5.2.1 Types of Load
Loads on structures are usually differentiated as ‘dead loads’ and ‘live loads’. These types of loads are referred to in some codes as ‘permanent actions’ and ‘imposed actions’, respectively.
Dead loads are loads which are always present, being the self‐weight of the building materials and any permanent fixtures. Live loads, or ‘occupancy loads’, are loads which may or may not occur at any time, from a wide variety of sources including the following:
• Human occupancy loads are from the weight of people. These may vary from zero to very high levels, especially where crowds can gather. Day‐to‐day loads are usually much less than the loads specified by structural design codes.
• Non‐human occupancy loads come from equipment, goods, and other moveable objects.
The weight of objects may be very low and variable in spaces like office buildings, or heavy and semi‐permanent in warehouses and libraries.
• Snow loads are seasonal, with large geographic differences. Some areas may expect heavy snow to remain for several months every year, whereas others may expect no snow, or very infrequent snow loads.
• Wind loads are experienced by most buildings. The probability of extreme wind loads varying greatly, depending on location and topography. Critical wind loads are usually lat- eral loads on walls or uplift loads on roofs.
• Major earthquakes are extreme events which do not occur often. Some areas expect no earthquakes, others may have many small earthquakes and others have a low but significant probability of a rare major earthquake. Earthquake loads are inertial loads acting at the centre of mass, mostly in the horizontal plane.
5.2.2 Load Combinations
The above loads never all occur at the same time. Structural design at normal temperatures requires investigation of several alternative load combinations as specified by national building codes. At the time of a fire the most likely load is the dead load and a part of the occupancy load.
5.2.3 Structural Analysis
Structural analysis is the process of assessing the load paths in a building, to understand ways in which applied loads on the floors or roofs or walls of the building ‘flow’ through the beams,
5.2.4 Non‐linear Analysis
Most simple structural analysis assumes that the structure behaves in a linear and elastic manner. A linear elastic structure is one where deformations are directly proportional to the applied loads and the structure reverts to its original shape when all loads are removed. The linear elastic assumption is good for most structures at low levels of load.
There are two main sources of non‐linearities in structural analysis. Geometrical non‐
linearities occur when deformations become so large that they induce additional internal actions, resulting in even larger deformations. Column buckling is the most common case of geometrical non‐linearity. Material non‐linearities occur when materials are stressed beyond the elastic range causing yielding or ‘plastic’ behaviour, in which case the structure will have permanent deformations after the loads are removed. Understanding of non‐linear behaviour becomes important if the ultimate strength of the structure is to be well understood.
The simplest computer programs for structural analysis consider only linear elastic behav- iour. More advanced programs can include both geometrical and material non‐linear analysis.
Non‐linear behaviour can be very important under fire conditions because deformations are larger and material strength is less than in normal temperature conditions. Computer programs for structural analysis in fire conditions are discussed in Chapter 11.
5.2.5 Design Format
The specification of design loads and material strength depends on the format of the national building code, which varies from country to country.
The traditional design format, still used in many countries, is working stress design or allowable stress design where calculated member stresses under the actual loads expected in the building are compared with the allowable or permissible stresses which are considered safe for the material under long term loads. There is usually a large safety factor built into the safe working stresses.
Modern design codes use the ultimate strength design format in which internal forces resulting from the maximum likely values of load (‘characteristic loads’) are compared with the expected member strength using the short term strength of the likely materials (‘characteristic strength’). This design format is known as limit states design in Europe and load and resistance factor design (LRFD) in North America. There are minor differences between these formats, but the principles are similar.
Limit states design clearly differentiates between the strength limit state (or ultimate limit state) and the serviceability limit state. The ‘strength’ or ‘ultimate’ limit state is concerned with preventing collapse or failure whereas the ‘serviceability’ limit state is concerned with controlling deflection or vibration which may affect the service of the building. The loads specified for the serviceability limit state are those load combinations which are expected to occur more frequently during the life of the building. Structural design for fire is mainly concerned with the ultimate limit state because it is strength and not deflection which is criti- cal to prevent collapse of buildings exposed to fire.
