Modeling Steel Frame Buildings in Three Dimensions. I: Panel Zone and Plastic Hinge Beam Elements

A procedure for efficient three-dimensional nonlinear time-history analysis of steel framed buildings is derived. It incorporates two types of nonlinear beam elements—the plastic hinge type and the elastofiber type—and nonlinear panel zone elements to model yielding and strain-hardening in moment-frames. Floors and roofs of buildings are modeled using 4-node elastic diaphragm elements. The procedure utilizes an iteration strategy applied to an implicit time-integration scheme to solve the nonlinear equations of motion at each time step. Geometric nonlinearity is included. An overview of the procedure and the theories for the panel zone and the plastic hinge elements are presented in this paper. The theory for the elastofiber element along with illustrative examples are presented in a companion paper. The plastic hinge beam element consists of two nodes at which biaxial flexural yielding is permitted, leading to the formation of plastic hinges. Elastic rotational springs are connected across the plastic hinge locations to model strain-hardening. Axial yielding is also permitted. The panel zone element consists of two orthogonal panels forming a cruciform section. Each panel may yield and strain-harden in shear.     

Transverse Natural Vibration of a Beam with an Internal Joint Carrying In-Plane Flexibilities

Existing research on semirigid jointed frame includes only the rotational flexibility of the joint without consideration of the flexibility in the direction of the transverse shear force. This omission would lead to inaccuracies in the dynamic response of structures, especially in the nonlinear analysis. This paper investigates the dynamic behavior of a bolted joint which has flexibility in both the tangential and rotational directions. The joint is prestressed with axial tension in the bolt shank. It is represented as a virtual connection spring element at the intersection between the beam and the supporting member. The formulation of the hybrid beam–column element including the end springs is presented, and the dynamic behavior of a cantilever beam with this nonlinear semirigid joint is studied. The natural frequencies and mode shapes remain relatively unchanged only for a limited range of the joint stiffnesses, and the eigenpair is influenced by the instantaneous stiffness of the joint defined at a point on the hysteretic loop, particularly when the initial moment stiffness is small.     

Mixed Finite Element for Three-Dimensional Nonlinear Dynamic Analysis of Rectangular Concrete-Filled Steel Tube Beam-Columns

A beam finite-element formulation following Euler-Bernoulli beam theory is presented for geometrically and materially nonlinear analysis of rectangular concrete-filled steel tube (RCFT) beam-columns. The formulation is geared for conducting transient dynamic analysis of composite steel/concrete frame structures. The element stiffness and internal forces were derived through adopting a mixed finite-element approach based on the Hellinger-Reissner variational principle. The load transfer between the steel and concrete constitutive materials was provided through steel and concrete interface via friction and interlocking. Six extra translational degrees-of-freedom (DOFs) were added to the conventional 12 DOF beam element to quantify the differential displacement between the two media. The formulation was verified for a range of geometrically nonlinear test problems and geometrically and materially nonlinear RCFT experimental test specimens from the literature. Strong correlation and convergence characteristics were achieved compared to the published results.     

Large-Deflection Stability of Slender Beam-Columns with Both Ends Partially Restrained against Rotation

The large-deflection analysis and postbuckling behavior of laterally braced or unbraced slender beam columns of symmetrical cross section subjected to end loads (forces and moments) with both ends partially restrained against rotation including the effects of out-of-plumbness is developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions utilized herein are those presented previously by the senior writer. The proposed method can be used in the large-deflection elastic analysis and postbuckling behavior of slender beam columns with rigid, semirigid, and simple flexural connections and both ends. Only bending strains are considered, i.e., the effects of axial and shear strains are neglected. An example is included that shows the effects of flexible connections at both ends on the large-deflection analysis and postbuckling behavior of slender beam columns.     

