Technical Discussion
R440 DESIGN OF PASSIVE FIRE PROTECTION Introduction Traditionally structures are designed assuming material temperatures to be at or near the ambient level. A Passive Fire Protection (PFP) coating is then applied such that the steel temperature during a postulated fire would not increase beyond a limiting or critical value at which the load carrying capacity of the structure first becomes insufficient. Steel design codes allow a fraction, generally 60%, of the yield stress of the material, to be utilised in order to attain an acceptable probability of failure. Furthermore, the present design practice uses the elastic method of analysis for calculating the internal stress distributions. However, if the material is ductile, members are compact and lateral supports and joints strength are adequate, then the structure could deform in-elastically. As a result, the internal forces redistribute and the structure shows more load carrying capacity than the elastic method would suggest. For an extreme event such as fire, all these reserves can be utilised so that the PFP requirement is optimised. In addition in deciding on the safety margins for operating loads one should consider the probability of both fire loads and the operating loads to be at their maximum possible value simultaneously. In the Appendix, a theorem is proved which shows the selflimiting nature of thermal stresses. It is shown that if the temperature is kept bounded (limited) during the required survival time, then the thermal stresses stop growing as the temperature reaches a limiting value. The temperature is required to be bounded to prevent material degradation beyond which the member could not sustain its self-weight and operating loads. It is always possible to slow down the temperature growth for a given survival time by providing adequate amount of PFP. In order to use this theorem for determining the limiting temperature of a structural system, one needs to determine the ultimate capacity of that system. The method uses a lower bound solution (or exact solution, if one can be found) to determine a structures reserve strength and the minimum strength to be retained to prevent collapse in a fire. Using this minimum required strength and steel yield strength at elevated temperatures, the limiting temperature is obtained. Results obtained with the Steel Temperatu re In C
20 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300
Table 440.1
proposed methods are in close agreement with test results and results from non-linear finite element analysis. It is also shown that results are in close agreement with those obtained by the British and European codes of practice.
Properties of steel at elevated temperature The yield strength of steel at elevated temperature is defined in terms of strain. The strain at yield is defined as the value consistent with the plateau for mild steel. Traditionally for steel without definable yield stress, 0.2% strain is used at room temperature to define yield strength. At elevated temperatures the yield plateau is quite extended. To make better use of such ductility, relatively larger strains can be allowed. Large strain means large deformations, which could cause cracks or even separation of the passive fire protection material. Thus, the level of strain needs to be limited in order to prevent the possible damage to fire coating. Some codes allow larger strain but require the adhesion of PFP at high strain to be proved by testing. BS5950 gives table of strength retention factors as function of temperature and strain, together with a recommendation on the maximum level of strain that could be considered for various structural members, i.e. beams, columns, etc. Table 440.1 shows the steel yield strength at elevated temperature for Grade 50 steel. Both strength and the modulus of elasticity reduce as the temperature increases. Reference 1 gives two methods for determining values of Youngs modulus and the yield stress at various temperatures. These references give values which show higher retention of properties by steel at elevated temperatures compared to other sources - for example see Reference 2. The enhancement is achieved by defining a higher strain level associated with the yield stress at any particular temperature.
