Advanced analysis for three dimensional semi rigid steel frames subjected to static and dynamic loadings

Motivation

Steel framed structures exhibit nonlinear behavior before reaching their ultimate load-carrying capacity, making second-order inelastic analysis or advanced analysis essential for accurately predicting their performance This advanced analysis effectively captures the ultimate strength and stability of the entire structural system, eliminating the need for separate member capacity checks outlined in conventional specifications Although it introduces greater complexity, it significantly reduces the effort required for design assessment, allowing for efficient final checks of both member and system limit states based on preliminary member sizing for serviceability Advanced analysis methods for steel framed structures are typically categorized into plastic hinge and plastic zone approaches, depending on the refinement level used to represent yielding.

The refined plastic hinge method simplifies the modeling of beam-column members by eliminating the need for further subdivision and lumping plastic hinges at both ends to represent inelastic material behavior Unlike traditional plastic zone analysis, this approach evaluates inelastic behavior using member forces and a yielding surface criteria Its primary advantages include straightforward formulation and implementation, along with enhanced accuracy in assessing the strength and stability of structural systems, often utilizing just one or two elements per member.

The plastic zone approach involves discretizing beam-column members into finite sub-elements along their length, with each sub-element's cross section divided into small fibers to monitor uniaxial stress-strain relationships during analysis This method effectively spreads plasticity throughout both the cross section and member length, providing an "exact" solution that accounts for critical factors like local, flexural-torsional, and lateral-torsional buckling in steel structures However, its high computational cost and resource demands due to the need for refined discretization limit its application mainly to research purposes or the analysis of specific structural details.

In the traditional analysis and design of steel frames, beam-to-column connections are typically simplified as either fully rigid or ideally pinned However, research has shown that these connections exhibit nonlinear behavior, falling between the extremes of perfect rigidity and frictionless pinning, indicating a degree of joint flexibility known as semi-rigid connections These semi-rigid connections significantly contribute to the nonlinearity observed in the structural behavior of steel frames under both static and dynamic loads Therefore, accurately simulating the actual behavior of these connections is crucial for effectively analyzing the overall responses of structural systems.

A Practical Advanced Analysis Program (PAAP) was developed by Kim et al (2006) for the inelastic static and dynamic analysis of space steel frames, demonstrating accuracy and efficiency for practical design However, subsequent developments by Thai and Kim (2009; 2011) introduced a nonlinear static algorithm and new truss and cable elements, yet the PAAP program still lacks consideration for nonlinear beam-to-column connections and the spread of plasticity, which are crucial for accurate strength and deformation analysis of steel structures Additionally, the program's dynamic analysis relies on the Newmark average accelerate method without numerical dissipation, necessitating improvements to ensure numerical stability in complex nonlinear dynamic problems This research addresses these limitations to enhance the PAAP program's capabilities.

Objective and Scope

This dissertation aims to create advanced analytical methods for accurately and efficiently assessing the strength and behavior of three-dimensional semi-rigid steel framed structures under static and dynamic loads The study incorporates three nonlinear elements, addressing both geometric and material nonlinearities, into the computer program PAAP: the plastic-hinge beam-column element, the plastic-fiber beam-column element, and the multi-spring connection element The proposed program offers three types of analysis to enhance structural evaluation.

The proposed programs facilitate a comprehensive assessment of steel framed structures and their components through nonlinear elastic and inelastic static analysis, as well as time-history analysis, accommodating various connection types such as rigid, linear semi-rigid, and nonlinear semi-rigid Additionally, they enable free vibration analysis, ensuring a realistic evaluation of both strength and behavior in structural engineering.

Specific objectives of the research are as follows:

Develope a plastic-zone method using Hermite interpolation functions for second- order inelastic analysis of plane semi-rigid steel frames subjected to dynamic loadings

A computer program named NSAP (Nonlinear Structural Analysis Program) is developed

Develope a plastic-fiber beam-column element using stability functions This beam-column element is capable of capturing accurately the second-order spread-of- plasticity behavior of space steel frames

Develop a versatile multi-spring element designed to accurately simulate the behavior of nonlinear steel beam-to-column connections This element is capable of representing various connection types, including rigid, pinned, linear semi-rigid, and nonlinear semi-rigid configurations.

Enhancing the nonlinear dynamic algorithm of the PAAP program by implementing the Hilber-Hughes-Taylor method (Hilber et al., 1977) offers significant advantages, including unconditional numerical stability and second-order accuracy This method also facilitates numerical damping in nonlinear solutions, a feature not achievable with the Newmark average acceleration method.

Current research focuses on steel framed structures with semi-rigid connections under both static and dynamic load conditions The proposed program (PAAP) features a limited element library that includes a "line" element, which encompasses three nonlinear elements: the plastic-hinge beam-column element, the plastic-fiber beam-column element, and the multi-spring connection element.

Organization of Dissertation

This dissertation includes five chapters Contents of these chapters are as follows: Chapter 1 introduces the motivation, objective, scope of this research and journal articles

Chapter 2 introduces a second-order distributed plasticity method for conducting nonlinear time-history analysis of planar semi-rigid steel frames, utilizing a custom-developed program called the Nonlinear Structural Analysis Program (NSAP), which is built on the C++ programming language.

Chapter 3 introduces a second-order plastic-hinge method for the nonlinear static and dynamic analysis of space semi-rigid steel frames, utilizing the PAAP program developed in FORTRAN 77 by the Bridge and Steel Structures Laboratory In Chapter 4, a second-order spread-of-plasticity approach is discussed for the same type of analysis, also employing the PAAP program.

In Chapter 5, summary and conclusions of the present work are made and directions for future work are recommended

T HIS D ISSERTATION IS S UMMARIZED FROM

Cuong N.-H., Phu-Cuong Nguyen and Seung-Eock Kim, Second-order plastic- hinge analysis of space semi-rigid steel frames, Thin-Walled Structures 60 (2012) 98-104.

Phu-Cuong Nguyen and Seung-Eock Kim, Nonlinear elastic dynamic analysis of space steel frames with semi-rigid connections, Journal of Constructional Steel Research 84 (2013) 72-81

Phu-Cuong Nguyen and Seung-Eock Kim, Nonlinear inelastic time-history analysis of three-dimensional semi-rigid steel frames, Journal of Constructional Steel Research 101 (2014) 192-206

Phu-Cuong Nguyen and Seung-Eock Kim, An advanced analysis method for three-dimensional steel frames with semi-rigid connections, Finite Elements in Analysis and Design 80 (2014) 23-32

Phu-Cuong Nguyen and Seung-Eock Kim, Distributed plasticity approach for time-history analysis of steel frames including nonlinear connections, Journal of Constructional Steel Research 100 (2014) 36-49

Phu-Cuong Nguyen and Seung-Eock Kim, Second-order spread-of-plasticity approach for nonlinear time-history analysis of space semi-rigid steel frames, Computer & Structures Under review

Chapter 2 S ECOND -O RDER S PREAD - OF -P LASTICITY

A PPROACH FOR N ONLINEAR D YNAMIC A NALYSIS OF

T WO -D IMENSIONAL S EMI -R IGID S TEEL F RAMES

Introduction

Conventional designs often assume beam-to-column connections are either fully rigid or ideally pinned, leading to inaccurate predictions of the seismic response in moment-resisting steel frames In reality, these connections exhibit a nonlinear moment-rotation relationship, classifying them as semi-rigid connections Dynamic tests have demonstrated the ductile and stable hysteretic behavior of steel frames, highlighting the significance of semi-rigid connections under cyclic and seismic loads (Azizinamini and Radziminski, 1989; Nader and Astaneh, 1991; Nader and Astaneh-Asl, 1992; Elnashai and Elghazouli, 1994; Nader and Astaneh-Asl, 1996; Elnashai et al., 1998).

To accurately predict the behavior of steel frames under severe loading conditions, advanced analysis methods are essential These analyses must consider key factors such as geometric nonlinearities, material plasticity, nonlinear connections, geometric imperfections, and residual stress Two primary approaches for analyzing steel frame structures are the plastic hinge approach and the distributed plasticity approach The plastic hinge method, which focuses on concentrated plasticity, forms a hinge at monitored points, providing a computationally efficient way to account for inelastic behavior; however, it may overestimate the limit strength of structures In contrast, the distributed plasticity approach allows yielding to occur throughout the entire length and depth of members, offering a more accurate representation of inelastic behavior in frame structures subjected to severe loads.

Over the past two decades, there has been a scarcity of analytical studies focusing on the second-order inelastic dynamic behavior of steel frames with nonlinear semi-rigid connections Notable contributions include Gao and Haldar's (1995) robust finite-element method for assessing nonlinear responses of space structures under dynamic loads, and Lui and Lopes' (1997) approach incorporating stability functions and bilinear models to address geometrical nonlinearities and connection flexibility Awkar and Lui (1999) expanded this methodology to multi-story semi-rigid frames, while Chan and Chui (2000) published a comprehensive analysis of semi-rigid steel frames, introducing a spring-in-series model for simulating material plasticity More recently, Sekulovic and Nefovska-Danilovic (2008) applied refined plastic hinge and spring-in-series methods for transient analysis, albeit neglecting P-small delta effects Despite these advancements, analytical research on the second-order distributed plasticity analysis of semi-rigid steel frames under dynamic loads remains limited.

This paper develops a sophisticated second-order spread-of-plasticity method for the nonlinear inelastic time-history analysis of plane semi-rigid steel frames, building on the foundational work of Foley and Vinnakota Utilizing an elastic-perfectly plastic model with linear strain hardening, the study establishes a new nonlinear element tangent stiffness matrix based on the principle of stationary potential energy To accurately capture second-order effects and plasticity spread, each frame member is divided into multiple sub-elements The tangent stiffness matrix incorporates geometric nonlinearity, gradual yielding, and the flexibility of nonlinear connections, which are modeled using zero-length rotational springs The analysis includes the movement of the strain-hardening and elastic neutral axis due to gradual yielding, along with considerations for bowing effects, geometric imperfections, and residual stress Additionally, the study integrates three major sources of damping: structural viscous damping, hysteretic damping from nonlinear connections, and material plasticity A numerical procedure combining the Newmark average acceleration method and the Newton-Raphson iterative algorithm is proposed to solve the nonlinear motion equations Several numerical examples demonstrate the accuracy and validity of the proposed second-order inelastic dynamic analysis procedure for steel frames with nonlinear flexible connections.