Composite Bars of Arbitrary Cross Section in Nonlinear Elastic Nonuniform Torsion by BEM

In this paper the elastic nonuniform torsion analysis of composite cylindrical bars of arbitrary cross section consisting of materials in contact, each of which can surround a finite number of inclusions, taking into account the effect of geometric nonlinearity is presented employing the boundary element method (BEM). All of the cross section’s materials are perfectly bonded together, that is separation is not allowed. The torque-rotation relationship is computed based on the finite displacement (finite rotation) theory, that is the transverse displacement components are expressed so as to be valid for large rotations and the longitudinal normal strain includes the second-order geometric nonlinear term often described as the “Wagner strain.” The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross section’s torsional rigidity is evaluated exactly without using the so-called Saint-Venant’s torsional constant. The torsional rigidity of the cross section is evaluated directly employing the primary warping function of the cross section depending on its shape. Three boundary value problems with respect to the variable along the beam axis angle of twist, to the primary and to the secondary warping functions are formulated. The first one, employing the Analog Equation Method (a BEM based method), yields a system of nonlinear equations from which the angle of twist is computed by an iterative process. The rest two problems are solved employing a pure BE method. Numerical results are presented to illustrate the method and demonstrate its efficiency and accuracy. The developed procedure retains most of the advantages of a BEM solution over a pure domain discretization method, although it requires domain discretization.     

Hybrid Beam-Column Element with End Frictional Joints

Most research on semirigid jointed frames include only the rotational flexibility of the joint without consideration of the flexibility in the direction of the shear force. This paper investigates the nonlinear frictional behavior of a bolted slotted joint which includes flexibility in the tangential direction. The contact behavior at the end of the slot is also studied. The joint is prestressed with axial tension in the bolt shank. A virtual connection spring element is included at the intersection point between the beam and the column members. The formulation of the hybrid beam-column element including the end springs is presented, and the elemental matrices are also given for easy and direct implementation into finite-element analysis for static and/or dynamic loading.     

Large Deflection Stability of Slender Beam-Columns with Semirigid Connections: Elastica Approach

The large-deflection elastic analysis of slender beam-columns of symmetrical cross sections with semirigid connections under end loads (forces and moments) including the effects of out-of-plumbness is developed in a classical manner. The classical theory of the “Elastica” and the corresponding elliptical functions are utilized in the proposed method which can be used in the large-deflection stability analysis of slender beam-columns with rigid, semirigid, and simple connections under any combination of end loads (conservative and nonconservative). The proposed method consisting of a closed-form solution of the Elastica can also be utilized in the large deflection analysis of beam-columns whose connections suffer from flexural degradation or, on the contrary, flexural stiffening. The main limitation of the Elastica is that only flexural strains are considered (the effects of axial and shear strains are neglected). Therefore results from the proposed method are theoretically exact from small to very large curvatures and transverse and longitudinal displacements for plane beam-columns under bending actions. The large-deflection analysis of a beam-column with flexible connections at both ends becomes a complex problem requiring the simultaneous solution of at least two highly nonlinear equations with elliptical integrals. The solution of this problem becomes even more complex when the end connections are nonlinear or the direction of the applied end load changes (like “follower” loads). The validity and effectiveness of the proposed method and equations are verified against available solutions of very large deflection elastic analysis of beam-columns. Four comprehensive examples are included for verification and easy reference.     

Finite Element Response Sensitivity Analysis of Steel-Concrete Composite Beams with Deformable Shear Connection