Some national codes are in transition from working stress design to ultimate strength design. It is possible to make a rough comparison (or soft conversion) between the two for- mats. The use of either design format should result in similar member sizes, especially for simple structural members.
5.2.6 Working Stress Design Format
The loads in working stress design or allowable stress design are the typical loads expected in normal use of the building. The dead load is the self‐weight of the structure estimated by the designer, and the live loads are specified by national design codes.
Considering dead loads and live loads, most codes specify only one load combination for the design load Lw given by:
Lw G Q (5.1)
where G is the dead load and Q is the live load.
Other combinations are given for situations including snow, wind or earthquake loading. In the structural design process, the load Lw is used to calculate internal forces (bending moment, axial force and shear force) in each structural member, then the resulting stresses are calcu- lated and these are compared with the allowable design strength for the material, which is considered to be the safe stress for long term loads.
For example, in the design of a tension member, the axial tensile force Nw (N) in the member is calculated from the above load combination. The resulting tensile stress f*t (MPa) is calcu- lated from:
ft* N Aw/ (5.2)
where A is the cross‐sectional area of the member (mm2).
The design equation which must be satisfied is:
*
t a
f f (5.3)
where fa is the allowable design stress in the code (MPa).
The actual level of safety is not clearly known in this format because the loads are not the worst loads that could occur, and the allowable stresses are known to be safe, but are not directly related to the failure stresses.
u k u k k
Lu 1 4. Gk or Lu 1 2. Gk 1 6. Qk (5.4b) Lu 1 35. Gk or Lu 1 2. Gk 1 5. Qk (5.4c) where Lu is the factored load combination, Gk is the characteristic dead load and Qk is the characteristic live load.
Of the two equations given in each row above, the first is the combination for dead load only and the second is the combination for dead load and live load combined. In the structural design process, the combination having maximum effect is used to calculate the internal forces (bending moment, axial force and shear force) in each structural member, to be compared with the load capacity of the proposed member. Additional combinations for use with wind, snow or earthquake loads can be found in the relevant national standards.
The load capacity is obtained from the short term characteristic strength specified in the material code. The characteristic stress is an estimate of the 5th percentile failure stress. The nominal load capacity is reduced by a strength reduction factor Φ which is intended to allow for uncertainty in the estimates of material strength and section size. The value of Φ is nor- mally in the range 0.7–0.9. In the European system, the strength reduction factor Φ is replaced by 1/γM where γM is the partial safety factor, analogous to the inverse of the strength reduction factor Φ, for each material.
Hence, verification of the design for strength requires that
U* R (5.5)
where U* is the internal force resulting from the applied load, R is the nominal load capacity and Φ is the strength reduction factor (1/γM).
The internal force U* may be axial force N*, bending moment M* or shear force V* occurring singly or in combination. The load capacity R will be the axial strength, flexural strength or shear strength, in the same combination.
For example, in the design of a tensile member, the axial force N* (N) obtained from the worst factored load combination in Equation 5.6 must not exceed the design capacity ΦNn so the design equation is:
N* Nn (5.6)
where Nn is the nominal axial load capacity (N) given by:
Nn f At (5.7)
where ft is the characteristic tensile strength (MPa) and A is the cross‐sectional area (mm2).
When comparing working stress design with ultimate strength design, note that the characteristic tensile strength ft is larger than the long term allowable tensile strength ftw with a corresponding difference between the loads N* and Nw.
5.2.8 Material Properties
The derivation of material design values for strength depends on the format of the design system in use. In the traditional system of working stress design, the design strength (or permissible stress) represents the stress which can be sustained safely under long duration loads. In the more modern system of limit states design or LRFD, the characteristic strength (or design strength) represents the stress at which the material will fail under short duration loads. In most countries the characteristic strength is the 5th percentile short term failure stress ( estimated with 75% confidence) for a typical population of material of the size and quality under consideration.
For modulus of elasticity two values are needed; the 5th percentile value (or ‘lower bound’
value) for buckling strength calculations, and the mean value for deflection calculations.