Nonlinear Finite Element Formulation

Second-Order Spread-of-Plasticity Beam-Column Element

The investigation of a typical beam-column member under load involves dividing its cross-section into small fibers, each characterized by specific material properties, geometric characteristics, area, and centroid coordinates This approach allows for the direct consideration of residual stress by assigning an initial stress value to each fiber Additionally, second-order effects are incorporated by utilizing multiple sub-elements per member, which enables the updating of the element stiffness matrix and nodal coordinates at each iterative step.

Fig 2.1 Beam-column element modeling under arbitrary loads

Fig 2.2 Meshing of beam-column element into n sub-elements

To optimize computational efficiency in assembling the structural stiffness matrix and solving nonlinear equations, n sub-elements are condensed into a typical beam-column member with six degrees of freedom at both ends, utilizing the static condensation algorithm developed by Wilson in 1974 Additionally, a reverse condensation algorithm is employed to determine displacements along the member length, allowing for the assessment of distributed plasticity effects and second-order effects.

In the development of the second-order spread-of-plasticity beam-column element, the following assumptions are made: (1) the element is initially straight and prismatic;

In this study, we analyze the behavior of structural elements under specific assumptions: plane cross-sections remain plane after deformation and normal to the deformed axis, while out-of-plane deformations and Poisson's effect are neglected Shear strains are considered negligible, and although member deformations are small, overall structure displacements can be significant Residual stress is uniformly distributed along the member length, and yielding of the cross-section is influenced solely by normal stress We adopt a linear strain-hardening elastic-perfectly plastic material model, as utilized by Toma and Chen (1992), where strain hardening begins at a strain of ε_sh = 10ε_y, with a modulus E_sh equal to 2% of the elastic modulus E The total internal strain energy of a beam-column element is then defined based on these parameters.

Fig 2.3 Illustration of meshing of element cross-section and states of fibers

The normal stresses corresponding to the strain state of fibers are calculated as follows:

  for elastic fibers for yielding fibers for hardening fibers y y y sh sh sh

Fig 2.4 Constitutive model is assumed for steel material

The total internal strain energy of a partially strain-hardening elastic-plastic beam- column element can be expanded as

In a cross section of a material, normal strain (ε) and normal stress (σ) are experienced by fibers, with the elastic modulus (E) and strain-hardening modulus (E sh) influencing their behavior The volume of fibers varies according to their states—elastic, plastic, and strain-hardening—indicated by the subscripts e, p, y, and sh Figure 2.3 depicts the partitioning of cross-section fibers, highlighting the shifts in the center of the initial neutral axis (d CGe) and the distance to the strain-hardening neutral axis (d CGsh) caused by fibers in the strain-hardening regime.

Replacing the integrations over the volume of the element in Eq (2.3) by integrating along the length and throughout the cross section of the element, Eq (2.3) is expressed as

1 2 e p p sh sh sh sh e y p y y p sh

L A L A L A L A y sh sh L A sh sh sh y y L A sh

U E dA dx dA dx dA dx dA dx

(2.4) where A e is the remaining elastic area, A p is the yielding area, A sh is the strain- hardening area within a cross section, and L is the length of the element

The normal strain of the assuming beam-column element can be predicted by the following strain-displacement relationship (Goto and Chen, 1987)

In this formulation, the longitudinal displacements along the element are described by the function u, while the transverse displacements are represented by the function v, with dx indicating an infinitesimal length of the element The analysis utilizes linear shape functions for longitudinal displacements and cubic Hermite shape functions for transverse displacements.

(2.7) Substituting Eq (2.5) into Eq (2.4), the total internal strain energy of the partially

U A S I dx dx dx dx dx

A S dx dx dx dx dx dx du d v P dv

1 2 sh L sh zsh zsh sh L sh sh zsh y sh sh sh

A S I dx dx dx dx dx

A S dx dx dx dx dx dx du dv

L L y sh sh zsh sh sh y y sh dx

(2.8) where characteristics of the cross section illustrated in Fig 2.3 as follows:

In the equation (2.11), the variables o, p, and q represent the quantities of elastic, yielding, and strain-hardening fibers, respectively The term I z j denotes the z-axis moment of inertia of the j fiber around its centroid Additionally, d CGe indicates the shift of the center of the initial neutral axis, while d CGsh refers to the distance from the initial neutral axis to the new strain-hardening neutral axis established by the fibers in the strain-hardening state.

The potential energy of the element with loads indicated in Fig 2.1 can be expressed as

The equation V = -∫ w(x)v(x)dx - P*v(P) - r^T*d represents the relationship between distributed loads and concentrated loads in structural analysis Here, w(x) denotes the function of the distributed load, P indicates the magnitude of the concentrated load, v(P) signifies the displacement at the point where the concentrated load is applied, r is the nodal load vector at both ends of the element, and d is the nodal displacement vector corresponding to those ends.

The total potential energy of the element is written as follows:

In equilibrium configurations, the principle of stationary potential energy indicates that the total potential energy remains unchanged for small variations in generalized coordinates Mathematically, this is expressed as the total change in potential energy equaling zero.

 Taking the partial derivatives of Eq (2.13), the set of equilibrium equations of the beam-column element can be given by

The equation r = [K Se]{d} + {EF} + {r p} + {r sh} represents the relationship between the element secant stiffness matrix [K Se] and the various forces acting on an element Here, {r} denotes the vector of element nodal forces, while {EF} represents the fixed end-forces resulting from the combination of distributed and concentrated loads Additionally, {r p} is the nodal load vector that corresponds to the yielding area of the cross section, and {r sh} accounts for the nodal load vector associated with the strain-hardening area of the cross section.

The element tangent stiffness matrix can be obtained by applying a truncated Taylor series expansion of the element equilibrium equations as follows:

The stiffness matrix equation for elastic and strain-hardening fibers is represented as K = K₀ + K₁ + K₂ + Kₚ + Kₕ₀ + Kₕ₁ + Kₕ₂ + Kₚₕ, where K₀ denotes the linear stiffness matrix for elastic fibers, K₁ and K₂ account for second-order and bowing effects on elastic fibers, Kₚ is the plastic stiffness matrix for yielding fibers, Kₕ₀ is the linear stiffness matrix for strain-hardening fibers, Kₕ₁ and Kₕ₂ consider second-order and bowing effects for strain-hardening fibers, and Kₚₕ represents the plastic stiffness matrix for strain-hardening fibers All component stiffness matrices in this equation are symmetric, with their non-zero terms detailed in Nguyen and Kim (2014).

2 sh sh sh 15 sh zsh

Nonlinear Beam-to-Column Connections

2.2.1 Modified Tangent Stiffness Matrix including Nonlinear Connections

Fig 2.5 Beam-column member including nonlinear connections with eight degrees of freedom

Fig 2.6 Modified beam-column member with conventional six degrees of freedom

Neglecting axial and shear deformations in connections, semi-rigid connections are represented by zero-length rotational springs at the ends of beam-column members The static condensation algorithm modifies the eight degrees of freedom of the beam-column member to the conventional six degrees of freedom while accounting for semi-rigid connections, facilitating the assembly of the structural stiffness matrix This approach is similar to that described in Chen and Lui (1987) A modified process is outlined for implementation.

Equilibrium equations of rotational spring elements are given by

      (2.19) where R k 1 and R k 2 are stiffness of rotational springs, and they are defined by the moment-rotation relationship of connections

Equilibrium equations for the beam-column member, including nonlinear connections, with eight degrees of freedom are written as:

The condensed displacement vector \( d_a \) encompasses two degrees of freedom, while \( d_b \) represents the displacement vector of a modified beam-column member with six conventional degrees of freedom The equivalent nodal forces \( P_{V1}, M_1 \) and \( P_{V2}, M_2 \) are generated by distributed and concentrated forces acting on the beam-column member between its ends.

Rewriting Eq (2.21) as algebraic equations

 K ba    d a  K bb      d b  r b (2.23) From Eq (2.22), we have

Eq (2.24) is used to solve the condensed displacements Substituting Eq (2.24) into Eq (2.23), equilibrium equations including the essential six degrees of freedom are written as

 K bb  K ab T K aa  1 K ab      d b  r b   K ab   T K aa   1   r a (2.25)

  K '     d b  r ' (2.26) Substituting  K aa   1 into Eq (2.25), we obtain the modified tangent stiffness matrix   K ' including nonlinear connections and the modified load vector   r ' which are shown in Ref (Nguyen and Kim, 2014) as follows:

2.2.2 Moment-Rotation Relationship of Nonlinear Connections

This study explores the nonlinear behavior of semi-rigid connections through a nonlinear moment-rotation curve, which is mathematically defined with parameters derived from curve fitting of experimental results The research utilizes the Richard-Abbott four-parameter model and the Chen-Lui exponential model to effectively represent the nonlinear moment-rotation characteristics of these connections.

In 1975, Richard and Abbott proposed a four-parameter model (Richard and Abbott, 1975) The moment-rotation relationship of the connection is defined by

1 ki kp r kp r n n ki kp r

In the context of connection mechanics, the moment (M) and rotation (θr) of the connection are influenced by several factors, including the shape parameter (n), initial connection stiffness (Rki), strain-hardening stiffness (Rkp), and reference moment (M0).

In 1986, Lui and Chen proposed the following exponential model (Lui and Chen, 1986):

The equation (2.28) describes the relationship between the moment (M) and the absolute value of the rotational deformation (θr) of a connection, incorporating various factors The scaling factor (α) and the strain-hardening stiffness (Rkf) play crucial roles, alongside the initial moment (M0), curve-fitting coefficient (Cj), and the number of terms (n) considered in the analysis.

2.2.3 Cyclic Behavior of Nonlinear Connections

The independent hardening model shown in Fig 2.7 is used to trace the cyclic behavior of nonlinear connections because of its simple application (Chen and Saleeb,

The virgin M-θr relationship is characterized by the models presented in Eqs (2.27) and (2.28) To determine the instantaneous tangent stiffness of connections, one must calculate the derivative of M with respect to θr from these equations Additionally, the hysteretic behavior of semi-rigid connections is outlined in this context.

1) If a connection is initially loaded, M M is positive and the M  r curve follows line OA with the initial stiffness R ki shown in Fig 2.7, the instantaneous dM

2) At point A, if the connection is unloaded, M M is negative and the M r curve goes back along line ABC with the initial stiffnessR ki

3) At point C, if the connection is continuously unloaded, M M is positive and the M r curve follows line CD with the initial stiffness R ki followed by the tangent stiffnessR kt

4) At point D, if the connection is reloaded, M M is negative and the M r curve follows the straight line DE with the initial stiffnessR ki

5) At point E, if the connection is continuously reloaded, the M  r curve follows the line EF which is similar to line OA

6) At point F, the connection shows a similar curve to steps 1) - 5)

Fig 2.7 The independent hardening model

Nonlinear Solution Procedures

A nonlinear algorithm utilizing the Newmark average acceleration method, established by Newmark in 1959, is developed to effectively solve the governing differential equations of motion due to its unconditional numerical stability and second-order accuracy This method allows for the incremental equation of motion of a structure to be expressed clearly.

