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- Second-order analysis for semirigid steel frame design

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The inelastic second-order behavior of steel structural columns under minor-axis bending is presented in this article. To study this behavior, a nonlinear frame element formulation is adopted in which the steel's plasticity process is accompanied at the nodal points of each finite element through the refined plastic-hinge method RPHM.

A tangent modulus approach is employed in order to consider the stiffness degradation in function of the internal forces' variation, and the second-order effects, residual stresses and geometric imperfections are considered in the modeling of column behavior.

As a criterium for defining the ultimate limit state of the column cross-section, strength surfaces are adopted. These surfaces describe the interaction between the axial force and bending moment N-M interaction diagrams. To solve the nonlinear equilibrium equation for the structural system, the Newton-Raphson method is used, coupled with continuation strategies. Columns with different slenderness, boundary and loading conditions are analyzed, and the results obtained are comparable to those found by other researchers.

The results lead to the conclusion that the numerical approach adopted in this study can be used for a better behavioral understanding of the steel column under weak-axis bending.

Nowadays, steel material is commonly employed in civil construction. Besides being a completely recyclable material, steel has important characteristics such as strength and durability, good ductility and speedy manufacturing and assembly times. For the steel member and frame structure analysis, highlighted herein is an advanced analysis that has the capacity to simultaneously evaluate its strength and stability.

In a nonlinear finite element context for steel member and frame numerical analysis, two inelastic analysis approaches are usually employed: the plastic zone method PZM and the refined plastic-hinge method RPHM. In the PZM Clarke, ; Alvarenga, , the cross-section of each finite element is discretized in fibers and the second order effects and residual stresses can be directly considered in the analysis. Another important characteristic is that with the stress state obtained in each fiber, it is possible to monitor the gradual yielding in the cross-section.

However, the PZM is not routinely used in engineering offices, since it requires an intense computational effort. At these nodes, the formation of plastic hinges can occur, characterizing the plasticity of the entire cross-section. This method is computationally more practical and captures, in an approximating manner, the advance of the plasticity in the element cross-sections before the formation of the plastic hinges.

Residual stresses effects can also be considered. The objective of this study is to use an RPHM that permits adequate modeling for the inelastic behavior of steel columns with type I compact sections under weak or minor-axis bending. Although unusual, the columns where the bending occurs under this axis can present important benefits such as the capacity to develop all of their plastic strength without lateral-torsional buckling BS , ; AISC, In this equation, the cross-section stiffness degradation varies in function of the axial force and bending moment around the minor axis.

The numerical formulation proposed also employs the strength surfaces McGuire et al. Validation of these strategies is made by analysis of the columns under various boundaries, slenderness and loading conditions. The results obtained are compared with analytical and numerical solutions obtained using the PZM. In this study, the second order effects are simulated, and the nonlinear solution methodology is based on the Newton-Raphson method coupled with continuation strategies.

The details of this methodology are presented on the next two sections. Section 4 presents three numerical examples.

The following assumptions are considered in the column modeling: all finite elements are initially straight and prismatic and their cross-sections remain plane after a deformation, the steel profiles are compact, rigid body large displacements and rotations are permitted, and the shear deformation effects are ignored. The finite element utilized is the frame element delimited by nodal points i and j, with fictitious springs at the ends, as illustrated in Fig.

Also in Fig. The objective of the RPHM is to capture the advance of the plastification at the nodal points of elements, from the beginning of the yielding to its total plastification with the plastic hinge formation. To achieve this goal, the inelastic formulation adopted in this study is based on the proposal made by Chan and Chui , where it is considered that the plasticity development in the structural members is simulated through fictitious springs Fig.

In Fig. For the internal beam element Fig. The terms k ii , k ij , k ji and k jj are responsible for simulating the bending stiffness and second-order effects.

They are defined as Yang and Kuo, :. Associating the Eqs. The tangent modulus E t captures, in an approximate manner, the reduction of the element's sectional stiffness due to the axial force. The strength equations for the columns of AISC define the variation of the tangent modulus as:.

These equations include the effect of the initial imperfections as well as the residual stresses in the columns. Ziemian and McGuire proposed a modification to the equations presented in the CRC Galambos, in which the tangent modulus varies in function of the axial force and the bending moment about the weakest axis.

In this case:. Using these equations in the analysis of structural members under minor-axis bending, Ziemian and McGuire obtained results that were compatible with those encountered when using the PZM.

Considering only the axial forces, Fig. To reach the proposed objective herein, the tangent modulus equations are utilized as indicated by Ziemian and McGuire , which depend on the axial force and moment under minor-axis bending. Since this moment is evaluated at the i and j ends of the finite element Fig. The terms k of the stiffness matrix in Eq. Using the appropriate interpolation functions, the first part of k ii , k ij , k ji and k jj are redefined considering E t , through the relationships:.

Resolving the previous integrals, the coefficients k ii , k ij , k ji and k jj become:. Equation 6 is valid until the internal forces reach the section plastic strength.

From then on, with the section having already suffered total plastification, the increase of the axial force, for example, creates a condition where the strength of the section is less than that of the forces acting upon it. As such, an alteration in the force-displacement relationship is necessary so that the strength plastic equations for the section are not violated.

This alteration and the transformation of Eq. In an inelastic analysis, it is necessary to accurately estimate the ultimate capacity of the structural members. To achieve this, criteria are established that define the plastic strength limit of the structural member. Amongst the various existing criteria von Mises, Tresca, etc. In the RPHM context, these surfaces are responsible for defining the instant in which the plastic hinge occurs cross-section total plastification.

