The stiffness reduction that results from yielding of the cross-section due to minor axis bending and axial compression is studied in detail for compact W-Shapes with a European Convention for Constructional Steelwork (ECCS) residual stress distribution pattern. For a given minor axis moment m (M/M_p), compressive load p (P/P_y), and residual stress ratio c_r (σ_r/σ_y), the distribution of stresses throughout the cross-section and associated reduced stiffness τ (EI_(ep)/EI) are evaluated using a fiber element model with 2,046 elements. A three-dimensional m-p-τ surface plot of a W8×31 with c_r = 0.3 is used to discuss the stress states and reduced stiffness for the m and p conditions around the perimeter of the surface. For these conditions, equations that predict the extent of yielding and the distribution of stresses in the flanges and web are provided for any given W-Shape and c_r condition. The ability to determine the stresses with analytical expressions leads to reduced stiffness equations that are no longer dependent upon empirical relationships. Seven elasto-plastic stress states have been identified for the m and p conditions on the interior of the surface. Discussion is provided on the development of similar analytical expressions for these conditions. Numerous figures are provided on the progression through the various stress states as m is increased from the initial yield condition up to the fully plastic state for p = 0.2, 0.5 and 0.75. The data from the m-p-τ surface plot for the W8×31 with minor axis bending are used to develop two nonlinear regression equations for use as tangent modulus expressions in MASTAN2. The El-Zanaty portal frame is modeled with p = 0.4 and 0.6. The maximum lateral load and deflection values compared very closely with published results of the same frame under similar conditions.
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