The equation (2.29) describes the relationship between incremental acceleration, velocity, and displacement vectors, denoted as {\Delta D \ddot}, {\Delta D \dot}, and {\Delta D}, respectively It incorporates mass, damping, and tangent stiffness matrices, represented by [M], [C_T], and [K_T], alongside the external incremental load vector {\Delta F_{ext}} The notation distinguishes values at time t and t + \Delta t Additionally, the structural viscous damping matrix [C_T] can be characterized using Rayleigh damping, as referenced by Chopra (2007).

  C T  M   M  K  K T  (2.30) where  M and  K are the coefficients of mass- and stiffness-proportional damping, respectively If both modes are assumed to have the same damping ratio, then

  (2.31) where 1 and 2 are the natural frequencies of the first and second modes of the frame, respectively

Using Newmark’s approximate equations in standard form as shown in (Newmark,

Transforming Eqs (2.32) and (2.33), the incremental velocity and acceleration vectors at the first iteration of each time step can be written as

Substituting Eqs (2.34) and (2.35) into Eq (2.29), the incremental displacement vector can be calculated from

  and    F ˆ are the effective stiffness matrix and incremental effective force vector, respectively, given as

The Newton-Raphson iterative method effectively eliminates unbalanced forces at each time step During the initial iteration, the total displacement, velocity, and acceleration at time t + Δt are updated using the incremental displacement {ΔD t + Δt}.

For the second and subsequent iterations of each time step, the structural system is solved under the effect of the unbalanced force vector   R as

  (2.42) where the effective stiffness matrix ˆ

   and the residual force vector   R k are calculated at the unbalanced iterative step k, respectively, as follows:

The equation R = F + Δ - F - F - F (2.44) represents the total external force vector {F ext t + Δ t} At the unbalanced iterative step k, the inertial force vector {F ine} k, the damping force vector {F dam} k, and the updated internal force vector {F int} k are defined accordingly.

At each iterative step, the characteristics of each beam-column element's cross-section and the stiffness of semi-rigid springs are updated to assemble a new structural stiffness matrix The structural response history is then saved for the next time step once the convergence criterion is met.

Numerical Examples and Discussions

Portal Steel Frame subjected to Earthquakes

Fig 2.9 Portal frame subjected to earthquakes

The geometry and material properties of a portal frame with lumped masses at the frame nodes are depicted in Fig 2.9 For accurate numerical modeling, each frame member is represented by forty discrete elements in both the proposed program and ABAQUS, utilizing the B22 Timoshenko beam element This approach ensures a precise capture of the nonlinear inelastic response of the frame, which would be compromised if fewer elements were employed Additionally, in the proposed program, all elements are divided into sixty-six fibers, with twenty-seven fibers allocated to both flanges and twelve fibers to the web of the cross-section.

The vibration analysis reveals the first two natural periods of the portal frame in the direction of the applied earthquake, as shown in Table 2.2 A strong correlation is observed between the natural periods generated by ABAQUS and those produced by the proposed program These natural periods are then utilized to estimate the Rayleigh damping matrix, assuming an equivalent viscous damping ratio of 5% for the subsequent time-history analysis.

Table 2.2 Comparison of first two natural periods (s) along the applied earthquake direction of portal frame

Table 2.3 Comparison of peak displacements (mm) of portal frame

Earthquakes Max/Min Analysis type ABAQUS Present Diff (%)

The second-order elastic and inelastic displacement histories of the frame are compared under the Loma Prieta and San Fernando earthquakes, as shown in Figures 2.10 and 2.11, without accounting for the initial residual stress.

Table 2.3 presents a comparison of peak displacements, revealing a maximum difference of 4.98% The results indicate that the proposed program and ABAQUS yield nearly identical outcomes across all scenarios, particularly in terms of permanent displacement drifts due to gradual yielding behavior in second-order inelastic analyses Notably, there is a clear discrepancy in displacement responses between second-order elastic (SE) and second-order inelastic (SI) analyses.

In considering the effect of initial ECCS residual stress (Kim, Ngo-Huu and Lee,

In 2006, Figure 2.12 illustrates a comparison of the second-order inelastic responses of the frame, revealing that the permanent drifts of the roof floor remain largely consistent across both scenarios Nevertheless, there is a notable difference in the permanent plastic deformation observed in fiber no.

1 in the cut section A-A illustrated in Fig 2.13 has significant differences as plotted in Fig 2.14 and Fig 2.15

(b) Second-order inelastic Fig 2.10 Displacement time-history responses of portal frame under Loma Prieta earthquake

(b) Second-order inelastic Fig 2.11 Displacement time-history responses of portal frame under San Fernando earthquake

Present without residual stress Present with residual stress

Di sp la cement ( mm)

(b) San Fernando Fig 2.12 Second-order inelastic responses of portal frame with and without residual stress

Fig 2.13 Cut section A-A and fiber no 1 is being monitored

The analysis of second-order inelastic responses of a frame subjected to the San Fernando earthquake reveals significant differences in computation time between the proposed program and ABAQUS Using the same personal computer configuration (AMD Phenom II X4 955 Processor, 3.2 GHz; 4.00 GB RAM), the proposed program completed the analysis in just 1 minute and 26 seconds, while ABAQUS took 25 minutes and 52 seconds—making it 18 times slower This stark contrast highlights the superior computational efficiency of the proposed program.

(b) San Fernando Fig 2.14 Plastic deformation and stress at fiber no 1 in cut section A-A in portal frame with and without residual stress

P las tic D ef o rm at ion

(b) San Fernando Fig 2.15 Plastic deformation and stress at fiber no 1 in cut section A-A

Two-Story Steel Frame with Nonlinear Connections

Fig 2.16 Two-story steel frame with nonlinear connections

Chan and Chui (2000) studied a single-bay two-story steel frame with flexible beam-to-column connections, as illustrated in Fig 2.16 The frame members are W8x48, with a Young’s modulus (E) of 205 x 10^6 kN/m² and a yielding stress (σy) of 235 MPa, including an initial ECCS residual stress distribution (Kim, Ngo-Huu, and Lee, 2006) An initial geometric imperfection of 1/438 is applied to the column Vertical static loads are introduced to account for second-order effects, followed by sudden horizontal forces applied at each floor over 0.5 seconds, as depicted in Fig 2.16 The model incorporates lumped masses of 5.1 and 10.2 tons at the tops of columns and mid-beams, respectively, with a time step (Δt) of 0.001 seconds, while ignoring viscous damping The Richard-Abbott model parameters for beam-to-column connections include Rki = 23,000 kN·m/rad and Rkp = 70 kN·m/rad.

The second-order elastic time-history responses predicted by the proposed program for rigid, linear semi-rigid, and nonlinear semi-rigid frames align closely with the findings of Chan and Chui (2000), as illustrated in Figures 2.17 and 2.19a However, the second-order inelastic responses, depicted in Figures 2.18 and 2.19b, reveal differences attributed to plasticity modeling; the proposed program effectively captures the spread of plasticity, while Chan and Chui employed a concentrated plastic hinge approach Additionally, the moment-rotation relationships at connection C are presented in Figure 2.19 for both second-order elastic and inelastic analyses.

Proposed, rigid connection Chan and Chui, rigid con.

La te ra l displaceme nt (m)

Chan and Chui, linear con.

La te ra l displaceme nt (m)

Chan and Chui, nonlinear con.

(c) Nonlinear semi-rigid connections Fig 2.17 Second-order elastic responses of two-story frame for various connections

La ter al d is p lacemen t (m )

Proposed, rigid connection Chan and Chui, rigid con.

Chan and Chui, linear con.

Chan and Chui, nonlinear con.

(c) Nonlinear semi-rigid connections Fig 2.18 Second-order inelastic responses of two-story frame for various connections

Proposed, rigid connection Chan and Chui, rigid con. Proposed, linear con Chan and Chui, linear con. Proposed, nonlinear con Chan and Chui, nonlinear con.

4.3 Vogel Six-Story Steel Frame with Nonlinear Connections – A Case Study

: Lumped mass due to vertical distributed static load.

49.1 kN/m 49.1 kN/m 49.1 kN/m 49.1 kN/m 49.1 kN/m 31.7 kN/m

IPE400 IPE360 IPE360 IPE300 IPE300 IPE240

HEB22 0 H EB2 2 0 H EB220 H EB2 20 H EB1 6 0 H EB160 HEB26 0 H EB260 HEB24 0 H EB2 40 H EB2 0 0 H EB200 6 x 3.75 m = 22 5 m

Fig 2.20 Vogel six-story steel frame with semi-rigid connections

Vogel (1985) introduced a two-bay, six-story steel frame as a calibration frame for second-order inelastic static analysis Chui and Chan (1996) developed semi-rigid beam-to-column joints to investigate the dynamic behavior associated with connection flexibility, as illustrated in Fig 2.20 An initial out-of-plumbness, denoted as ψ, was also considered in their study.

In this study, a frame with various connection types was analyzed under different horizontal loads, F1(t) and F2(t) The parameters for the Chen-Lui exponential model for a flush end plate connection were determined, including R ki at 12340.198 kN.m/rad and R kf at 108.924 kN.m/rad The Young's modulus was set at 205 × 10^6 kN/m², and viscous damping was neglected Static loads of 31.7 kN/m² and 49.1 kN/m² were uniformly distributed on beams, while the self-weight density was 7.8 kN/m³, leading to the conversion of these loads into lumped masses at frame joints The fundamental natural frequencies were calculated to be 2.41 rad/sec for fully rigid connections and 1.66 rad/sec for linear semi-rigid connections.

Table 2.4 Periods and Rayleigh damping coefficients of Vogel frame

Frame types 1 st period (s) 2 nd period (s)   

The accuracy of the proposed program for second-order elastic response analysis of various frames under dynamic loadings at frequencies of 1.66 and 2.41 rad/sec is demonstrated by comparing results with those of Chan and Chui (1996), showing strong alignment when ignoring the bowing matrix Resonances occur in linear semi-rigid and rigid frames at the fundamental natural frequency, while frames with nonlinear connections exhibit no resonance due to hysteretic damping When incorporating the bowing matrix, the displacement responses of linear semi-rigid and rigid frames differ significantly from those without bowing effects Ultimately, the bowing effect increases frame deflection, and the results align with those of Chan and Chui, confirming the program's validity.

