Reduced cross-section method

The reduced cross-section method is a calculation method for the fire resistance of timber members. The charred layer and an extra zero-strength layer are removed from the cross-section, and the remaining section is checked with the strength and stiffness of wood at room temperature. It is also known as the effective cross-section method, the name used in the version of Eurocode 5 described by White.

The method rests on two observations. Char has virtually no load-bearing capacity, and the wood just below the char is heated and weakened. Instead of modelling this weakened zone, the method treats it as an equivalent layer of wood without any strength, 7 mm thick in the European pre-standard ENV 1995-1-2:1994. The method needs only a charring rate, the original dimensions of the member and its room-temperature strength. It is therefore one of the simpler ways to calculate the behaviour of timber in fire. Methods of this kind are generally applied to large members, such as sawn or glued-laminated beams and columns.

Charring and the residual cross-section

When wood is heated, it decomposes by pyrolysis: large molecules break down into smaller ones and form combustible pyrolysis gases. The solid residue is char, which consists mainly of carbon and has a black, porous structure. Because wood turns into char and gas, its density drops. In a fire, the pyrolysis gases burn as flames at the surface, while glowing combustion and mechanical breakdown eventually erode the outer char layer. The boundary between char and uncharred wood is distinct and lies at a temperature of about 300 °C.

The charring rate is the linear rate at which wood is converted to char. Under standard fire exposure, it is fairly constant after a higher rate at the start. The empirical model used most widely assumes a rate of 0.6 mm/min perpendicular to the grain for all woods in the ASTM E119 standard fire test. Actual rates differ between species, with density, chemical composition, permeability and moisture content. Denser wood chars more slowly, and moisture slows charring but also lowers the strength of the wood. Parallel to the grain, wood chars about twice as fast as across it.

Charring removes material from every exposed face, so the cross-section shrinks as the fire goes on. The International Fire Engineering Guidelines describe this as a reduction of the effective section dimensions with a consequent loss of load-bearing ability. The corners of a charring beam or column become rounded, with a radius about equal to the char depth on the sides. In ENV 1995-1-2:1994, the 1994 pre-standard for timber structures in fire within the Eurocodes, this rounding is covered by a higher design charring rate. The design rates in that document are 0.7 mm/min for glued laminated softwood and 0.8 mm/min for solid softwood, both with a characteristic density of 290 kg/m³. For hardwood with a characteristic density of 450 kg/m³, the design rate is 0.5 mm/min.

The heated layer below the char

The uncharred wood next to the char is not at room temperature. Because wood conducts heat poorly, the temperature gradient below the char is steep. Once charring has become steady, the wood about 6 mm below the char base is still at about 180 °C. The moisture content peaks where the wood is at about 100 °C, roughly 13 mm below the char base. In this zone, the strength and stiffness of the wood are reduced, which is a form of strength reduction at elevated temperatures. Tests on softwood show the loss of stiffness, compressive strength and tensile strength fading out over a few centimetres below the char layer.

White gives a simple profile for the temperature in a thick piece of wood below the char front. It is based on a power term, and it is one of two common forms, the other being an exponential term. The profile reads: $$T=T_i+(300-T_i){(1-\frac{x}{d_p})}^2$$ Here the char front sits at 300 °C, and the temperature falls to the initial value at the thermal penetration depth. In White's tests, the penetration depth averaged 33 mm, and European tests led to a more conservative recommended value of 40 mm. With an initial temperature of 25 °C and a penetration depth of 40 mm, the formula gives about 94 °C at 20 mm and 42 °C at 30 mm below the char front. The profile does not show the plateau near 100 °C that often occurs in moist wood.

The heated zone is therefore several centimetres deep, and the loss of strength within it is gradual rather than complete. The reduced cross-section method replaces the gradual loss of strength by an equivalent layer of zero strength. In a model for glued-laminated beams, Schaffer and colleagues estimated this layer at about 8 mm and added it to the char depth. The same approach was used in a reliability-based model of glued-laminated beams made of laminates of different grades.