Utilisation of reserve strength
Within the design framework, the structural design is governed by one of several extreme load combinations, such that the probability of the resultant maximum stresses exceeding the failure limit is acceptably low. It is obvious that the probability of extreme design loads BS 5950: Part 8 EC: Part 10 and extreme fire at the same time is quite remote. Consequently for the safety Strength Retention Factor at Strain (in % ) Effective Yield Stresses assessment of a structure under fire attack, of: as a proportion of load factors on the material and loads can 0.5% 1.5% 2.0% be reduced or removed. There are other 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.970 sources of strength reserves, such as the 1.000 1.000 1.000 0.946 ability of the structure to deform plastically 1.000 1.000 1.000 0.854 and redistribute internal forces. 1.000 0.971 0.956 0.798 0.622 0.378 0.186 0.071 0.030 0.0206 0.0137 0.0069 0.0000
0.756 0.460 0.223 0.108 0.059 0.0394 0.0263 0.0131 0.0000
0.776 0.474 0.232 0.115 0.062 0.0446 0.0297 0.0149 0.0000
0.780 0.470 0.230 0.110 0.060 0.040 0.020 0.0000 0.0000
The theorem given in the Appendix shows that the reserve strength of a member determines how hot that member can get before collapse. To determine the reserve strength, either the exact capacityaccording to the plasticity theory, see
Steel yield strength at elevated temperature for Grade 50 steel
Issue 32 - May 2002
Page 9
Technical Discussion Reference 4 - or a lower bound estimate of the capacity is required. Constructing an exact (or close) lower bound solution for a complex large structure is not easy. Codes of practice give simplified methods (known as the sub-framing, see BS5950 Section 5.7) for the plastic design of multi-storey rigid frame. Non-linear finite element analysis gives a close lower bound solution, since at all stages of loading the system is in equilibrium and the yield condition of plasticity is also satisfied. Most of the commercially available software packages use distributed plasticity and could also account for instability and change of geometry. The advantage of non-linear finite element analyses is the ability to discriminate between members, i.e. let some members degrade faster than the others and shed their load to the surviving members. Very good estimates of limiting temperature can also be obtained by approximate methods of estimating the reserve strength. For instance, the elastic analysis result, with all material and load factors set to one, is a lower bound solution. BS5950 Section 8 gives an approximate method for determining the capacity (utilisation ratio) on a member by member basis. This approach is useful when every member in the fire-affected area is protected.
of the first hinge corresponds to exhaustion of the reserve strength. The above statement is true when yielding governs. However, if bucking intervenes, then some of the extra margin of strength may not be present. BS5950 Section 5.7 describes a sub-framing method to estimate the load carrying capacity of multi-storey rigid frames.
Example 1 Consider a rigidly supported uniform I-beam of length L subjected to a uniformly distributed load p per metre length. Assume that the section is compact and adequate number of lateral braces prevents lateral buckling. Suppose that this beam has already been designed to remain elastic for the operating loads using a current practice (e.g. AISC- WSD), then:
( M e )max
Me pL2 S = Section Modulus = (F S. F .) = 12(F S. F .) Y Y where:
Manual application of the theorem A topside structure can be categorised as a braced frame structure, which means that all resistance to lateral forces, sway and frame instability is provided by the bracing system. This bracing system is in the form of an intersecting framework. The required member sizes for the beams and columns in such frame are often governed by the gravity load. However, as all joints are fully welded, web members of the truss-work will take part in carrying some of the gravity loads. The purpose is to determine the approximate ultimate capacity of structural members for the calculation of the critical temperature. In order to make a safe estimate of critical temperature the lower bound approach of plasticity theory is used. It is generally better for the beams to fail first, as the failure of columns would cause the total collapse of the structure. Consequently, all girders are assumed to fail by forming threehinged beam mechanisms. With the reserve strength known, the limiting temperature can be determined by using the material strength retention table at elevated temperatures. Columns in the framework must retain sufficient strength to resist the axial force from the floors above and bending moments transmitted to the joints by girders framing into them. As the column section sizes are already selected for the operational load, it remains to calculate its reserve strength. This reserve strength is a function of the column slenderness ratio and the axial force that it carries, as well as if it buckles in single or double curvature. For slenderness ratios up to 50 and an axial force up to 0.6 PY and double curvature buckling, the reduction in the moment capacity M P is negligible. However for higher axial load (up to 0.9 PY ) the maximum moment will be somewhat reduced. Beam-columns in topside structures generally fall within the category for which the maximum resisting moment can be used. But, it will be conservatively assumed that the formation
Page 10
pL2 = . Thus, 12
(1) Yield strength of the steel at room temperature FY = A safety factor to give allowable bending S. F .= stress; typically 1.67 (Ref. 5) M e = The maximum elastic moment
It is assumed that this member is fully utilised according to the design code. If not then there is some margin that can be used. Suppose that this beam is subjected to a fire, which cause the steel temperature to rise to Tl at time t e . It is necessary to construct a solution in the following form:
(
mi + M i* ≤ M PTl where:
(
)
(2)
i
)
M PTl denotes the plastic moment at the limiting i temperature Tl .
mi can be any self-equilibrating stress field, including mi = 0 everywhere. M i* can be any solution including the
Since
elastic solution or the exact collapse solution, i.e. M collapse = M c = pL2 16 , then:
Z = M c FYT = pL2 16 FYTl where:
(3)
FYTl refers to yield strength at temperature Tl .
