[BACKGROUND ART] Over the past twenty years, research has developed and validated various methods of performing second-order inelastic analysis on steel frames. Most of these studies can be categorized into two main types: (1) plastic zone and plastic hinge. In the plastic zone method, a frame member is subdivided into several finite elements, and the cross-section of each finite element is further subdivided into many fibers. Although the plastic zone solution is known as the"exact solution", it is not be used in daily engineering design, because it is too intensive in computation. In the plastic hinge method, only one beam-column element per member can capture the second-order effect. Therefore the plastic hinge method has a clear advantage over the plastic zone method. The plastic hinge method, however, generally over predicts the actual strength and stiffness of member.

Chen and Kim extended the modified plastic hinge analysis including gradual yielding and second-order effects. The analysis provides very good agreements with plastic zone method, but it can be used for only the two-dimensional structure.

Several nonlinear analysis methods for the three-dimensional structure have been developed by Orbison, Prakash, Powell, and Liew. Orbison's

method is an elastic-plastic hinge analysis without considering shear deformations. The material nonlinearity is considered by the tangent modulus Et and the geometric nonlinearity is by a geometric stiffness matrix. DRAIN-3DX was developed by Prakash and Powell. The material nonlinearity is considered by the stress-strain relationship of the fibers in a section. The geometric nonlinearity caused by axial force is considered by the use of the geometric stiffness matrix, but the nonlinearity caused by the interaction between the axial force and bending moments is not considered.

Liew's analysis is a refined plastic hinge method considering shear deformations. Liew's method overestimates the strength and stiffness of the member subjected to significant axial force.

Thus a new analysis method having the advantage of existing analysis as well as reducing analysis time is required.

[DISCLOSURE OF INVENTION] To achieve the objectives mentioned above, stability functions accounting for nonlinearity, softening plastic hinge, modified stiffness matrix accounting for shear deformation of the member are used in this software.

Stability Functions Accounting for Second-Order Effect To capture second-order (large displacement) effects, stability functions are used to minimize modeling and solution time. Generally only one or two elements are needed per a member. The simplified stability functions reported by Chen and Lui (1992) are used here. Considering the prismatic beam-column element in Fig. 1, the incremental force- displacement relationship of this element may be written as

where S1, S2 = stability functions = incremental end moments p = incremental axial force OA, OB = incremental joint rotations e = incremental axial displacement A, , I , L = area, moment of inertia, and length of beam-column element E = modulus of elasticity.

The stability functions given by Eq. (1) may be written as where # = P/(#2 EI/L2), P is positive in tension.

The force-displacement equation may be extended for three-dimensional beam-column element as

where P, MyA, MyB, MzA, MzB, and, T are axial force, end moments with respect to Y and z axes and torsion respectively. 5 , SYB OZAX O-B, and, 0 are the axial displacement, the joint rotations, and the angle of twist. s1, s2, s3, and S4 are the stability functions with respect to Y and Z axes, respectively.

CRC Tangent Modulus Model Associated with Residual Stresses The CRC tangent modulus concept is used to account for gradual yielding (due to residual stresses) along the length of axially loaded members between plastic hinges. The elastic modulus E (instead of moment of inertia I) is reduced to account for the reduction of the elastic portion of the cross-section since the reduction of the elastic modulus is easier to implement than a new moment of inertia for every different section. The rate of reduction in stiffness is different in the weak-and strong-directions, but this is not considered since the dramatic degradation of weak-axis stiffness is compensated for by the substantial weak-axis plastic strength (Chen and Kim 1997). This simplification makes the present methods practical. From Chen and Lui (1986), the CRC Et is written as

Parabolic Function for Gradual Yielding due to Flexure The tangent modulus model is suitable for the member subjected to axial force, but not adequate for cases of both axial force and bending moment. A gradual stiffness degradation model for a plastic hinge is required to represent the partial plastification effects associated with bending. We shall introduce the softening plastic hinge model to represented the transition from elastic to zero stiffness associated with a developing hinge. When softening plastic hinges are active at both ends of an element, the slope-deflection equation may be expressed as

The terms #A and #B is a scalar parameter that allows for gradual inelastic stiffness reduction of the element associated with plastification at end A and B. This term is equal to 1.0 when the element is elastic, and zero when a plastic hinge is formed. The parameter 71 is assumed to vary according to the parabolic function: 77 1. 0 for α # 0. 5 (8a) 1 = 4) for a > 0. 5 (8b) where a is a force-state parameter that measures the magnitude of axial force and bending moment at the element end. The term a may be expressed by AISC-LRFD and Orbison, respectively: Shear deformation To account for transverse shear deformation effects in a beam- column element, the stiffness matrix may be modified as