The normal temperature properties of steel, concrete and timber are compared briefly as an introduction to elevated temperature design in subsequent chapters, using Figure 5.1 and Figure 5.2. Figure 5.1 shows a simply supported beam with two point loads. The bending moment at mid‐span produces the internal strain distribution as shown with tensile strains at the bottom and compressive strains at the top. Figure 5.2 shows typical stress–strain relation- ships for the three materials. These are not drawn at the same scale because the yield strain for steel is much greater than the crushing strength of concrete or wood which are similar.
It can be seen that typical steel has the same properties in both compression and tension, with elastic behaviour to a well‐defined yield point, followed by very ductile behaviour.
Concrete has very little dependable tensile strength, but is strong in compression, with limited ductility. The ductility of reinforced concrete can be substantially increased by confining the compression zone with stirrups. Wood is ductile in compression but exhibits brittle failure in tension. In Figure 5.2 the solid line shows parallel to grain behaviour where the tensile strength is very high, and the dotted line shows perpendicular to grain behaviour where the tensile strength is very weak (splitting failure).
Compression
Tension
Figure 5.1 Internal strains in a simply supported beam
All three materials show some non‐linear material behaviour, especially in compression.
This non‐linear behaviour under increasing load is often termed ‘plasticity’.
The lower parts of Figure 5.2 show flexural stresses in beams of typical cross sections, based on the stress–strain relationships shown above. Internal stresses are shown twice: initially for beams lightly stressed in the elastic range; and secondly for beams stressed to near failure in the inelastic range. The steel beam develops plastic yielding over most of the cross section when approaching its ultimate flexural strength, depending on the amount of curvature. The reinforced concrete beam has a parabolic stress distribution in compression at ultimate strength, with the resulting compressive force equal to the yield force of the reinforcing bars which are yielding in tension. The parabolic compression block is approximated by the dotted rectangle for design purposes (Park and Paulay, 1975). The internal stress distribution for timber depends on the material properties. Commercial quality timber usually has low tensile strength due to defects, so it fails when the stress distribution is in the linear elastic range. For high quality timber with no defects in the tension zone, ductile yielding occurs in compression as shown, leading to lowering of the neutral axis and causing very high tensile stresses.
5.2.9 Probability of Failure
The objective of structural design is to provide buildings with an acceptably low probability of failure under extreme loading conditions. Probabilities of failure are not usually stated in design codes, but they have been used by the writers of ultimate strength design codes to establish the necessary strength reduction factors to give a target level of safety for all antici- pated conditions, using characteristic values of load and resistance.
fy
fy fy
fc
ft f´c
Figure 5.2 Stress–strain relationships and internal flexural stresses for steel, concrete and timber beams
Figure 5.3(a) shows schematically that load U and resistance R are both probabilistic quan- tities, with a distribution of values about a mean. There is a small probability of failure which can be calculated from the area of overlap between the two curves if their distributions are known. The characteristic value of member resistance usually represents the lower 5th percen- tile tail of the strength distribution, and the design load represents a high percentile of likely loads for a given return period.
When considering Figure 5.3(a) for fire design, the load and resistance curves can be quan- tified in the time domain, the temperature domain or the strength domain (as shown in Chapter 4). If Figure 5.3(a) represents load and resistance at room temperatures, both the curves will shift to the left under fire conditions because the expected loads are less and the strength decreases due to elevated temperatures.
The ultimate limit state representing failure occurs if R < U, so the likelihood of failure is related to the difference R − U. Figure 5.3(b) shows the frequency distribution of R − U. The probability R − U < 0 is given by the shaded area under the distribution. Limit state design codes are usually calibrated to give a certain reliability index β, which is the number of stan- dard deviations of the mean value of R − U above zero, as shown in Figure 5.3(b). For given distributions, the strength reduction factor Φ is derived by code writers to give a target reli- ability index β, in the range between 2 and 3 (roughly equivalent to a probability of failure between 10–2 and 10–3).
The above discussion shows that although there is a probabilistic framework behind the ultimate strength code formats, day‐to‐day design is a deterministic process.
Structural design for fire safety has far more uncertainly than structural design for normal temperature conditions. This book considers structural fire safety in a deterministic frame- work, rather than a probabilistic framework. The science of structural reliability is rapidly developing, but applications to structural fire safety are still in their infancy despite pioneering work many years ago by Magnusson (1972) and Schleich (1999).