It should be mentioned that p does not require a load factor for an extreme event such as fire, since the probability of simultaneous occurrence of extreme gravity load and the maximum credible fire is quite remote. Dividing Equation (3) by Equation (2) gives
(
Z S = (3 4 S . F .) × FY FYTl
)
(4)
Issue 32 - May 2002
Technical Discussion Substituting for Z S = 112 . and rearranging gives FY FYTl = 2.494 or
S. F . = 167 . , and FYTl FY = 0.4
Using Table 1 and 1.5% strain (allowed for beam) for the required strength retention factor 0.4 gives Tl = 625o C . If the level of strain is allowed to be 2%, then Tl = 670 o C . This should be compared with BS5950 recommended value, which ranges from 590 to 680°C depending on floor arrangement. If supports were not fixed, but pinned, then the reserve strength . × 167 . = 187 . , i.e. the of the member is FY FYTl = 112 required strength retention is FYTl FY = 0.535 . Using this value and 1.5% strain, the limiting temperature is 575°C, which is the same as given BS5950. If this member were under-utilised, the reserve of strength would have been more. As a result, the limiting temperature would have been higher. These values compare very well with experimental results, see for instance Reference 5. It should be noted that the critical temperature is a function of reserve strength, the higher the reserve strength, the higher the limiting temperature would be. As the current practice uses elastic analysis methods, the reserve strength varies from member to member, depending on the load pattern as well as on the structural action, and end fixity.
Example 2 Consider a column within a topside offshore structure, which is made of Grade 50 steel, and is designed according to a code, such that the section is fully utilised. Furthermore, suppose that its slenderness ratio is less than 50 and the axial load is less than 0.6 PY . It is required to compute its limiting temperature. Buckling would not reduce the capacity of such column to a great extent. Consequently, the capacity corresponding to formation of the first hinge can be realised. Thus, a lower bound of reserve strength is FY FYTl = 112 . × 167 . = 187 . so that the required strength retention is FYTl FY = 0.535 . Using this value and 0.5% strain, the limiting temperature is 535°C, which is the same as given BS5950. The maximum strain is limited to 0.5% to reflect its importance to the rest of the topside structure.
Non-linear finite element method An alternative method is to use non-linear finite element (FE) method. The FE method is by far the most flexible tool as it allows to optimise the amount PFP (both the extent and the film thickness). Although the principles established in the Appendix still applies, there is no need to determine a closed form solution, or approximately determine the utilisation. Furthermore FE allows considering the complete system. The design fires are generally expressed in terms of heat fluxes (convective or radiative) and gas temperature in the fire affected areas. The mode of the heat transfer through the structure is by conduction. Heat conduction is a slow process. As a result the rise in temperature away from the fire-affected area is dependent on member sizes and the heating environment, i.e. heat fluxes and gas temperatures. Previous experience with similar situations indicates that for a typical topside under a fire attack, it is adequate to consider the area immediately adjacent to the fire affected zone only. Based on this finding, heat transfer models of the topside consists of the fire-affected zone and its immediate neighbouring areas only. If the intention is only to determine the limiting temperature, then a heat transfer analysis of the unprotected structure is performed to determine the temperature time-histories of all fireaffected members. In the first step of a two-step non-linear structural analysis the operational loads are applied. In the second step, temperature time histories at the nodal points will be applied. At this stage zero time signifies the start of the fire. By marching through the time from the start of the fire, the time to collapse will be determined. The temperature associated with the time to collapse is the limiting temperature. By judicially adding PFP to those members that failed earlier, an optimal solution would be obtained for a particular fire affected zone. Generally several runs may be required. For the optimal design of PFP a decision should be made on the type of PFP to be used, since the temperature rise in a member is dependent on the type of PFP. Various types of PFP provide protection on a different basis. Each scheme of PFP (i.e. the extent of protection, the type of PFP and thickness) requires one set of heat transfer analysis.
