Calculation of structural responses using simple approaches For some simple structural members such as beams and columns if the

fire scenarios are identical or very much similar along the longitudinal direction of the member, then the temperature field can be assumed to be independent of the longitudinal coordinate and determined based on a two-dimensional heat transfer problem within the cross section of the member. Once the temperature field has been determined, the response of the member to the applied loading can be calculated based on the simple bending theory of Bernoulli beams.

The main assumption of the Bernoulli beam is the linear distribution of the axial strain in the cross section. Under this assumption, the axial strain at any coordinate point of the cross section can be expressed as the sum of a membrane strain and two bending strains as follows

ε(y, z)= ε₀+ yκxy+ zκxz (6.21) where ε₀ is the membrane strain, κxy and κxz are the curvatures of the beam in the xy- and xz-planes, respectively. On the other hand, the total strain can be decomposed in terms of the components generated by individual actions (Li and Purkiss, 2005)

ε(y, z)= εσ(σ , θ)+ εcr(σ , θ, t)+ εtr(σ , θ)+ εth(θ) (6.22) where σ is the stress, θ is the temperature, t is the time, εσ is the stress-induced strain which is the function of stress and temperature, εcr is the classical creep strain which is the function of stress, temperature and time, ε_tr is the transient strain which is the function of stress and temperature and exists only for concrete material and εth is the thermal strain which is the function of temperature. Expressions for εσ, εcr, εtr and εthfor steel and concrete materials can be found in Chapter 5. It is known that, when the strain involves the plastic strain, the stress–strain relation is usually expressed in the increment form. According to Eq. (6.22), the increment of the axial strain can be expressed as

ε= fσ(σ , θ, t)σ+ fθ(σ , θ, t)θ+ ft(σ , θ, t)t (6.23)

where ε, σ , θ and t are the increments of strain, stress, temperature and time, respectively. Similarly, the increment form of Eq. (6.21) can be expressed as,

ε= ε₀+ yκxy+ zκxz (6.24) where ε₀is the increment of membrane strain, κxy and κxz are the increments of curvatures. Substituting Eq. (6.23) into Eq. (6.24) yields

σ = 1 fσ

(ε₀+ yκxy+ zκxz− fθθ− ftt) (6.25)

Let Nx be the axial membrane force and My and Mz be the bending moments about y- and z-axes. Their increments thus can be expressed as

Nx=

Substituting Eq. (6.25) into Eq. (6.26) yields,

Nx= K₁₁ε₀+ K₁₂κ_xy+ K₁₃κ_xz− F₁

Equation (6.27) can be rewritten into the matrix relationship between the increments of generalized strains and generalized forces

⎡

Equation (6.28) is the generalized form of the bending equation of Bernoulli beams, in which the increments of generalized forces Nx, Mz

and Mycan be expressed in terms of the increments of externally applied mechanical loads and reaction forces at boundaries (if it is a statically inde-terminate structure) through the use of equilibrium equations. The incre-ments F₁, F₂and F₃are generated due to the temperature and time increments. In the case of ambient temperature, F₁, F₂and F₃remain zero and thus Eq. (6.28) reduces to the conventional incremental form of the bending equation of beams. Further, if the material constitutive equa-tion is linear, then the stiffness coefficients Kij will be independent of stresses and strains and thus the relationship between generalized strains and generalized forces will be the same as that between their increments.

In the case where the temperature and internal forces are independent of the longitudinal coordinate, Eq. (6.28) can be solved directly based on increment steps. Examples of this include the column subjected to pure compression and the beam subjected to pure bending. Otherwise, the membrane strain and curvatures must be solved by considering the com-patibility along the longitudinal direction of the member with imposed or calculated end conditions (Purkiss and Weeks, 1987; Purkiss, 1990a).

Note that

where u is the increment of axial displacement, v and w are the increments of deflections in y- and z-directions, qx is the increment of axial distributed load (qx is positive if it is in x-axis direction),

qy and qz are the increments of transverse distributed loads in y-and z-directions (qy and qz are positive if they are in y- and z-axis directions). For statically determinate beams, Nx, Mzand Mycan be

determined from Eq. (6.30) or from static equilibrium equations directly.

For statically indeterminate beams, Nx, Mz and My cannot be determined directly from static equilibrium equations and will involve some of unknown reaction forces, which need to be determined from displacement boundary conditions. Substituting Eq. (6.29) into Eq. (6.28) yields

Equation (6.31) can be solved using various discrete methods along the longitudinal direction to convert the differentiation equations into algebraic equations. Because of the non-linearity involved in the material constitutive equation, the stiffness coefficient K_ijin Eq. (6.31) are not only temperature dependent but also stresses and strains dependent. Thus, iterations are required in solving the equations at each time step to cor-rect for non-linearities. This finally leads to a complete deformation time history for the given externally applied loads. Such calculations are only amenable to computer analysis, examples of which include FIRES-RC (Becker and Bresler, 1972), CEFFICOS (Schleich, 1986, 1987), CONFIRE (Forsén, 1982). Figure 6.2 provides a flowchart for this kind of calcu-lations. The method can be applied to steel, concrete and composite steel–concrete members. For timber, which chars substantially when sub-jected to heat leaving a relatively unaffected core, calculations can be undertaken using normal ambient methods with the use of temperature-reduced strengths if considered appropriate on the core after the parent section has been reduced by the appropriate depth of charring. Hosser, Dorn and Richter (1994) provided a very useful overview and assessment of some of available simplified, i.e. non-computer code, design methods.

6.3.2 Calculation of structural responses using finite element