Lat eral dis pl a cem en t (m)

Time (s) Proposed, rigid connection Chan and Chui, rigid con.

Proposed, linear con Chan and Chui, linear con.

Proposed, nonlinear con Chan and Chui, nonlinear con.

(b)  = 2.41 rad/sec Fig 2.21 Second-order elastic displacement responses at roof floor of Vogel frame under forced loadings – without bowing effects

L atera l di sp laceme nt (m)

(b)  = 2.41 rad/sec Fig 2.22 Second-order elastic displacement responses at roof floor of Vogel frame under forced loadings – with bowing effects

Chui and Chan, nonlinear con.

Fig 2.23 Moment-rotation responses at connection C of Vogel frame under forced loadings with  = 1.66 rad/sec in the second-order elastic analysis

La ter al d isp lac ement (m)

Case 1 - w/o RS+IMGI Case 2 - RS

Case 3 - IMGI Case 4 - RS+IMGI

Late ral d isp lacem en t (m)

La teral disp lace ment (m)

(c) Nonlinear connections Fig 2.24 Second-order inelastic displacement responses at roof floor of Vogel frame under El Centro earthquake considering geometric imperfections

Fig 2.25 Moment-rotation responses at connection C of Vogel frame under El Centro earthquake considering geometric imperfections in the second-order inelastic analysis

This case study examines the impact of inelastic hysteretic damping, semi-rigid connection hysteresis loops, residual stress, and initial geometric imperfections on structural responses Utilizing a yielding stress of 300 MPa and initial residual stress as outlined by ECCS (Kim, Ngo-Huu, and Lee, 2006), the study incorporates the reduced tangent modulus method to account for initial member out-of-straightness (Kim and Chen, 1996; Chen and Kim, 1997) Young’s modulus is set at 0.85×E for all steel members to address these imperfections Prior to conducting second-order inelastic dynamic analysis, the frames are subjected to distributed loadings on the beams Rayleigh-type viscous damping is applied, with coefficients detailed in Table 2.4 The analysis includes a frame with various connections under the El Centro earthquake, evaluating four scenarios of geometric imperfections: case 1 without residual stress and initial out-of-straightness, case 2 with only residual stress, and others.

The second-order inelastic dynamic responses of the frame with different connections exhibit notable variations, as illustrated in Fig 2.24 While the inclusion of initial residual stress does not significantly affect these responses, the initial member out-of-straightness has a substantial impact on the frame's final behavior during the period from 5 to 30 seconds.

The initial member out-of-straightness significantly influences frame behavior, surpassing the impact of residual stress Figure 2.25 illustrates the nonlinear moment-rotation responses at connection C for the four analyzed cases.

It can be seen that hysteresis loops are unstable due to member-force redistribution caused by gradual yielding of framed members.

Summary and Conclusions

Second-Order Plastic-Hinge Beam-Column Element

2.1.1 Stability Functions accounting for Second-Order Effects

To effectively analyze the interaction between axial force and bending moment, stability functions are employed to reduce modeling and solution time Typically, a single element per member suffices to accurately capture the P-δ effect According to Kim et al (2001), the incremental relationship between basic force and deformation in a three-dimensional beam-column element can be articulated as follows.

0 0 0 0 0 y y y y yA yA y y y y yB yB zA z z zA z z zB zB z z z z

 (3.1) where P, M yA , M yB , M zA , M zB , and T are incremental axial force, end moments with respect to y and z axes, and torsion, respectively;  ,  yA ,  yB ,  zA ,  zB , and

 are the incremental axial displacement, the joint rotations, and the angle of twist; A ,

The parameters I_y and I_z represent the area and moment of inertia concerning the y and z axes, while J denotes the torsional constant and L signifies the length of the beam-column element Additionally, E and G refer to the elastic and shear modulus of the material, respectively The stability functions S_1n and S_2n are defined with respect to the n axis.

 n  y z ,  given by Chen and Lui (Chen and Lui, 1987) as

The solutions derived from Eqs (3.2-3.3) become indeterminate when the axial force is zero To address this issue and to eliminate the need for different expressions for S1n and S2n based on the axial force's sign, Lui and Chen (1986) proposed a set of power-series expansions to approximate the stability functions These power-series expressions have demonstrated a high degree of accuracy, converging effectively within the first ten terms of the series Additionally, if the axial force in the member is within the range of -2.0, the proposed method remains valid and effective.

 n ≤ 2.0 (n y z, ), the following simplified expressions may be used to closely approximate the stability functions

For most practical applications, it gives an excellent correlation to the "exact" expressions given by Eqs (3.2-3.3) However, for  n other than the range of -2.0 ≤

 n ≤ 2.0, the conventional stability functions in Eqs (3.2-3.3) should be used

2.1.2 Refined Plastic Hinge Model accounting for inelastic effects

Material nonlinearity in steel structures involves gradual yielding influenced by residual stresses and flexural effects The Column Research Council (CRC) tangent modulus concept, denoted as E t, is used to address the yielding associated with residual stresses, while flexural yielding is modeled using a parabolic function Consequently, the interaction between force and deformation in a 3-D beam-column is adjusted to incorporate these inelastic effects.

0 0 0 0 / t yA iiy ijy yA yB ijy jjy yB iiz ijz zA zA ijz jjz zB zB

From (Chen and Lui, 1987), the CRC tangent modulus E t accounting for the effects of residual stresses and gradual yielding due to axial force is written as t 1.0

The scalar parameters  A and  B facilitate a gradual reduction in inelastic stiffness associated with plastification at the ends A and B of an element These parameters equal 1.0 in the elastic state and drop to zero upon the formation of a plastic hinge It is assumed that the parameter  changes according to a parabolic function.

The force-state parameter, denoted as α, quantifies the magnitude of axial force and bending moment in an element, with a value of α equal to 0 for α greater than 1 This parameter can be represented using AISC-LRFD or Orbison yield surfaces, as illustrated in Figure 3.1.

For AISC-LRFD plastic strength surface

For Orbison plastic strength surface (Orbison, McGuire and Abel, 1982)

      (3.20) where pP P/ y , m z M z /M pz (strong-axis), m y M y /M py (weak-axis); P y , M yp , and M zp are squash load, and plastic moment capacity of the cross-section about the y

When the force point is within the initial yield surface (α ≤ 0.5), the element remains fully elastic with no reduction in stiffness However, if the force point exceeds the initial yield surface and enters the full yield surface (0.5 < α ≤ 1.0), the element's stiffness is reduced to reflect the effects of plastification at the ends This reduction in stiffness is modeled using a parabolic function as indicated in Eq (3.16) If the member forces exceed the plastic strength surface, adjustments must be made accordingly.

   1.0 , the member forces will be scaled down to move the force point return the yield surface based on the incremental-iterative scheme

To accurately incorporate the effects of transverse shear deformation in a beam-column element, it is essential to adjust the stiffness coefficients The flexibility matrix can be derived by inverting the flexural stiffness matrix.

2 2 jj ij ii jj ij ii jj ij

MB ij ii B ii jj ij ii jj ij k k k k k k k k M k k M k k k k k k

The stiffness matrix components for a planar beam-column are represented by k ii, k jj, and k ij Additionally, the slopes of the neutral axis resulting from the bending moment are denoted as θ MA and θ MB The flexibility matrix that accounts for shear deformation can be expressed accordingly.

(3.22) where GA S and L are shear rigidity and length of the beam-column element, respectively The total rotations at the two ends A and B are obtained by combining Eqs (3.21) and (3.22) as

The basic force and deformation relationship including shear deformation is derived by inverting the flexibility matrix as

2 2 ii jj ij ii s ii jj ij ij s ii jj ij s ii jj ij s

B ii jj ij ij s ii jj ij jj s B ii jj ij s ii jj ij s k k k k A GL k k k k A GL k k k A GL k k k A GL

The member basic force and deformation relationship can be extended for three- dimensional beam-column element as

0 0 0 0 0 t yA iiy ijy yA yB ijy jjy yB zA iiz ijz zA zB ijz jjz zB

2 iiy jjy ijy iiy sz iiy iiy jjy ijy sz k k k k A GL

2 iiy jjy ijy ijy sz ijy iiy jjy ijy sz k k k k A GL

2 iiy jjy ijy jjy sz jjy iiy jjy ijy sz k k k k A GL

2 iiz jjz ijz iiz sy iiz iiz jjz ijz sy k k k k A GL

2 iiz jjz ijz ijz sy ijz iiz jjz ijz sy k k k k A GL

2 iiz jjz ijz jjz sy jjz iiz jjz ijz sy k k k k A GL

    (3.31) where A sy and A sz are the shear areas with respect to y and z axes, respectively

2.1.4 Element Stiffness Matrix accounting for P  Effect

This section discusses the assembly of the member's basic force and deformation relationship into the structural stiffness matrix, which connects nodal forces and displacements Incremental end forces and displacements are illustrated in Fig 3.2a, while Fig 3.2b presents the sign convention for the positive directions of these forces and displacements in a beam-column element By analyzing both figures, the equilibrium and kinematic relationships can be symbolically represented.

  (3.33) where   f n and   d L are the incremental nodal force and displacement vectors of a beam-column member expressed as

Fig 3.2 Element end force and displacement notations and   f e and   d e are the incremental basic force and displacement vectors of a r 2, d 2 r 1, d 1 r 3, d 3

  d e T   yA  yB  zA  zB  (3.37) and   T 6 12  is a transformation matrix written as

Using the transformation matrix by equilibrium and kinematic relations, the force- displacement relationship of a beam-column member may be written as

  f n  K n    d L (3.39) where   K n is the element stiffness matrix expressed as

Eq (3.40) can be sub-grouped as

, , , , , iiz ijz jjz iiz ijz t iiy ijy jjy iiy ijy iiy iiz ijy ijz jjy jjz

Equation (3.39) is employed to prevent side-sway in the member Allowing sway can lead to extra axial and shear forces being introduced into the member These additional forces, resulting from member sway, can be correlated with the end displacements of the member.

By combining Eqs (3.39) and (3.46), the general force-displacement relationship of a beam-column element obtained as

We obtain the stiffness matrix   K t that includes the terms representing axial and flexural actions simultaneously.