BS5950 Section 8 BS5950 allows using the limiting temperature method to determine the behaviour of hot finished steel member in fire. Table 5 of BS5950 lists the limiting temperature as a function of member type (beam, column or tension members), the slenderness ratio and the load ratio. Expressions for calculations of the load ratio are given in Sections 4.4.2.2 and 4.4.2.3. The load ratio is very similar to the Interaction Ratio (IR), or the utilisation ratio. The theorem can be used to explain the BS550, as explained in Examples 1 and 2. For the first iteration of the procedure outline in Section 6, the IR (after resetting load and resistant factors) can be used as surrogate for the load ratio. This enables to determine the approximate limiting temperature and hence the film thickness.
Issue 32 - May 2002
An alternative way for determining an approximate limiting temperature for the first iteration is to use BS 5950:Part 8 Tables. Using certified fire test for the chosen PFP, the film thickness is determined. The calculated thickness and the material behaviour of PFP are then used in the heat transfer model to determine the temperature time histories of all points on the structure. Yet another alternative method of determining an initial limiting temperature is by ramping the steel temperature of the entire model uniformly in a non-linear analysis. This eliminates the need for the heat transfer analysis of un-protected steelwork. The temperature at which the software fails to converge would be an average critical temperature for the whole structure. For a structure designed to present Codes of Practices, such BS 5950
Page 11
Technical Discussion or AISC, the structure can be heated up to more than 600°C (and more) before instability sets in.
Determining the average limiting temperature using design software Design software can be used to construct a lower bound solution. Elastic design methods give a lower bound solution. This is due to the fact that the external loads are in equilibrium with the internal forces and the code check ensures that the yielding condition is not violated. Firstly, set all material and load factors to one (or to another value commensurate with simultaneously occurrence of maximum operation load and fire loads). For the first iteration assume that the average critical temperature for the whole system is, say 400°C. Define the modulus of elasticity and material yield compatible with this temperature. As shown in the Appendix the effect of thermal expansion due to temperature rise can be ignored in determining the limiting temperature. The member unity check (or utilisation factor) will indicate if the structure would survive this level of temperature. Increase (or decrease) the stipulated temperature and repeat the analysis/design cycle until an adequate number of members fail the unity check. Since the effect of redistribution of internal forces cannot be accounted, results are always a poor lower bound solution. This approach would only give a minimum estimate of the average limiting temperature for the whole system. There are ways to improve the accuracy of this approach.
Limit State of Insulation: This limit state is reached when heat transfer through the structure or vessel raises the temperature of the unexposed face to a level considered unsafe for combustible materials in contact with that face. For instance, The temperature of unexposed surface must not increase more than an average of, say, 140°C or a maximum of 180°C at any point, above the initial temperature. Although the theorem of the Appendix is proved in the context of a beam, it is also valid for a continuum. Any of the above mentioned methods can be used for determining the limiting temperature of a vessel. Although the last Limit State is more relevant for design of vessels, but all three limit states should be considered for vessels.
Conclusions The self-limiting nature of thermal stresses in a steel-framed structure under fire heating is proved in this paper. It is also shown the way in which the theorem for determining the limiting temperature of structural members can be used. It is required to determine a lower bound solution. The closeness of this lower bound to the exact solution determines the accuracy of the calculated limiting temperature.
References [1]
PFP Design for vessels The above discussion was centred on calculating the limiting temperature. The limiting temperature governs the Stability Limit State. However, other limit states need to be considered for walls and floor, as well as vessels containing volatile material. The fundamental to fire protection is the ability of fire barriers to prevent the transfer of heat, flames or hot gases through the structure such that the ignition of combustible materials on the non-fire side is prevented. Thus, the formation of openings through which gas transfer or passage of flame can take place should be prevented. In addition, temperature rise on the unheated face should be limited so as to eliminate the ignition possibility. The above performance requirement imposes three design conditions which are known as the stability, integrity and insulation limit states. A brief description of these limit states and the limiting conditions are given below. Stability Limit State: This limit state is reached when the total collapse or unacceptable deformation occurs. Load carrying members are expected to resist failure during heating period as well as during the cooling phase. Integrity Limit State: This limit state is reached when a breach occurs in fire barrier through which passage of flame or hot gases becomes possible. This limit state is mainly relevant to separating constructions, namely floors and walls. In fire test it is judged by the application of a cotton-fibre pad to the suspect opening on the unheated face. Vessels should also comply with this requirement.
