strength failures. Many tools are available for calculating the structural fire performance of load‐bearing construction, ranging from hand calculations and design charts to a variety of computer programs discussed in Chapter 11.
For most materials, stress–strain relationships at elevated temperatures can be obtained directly from steady‐state tests at certain elevated temperatures (regime 1), or they can be derived from the results of transient tests. Anderberg (1988) compares stress–strain relation- ships obtained in both ways and points out that there are differences due to the effect of creep.
For most materials, yield strength and modulus of elasticity both decrease with increasing temperature.
5.4.2 Components of Strain
Analysis of a structure exposed to fire requires consideration of the deformation of the struc- tural under the fire limit state loads. The deformation of materials at elevated temperature is usually described by assuming that the change in strain Δε consists of four components:
i ,T th T cr , ,T t tr , (5.12)T
T = const
σ= const σ= stress, T= temperature
ε= strain, ( ) = differential with respect to time Steady state tests
Steady state tests 1. Stress–strain relationship
strain-controlled 2. Stress–strain relationship stress-controlled
ε= const
Figure 5.5 Testing regimes for determining mechanical properties of materials at elevated tempera- tures. Reproduced from Anderberg (1988) by permission of Elsevier Science
where ε is the total strain at time t, εi is the initial strain at time t = 0, εσ(σ,T) is the mechanical, or stress‐related strain, being a function of both the applied stress σ and the temperature T, εth(T) is the thermal strain being a function only of temperature T, εcr(σ,T,t) is the creep strain, being additionally a function of time t, and εtr(σ,T) is the transient strain which only applies to concrete.
5.4.2.1 Stress‐related Strain
The stress‐related strain (or mechanical strain) refers to the strain which results in stresses in the structural members. These stresses are based on the stress–strain relationships shown in Figure 5.2, used for the structural design of all materials. For fire design of individual struc- tural members such as simply supported beams which are free to expand on heating, the stress‐related strain is the only component of strain that needs to be considered. If the reduction of strength with temperature is known, member strength can easily be calculated at elevated temperatures using simple formulae such as those given in this chapter.
The stress‐related strains in fire‐exposed structures may be well above yield levels, resulting in extensive plastification, especially in steel buildings with redundancy or restraint to thermal expansion. Computer modelling of fire‐exposed structures requires knowledge of stress–strain relationships not only in loading, but also in unloading, as members deform and as structural members heat up and cool down in real fires (Franssen, 1990, El‐Rimawi et al., 1996).
5.4.2.2 Thermal Strain
Thermal strain is the well known thermal expansion that occurs when most materials are heated, with expansion being related to the increase in temperature. Thermal strain is not usually important for fire design of simply supported members, but must be considered for frames and complex structural systems, especially where members are restrained by other parts of the structure and the thermal strains can induce large internal forces.
5.4.2.3 Creep Strain
Creep is the term which describes long term deformation of materials under constant load.
Under most conditions, creep is only a problem for members with very high permanent loads.
If the load is removed there will be slow recovery of some of the creep deformations, as shown in Figure 5.6(a). Creep becomes more important at elevated temperatures because creep can accelerate as load capacity reduces, leading to secondary and tertiary creep as shown in Figure 5.6(b). ‘Relaxation’ is the complementary term which describes the reduction of stress in materials subjected to constant deformation over a long period of time.
Creep is relatively insignificant in structural steel at normal temperatures. However, it becomes very significant at temperatures over 400 or 500 °C and is highly dependent on stress level. At higher temperatures the creep deformations in steel can accelerate rapidly, leading to plastic behaviour and ‘runaway’ failure. Creep in wood is complicated by changes in moisture content such that creep deformations tend to be larger in environments where the moisture content of the wood fluctuates over time, hence creep can become a major concern in fire‐
exposed wood which is at temperatures around 100 °C.
Creep strain is not usually included explicitly in fire engineering calculations because of the added complexity and the lack of sufficient input data. This applies to both hand and computer methods. Any structural analysis computer program for elevated temperature is already very complex without have to explicitly include the effects of time‐dependent behaviour. The effects of creep are usually allowed for implicitly by using stress–strain relationships which include an allowance for the amount of creep that might usually be expected in a fire‐exposed member.
5.4.2.4 Transient Strain
Transient strain is caused by expansion of cement paste when it is heated for the first time under load. Transient strain is often included in analytical models for predicting the behaviour of reinforced concrete structures exposed to fire. See Chapter 7.
5.4.2.5 Effect of Strain Components
Equation 5.11 can be simplified, ignoring the last two terms to give
total th (5.13)
Elastic deformation Load
applied Time
Deformation
Primary creep
Secondary creep
Tertiary creep (b)
Figure 5.6 Creep in structural materials: (a) creep under normal conditions; (b) creep at elevated temperatures
where εσ is the stress‐related strain and εth is the thermal strain resulting from thermal expan- sion. This is a key relationship for understanding the structural behaviour of fire‐exposed structures, because structural engineers are interested in stresses and deformations in struc- tures. The deformations in the structure depend on the total strain εtotal and the stresses in the structure depend on the stress‐related strain εσ.
Rotter et al. (1999) explain this further by considering two contrasting types of structure.
For a lightly loaded structure in which there is no resistance to thermal expansion, the total strain is dominated by the thermal strain and the mechanical strain is very low, and hence most deformations (bowing or elongation) result from the thermal strain which is only a function of temperature. In a very different type of structure where there is severe restraint to thermal expansion, there can be no elongation, so εtotal = 0, hence the thermal and mechanical strains are approximately equal and opposite. Both may be very large, resulting in high levels of plastification and high stresses (with much yielding) because of the high mechanical strains.
These aspects of structural behaviour under fire conditions would not be intuitively expected by most structural engineers. The design of simple members exposed to fire is not difficult, as described in this book, but highly redundant structures must be analysed with sophisticated computer programs in order to quantify these effects.