This paper presents an efficient algorithm for both limit and shakedown analysis of 3-D steel frames by kinematical method using linear programming technique. Several features in the application of linear programming for rigid-plastic analysis of three-dimensional steel frames are discussed , as: change of the variables, automatic choice of the initial basic matrix for the simplex algorithm, direct calculation of the dual variables by primal-dual technique. | Vietnam Journal of Mechanics, VAST, Vol. 29, No. 3 (2007), pp. 337 (N, Afy, l\I2 ) ~ 0, with N is the normal force and !Vly, !Vlz are respectively bending moments about toy and z axes. The plastic hinge modelling is described by the choice of net displacement (relative) - force relationship at the critical sections. In present work, the normality rule is adopted: or, symbolically: ei = Ai Ne, () where Ai is the plastic deformation magnitude; ei is the vector of longitudinal displacement and two rotations of ith section; is a gradient vector of the yield surface . - Bd' = - Bdo (rd' = ~ + fT do () ,\ , d' ~ 0 Therefore , the vector of variables, matri x of constraint, vector of seco nd member corresponding to the problem of Eq. (3 .3) for limit analysis are given below: x* T = [z xT b*T = [0 W' [ 77] = [z bT] = [0 d'T - Bdo ,\T ~ IJ] + (f'do] ~ where 77 is an a rtifi cial variable vvhi ch must be taken out of the basic vector int he simplex process;~ is a constant chosen in relat ion with the value of do [12]. The use of simplex technique we need to find an initial admissible solution s11cl1 that the iuiti al value of any variable (except the objective fu11 ct ion) rnust be non-negat ive. To satisfy t his requirement, it appears that the following a rrangement leads to goo d behav iours of t he a utomat ic calculat ion: The linearized cond ition of plast ic admissibility for the 'i 1h section (Eq. ()) may be expanded as follows: N>O N > > > > > > a~. Si0 S0 S0 S0 S0 S0 a3 { Ni } -a~. l\f~ Mzi ~ Sb Sb S0 . S0 Si0 Sb S0 Si0 S0 S0 () -a(; ai6 · -a~ a6 -a4 a(; -ai4 -ai6 a~ a6 -a4 -a(; The Fig. 1 describes the projection of 16 planar facets of the polyhedral stress-resultant 10 11 12 13 14 15 16 a4 ai4 341 yield surface corresponding to the 16 inequalities numbered on the Eq. (). According to Eqs. (), (), we see that Eq. () can be written under symbolic formulation: () Put: ai1 - a i2 [ -a3 Let