Page 12
[2]
[3]
[4]
[5]
The Steel Construction Institute Fire Resistant Design of Steel Structure - A Handbook to BS5950: Part 8, SCI Publication 080, 1990. European Convention for Constructional Steelwork European Recommendations for the Fire Safety of Steel Structures, Elsevier 1983. American Iron and Steel Institute, Plastic Design of Braced Multi-storey Steel Frames, Published by Committee of Structural Steel Produces in co-operation with AISC, 1968. Horne M. R. and Morris L. S., Plastic Design of Low-Rise Frames, Constrado Monograph, Collins, London, 1981. BS 5950, Part 1: 1990. British Standard Institute.
Appendix Description of Fire Loads Fires impose some level of thermal loading onto a structure and heat it up in a certain manner. The way a structure heats up by a fire can be called a heating programme. The heating environment causes the steel temperature to rise above the ambient level. The question that we need to answer is how hot the structural members can get before the load carrying capacity becomes inadequate. Generally, there are a large number of possible scenarios, which could lead to a sustained fire. If a number of such scenarios can be distinguished as to be severe enough which require attention, then temperature time history of the structural members due to the fire heating can be expressed in the following form
Ts (x, t ),
for s = 1,2, ... , n
where x denotes an arbitrary point on the structure, t is the accumulated time from the start of the fire, and s characterises s -th fire scenario.
Issue 32 - May 2002
Technical Discussion The range of possible variations of the heating environment imposed on the structure is defined by T(x, t ) , such that at any instant T(x, t ) ≥ Ts (x, t ), for s = 1,2, ... , n , that is T(x, t ) is a design heating programme which encompasses all possible fires in an identifiable area. Since this design fire which envelopes all fires in an area at any given point and at any instant of time, the actual temperature would not exceed T(x, t ) , the limit on temperature. This limiting temperature will be set such that the strength retained at the end of survival time is just enough to sustain the member loads. If required, PFP can be applied to control the rate of temperature rise, such that the heated material would retain sufficient strength to carry its loads. Thermal Stresses in a Fire Consider an ideal elastic-plastic structure experiencing a set of mechanical loads, which are assumed to remain un-changed. Furthermore, assume that the local buckling of the frame members is prevented and the deformations are small enough for changes in geometry to be negligible. This structure which is designed to a certain code is then subjected to an arbitrary temperature field T = T(x, t ) as a result of a fire. This temperature field increases as a function of time from the start of the fire. The instantaneous bending moment at cross-section i due to only mechanical (operating) load combination is denoted by M i* . This moment is computed on the assumption that the structure carries this load combination wholly by elastic action (for the instantaneous values of the Youngs Modulus at the same point of the heating Programme). Heating would cause some thermal stresses to develop in the structure. At the same instance of the time when M i* was calculated, let the actual bending moment at the same crosssection i in the actual elasto-plastic state to be M i . Let mi be the bending moment in the cross-section i due to the thermal effect. This is given by the difference
mi = M i − M i*
(A1)
The actual set of the bending moments M i and M i* (the bending moment due to operating loads only and assuming perfect elastic action) at cross-section i must satisfy the equilibrium equation for the given loads. Since mi is due to the heating, it must satisfy the equation of equilibrium for zero external loads. This seems to imply that the mechanical loads and temperature rise cause only bending moments, which in general is not true, since there will be some axial force. This assumption is not necessary for general solid. Here the context of a framed structure is used to emphasise the practical implication of the theorem. For a framed structure, if the axial component of the stresses is substantial, then buckling may intervene before yielding could take place. It is further assumed that the steel temperature at the time t = t e is limited to (A2) Tl = T (x, t e ) where Tl is the critical (or limiting) temperature that can be allowed a structural member to reach before exhaustion of its