Modification of Element Stiffness for the Presence of Plastic Hinges If the state of forces at any cross-section equals or exceeds the plastic section capacity, a plastic hinge is formed where a slope discontinuity occurs. Therefore, the slope-deflection equation need to be modified to reflect the change in element behavior due to formation of plastic hinges at the element ends. If a plastic hinge forms in an element at end A, the slope-deflection equation from Eq. (14) is written as where #MypcA and #MxpcA are the changes in plastic moment capacity at

the end A as P changes. SyA and SZA can be solved from the second and fourth rows of Eq. (11) as Back-substituting Eq. (12a) and (12b) into the first, third, fifth, and sixth rows of Eq. (18), the modified slope-deflection equation can be expressed as

Eq. (13) represents the modified slope-deflection equation of a frame element with a plastic hinge formed at end A. If a plastic hinge forms at end B, a similar approach can be followed. The corresponding slope-deflection equations If plastic hinges form at both ends, 0. and OB can be written in terms of the change in moment at the respective end of the element. The resulting slope-deflection equation is

where AMY,,, AMypcB #MzpcA, and, #MzpcB are the changes in the plastic moment capacity at the respective end of the members as P changes.

Eq. (13) through (15) account for the presence of plastic hinge (s) at the element end (s). They may be written symbolically as where [Ke, 1] is the modified basic tangent stiffness matrix due to the presence of plastic hinge (s). {teh} is an equilibrium force correction vector that results from the change in moment capacity as P changes.

Since the force-point movement remains on the plastic strength

surface of a member, the plastic strength surface requirement of a section is not violated by the change of member forces after the full plastic strength of a cross-section is reached. In Fig. 4, after the force- point reaches point Q on the plastic strength surface and the member axial force increases, a corresponding moment decrease results in the member. The force-point then moves from Q to R. The moment at a plastic hinge increases from point R, which decreases the axial forces causing the force-point to relocate to Q.

Element Stiffness Matrix The end forces and end displacements used in Eq. (9). The sign convention for the positive directions of element end forces and end displacements of a frame member is shown in Fig. 5 (b). By comparing the two figures, we can express the equilibrium and kinematic relationships in symbolic form as {fn} and {dL} are the end dorce and displacement vector sof a frame member expressed as {fn}T = {rn1 rn2 rn3 rn4 rn5 rn6 rn7 rn8 rn9 rn10 rn11 rn12} (19a) {dL}T = {d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12} (19b) {fe} and {de} are the end dorce and displacement vectors in Eq. (13).

[T]6#12 is a transformation matrix written as

Using the transformation matrix by equilibrium and kinematic relations, the force-displacement relationship of a frame member may be written as [Kn] is the element stiffness matrix expressed as Eq. (22) can be subgrouped as

Eq. (23) is used to enforce no sidesway in the member. If the member is permitted to sway, an additional axial and shear forces will be induced in the member. We can relate this additional axial and shear forces due to a member sway to the member end displacements as where {fs}, {dL} and [Ks] are end force vector, end displacement vector, and the element stiffness matrix. They may be written as {fs}T = {rs1 rs1 rs3 rs4 rs5 rs6 rs7 rs8 rs9 rs10 rs11 rs12} (27a) {dL}T = {d1 d2 d3 d4 d5 d6 d7 d8 d9 d10 d11 d12} (27b)

By combining Eq. (21) and Eq. (26), we obtain the general beam- column element force-displacement relationship as {} = [L} (30) where M=M+ {} (31) Kroar-LKn, +LKs (32) Numerical Implementation The simple incremental method, as a direct nonlinear solution technique, is used in the analysis. Its numerical procedure is straightforward in concept and implementation. The advantage of this method is its computational efficiency. This is especially true when the structure is loaded into the inelastic region since tracing the hinge-by- hinge formation is required in the element stiffness formulation. For a finite increment size, this approach only approximates the nonlinear structural response, and equilibrium between the external applied loads

and the internal element forces is not satisfied. To avoid this, an improved incremental method is used in this program. The applied load increment is automatically reduced to minimize the error when the change in the element stiffness parameter (A77) exceeds a defined tolerance. To prevent plastic hinges from forming within a constant- stiffness load increment, load step sizes less than or equal to the specified increment magnitude are internally computed so plastic hinges form only after the load increment. Subsequent element stiffness formations account for the stiffness reduction due to the presence of the plastic hinges. For elements partially yielded at their ends, a limit is placed on the magnitude of the increment in the element end forces.