Semi-rigid Connection Element

A newly developed independent zero-length element features three translational and three rotational springs to effectively simulate general connections in 3D-framed analysis This multi-spring element links two nodes with coincident coordinates, as illustrated in Fig 3.3 In this study, the translational and torsional springs exhibit linear stiffness and are fully rigid, whereas the rotational springs along the y and z axes may possess either linear or nonlinear stiffness Notably, the coupling effects among the six springs within the connection are disregarded.

Fig 3.3 Space connection element model with zero-length

The relation between the incremental force vector   F S and displacement vector

  U S of the multi-spring element corresponding to six degrees of freedom is as follows:

(3.54) where  K S  is the diagonal tangent stiffness matrix for each multi-spring element, tra

R n and R n rot are the component stiffness for the translational and rotational springs with respect to the n axis  n  x y z , , 

2.2.2 Semi-Rigid Connection Models for Rotational Springs

This study employs the Kishi-Chen three-parameter power model and the Richard-Abbott four-parameter model to analyze the behavior of semi-rigid connections Additionally, the independent hardening model is utilized to accurately predict the cyclic behavior of these connections.

The Kishi-Chen model, developed by Kishi and Chen in 1987, is widely recognized for its effectiveness in modeling semi-rigid connections, requiring only three parameters to accurately represent the moment-rotation curve while consistently ensuring positive stiffness The moment-rotation relationship for this connection is defined by the framework established by Chen and Kishi.

(3.55) where M and  r are the moment and the rotation of the connection, n is the shape parameter,  0 is the reference plastic rotation, and R ki is the initial connection stiffness

Richard and Abbott proposed a four-parameter model (Richard and Abbott, 1975) The moment-rotation relationship of the connection is defined by

In the analysis of connections, the moment (M) and rotation (θr) are crucial parameters, while the shape is defined by the parameter (n) Additionally, the initial connection stiffness (Rki) and strain-hardening stiffness (Rkp) play significant roles, with M0 serving as the reference moment for comparison.

Lui and Chen (Lui and Chen, 1986) proposed the following exponential model:

The equation (3.57) describes the relationship between the moment (M) and the absolute value of the rotational deformation (θr) of a connection, incorporating a scaling factor (α), the strain-hardening stiffness (Rkf), the initial moment (M0), the curve-fitting coefficient (Cj), and the number of terms (n) considered.

In 1943, Ramberg and Osgood proposed a nonlinear stress-strain relationship model (Ramberg and Osgood, 1943) Ang and Morris standardized this model (Ang and Morris, 1984) as

In the context of connection analysis, M represents the moment while θ denotes the rotation of the connection The reference moment and rotation are indicated as Mo and θo, respectively The parameter n is crucial as it defines the curve's shape, and Rki signifies the initial stiffness of the connection.

2.2.3 Cyclic Behavior of Rotational Springs

The independent hardening model shown in Fig 3.4 is used to represent for the

The virgin M-θr relationship is characterized by the connection models outlined in Eqs (3.55-3.58) The instantaneous tangent stiffness of these connections is derived from the derivatives of these equations Additionally, the hysteretic behavior of semi-rigid connections is described in this context.

1) If a connection is initially loaded, M M is positive and the M  r curve follows the line OA with the initial stiffness R ki shown in Fig 3.4 The instantaneous tangent stiffness will be kt r

2) At point A, if the connection is unloaded, M M is negative and the M r curve goes back along the line ABC with the initial stiffnessR ki

3) At point C, if the connection is continuously unloaded, M M is positive and the M  r curve follows the line CD with the initial stiffness R ki followed by the tangent stiffnessR kt

5) At point E, if the connection is continuously reloaded, the M  r curve follows the line EF which is similar to the line OA

Nonlinear Static Algorithm

The Generalized Displacement Control (GDC) method, introduced by Yang and Shieh in 1990, is recognized as a highly robust and effective approach for addressing nonlinear problems with multiple critical points Its strength lies in its general numerical stability and efficiency, making it a preferred choice in various applications The incremental equilibrium equation for a structure can be reformulated for the j-th iteration of the i-th incremental step.

  (3.59) where K i j  1  is the tangent stiffness matrix, D i j  is the displacement increment vector,   P ˆ is the reference load vector,  R i j  1  is the unbalanced force vector, and  i j is the load increment parameter

Fig 3.5 General characteristic of nonlinear systems

According to (Batoz and Dhatt, 1979), Eq (3.58) can be decomposed into the following equations:

Snap-back points Limit point

Once the displacement increment vector   D i j  is determined, the total displacement vector   D i j of the structure at the end of j th iteration can be accumulated as

The total applied load vector   P j i at the j th iteration of the i th incremental step relates to the reference load vector   P ˆ as

  P j i   i j   P ˆ (3.64) where the load factor  i j can be related to the load increment parameter  i j by

The load increment parameter \( \lambda_{ij} \) is an unknown value defined by a constraint condition In the initial iterative step (\( j = 1 \)), this parameter is calculated using the Generalized Stiffness Parameter (GSP).

  (3.66) where  1 1 is an initial value of load increment parameter, and the GSP is defined as

For the iterative step  j  2 , the load increment parameter  i j is calculated as

The displacement increment generated by the reference load at the first iteration of the previous incremental step is denoted as \( \{\Delta D^{\hat{1}}_{i-1}\} \) In contrast, \( \{\Delta D^{\hat{i}}_j\} \) and \( \{\Delta D_{ij}\} \) represent the displacement increments produced by the reference load and unbalanced force vectors, respectively, during the j-th iteration of the i-th incremental step, as outlined in Equations (3.60) and (3.61).

The major characteristics of the GSP defined in Eq (3.67) are as follows:

The GSP, or Generalized Stiffness Parameter, is the ratio of norm displacement increments from the first incremental step to the current step, indicating the structure's stiffness at the current step compared to the initial one Unlike the Current Stiffness Parameter (CSP) introduced by Bergan, the GSP remains bounded near snap-back points, offering a significant advantage in structural analysis.

The sign of the Generalized Sign of Plasticity (GSP) is determined by two vectors, as illustrated in Eq (3.67) and Fig 3.6 Generally, the GSP remains positive, except during the incremental steps that occur right after limit points When transitioning past a limit point, a negative GSP is reversed by multiplying it with a minus sign to change the loading direction.

The GSP starts at a value of unity and approaches zero at the limit point It increases during the stiffening phase of the structure and decreases during the softening phase.

The following lists the essential steps in the application of the generalized displacement control method:

Step 1 Select the initial load increment  1 1 , the total number of incremental steps

N, and the allowable load factor  max

Step 2 Initialize S 1 n 4, S 2 n 2,   P 0 1  0 ,   R 0 i  0 ,   D 0 1  0 ,   1 0  0 Step 3 For the first iteration  j  1  at each increment step i :

(a) Form the structure stiffness matrix K 0 i 

(c) Calculate the load increment parameter  1 i : For i1, set  1 i  1 1 ; for i2, use

Eq (3.67) to determine GSP and calculate  1 i using Eq (3.66) Further, let  1 i be of the same sign as  1 i  1 If GSP is negative, multiply  1 i by 1 to reverse the direction of loading

Step 4 For the remaining iterations  j  2  :

To update the structure stiffness matrix \([K_{ij} - 1]\) for frame-type structures, it is essential to modify the stability functions using the current axial force as outlined in Eqs (3.2) and (3.3) Additionally, the basic stiffness matrix in Eq (3.25) should be updated to incorporate inelastic effects.

(b) Solve Eqs (3.60) and (3.61) for   D ˆ i j  and   D i j , respectively

(c) Determine the load increment parameter  i j using Eq (3.68)

Step 5 Compute   D i j  and  i j using Eqs (3.62) and (3.65), respectively

Step 6 Update total displacement   D i j and external load   P j i using Eqs (3.63) and (3.64), respectively

Step 7 Calculate the internal force   F j i for the beam-column element:

(a) Extract the element end displacement d i j  from D i j 

(b) Update the nodal coordinates and the transformation matrix

(c) Calculate the incremental basic displacement d e  using Eq (3.33)

(d) Calculate the incremental basic force f e  using Eq (3.25)

(e) Update the element member basic force using    1    i i j e j e e f  f   f

(f) Check the yield condition: If the member basic force violates the yield surface, it is scaled down to return the force point to the yield surface

(g) Form the element force in local coordinates using   f j i l    T T   f j i e

(h) Assemble the global internal force vector   F j i

Step 8 Calculate unbalanced force vector using       R i j  P j i  F j i

Step 9 Check the convergence: If the ratio of the norm of the unbalanced force

  R i j to the norm of the applied force   P j i is smaller than a preset tolerance, go to step 10 Otherwise, let j j1 and return to step 4

Step 10 Check the termination: Whether the total number of steps is smaller than the preset number N or whether the load factor  i j is smaller than the allowable value max , let i i 1 and go to step 3 Otherwise, stop the procedure

A flow chart of the GDC algorithm is illustrated in Fig 3.7

Fig 3.7 Flow chart of the GDC algorithm

Nex t in cr em en t ( i = i+1 )

Nonlinear Dynamic Algorithm

A novel algorithm combining the Hilber-Hughes-Taylor (HHT) method, known for its unconditional stability and second-order accuracy, with the Newton-Raphson method is introduced for numerically integrating nonlinear motion equations This approach not only enhances stability but also introduces numerical damping in nonlinear solutions, a feature absent in the traditional Newmark method The incremental equation of motion for structures can be effectively modified using this algorithm.

In the analysis of dynamic systems, the vectors of incremental acceleration, velocity, and displacement are denoted as {\(\Delta D\)¨}, {\(\Delta D\)¨}, and {\(\Delta D\)}, respectively The mass, damping, and tangent stiffness matrices are represented by [M], [C\(_L\)], and [K\(_T\)], while the external incremental load vector is indicated as {\(\Delta F_{ext}\)} To differentiate between values at time t and t+Δt, superscripts t and t+Δt are utilized The viscous damping matrix [C\(_L\)] can be characterized as a Rayleigh damping matrix, as described by Chopra (2007).

  C L  M   M  K  K L  (3.70) where  M and  K are mass- and stiffness-proportional damping factors, respectively

In this paper, the coefficients used in nonlinear dynamic time-history analysis are treated as constant The last-committed stiffness matrix, denoted as [K L], represents the tangent stiffness matrix at the beginning of a new time step and remains unchanged throughout the unbalanced iterative procedure for each time step If both modes share the same damping ratio (ξ), the natural frequencies of the first and second modes of the analyzed frame are represented as ω1 and ω2, respectively, corresponding to the primary direction of dynamic loading.

