Beams

5.5 Design of Individual Members Exposed to Fire

5.5.3 Beams

5.5.3.1 Flexural Design

The process shown above for tensile members can be applied to bending members. This sec- tion refers only to simply supported beams. Beams with continuous supports, axial restraint, or larger frames are described in a later section.

Figure 5.8 shows loads and bending moments for a simply supported roof beam designed to support both dead load and snow load. Note that in this book, bending moment diagrams are plotted on the tension side of flexural members, following the European convention which is opposite to that used in North America. A positive bending moment is one which causes

tension on the underside of the beam (a ‘sagging’ moment). A negative bending moment is one which causes tension on the top of the beam (a ‘hogging’ moment).

Under factored design loads of self‐weight and snow load, the bending moment diagram is shown by the solid curve, where the mid‐span bending moment M^*_cold (kNm) is given by:

M_cold^* w L_c ²/ (5.17)8

where w_c is the uniformly distributed factored load (kN/m) and L is the span (m).

Crushing strength (a)

(b) N_c

N_crit

N_crit N_crit

Axial load capacity

Euler curve

L

L

Length

Figure  5.7 Column buckling. (a) Effect of member length on compressive load capacity. (b) Steel column which has buckled during fire exposure. Reproduced from HMSO (1961) by permission of Her Majesty’s Stationery Office

Under normal temperature conditions a member size must be selected with a sufficiently large section modulus Z to satisfy the design equation:

*

cold n

M M (5.18)

where M_n f Z_b

and Φ is the strength reduction factor, f_b is the characteristic flexural strength at normal tem- peratures and Z is the section modulus of the cross section.

For materials such as wood, where design is based on elastic behaviour, Z is the elastic section modulus. For rectangular sections Z bd²/ . For ductile materials like steel, the 6 plastic section modulus S may be used instead of Z, giving slightly higher design strengths.

(Note that the symbols Z and S are used with reversed meanings in some countries.)

The resulting short term flexural capacity under cold conditions is R_cold f Z_b , shown by the lower straight line in Figure 5.8. R_cold is greater than M_cold^* because of the following factors in the design process:

1. The strength reduction factor Φ is always less than 1.0 in normal temperature conditions.

2. The size of the selected member may be larger than exactly needed (because of steps in avail- able sizes or because the size was chosen to control deflections, or for architectural reasons).

Bending moment

M^*_fire

M^*cold

R_cold

Figure 5.8 Bending moment diagrams for a simply supported beam

3. For some materials such as timber, the strength depends on the duration of the load, so there will be a difference between the short term capacity and long term load demand.

If a fire occurs when there is no snow on the roof, the bending moment at mid‐span will be less, as shown by the dotted curve. The mid‐span bending moment M^*_fire (kNm) is given by:

M^*_fire w L_f ²/ (5.19)8

where w_f is the uniformly distributed factored load for fire conditions (kN/m).

It can be seen that for failure to occur as a result of the fire, the flexural capacity would have to drop from R_cold to M^*_fire. In this case the design equation becomes:

*

fire f

M M (5.20)

where M_f f_{b f,} Z_f

and f_b,f is the characteristic flexural strength of the material at elevated temperature and Z_f is the appropriate section modulus of the cross section (possibly reduced by fire exposure).

The ratio M^*_fire/R_cold is the ‘load ratio’. This example demonstrates the important principle that if the load ratio is low, the necessary drop in strength for failure to occur is large, hence the greater the fire resistance.

5.5.3.2 Lateral Buckling of Beams

The above equations for bending assume that the beam will fail by flexural yielding. Slender beams with no lateral restraint may fail by lateral torsional buckling at a load less than the flexural load capacity. This can only happen if the compression edge of the beam is free to buckle sideways. Lateral buckling is more of a problem for slender beams of materials like steel than for compact materials like concrete. At normal temperatures the critical buckling load can be calculated by using formulae from structural design codes. Chapter  6 gives guidance for checking lateral buckling of steel beams under fire exposure.

Any members providing lateral bracing to beams or columns must have at least the same fire resistance as the main members. This can be difficult to calculate if the bracing members are not actually load bearing, but they only provide bracing. A common rule‐of‐thumb is that bracing members should be designed to resist 2½% of the axial force in the braced member, in addition to any applied loads and self‐weight. This can also be used in fire design. The hierarchy of lateral support must be followed through carefully. For example, main roof beams may rely on secondary beams for lateral stability, the secondary beams may rely on purlins and the purlins may relay on the roofing material. If the main beams are to resist a design fire, then all of the related materials must remain in place for the duration of the fire exposure.

5.5.3.3 Shear Design

Design for shear can be handled in the same way if sufficient information is available on the shear resistance of materials and members at elevated temperatures.

building. In general, the continuity of moment‐resisting connections enhances fire resistance of members in frames, so that design of individual members using the methods described above is conservative. Special purpose computer programs should be used when assessing the expected fire performance of large or special structures including multi‐bay frames. Design of unbraced frames is more difficult than braced frames because lateral deformations and the resulting P–Δ (P–Delta) effects must also be considered.