Issue 32 - May 2002
capacity at this temperature. The basic assumption is that, even though the steel temperature is rising, it will not be permitted to increase beyond this limiting value (to be established) during survival time t e . The fire protection of a structure is then reduced ensuring this limiting temperature is not reached during the required survival time. By applying an appropriate amount of a suitable PFP to the member. However, the steel temperature could drop or fluctuate, provided it remains below Tl during the time period t e . Suppose that a distribution of the thermal bending moment mi is found which is statically admissible and satisfies at every cross-section i the condition
mi + M i ≤ ( M PT )i (A3) where ( M PT )i denotes the plastic moment at temperature T for cross-section i . The temperature dependence of the yield strength is accounted for by allowing the yield condition to contract as the temperature rises. The value of the yield surface contraction (or yield strength retention) is given in Table 1. The bending moment mi may be any system of thermal bending moment, whether real or hypothetical which satisfies the appropriate equations of equilibrium for zero external load. It will now be shown that as the temperature rises (up to Tl ) the thermal stresses stop growing, that is, a limiting state will be reached. However, the actual distribution at this stage will not necessarily be the assumed distribution mi . A structure that satisfies the condition (A3) for a given set of mechanical loads and heating Programme Equation (A2) has no possible mode of collapse. Proof of the above stated theorem is along the line given by Horn and others for the shakedown theorem. Consider now the fictitious elastic energy difference mi − mi ,
(
~ Π of the moment
)
2 ~ 1 Π = ∫ 1 (EI )i × (mi − mi ) dsi (A4) 2 in which mi represent the actual thermal moments (stresses) at any cross-section i of the frame at any stage of the heating process, dsi is an element of length of the member at section i , (EI )i is the flexural rigidity at this section, and integration ~ is a positive extends over all members of the structure. Clearly Π
quantity and is a measure of the difference between the actual and hypothetical thermal moment distribution, mi and mi .
(
)
The moment differences mi − mi are connected with the
δmi (EI )i mi and mi satisfy
curvature change
by Hooks law. Since both
the condition of equilibrium distribution with zero external loads, then mi − mi satisfy these conditions. The derivative of energy with respect to time, noting that mi are by definition independent of time, is 2 ~ dΠ dt = ∫ 1 (EI )i (mi − mi )δmi dsi + ∫ (mi − mi ) δ 1 (EI )i dsi
(
)
(
)
(
)
(A5).
Page 13
Technical Discussion Since
E i is a decreasing function with time, and then
δ (1 (EI )i ) is a negative quantity, hence the second integral of Equation (5) is either zero or negative. Thus
(
)
~ dΠ dt ≤ ∫ 1 (EI )i (mi − mi )δmi dsi
(
(A6)
)
Now consider mi − mi as the system of bending moments in equilibrium with zero external load, and changes of curvature
δmi
(EI )i
and hinge rotation Θ i associated with increments
δmi (EI )i of the thermal moments, as compatible displacements. The equation of virtual work gives
∫ (EI ) (m −1
i
i
− mi )δmi dsi + ∑ (mi − mi )Θ i = 0
(A7)
Combining equation (A6) and (A7) gives
~ dΠ dt ≤ − ∑ (mi − mi )Θ i
(A8)
Since the bending moments mi reach yield surface bending moments mi lie inside yield surface, therefore
(mi − mi )Θ i > 0
(A9)
The thermal moments will no longer change with temperature. Thus the system of thermal moments mi existing in the structure in the shakedown condition will differ from mi but mi will settle down to a constant distribution. Since the conditions of the theorem proved above includes those of the lower bound theorem of plastic collapse, the limiting load in the presence of heating cannot exceed the load corresponding to static collapse.
Further information For further details, please contact: Sirous Yasseri Kellogg Brown & Root (KBR) South Point 6-14 Sutton Court Road Sutton Surrey SM1 4TY United Kingdom Tel: +44 (0) 1372 865 226 Fax: +44 (0) 1372 865 111 E-mail:
[email protected]
~
Thus dΠ dt < 0 as long as Θ i ≠ 0 . Since the elastic energy ~ Π is non-negative, a time will be reached when hinge rotation ceases
(i.e.Θ
Page 14
i
)
~ = 0, dΠ dt = 0 .
Issue 32 - May 2002