The behavior of steel-concrete composite beams is strongly influenced by the type of shear connection between the steel beam and the concrete slab. For accurate analytical predictions, the structural model must account for the interlayer slip between these two components. In numerous engineering applications (e.g., in the fields of structural optimization, structural reliability analysis, and finite element model updating), accurate response sensitivity calculations are needed as much as the corresponding response simulation results. This paper focuses on a procedure for response sensitivity analysis of steel-concrete composite structures using displacement-based locking-free frame elements including deformable shear connection with fiber discretization of the cross section. Realistic cyclic uniaxial constitutive laws are adopted for the steel and concrete materials as well as for the shear connection. The finite element response sensitivity analysis is performed according to the direct differentiation method. The concrete and shear connection material models as well as the static condensation procedure at the element level are extended for response sensitivity computations. Two steel-concrete composite structures for which experimental test results are available in the literature are used as realistic testbeds for response and response sensitivity analysis. These benchmark structures consist of a nonsymmetric, two-span continuous beam subjected to monotonic loading and a frame subassemblage under cyclic loading. The new analytical derivations for response sensitivity calculations and their computer implementation are validated through forward finite difference analysis based on the two benchmark examples considered. Selected sensitivity analysis results are shown for validation purposes and for quantifying the effect and relative importance of the various material parameters in regards to the nonlinear monotonic and cyclic response of the testbed structures.     

Behavior of Corrugated Web I-Girders under In-Plane Loads

A theoretical formulation of the linear elastic in-plane and torsional behavior of corrugated web I-girders under in-plane loads is presented. A typical corrugated web steel I-girder consists of two steel flanges welded to a corrugated steel web. Under a set of simplifying assumptions, the equilibrium of an infinitesimal length of a corrugated web I-girder is studied, and the cross-sectional stresses and stress resultants due to primary bending moment and shear are deduced. The analysis shows that a corrugated web I-girder will twist out-of-plane simultaneously as it deflects in-plane under the action of in-plane loads. In the paper, the in-plane bending behavior is analyzed using conventional beam theory, whereas the out-of-plane torsional behavior is analyzed as a flange transverse bending problem. The results for a simply supported span subjected to a uniformly distributed load are presented. Finally, finite element analysis results are presented and compared to the theoretical results for validation.     

Time-Dependent Analysis of Shear-Lag Effect in Composite Beams

Taking into account the long-term behavior of the concrete, a model for analyzing the shear-lag effect in composite beams with flexible shear connection is proposed. By assuming the slab loss of planarity described by a fixed warping function, the linear kinematics of the composite beam is expressed by means of four unknown functions: the vertical displacement of the whole cross section; the axial displacements of the concrete slab and of the steel beam; and the intensity of the warping (shear-lag function). A variational balance condition is imposed by the virtual work theorem for three-dimensional bodies, from which the local formulation of the problem, which involves four equilibrium equations with the relevant boundary conditions, is achieved. The assumptions of linear elastic behavior for the steel beam and the shear connection and of linear viscoelastic behavior for the concrete slab lead to an integral-differential type system, which is numerically integrated. The numerical procedure, based on the step-by-step general method and the finite-difference method, is illustrated and applied to an example of practical interest.     

Finite element analysis of the end cutting process for U-section parts with high-strength steel

This paper presents a constitutive framework for finite element analysis of the truck beam end cutting process.For this purpose,a finite strain anisotropic elasto-plastic model,which takes nonlinear kinematic and isotropic hardening into account,is presented.Three factors are investigated to determine the effect on cutting quality:radius of cutting tools,strength of materials and relative clearance in cutting.The recommendations made herein are based on the simulation results.     

Comprehensive Composite Inelastic Fiber Element for Cyclic Analysis of Concrete-Filled Steel Tube Columns

This paper presents the development of a comprehensive composite beam-column fiber element for large displacement nonlinear inelastic analysis of concrete-filled steel tube (CFT) columns. The bond/slip formulation represents the interaction between concrete and steel over the entire contact surface between the two materials. Thus, the modeling accounts for the two factors that cause the slippage between steel shell and concrete core. The first factor is the difference between axial elongation of the steel shell and the concrete core, and the second is the difference between curvatures in the cross section for the concrete core and the steel shell. These effects are integrated over the perimeter and are added to the virtual work expression of the basic element. Furthermore, the constitutive models employed for concrete and steel are based on the results of a recent study and include the confinement and biaxial effects. A 13 degree of freedom (DOF) element with three nodes, which has five DOF per end node and three DOF on the middle node, has been chosen. The quadratic Lagrangian shape functions for axial deformation and the quartic Hermitian shape functions for the transverse directions are used. The model is implemented to analyze several CFT columns under constant concentric axial load and cyclic lateral load. The effect of semi- and perfect bond is investigated and compared with experiments. Good correlation has been found between experimental results and theoretical analyses. The results show that the use of a studded or ribbed steel shell causes greater ultimate strength and higher dissipation of energy than the columns with nonstudded steel shells.     