Calculation steps

The calculation starts from the char depth, which equals the charring rate multiplied by the time of fire exposure. The method then adds the zero-strength layer to obtain the effective depth that is removed from each exposed face. In ENV 1995-1-2:1994, as described by White, the zero-strength layer is 7 mm, reached as a linear fraction of 7 mm during the first 20 minutes of exposure. $$d_{\mathrm{e}\mathrm{f}}=\beta t+\delta$$ For a rectangular beam heated on both sides and on the bottom, the breadth loses two layers and the depth one. For a member heated on all four sides, the depth also loses two layers. White writes the section modulus of the charred member in this form, and the method uses it with the effective depth in place of the char depth: $$S_{\mathrm{e}\mathrm{f}}=\frac{1}{6}(B-2d_{\mathrm{e}\mathrm{f}}){(D-j{d}_{\mathrm{e}\mathrm{f}})}^2$$ The rest of the section keeps its room-temperature properties. The member therefore passes the check as long as the bending moment in the fire does not exceed the room-temperature bending strength times the effective section modulus. In symbols, the condition reads: $$M\le f_m{S}_{\mathrm{e}\mathrm{f}}$$

Symbol	Description	Unit
$d_{\mathrm{e}\mathrm{f}}$	effective depth removed from each exposed face	mm
$\beta$	charring rate perpendicular to the grain	mm/min
$t$	time of fire exposure	min
$\delta$	thickness of the zero-strength layer	mm
$B$	original breadth of the member	mm
$D$	original depth of the member	mm
$j$	number of exposed faces across the depth: 1 for three-sided, 2 for four-sided exposure	–
$S_{\mathrm{e}\mathrm{f}}$	elastic section modulus of the effective cross-section	mm³
$M$	bending moment from the loads during the fire	N·mm
$f_m$	bending strength of the wood at room temperature	N/mm²
$T$	temperature of the wood	°C
$T_i$	initial temperature of the wood	°C
$x$	distance below the char front	mm
$d_p$	thermal penetration depth	mm

The strength value must match the method. Calculations of the residual section are normally made with ultimate strength values, while Eurocode 5 uses design or characteristic values. White warns that design values and methods must be compatible. The performance of the member depends on the ratio between the applied load and the ultimate capacity of the residual member.

Worked example

White's chapter contains a glued-laminated beam of 130 mm × 533 mm, exposed to fire on three sides. The example here applies the reduced cross-section method to the same beam, with the ENV 1995-1-2:1994 design charring rate of 0.7 mm/min for glued laminated softwood and a zero-strength layer of 7 mm. Before the fire, the section modulus is 130 × 533² / 6 = 6.16 × 10⁶ mm³.

After 30 minutes, the effective depth is 0.7 × 30 + 7 = 28 mm. The effective breadth is then 130 − 2 × 28 = 74 mm, and the effective depth of the section is 533 − 28 = 505 mm. The effective section modulus is 74 × 505² / 6 = 3.15 × 10⁶ mm³, or 51 % of the original value. Without the zero-strength layer, 62 % would remain, so the 7 mm layer removes about 18 % of the remaining bending capacity.

After 60 minutes, the effective depth is 0.7 × 60 + 7 = 49 mm, and only 32 mm of the original 130 mm breadth still counts. The effective section modulus drops to 1.25 × 10⁶ mm³, or 20 % of the original value. A constant charring rate is reasonable only while a member behaves as a thick slab. Once the temperature at the centre of the member starts to rise, the charring rate increases, so results for such thin remaining sections are doubtful.

Suppose the bending moment during the fire is half of the moment the original section could carry, and the same bending strength applies before and during the fire. The beam then fails when the effective section modulus falls below half of its original value. The calculation gives 31.0 minutes, and rounded down to the safe side, the beam has a fire resistance of 30 minutes under these assumptions. White stresses that the outcome depends on exactly this ratio between the applied load and the capacity of the residual member.

Alternative methods

White distinguishes two ways to assess the load-carrying capacity of the wood that remains. The remaining section can be treated as a single homogeneous material, or as a composite of layers with different properties. In the homogeneous approach, the analyst uses either reduced material properties or room-temperature properties. If the properties are not reduced, a larger part of the cross-section has to be removed, which is the role of the zero-strength layer.