For this study, two strength surfaces are adopted where the interaction between axial force and the bending moment in the transversal sections are evaluated. The behavior of the structural member when the bending moment acts on the weakest axis is illustrated in Fig. The first strength surface, proposed by McGuire et al. The British Standard BS supplies expressions for the reduced plastic moment M pr for compact I or H profiles in the presence of axial force. In relation to the minor axis bending, these equations are given by:.

The term Z y is the minor-axis plastic section modulus. Figure 4 illustrates the two strength surfaces obtained by the equations previously presented. The effects caused by residual stresses are considered in an implicit manner on the tangent modulus. However, these stresses are expressed in an explicit form to determinate the bending moment when yield begins Eq.

It is worth mentioning that in members under weak-axis bending, the effect of residual stresses is more pronounced, since they affect the extreme fibers more. In this section, the adopted numerical strategies for the inelastic second-order analysis of columns under minor-axis bending are evaluated. Specifically, the importance of considering the modified tangent modulus model proposed by Ziemian and McGuire to simulate the stiffness degradation of the cross-section will be highlighted.

Also studied are the two plastic strength surfaces that were previously described. To demonstrate this, three columns with different load and boundary conditions are studied and the results compared with those found in literature. As nonlinear solver strategy, the generalized displacement strategy load-increment strategy and the minimum norm of residual displacement strategy with the Newton-Raphson method iterative strategy are adopted Silva, A tolerance factor of 10 -4 is used.

To facilitate the presentation of the results, a simplified notation is used, given in Table 1 , to refer to the different tangent modulus models. As indicated, the use of the constant modulus of elasticity is also possible.

Consider a fixed-free column submitted to a permanent vertical load P and a variable horizontal load H applied perpendicularly to the minor axis as illustrated in Fig. This example was previously analyzed by Zubydan to validate his numerical formulations. This figure also includes the member material and geometry data, as well as the finite element discretization.

By controlling the horizontal displacement u at the top of the column, the equilibrium paths illustrated in Fig. The values obtained for the horizontal load at collapse are summarized in Table 2. The results obtained by Zubydan , who utilized the PZM, are used for comparison. The residual stresses are defined according to recommendations from ECCS The strength surfaces proposed by McGuire et al. By analyzing the load-displacement curves and the collapse loads in Table 2 , it is possible to conclude that the tangent modulus E t3 is more efficient in forecasting the load limit.

Now, maintaining the tangent modulus E t3 , another analysis involving the variation of the strength surfaces is performed. Figure 8 displays the equilibrium paths obtained. Notice the good agreement between the two strength surfaces studied. Assuming that there is a permanent axial load P acting at the ends of the pinned column, two loading conditions are evaluated: the variable bending moment acting at the two ends and the variable horizontal load applied to the center of the column, as shown in Fig.

The column bending in both loading cases occurs under the weak-axis. Ten finite elements are used in the discretization of the member. The cross-section and the material data used are also presented in Fig. In the analysis, the strength surface used is that which is recommended in BS and the residual stresses indicated in ECCS are adopted.

The interaction curves axial force x bending moment for each of these investigated columns are presented in Fig. The analyses adopted E t1 and E t3 for the tangent modulus. With the finality of validating results, analytical solutions developed by Kanchanalai and Lu were employed. For the case of the column with moments at the two ends, the results obtained by Zubidan are also used. It is observed in Fig. However, for less slender parameters, the strength for the column is overestimated.

In turn, the tangent modulus E t3 furnished conservative curves for all of the situations related to the analytical solutions, and it is more precise when compared to the solutions by Zubidan , who used the PZM.

Du kanske gillar. Ladda ned. Spara som favorit. Skickas inom vardagar. Plastic Design of Steel Frames assesses the current status and future direction of computer-based analyses of inelastic strength and stability for direct frame design.

Plastic Design of Steel Frames assesses the current status and future direction of computer-based analyses of inelastic strength and stability for direct frame design. It shows how design rules are used in practical frame design and provides an introduction to the second-order theory of inelastic frame design. The book includes two computer programs on a diskette: one for the first-order analyses and the other for the second-order plastic hinge analysis of planar frame design. The second-order program can be used to predict realistic strengths and stabilities of planar frames, thereby eliminating the tedious task of estimating factors for individual member capacity checks. Both programs include clear input instructions. The diskette also contains the Fortran source-code listing for the second-order plastic-hinge analysis, enabling the user to customize the program. Springer Professional.

Chapter 7 presents a computer-based method for the first-order plastic hinge-by-hinge analysis for frame design. The computer program FOPA.

Structural analysis is the process of calculating the forces, moments and deflections to which the members in a structure are to be subjected. There is a vast range of analysis tools offering speed, precision and economy of design ; 3-D, FE modelling, bespoke portal frame , cellular beam or plate girder design software are now widely available. Modelling catenary actions, cold formed member performance or grillage analysis - all these are now commonplace for structures, where hand analysis is impossible.

It seems that you're in Germany. We have a dedicated site for Germany. Plastic Design of Steel Frames assesses the current status and future direction of computer-based analyses of inelastic strength and stability for direct frame design. It shows how design rules are used in practical frame design and provides an introduction to the second-order theory of inelastic frame design.

*Design of steel frames by second-order P- analysis fulfilling code requirements 3.*

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## Germaine Г.

The inelastic second-order behavior of steel structural columns under minor-axis bending is presented in this article.