The applied load increment in the above solution procedure may be reduced for any of the following reasons: (1) Formation of new plastic hinge (s) prior to the full application of incremental loads ; (2) The increment in the element nodal forces at plastic hinges is excessive ; (3) Non-positive definiteness of the structural stiffness matrix. As the stability limit point is approached in the analysis, large step increments may overstep a limit point. Therefore, a smaller step size is used near the limit point to obtain accurate collapse displacements and second- order forces.

[BEST MODE FOR CARRING OUT THE INVENTION] Column with Three-Dimensional Degree of Freedom A simply supported column with three-dimensional degree of

freedom is shown in Fig. 6. W8X3l column of A36 steel is used for the analysis. The column strength calculated by the proposed analysis, Euler solution, and DRAIN-3DX based on the slenderness parameter A, are compared in Fig. 7.

Table 1 Comparison of analysis result for a 3-D column L (m) Xc (weak axis) Euler (P/Py) AISC-LRFD Orbison 0.00 0.000 1. 000 1. 000 0. 993 22.51 0.125 1. 000 0. 972 0.928 45.03 0.250 1. 000 0.942 0.915 67.54 0.375 1. 000 0.911 0.897 90.06 0.500 1.000 0.879 0.876 112.57 0. 625 1. 000 0.843 0.851 135.09 0. 750 LOOP 0.805 0. 821 157.60 0.875 1. 000 0. 763 0.785 180.11 1.000 0.996 0.715 0.739 225.14 1.250 0.637 0.599 0.608 270.17 1.500 0.443 0. 443 0. 443 315.20 1. 750 0. 325 0. 326 0.326 360.23 2.000 0.249 0.249 0.235 405.26 2.250 0.197 0.197 0.197 450.29 2.500 0.159 0. 160 0. 160 495.05 2.750 0.132 0.132 0.132 540.34 3.000 0.111 0.111 0.111 585.37 3.250 0.094 0.095 0.095 630.40 3. 500 0. 081 0.082 0.082

The strength of the proposed analysis compares well with Euler's theoretical solution. The maximum error from the proposed analysis is 1.31% for the practical range of columns (/L2. 0). However, DRAIN- 3DX produces the maximum error of 21. 16%. The large error value is a result of not considering the interaction of the axial force and bending

moments when considering geometric nonlinear effect.

Where Table 1. L is length of column; Ac is slenderness parameter about weak axis is written as Euler buckling strength for weak axis written as Orbison's six-story space frame ignoring lateral torsional buckling Fig. 8 shows Orbison's six-story space frame (Orbison, 1982). The yield strength of all members is 250 MPa (36 ksi) and Young's modulus is 206,850 MPa (30,000 ksi). Uniform floor pressure of 4.8 kN/m2 (100 psf) is converted into equivalent concentrated loads on the top of the columns.

Wind loads are simulated by point loads of 26.7 kN (6 kips) in the Y- direction at every beam-column joints.

The load-displacement results calculated by the proposed analysis compare well with those of Liew and Tang's (considering shear deformations) and Orbison's (ignoring shear deformations) results (Table 2, Table 3, and Fig. 9). The ultimate load factors calculated from the proposed analysis are 2.057 and 2.066. These values are nearly equivalent to 2.062 and 2.059 calculated by Liew, Tang and Orbison, respectively.

Table 2 Analysis result considering shear deformation Method Proposed Liew's Plastic strength surface LRFD Orbison Orbison Ultimate load factor 1.990 2.057 2.062 Displacement at A in Y-208 mm 219 mm 250 mm Table 3 Analysis result ignoring shear deformation Method Proposed Orbison's Plastic strength surface LRFD Orbison Orbison Ultimate load factor 1.997 2.066 2. 059 Displacement at A in Y-199 mm 208 mm 247 mm

[BRIEF DISCRIPTION OF DRAWINGS] Figure 1 is a simply supported 3-dimension column.

Figure 2 is a graph of modified CRC tangent modulus.

Figure 3 is a graph which shows parabolic function represents refined plastic hinge Figure 4 is a graph of nonlinear analysis procedure.

Figure 5 is a program flow diagram Figure 6 is exampled 3D column for buckling analysis Figure 7 is a column strength curve of Figure 6 Figure 8 is exampled 3D asymmetric 6-story frame Figure 9 is load-displacement curve of Figure 8 [INDUSTRIAL APPLICABILITY] (1) Usage in the analysis and design of civil steel structures (2) Usage in the analysis and design of architectural steel structures