The reduced properties approach keeps the charred section but multiplies the room-temperature strength and stiffness of all uncharred wood by a factor α. Proposed values of α ranged from 0.5 in New Zealand to 0.83 in France, reflecting uncertainty, different design loads and different levels of safety. Eurocode 5 calls this the reduced strength and stiffness method, and its reduction factors depend on the fire-exposed perimeter of the residual section divided by its area. In light-frame members, α would be substantially lower than in large members.

Other simplified approaches handle the heated layer differently. Technical Report 10 of the American Wood Council does not use a zero-strength layer but increases the charring rate by 20 %. For large beams and columns, Lie's formulas give the fire resistance directly from the member size and a load factor, and they are recognised by building codes in the United States and Canada. For White's 130 mm × 533 mm beam at 75 % of its allowable load, Lie's formula gives 53.8 minutes. For members damaged in a fire, Williamson recommended a zero-strength layer of 6 mm where compression governs and 16 mm where tension governs.

The most complex approach divides the uncharred wood into layers or elements at different temperatures and moisture contents. Each layer receives its own strength and stiffness, and Eurocode 5 refers to such models as general calculation methods. White notes that Eurocode 5 offers calculation methods at three levels of complexity.

Use in codes and design

Fire resistance ratings of timber members have traditionally come from furnace tests to ASTM E119, and listed ratings apply only to the assembly that was tested. Calculation procedures for wood beams and columns have been accepted by building codes in the United States and Canada, and Eurocode 5 provides calculation methods in Europe. White identifies three factors that set the fire endurance of a wood member: the protective membrane, if there is one, the extent of charring, and the load-carrying capacity of the uncharred wood. The reduced cross-section method deals with the last two.

The International Fire Engineering Guidelines (2005) define heavy timber as large sawn or glue-laminated members with a smallest dimension of at least 80 mm, citing Buchanan. For such members, the guidelines refer to White's chapter and to the Australian Standard AS 1720.4 for methods to determine fire resistance. In Australia, the data sheets of the Fire Safety Verification Method (2020) list AS 1720.4 among the structural design codes that provide methods for calculating the fire resistance level of an element. They cite the 2006 edition of this standard. The method therefore suits the large beams and columns of heavy timber and mass timber buildings better than light-frame construction, where strength losses are much larger.

Connections need separate attention. Where a fire endurance of 1 hour is required, connectors and fasteners must be protected by 38 mm of wood, fire-rated gypsum board of suitable thickness, or a coating approved for a 1-hour rating. Glued-laminated beams often have a high-strength tension laminate at the bottom. Under the North American procedure for exposed members with a 1-hour rating, a core lamination is removed and an extra outer tension lamination added. A high-strength laminate then remains after charring.

Limits and uncertainty

The charring rates and the zero-strength layer apply to standard fire exposure. In a real fire, the charring rate depends on the heat flux to the surface. In tests on spruce, rates of 0.56, 0.80 and 1.02 mm/min were measured at external heat fluxes of 25, 50 and 75 kW/m². For natural fires, Eurocode 5 gives a parametric charring rate, and the International Fire Engineering Guidelines point to the 2003 final draft of Eurocode 5 for charring in realistic parametric fires.

The cooling phase of a natural fire brings a further complication. Tests on glued-laminated beams showed that the temperature of the residual cross-section keeps changing during the cooling period. The influence of high temperature is then no longer concentrated in the outer layer of the residual section, unlike under standard exposure. The Australian data sheets add a related caution about simple equivalence methods. Their empirical formulae were mostly validated on steel members with insulation that stays in place, so they should be applied with caution to exposed timber.

The value of the zero-strength layer itself varies between models, from 7 mm in ENV 1995-1-2:1994 to about 8 mm in the model of Schaffer and colleagues. Data on wood properties at high temperature are limited, as most measurements were made below 100 °C. The simple methods therefore suit large members best. In small members or thin remaining sections the charring rate increases, and in light-frame members the strength reduction is much larger.

See also

• Fire resistance
• Fire resistance criteria
• Parametric fire curve
• Standard fire curve
• Time equivalence

References

• Australian Building Codes Board. (2020a). Data sheets: Fire Safety Verification Method (Handbook annex, Print version 1.1).
• 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.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.
• White, R. H. (2002). Analytical methods for determining fire resistance of timber 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. 4-257–4-273). National Fire Protection Association; Society of Fire Protection Engineers.