Model for the Force–Displacement Relationship of Wire Rope Springs

A mathematical model for the restoring force of the wire rope spring is presented. The model is semianalytical in nature and is fully defined by the dimensions and properties of the spring. Emphasis is placed on the tension-compression mode of deformation. An experimental investigation is described in which the force–displacement relationships, for a number of springs, were obtained. For modeling purposes, the restoring force is decomposed into an elastic force and a damping force. The elastic force is modeled by a geometric nonlinear finite beam element. The equivalent cross section is calculated assuming full slip among the individual wires and strands. The damping force is modeled by a constant and a displacement-dependent forces. Based on energy dissipation equivalence, an empirical expression for the damping force is developed.     

Improved Decentralized Method for Control of Building Structures under Seismic Excitation

A decentralized control method with improved robustness and design flexibility is proposed for reducing vibrations of seismically excited building structures. In a previous study, a control scheme was developed for multistory building models using nonlinear, decentralized control theory. This control method has now been improved in this study in that less information about material properties and geometrical parameters of the building is needed and the selection of control design parameters is more flexible. The nonlinear behavior of the proposed control system is studied and its stability property is proven mathematically. To evaluate the effectiveness and robustness of the proposed method, three illustrative structural models, i.e., an eight-story elastic shear beam model, a two-story nonlinear elastic shear beam model, and a 20-story elastic benchmark model are studied. The 1940 El Centro and the 1995 Kobe earthquakes are used in these examples. The performance of the current control design, as applied to these examples, has shown to be more effective in reducing structural responses and improving robustness.     

Dynamic Modal Analysis and Stability of Cantilever Shear Buildings: Importance of Moment Equilibrium

The dynamic modal analysis (i.e., the natural frequencies, modes of vibration, generalized masses, and modal participation factors) and static stability (i.e., critical loads and buckling modes) of two-dimensional (2D) cantilever shear buildings with semirigid flexural restraint and lateral bracing at the base support as well as lumped masses at both ends and subjected to a linearly distributed axial load along its span are presented using an approach that fulfills both the lateral and moment equilibrium conditions along the member. The proposed model includes the simultaneous effects and couplings of shear deformations, translational and rotational inertias of all masses considered, a linearly applied axial load along the span, the shear force component induced by the applied axial force as the member deforms and the cross section rotates, and the rotational and lateral restraints at the base support. The proposed model shows that the stability and dynamic behavior of 2D cantilever shear buildings are highly sensitive to the coupling effects just mentioned, particularly in members with limited rotational restraint and lateral bracing at the base support. Analytical results indicate that except for members with a perfectly clamped base (i.e., zero rotation of the cross sections), the stability and dynamic behavior of shear buildings are governed by the flexural moment equation, rather than the second-order differential equation of transverse equilibrium or shear-wave equation. This equation is formulated in the technical literature by simply applying transverse equilibrium “ignoring” the flexural moment equilibrium equation. This causes erroneous results in the stability and dynamic analyses of shear buildings with base support that is not perfectly clamped. The proposed equations reproduce, as special cases: (1) the nonclassical vibration modes of shear buildings including the inversion of modes of vibration when higher modes cross lower modes in shear buildings with soft conditions at the base, and the phenomena of double frequencies at certain values of beam slenderness (L/r); and (2) the phenomena of tension buckling in shear buildings. These phenomena have been discussed recently by the writer (2005) in columns made of elastomeric materials.     