Using Newmark’s approximate equations in standard form (Newmark, 1959) as:

Transforming Eqs (3.72) and (3.73), the incremental velocity and acceleration at the first iteration of each time step can be written as

Substituting Eqs (3.74) and (3.75) into Eq (3.69), the incremental displacement can be calculated from

  and    F ˆ are the effective stiffness matrix and incrementally effective force vector, respectively, given as

The Newton-Raphson iterative procedure effectively eliminates residual forces at each time step During the initial iteration, the total displacement, velocity, and acceleration at time \( t + \Delta t \) are updated using the incremental displacement \( \{ \Delta D_{t + \Delta t} \} \).

For the second and subsequent iterations of each time step, the structural system is solved under the effect of the residual force vector   R as

  (3.82) where the effective stiffness matrix ˆ

   and the residual force vector   R k are calculated at the unbalanced iterative step k, respectively, as follows

The equation R = F + Δ - F - F - F (3.84) represents the relationship between the total external force vector {F ext t + Δ t} and various force components Specifically, it defines the inertial force vector {F ine} k, the damping force vector {F dam} k, and the updated internal force vector {F int} k at the unbalanced iterative step k.

 F int  k  F int   D t  t  k   (3.87) Once the convergence criterion is satisfied, the structural response is updated for the next time step as

As indicated in (Hughes, 2000), the HHT method will possess the unconditional stability and second-order accuracy when 1 , 0

A smaller value of α increases numerical damping in solutions, which is essential for achieving convergence in complex nonlinear problems Notably, setting α to 0 results in the average acceleration method from the Newmark family (with γ = 0.5 and β = 0.25), characterized by zero numerical damping.

The details of procedure for the application of the Hilber-Hughes-Taylor method and the Newton-Raphson iteration method are as follows:

(a) Form the effective stiffness matrix  Kˆ

  (b) Form the effective force vector Fˆ

Step 2 Corrector phase (force recovery)

(a) Update structural configuration and motion

In the plasticity check, if the member force exceeds the yield surface, it is reduced to align with the yield surface If the force remains within the yield limits, the process proceeds to the next step, which is convergence.

To ensure convergence, assess whether it exists; if so, update the structural configuration and motion at time \( t + \Delta t \) and proceed to the next time step If convergence is not achieved, apply the unbalanced force to the structural system and return to step 1.

A flow chart of the above procedure is illustrated in Fig 3.8

Fig 3.8 Flow chart of the proposed algorithm for dynamic analysis

Static Problems

Vogel (1985) introduced the portal rigid frame as the European calibration standard for static inelastic analysis, incorporating assumptions of initial out-of-plumb straightness and the ECCS residual stress model for the frame and its components, as illustrated in the accompanying figure.

Solve for the increment displacement

Update element force Form the tangent stiffness matrix

Cur re nt time step

The article discusses the update of structural response D D D, focusing on the second-order inelastic behavior while accounting for connection rigidity, as proposed by Chen and Kim in 1997 through the plastic-hinge method It presents the values of three parameters for the Kishi-Chen power model related to these semi-rigid connections.

The study utilizes the Orbison yield surface to analyze the second-order inelastic response of frames with various beam-to-column connections, including rigid, semi-rigid, and hinged types The parameters defined are R ki = kip in rad, M u = 1,250 kip in·, and n = 0.98 The findings indicate that the nonlinear load-displacement curves produced by the proposed program align closely with those from existing studies, highlighting the effectiveness of NASF, a nonlinear finite element program designed for second-order spread-of-plasticity analysis in semi-rigid planar steel frames (Nguyen, 2010).

Fig 3.9 Vogel portal frame with semi-rigid connections

4.1.2 Stelmack Experimental 2-D Two-Story Steel Frame

The one-bay two-story steel frame, tested by Stelmack (1982) and selected as a benchmark in this study, features an A36 W5x16 hot-rolled profile for all members The frame utilizes bolted connections with L4x4x1/2 angles made of A36 and A325 ⅝-inch diameter bolt fasteners, with its experimental moment-rotation relationship illustrated in Fig 3.12 Initially, gravity loads were applied at third points of the first-floor beam, followed by the application of lateral loads as the second phase of testing.

PZ, Vogel - rigid Proposed - semi-rigid

PZ, NASF - semi-rigid Proposed - hinge

Fig 3.10 Load – displacement response of Vogel portal frame

(PZ: plastic-zone method, PH: plastic-hinge method)

The Kishi-Chen power model's three parameters—R ki = 37,000 kip in rad, M u = 225 kip in, and n = 0.90—are established through curve-fitting Experimental data and curve-fitting show a strong correlation in the moment-rotation relationship, as illustrated in Fig 3.12 Additionally, the lateral load-displacement curves generated by the proposed program using the Orbison yield surface align well with experimental findings, as depicted in Fig 3.13 Consequently, the proposed analysis effectively predicts the behavior and strength of semi-rigid connections.

Fig 3.11 Stelmack experimental two-story frame

Fig 3.12 Moment – rotation behavior of Stelmack two-story frame

Lateral displacement at 1st floor (mm)

Fig 3.13 Load – displacement response of Stelmack two-story frame

4.1.3 Liew Experimental 2-D Portal Steel Frame

Liew et al (1997) conducted extensive tests on various portal frames and their joints to provide calibration data for the analysis and design of semi-rigid steel frames This research specifically utilizes the SRF3 portal semi-rigid frame under non-proportional loading to validate the accuracy of a proposed program in predicting the nonlinear behavior of steel frames Initially, vertical and horizontal loads of P = 612 kN and H = 29 kN are applied proportionally, followed by an increase in the horizontal load until frame collapse occurs, while maintaining constant gravity loads Using moment-rotation data from column-to-base and beam-to-column joints, two sets of parameters for the Kishi-Chen power model are derived through curve-fitting techniques A comparison of the test data and the proposed curve-fitting results illustrates the effectiveness of the model.

Fig 3.14 Liew SRF3 portal frame

Table 3.1 Curve-fitting connection parameters of Liew SRF3 portal frame

Connection M u (kNm) R ki (kNm/rad) n

Liew's test - Column Con CB2 Curve fitting - Column Con. Liew's test - Beam Con JSRF3 Curve fitting - Beam Con.

Fig 3.15 Moment – rotation relations of SRF3 portal frame

Liew's test - SRF3 Proposed - SRF3 - Curve Fitting

Fig 3.16 Load – displacement response of Liew SRF3 portal frame

4.1.4 Orbison 3-D Six-Story Steel Frame

The analysis of a six-story rigid frame, previously conducted by Orbison et al (1982) using the plastic hinge approach and more recently by Chiorean (2009) through the beam-column method, is illustrated in Fig 3.17 The frame employs A36 steel, characterized by a yield stress of 250 MPa and a Young’s modulus of 206,850 MPa across all members A uniform floor load of 9.6 kN/m² is translated into equivalent concentrated loads placed atop the columns, while wind loads are represented as point loads of 53.376 kN acting in the Y-direction at each beam-column connection.

Gr ou nd m oti on

(b) Perspective view Fig 3.17 Orbison six-story space frame with semi-rigid connections

Appli ed lo ad fa c tor

Proposed - rigid Chiorean - rigid Proposed - linear semi-rigid Chiorean - linear semi-rigid Proposed - nonlinear semi-rigid Chiorean - nonlinear semi-rigid

Fig 3.18 Load – displacement response at Y direction of Orbison six-story frame

Aside from the rigid frame case, two more cases of the frames with linear and nonlinear semi-rigid connections are proposed and investigated by Chiorean (Chiorean,

In the analysis of beam-to-column connections within the frame, bolted top and seat angles were utilized, with parameters derived from the Kishi-Chen power model For W12x53 and W12x87 beams aligned with the major-axis of the columns, the fixity factor is 0.86, the ultimate moment (M_u) is 300 kN·m, and the shape factor is 1.57 Conversely, for W12x26 beams aligned with the weak-axis of the columns, the fixity factor remains 0.86, but the ultimate moment (M_u) is reduced to 200 kN·m, with a shape factor of 0.86 The initial connection stiffness is subsequently calculated using a specific formulation (Chiorean, 2009).

 (3.93) where I 0 is the moment of inertia of the beam cross-section

The load-displacement curves at point A on the roof, as shown in Fig 3.18, demonstrate that the proposed program effectively utilizes the Orbison yield surface for predicting the behavior of rigid, linear semi-rigid, and nonlinear semi-rigid frames This aligns well with the findings of Chiorean (2009), confirming that the proposed program is reliable for accurately forecasting the nonlinear inelastic behavior of space steel semi-rigid frames.

Dynamic Problems

A computer program developed in FORTRAN utilizes the HHT method and the Newton-Raphson method, as illustrated in the flow chart in Fig 3.8 This study employs specific integration coefficients for the HHT method: α = 0, γ = 0.5, and β = 0.25, and analyzes three earthquake records.

El Centro, Northridge, and San Fernando, as illustrated in Fig 2.8, serve as ground excitation sources in dynamic analysis, specifically assessing the peak ground accelerations and time response of semi-rigid steel frame structures under dynamic loads To validate the predictions made by the proposed program, a comparison is conducted with existing literature results and those generated by SAP2000 It is important to highlight that SAP2000 lacks a semi-rigid element that accounts for the nonlinear effects of beam-to-column connections, whereas the proposed element effectively incorporates these critical effects.

4.2.1 Chan 2-D Two-Story Steel Frame

Fig 3.19 Chan 2-D two-story steel frame

Chan and Chui (2000) conducted a study on a single-bay two-story 2-D steel frame featuring flexible beam-to-column connections The frame, composed of W8x48 members, has a Young’s modulus (E) of 205 x 10^6 kN/m² and a yielding stress (σy) of 235 MPa An initial geometric imperfection of the column, represented as ψ of 1/438, was taken into account Vertical static loads were applied to the frame to assess second-order effects, followed by sudden horizontal forces applied at each floor over a duration of 0.5 seconds Additionally, lumped masses of 5.1 and 10.2 tons were modeled at the top of the columns and the midpoint of the beams, respectively.

In the dynamic analysis, a time step of 0.001 seconds is selected, and viscous damping is disregarded The Richard-Abbott model for semi-rigid connections utilizes four key parameters: \( R_{ki} = 23,000 \, \text{kN m/rad} \), \( R_{kp} = 70 \, \text{kN m/rad} \), \( M_o = 180 \, \text{kN m} \), and \( n = 1.6 \) Table 3.2 presents the peak displacements generated by the proposed program, as well as those from Chan and Chui (2000) and SAP2000 The results of the elastic responses from the proposed element align closely with the predictions of Chan and Chui, as well as those from SAP2000 However, in the case of inelastic responses, the primary differences arise from the modeling of material nonlinearity; the proposed program employs a refined plastic hinge method based on stability functions, while Chan and Chui and SAP2000 utilize a lumped plastic hinge method Additionally, the moment-rotation curves at connection C are illustrated in Fig 3.22.

