whole floors or frames. Milke also cites a review that found important simplifications in the finite element models for structural fire protection available at the time. These included perfect composite action between steel and concrete and the neglect of torsion, and large displacements were not accurately modelled.<ref name="MILKE-235" /> Large displacements are exactly what membrane action relies on.

The NIST investigations show what a model needs to include. In the truss model of the towers, the concrete slab was given different strengths in tension and compression. At the interior end of the slab, separate break elements represented the temperature-dependent tensile capacity of its steel reinforcement.<ref name="ZB-xxxviii" /> The model of the WTC 7 floor likewise included the contribution of the welded wire fabric to the tensile capacity of the slab.<ref name="McA-350">McAllister et al. (2008), p. 350.</ref> The behaviour of [[concrete in fire]] in tension, and of its reinforcement, therefore becomes as important as the behaviour of the steel members.

The NIST studies describe one-way catenary action and full-scale observations, but they give no design method for membrane action in floor slabs. The worked example shows the scale of the forces involved rather than a design check. Both Milke and NIST treat membrane action as a property of the whole floor system, with its supports and connections, rather than of single members.<ref name="MILKE-235" /><ref name="GM-296" />

== See also ==
* [[Finite element analysis]]
* [[Critical steel temperature]]
* [[Fire resistance]]
* [[Strength reduction at elevated temperatures]]

== Notes ==
<references />

== References ==
* 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.
* 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).
* Zarghamee, M. S., Bolourchi, S., Eggers, D. W., Erbay, Ö. O., Kan, F. W., Kitane, Y., Liepins, A. A., Mudlock, M., Naguib, W. I., Ojdrovic, R. P., Sarawit, A. T., Barrett, P. R., Gross, J. L., & McAllister, T. P. (2005). ''Component, connection, and subsystem structural analysis'' (NIST NCSTAR 1-6C). National Institute of Standards and Technology.
* Zarghamee, M. S., Kitane, Y., Erbay, Ö. O., McAllister, T. P., & Gross, J. L. (2005). ''Global structural analysis of the response of the World Trade Center towers to impact damage and fire'' (NIST NCSTAR 1-6D). National Institute of Standards and Technology.

[[Category:Structural Fire Engineering]]