Numeric-Analytic Form of the Adomian Decomposition Method for Two-Point Boundary Value Problems in Nonlinear Mechanics

A new numeric-analytic technique is developed for two-point nonlinear boundary-value problems (BVPs) of engineering interest. The analytic part of the method is based on a conventional Adomian decomposition method (ADM). However, given a discretization of the one-dimensional domain, the present algorithm applies the ADM, repetitively over successive intervals and exploits a shooting algorithm to solve the BVPs. Apart from a very high rate of convergence as the discretization is made finer, yet another significant advantage of the method is that it provides the solution in a piecewise functional form and one can finally arrive at a continuous form of the global solution. The procedure is used to study planar, large-deflection (Elastica) problem of a cantilever beam subjected to a transverse, concentrated load, at its free end. Moreover the elastoplastic behavior of a cantilever is also studied. Comparisons with exact solutions as well as with results via a few other competing algorithms demonstrate the remarkable accuracy of the proposed method.     

In-Plane Elastic Stability of Arches under a Central Concentrated Load

This paper is concerned with the in-plane elastic stability of arches with a symmetric cross section and subjected to a central concentrated load. The classical methods of predicting elastic buckling loads consider bifurcation from a prebuckling equilibrium path to an orthogonal buckling path. The prebuckling equilibrium path of an arch involves both axial and transverse deformations and so the arch is subjected to both axial compression and bending in the prebuckling stage. In addition, the prebuckling behavior of an arch may become nonlinear. The bending and nonlinearity are not considered in prebuckling analysis of classical methods. A virtual work formulation is used to establish both the nonlinear equilibrium conditions and the buckling equilibrium equations for shallow arches. Analytical solutions for antisymmetric bifurcation buckling and symmetric snap-through buckling loads of shallow arches subjected to this loading regime are obtained. Approximations for the symmetric buckling load of shallow arches and nonshallow fixed arches and for the antisymmetric buckling load of nonshallow pin-ended arches, and criteria that delineate shallow and nonshallow arches are proposed. Comparisons with finite element results demonstrate that the solutions and approximations are accurate. It is found that the existence of antisymmetric bifurcation buckling loads is not a sufficient condition for antisymmetric bifurcation buckling to take place.     

Nonlinear Analysis of Angle‐Ply Composite and Sandwich Laminates

A refined higher order shear deformation theory for linear and geometrically nonlinear behavior of fiber‐reinforced angle‐ply composite and sandwich laminates is established. Laminae material is assumed to be linearly elastic, homogeneous and isotropic/orthotropic. The theory accounts for nonlinear quadratic variation of transverse shear strains through the thickness of the laminate and higher order terms in Green's strain vector in the sense of von Karman. A simple C0 finite‐element formulation of this theory is then presented with a total Lagrangian approach, and a nine node Lagrangian quadrilateral element is chosen with nine degrees of freedom per node. Numerical results are presented for linear and geometrically nonlinear analyses of multilayer angle‐ply composite and sandwich laminates. The theory is shown to predict displacements and stresses more accurately than first‐order shear deformation theory. The results are compared with available closed‐form and numerical solutions of plate theories and three‐dimensional finite‐element solutions. New results are also generated for future evaluations.     

Stability and Collapse of Semirigidly Connected Portal Frames

A numerical procedure for the nonlinear elastic‐plastic instability analysis and collapse of semirigidly connected portal frames, with elastic rotational restraints at the supports, is presented. The procedure is based on nonlinear kinematic relations and linearly elastic material behavior except at the plastic regions (concentrated plasticity). The nonlinear flexible connections are represented by polynomial models. A computational technique for incorporating the stability and strength into the analysis is described in detail. It is found that several important parameters affect the failure modes and consequently the critical loads. These parameters are the slenderness ratio, support restraints, type of connections, and the loading conditions. It is also demonstrated that the connection flexibility has considerable effect on the critical load and the deformation. It is further concluded that for design application the assumption of linear (instead of nolinear, polynomial) connection behavior is adequate for portal frames only if the loading conditions do not produce a significant amount of bending moment at the joints.