Table 3.2 Peak displacements (mm) of 2-D two-story steel frame

Frame type Max/Min Analysis type Present Chan and Chui SAP2000

Latera l di s pl ac em e nt (m )

Proposed Chan and Chui SAP2000

Lateral di sp la cem e nt (m )

Proposed Chan and Chui SAP2000

Lateral di spla c em ent (m )

(c) Nonlinear semi-rigid connections Fig 3.20 Second-order elastic responses of 2-D two-story frame

La teral d is p lace m en t (m )

Proposed, refined plasic hinge Chan and Chui, lumped plasic hinge SAP2000, FEMA356 - plasic hinge

Late ral disp lac em e nt (m )

Proposed, refined plasic hinge Chan and Chui, lumped plasic hinge SAP2000, FEMA356 - plasic hinge

L at e ra l displ acem e nt (m )

Proposed, refined plasic hinge Chan and Chui, lumped plasic hinge

(c) Nonlinear semi-rigid connections Fig 3.21 Second-order inelastic responses of 2-D two-story frame

Rotation (rad) Proposed, rigid con Chan and Chui, rigid con. Proposed, linear con Chan and Chui, linear con. Proposed, nonlinear con Chan and Chui, nonlinear con.

(b) Inelastic responses Fig 3.22 Hysteresis loops at the connection C for

4.2.2 Vogel 2-D Six-Story Steel Frame

: Lumped mass due to vertical distributed static load.

49.1 kN/m 49.1 kN/m 49.1 kN/m 49.1 kN/m 49.1 kN/m 31.7 kN/m

IPE400 IPE360 IPE360 IPE300 IPE300 IPE240

HEB220 HEB220 HEB220 HEB220 HEB16 0 H E B 1 60 HEB260 HEB260 HEB240 HEB240 HEB20 0 HEB20 0 6 x 3 75 m = 22.5 m

Fig 3.23 Geometry and loads of Vogel six-story frame

Vogel (1985) introduced a two-bay six-story rigid frame model for static inelastic analysis Building on this, Chui and Chan (1996) incorporated semi-rigid joints at the beam ends to investigate the dynamic behavior influenced by connection flexibility, as illustrated in Fig 3.23 The study also accounted for an initial geometric imperfection of 1/450 in the column members, with Young’s modulus set at 205 x 10^6 kN/m² and a Poisson’s ratio of 0.3.

The curve fitted parameters of the Chen-Lui exponential model for a flush end plate connection were as follows: R = 12,340.198kN.m/rad; ki R kf = 108.924kN.m/rad; M = o 0.0kN.m; = 0.00031783; C 1 = -28.286; C 2 = 573.189; C 3 = -3,433.98; C 4 = 8,511.3;

In this analysis, the material was assumed to be elastic, ignoring viscous damping, with static loads of 31.7 and 49.1 kN/m² converted to lumped masses at nodal points (Ostrander, 1970) Chui and Chan (1996) utilized four elements per beam member and one per column member, whereas the proposed method simplifies this to one element per member for both beams and columns The study investigates rigid, linear semi-rigid, and nonlinear semi-rigid connections, with natural frequencies for fully rigid and linear semi-rigid connections aligning closely with the findings of Chui and Chan, as detailed in Table 3.3.

Table 3.3 Comparison of fundamental natural frequencies (rad/s)

Case Chui and Chan Proposed Diff (%)

The dynamic analysis reveals that time-displacement responses for frequencies of 1.00, 1.66, 2.41, and 3.30 rad/s align closely with the findings of Chui and Chan (1996), particularly demonstrating resonance in linear semi-rigid and rigid joint types when the dynamic force frequency matches the frame's fundamental natural frequency In contrast, the nonlinear semi-rigid frame exhibits no resonance due to energy dissipation from hysteretic damping in nonlinear connections Additionally, the displacement amplitude of the nonlinear semi-rigid frame significantly decreases, while the linear semi-rigid and rigid frames remain unaffected This highlights the critical influence of hysteretic damping in semi-rigid frames.

This study examines the impact of various connection models, specifically the Richard-Abbott and Ramberg-Osgood models, on structural response The Richard-Abbott model parameters include R ki = 12,336.86 kN.m/rad, R kp = 112.97 kN.m/rad, M 0 = 96.03 kN.m, and n = 1.6 (Chui and Chan, 1996) In contrast, the Ramberg-Osgood model is defined by θ 0 = 0.00609 rad, M 0 = 75.1293 kN.m, and n = 5.5 (Chui and Chan, 1996) Although the moment-rotation curves at connection C reveal slight differences between the models, the time-displacement responses remain consistent, as illustrated in Fig 3.29 Consequently, it can be concluded that varying models do not significantly influence frame behavior.

Proposed, rigid con Chui & Chan, rigid con.

Proposed, linear semi-rigid con Chui & Chan, linear semi-rigid con.

Proposed, nonlinear semi-rigid con Chui & Chan, nonlinear semi-rigid con.

Fig 3.24 Time-displacement response by second-order elastic analysis (= 1.00 rad/s)

Proposed, linear semi-rigid con Chui & Chan, linear semi-rigid con. Proposed, nonlinear semi-rigid con Chui & Chan, nonlinear semi-rigid con.

Late r al di splacem ent (m )

Fig 3.26 Time-displacement response by

L ate ral d isp lace m en t (m )

Time (s) Proposed, rigid con Chui & Chan, rigid con.

L ate ral d isp la ce m ent (m )

Time (s) Proposed, rigid con Chui & Chan, rigid con.

Fig 3.28 Time-displacement response by second-order elastic analysis under sudden load during 1s: F1(t) = 10.23kN, F2(t) = 20.44kN

L at eral displa ce m ent ( m )

Proposed, Exponential model Chui & Chan, Exponential model

Proposed, Ramberg-Osgood model Chui & Chan, Ramberg-Osgood model Proposed, Richard-Abbott model Chui & Chan, Richard-Abbott model

Fig 3.29 Comparing time-displacement response of the three models (= 1.66 rad/s)

Rotation (rad) Proposed, rigid con Chui & Chan, rigid con.

Proposed, linear semi-rigid con Chui & Chan, linear semi-rigid con.

Proposed, Exponential model Chui & Chan, Exponential model

Proposed, Richard-Abbott model Chui & Chan, Richard-Abbott model

Proposed, Ramberg-Osgood model Chui & Chan, Ramberg-Osgood model

4.2.3 Chan 3-D Two-Story Steel Frame subjected to Impulse Forces

Fig 3.31 Dimensions and properties of Chan space two-story frame

This study investigates the nonlinear dynamic responses of two-story space frames with different connection types—fully rigid, linear semi-rigid, and nonlinear semi-rigid—when subjected to a 100 kN impulse force The frame's member sizes and properties are illustrated in Fig 3.31 (Chan and Chui, 2000), with a consistent Young’s modulus of 205,000 MPa for all members To account for P-Δ and P-δ effects, static vertical loads of 36.9 kN and 46.1 kN are applied as lumped masses at the nodes The Chen-Lui exponential model is utilized for the flush end plate connection of semi-rigid joints, characterized by specific parameters including R ki of 12,340.198 kN.m/rad and R kf of 108.924 kN.m/rad, among others (Ostrander, 1970) Additionally, the connection stiffness about the weak-axis is assumed to be one-fifth of that about the strong-axis.

Two elements per beam member and one per column are used for this analysis A time step t of 0.005 s is chosen

Fig 3.32 Time-displacement response at node A in nonlinear elastic analysis

Figures 37-39 illustrate the time-displacement response at point A and the hysteresis loops of moment-rotation at joint C The outcomes for both rigid and linear semi-rigid connection scenarios align closely with the results obtained from SAP2000 software.

The SAP2000 software is unable to analyze nonlinear semi-rigid frames, which exhibit larger displacements and longer periods due to their increased flexibility from semi-rigid connections As illustrated in Fig 3.32, the response of these frames shows a displacement shift caused by permanent rotational deformation at the connections Additionally, the presence of nonlinear connections contributes to reduced deflection through energy dissipation.

Fig 3.33 Moment-rotation curve of nonlinear strong-axis spring at connection C

Fig 3.34 Moment-rotation curve of nonlinear weak-axis spring at connection C

4.2.4 3-D Two -Story Steel Frame subjected to Earthquakes

Fig 3.35 3-D two-story frame subjected to earthquakes

The study investigates the nonlinear inelastic dynamic response of a two-story 3-D steel frame with various connection types—fully rigid, linear semi-rigid, and nonlinear semi-rigid—under two distinct earthquakes, El Centro and Northridge The goal is to validate the accuracy of the beam-column element in predicting nonlinear effects in 3-D framed structures All members are specified as W8×31, with material properties including a Young’s modulus of 200 GPa and a Poisson ratio of 0.3 A yield stress of 500 MPa is assumed for the nonlinear inelastic time-history analysis For the semi-rigid joints, the Chen-Lui exponential model is applied, characterized by parameters: R ki = 12,340.198 kN.m/rad, R kf = 108.924 kN.m/rad, and M o = 0.0 kN.m.

The connection stiffness along the strong-axis is characterized by parameters such as α = 0.00031783, C1 = -28.286, C2 = 573.189, C3 = -3,433.98, C4 = 8,511.3, C5 = -9,362.567, and a constant C = 3,832.899 kN.m (Ostrander, 1970) The framed nodes have lumped masses estimated at 50 Ns²/mm, with earthquake excitations applied in the X-direction The first two periods for the frame types are compared to those from SAP2000, as detailed in Table 3.4, and these periods are utilized to predict the structure's viscous damping with a damping ratio (ξ) of 0.05.

In the proposed numerical modeling program, only one element per member is initially utilized; however, this approach fails to accurately capture second-order effects To enhance precision, each member is divided into six elements during the modeling of the framed structure in SAP2000 Additionally, it is important to note that SAP2000 lacks a beam-to-column connection element that accounts for the nonlinearity of the moment-rotation relationship.

