ABSTRACT
While at elevated temperatures, the mechanical properties of steel are obtained from
and
in which E0 and σ0 are the modulus of elasticity and yield strength of steel material at ambient state.
3 THEORETICAL FORMULATION
The normal strain at a typical steel cross-section can be represented by
The alternative forms of Eq. (6) are
In Eqs. (6) and (7) y¯ is the location of the elastic centroid of the steel section below the local reference level,which is assumed to be coincident with the top fibre of the steel cross-section, yn is the location of the neutral fibre, εm, εe and εθ are the appropriate membrane, mechanical and thermal strains which are obtained from
For a given cross-section , the magnitude of the total axial force and internal bending moment can be obtained from
in which the subscripts “e” and “pi” indicate the internal actions which act on the elastic part (e) and on each of the two plastic parts (p) of the cross-section, respectively and are determined from
Substituting Eqs. (1) and (7) into Equation (11) leads to
and ωA, ωB and ωI depend on the mechanical properties of the steel material. Equation (12) can then be rearranged so that the curvature κ at a cross-section is written as a function of the internal actions in the form
Using the equilibrium of the steel element in its deformed configuration, the magnitude of the internal bending moment M int is written as a function of bending moment, shear force and axial force at the left side of the element such that
In Eq. (14) f id is the total inertia and damping forces which is given by
The relationship between the slopes at the ends of the element can be numerically found by integration of the curvature in Eq. (13) so that
Similarly, the difference in the deflections at two ends of element can be determined from Eq. (16) such that
Using Eqs. (7) and (8) the relationship between the axial deformations at the two ends of the steel element can be expressed as
At the end of the member,
For this boundary value problem Eqs. (16) to (18) leads to a system of three equations which must be solved simultaneously for M 0, R0 and N 0; this is done herein using the mathematical package MAPLE.