Table 3.4 Comparison of fundamental natural frequencies (rad/s)

Frame type Mode SAP2000 Present Diff (%)

The displacement responses at the roof of rigid and semi-rigid frames, analyzed using both the proposed program and SAP2000, demonstrate consistent results during nonlinear elastic analysis for the El Centro and Northridge earthquakes Notably, the nonlinear semi-rigid frame exhibits displacement drift due to permanent rotational deformation at connections In nonlinear inelastic analysis, slight discrepancies arise between the proposed program and SAP2000 results, attributed to differences in the steel yielding model and structural viscous damping estimations While SAP2000 employs a lumped plastic hinge model per FEMA 356 specifications and calculates the damping matrix using the initial structural stiffness matrix, the proposed program utilizes a refined plastic hinge model and recalculates the damping matrix at each time step based on the structure's tangent stiffness matrix.

L ate ral dis place m e nt (m m )

(a) Second-order elastic responses of rigid frame

Lat e ra l dis place m e nt (m m )

Proposed, linear semi-rigidSAP2000, linear semi-rigidProposed, nonlinear semi-rigid

Lat er a l displ ac e m e nt (m m )

Proposed, rigid con. SAP2000, rigid con.

(c) Second-order inelastic responses of rigid frame

Proposed, linear semi-rigid SAP2000, linear semi-rigid Proposed, nonlinear semi-rigid

(d) Second-order inelastic responses of semi-rigid frame

Fig 3.36 Nonlinear time-history responses of 3-D two-story frame under El Centro earthquake

Latera l dis p la c em ent (m m )

(a) Second-order elastic responses of rigid frame

La te ral di s p la c e m en t (m m )

(b) Second-order elastic responses of semi-rigid frame

(c) Second-order inelastic responses of rigid frame

Lat eral d is p la cem ent (m m )

(d) Second-order inelastic responses of semi-rigid frame

Fig 3.37 Nonlinear time-history responses of 3-D two-story frame under Northridge earthquake

4.2.5 Orbison 3-D Six-Story Steel Frame – A Case Study

Nonlinear Beam-to-Column Connection Element

A newly developed independent zero-length multi-spring element features three translational and three rotational components, designed to effectively simulate the connections between various elements This innovative multi-spring element facilitates the connection of two nodes sharing identical coordinates.

The study aims to verify the bending moment transfer in both the major and minor axes of beam-column elements To achieve this, fully rigid springs are modeled in all numerical examples to eliminate translational and torsional deformation components at the connection Additionally, the coupling effects among the six springs in the connection are disregarded.

The relation between the incremental force vector F s  and incremental deformation vector u s  of the multi-spring connection element is given by

The diagonal stiffness matrix of the connection element, denoted as [K es], is integral to understanding the behavior of structural connections It relates to the axes of n (where n represents x, y, or z) and incorporates incremental spring forces (ΔP n), moments (ΔM n), axial deformations (Δδ n), and bending deformations (Δθ n) along these axes Additionally, the instantaneous tangent stiffness of the major and minor-axis rotational springs, represented by R z θ and R y θ, plays a crucial role in analyzing the connection's response under various loading conditions.

The moment M is a nonlinear mathematical function that depends on the relative rotation θ r at the connection, as described by the Kishi-Chen three-parameter power model and the Richard-Abbott four-parameter model.

Fig 4.3 Modeling of space connection element with zero-length

The Kishi-Chen model, introduced by Kishi and Chen in 1987, is widely recognized for its effectiveness in modeling semi-rigid connections This model simplifies the analysis by requiring only three parameters to accurately represent the moment-rotation curve, ensuring a consistently positive stiffness in the connection The moment-rotation relationship is detailed by Chen and Kishi, highlighting its significance in structural engineering.

(4.36) where M and  r are the moment and the rotation of the connection, respectively, n is the shape parameter,  0 is the reference plastic rotation, and R ki is the initial connection stiffness

Richard and Abbott proposed a four-parameter model (Richard and Abbott, 1975) The moment-rotation relationship of the connection is defined by

The equation (4.37) defines the relationship between the moment (M) and the rotation (θr) of a connection, incorporating the shape parameter (n), initial connection stiffness (Rki), strain-hardening stiffness (Rkp), and reference moment (M0).

Lui and Chen (Lui and Chen, 1986) proposed an exponential model as follows:

The equation (4.38) describes the relationship between the moment (M) and the absolute value of rotational deformation (θr) of a connection, incorporating key variables such as the scaling factor (α), strain-hardening stiffness (Rkf), initial moment (M0), curve-fitting coefficient (Cj), and the number of terms (n) considered in the analysis.

The incremental connection element deformation u s  in Eq (4.34) can be obtained from the incremental connection element displacement U s  as

The tangent stiffness matrix of a nonlinear connection element used for assembling the global structural stiffness matrix is obtained as follows:

2.2.2 Cyclic Behavior of Rotational Springs

The independent hardening model proposed by Chen and Saleeb in 1982 effectively represents the cyclic behavior of semi-rigid connections due to its straightforward application The virgin M-θ r relationship is defined by the connection models outlined in Eqs (4.36-4.38), and the instantaneous tangent stiffness of these connections is calculated by deriving these equations Additionally, the hysteretic behavior of semi-rigid connections is characterized in this model.

1) If a connection is initially loaded, M M is positive and the M  r curve follows the line OA with the initial stiffness R ki shown in Fig 4.4 The instantaneous tangent stiffness will be kt r

2) At point A, if the connection is unloaded, M M is negative and the M r curve goes back along the line ABC with the initial stiffnessR ki

3) At point C, if the connection is continuously unloaded, M M is positive and the M  r curve follows the line CD with the initial stiffness R ki followed by the tangent stiffnessR kt

5) At point E, if the connection is continuously reloaded, the M  r curve follows the line EF which is similar to the line OA

This section introduces a numerical approach for addressing static nonlinear equations in 3D framed structures One of the notable methods discussed is the Generalized Displacement Control (GDC) method, developed by Yang and Shieh.

The method introduced in 1990 is recognized as a highly effective approach for addressing static nonlinear problems with multiple critical points, thanks to its strong numerical stability and efficiency By reformulating the incremental form of the equilibrium equation, this technique enhances the solution process for complex scenarios.

  (4.42) where K i j  1  is the tangent stiffness matrix of a structure, D i j  is the incremental displacement vector,   P ˆ is the reference load vector,  R i j  1  is the unbalanced force vector, and  i j is the incremental load parameter

Eq (5.41) can be decomposed into the following equations

D i j  i j D ˆ i j   D i j  (4.45) The incremental load parameter  i j is an unknown It is determined from a constraint condition For the first iterative step  j  1 , the incremental load parameter i

 j is determined based on the Generalized Stiffness Parameter  GSP  as

  (4.46) where  1 1 is an initial value of incremental load parameter, and the GSP is defined as

For the following iteration  j  2 , the incremental load parameter  i j is given by

The incremental displacement generated by the reference load at the first iteration of the previous incremental step is denoted as \( \{ \Delta D^{(i-1)} \} \) In contrast, \( \{ \Delta D^{ij} \} \) and \( \{ \Delta D_{ij} \} \) represent the incremental displacements produced by the reference load and unbalanced force vectors, respectively, during the j-th iteration of the i-th incremental step, as outlined in Eqs (4.43) and (4.44).

The following is a step-by-step summary of solution algorithm focused on the element state determination process of a single iteration

Step 1 Solve the global structure equilibrium equation and update the incremental beam-column element and connection element displacements,   D  and U s , respectively

Step 2 Compute the incremental element deformation,    d and u s , using Eq (4.28) and Eq (4.39)

Step 3 Compute the incremental element force,   F  and F s , using Eq (4.11) and Eq (4.31) based on the element stiffness matrix,   K e 6 6  and  K es  6 6  of the previous step, respectively

Step 4 Compute the incremental section force   Q  using Eq (4.16)

Step 5 Compute the section stiffness  k sec  using Eq (4.21)

Step 6 Compute the incremental section deformation    q using Eq (4.20) Step 7 Compute the incremental fiber strain    e using Eq (4.22) and update fiber strain   e

Step 8 Compute the incremental fiber stress     using E i  e and update fiber stress    If  i  y assign  i  y for the elastic-perfectly plastic material model of steel

Step 9 If any fiber is in yielding state  i  y assign the fiber elastic modulus to be equal to zero, E i 0

Step 10 Compute the section resisting force   Q R using Eq (4.24)

Step 11 Update stiffness of beam-column and connection elements,   K e 6 6  and

 K es  6 6  Update tangent stiffness of beam-column and connection elements,   K 12 12  and  K S  12 12 

Step 12 Assemble the structure resisting force and structure stiffness matrix Step 13 Compute structure unbalanced forces

Step 14 Check for the structure convergence: If the structure unbalanced forces satisfy the specified tolerance (i.e., convergence is achieved), go to the next incremental load step Otherwise, return to step 1 for the next iteration to eliminate the structure unbalanced forces

A nonlinear algorithm utilizing the Hilber-Hughes-Taylor (HHT) method, also known as the alpha method, is designed to solve governing differential equations of motion due to its unconditional numerical stability and second-order accuracy This method uniquely allows for numerical damping in nonlinear solutions, a feature not achievable with the Newmark-beta method The incremental equation of motion for a structure can be effectively modified using this approach.

 (4.49) where the dissipation coefficient of 1 , 0

  for accuracy and numerical stability;

The vectors of incremental acceleration, velocity, and displacement are represented as {ΔḊ}, {ΔD}, and {ΔD}, respectively The mass, damping, and tangent stiffness matrices are denoted as [M], [C], and [K_T] The external incremental load vector is indicated by {ΔF_ext}, while the superscripts t and t+Δt differentiate the values at time t and t+Δt The structural viscous damping matrix [C] is defined using Rayleigh damping principles as outlined by Chopra (2007).

  C  M   M  K   K (4.50) where  M and  K are the coefficients of mass- and stiffness-proportional damping, respectively If both modes are assumed to have the same damping ratio, then

  (4.51) where 1 and 2 are the natural frequencies of the first and second modes of the frame, respectively

Using Newmark’s approximate equations in standard form as shown in (Newmark,

Transforming Eqs (4.52) and (4.53), the incremental velocity and acceleration vectors at the first iteration of each time step can be written as

Substituting Eqs (4.54) and (4.55) into Eq (4.49), the incremental displacement vector can be calculated from

  and    F ˆ are the effective stiffness matrix and incremental effective

The Newton-Raphson iterative method effectively eliminates unbalanced forces at each time step During the initial iteration, the total displacement, velocity, and acceleration at time t + Δt are updated using the incremental displacement {ΔD t + Δt}.

For the second and subsequent iterations of each time step, the structural system is solved under the effect of the unbalanced force vector   R as

  (4.62) where the effective stiffness matrix ˆ k

  and the residual force vector   R k are calculated at the unbalanced iterative step k, respectively, as follows:

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