Distributed plasticity approach for time-history analysis of steel frames including nonlinear connections

Journal of Constructional Steel Research 100 (2014) 36–49

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Journal of Constructional Steel Research

Distributed plasticity approach for time-history analysis of steel frames including nonlinear connections Phu-Cuong Nguyen, Seung-Eock Kim ⁎ Department of Civil and Environmental Engineering, Sejong University, 98 Gunja-dong Gwangjin-gu, Seoul 143-747, South Korea

a r t i c l e

i n f o

Article history: Received 10 December 2013 Accepted 3 April 2014 Available online xxxx Keywords: Advanced analysis Distributed plasticity Geometric imperfections Second-order effects Semi-rigid connections Steel frames Time-history analysis

a b s t r a c t This paper presents a displacement-based finite element procedure for second-order spread-of-plasticity analysis of plane steel frames with nonlinear beam-to-column connections under dynamic and seismic loadings. A partially strain-hardening elastic–plastic beam-column element, which directly takes into account geometric nonlinearity, gradual yielding of material, and flexibility of nonlinear connections, is proposed. Three major sources of damping are considered at the same time. They are structural viscous damping, hysteretic damping due to inelastic material, and hysteretic damping due to nonlinear connections. A nonlinear solution procedure based on the combination of the Hilber–Hughes–Taylor method and the well-known Newton–Raphson equilibrium iterative algorithm is proposed for solving differential equations of motion. The dynamic behavior predicted by the proposed program compares well with those given by the commercial finite element software ABAQUS and previous studies. Coupling effects of three primary sources of nonlinearity, the bowing effect, geometric imperfections, and residual stress are investigated and discussed in this paper. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction Conventional designs usually assume that beam-to-column connections are fully rigid or ideally pinned. This assumption causes an inaccurate prediction of the seismic response of moment-resisting steel frames because the real moment–rotation relationship of connections is a nonlinear curve, and such connections are called semi-rigid connections. Several dynamic tests were carried out to investigate the ductile and stable hysteretic behavior of steel frames, which is one of the important features of semi-rigid connections under cyclic and seismic loadings [1–6]. In order to predict actual behavior of steel frames, especially in severe loading conditions, advanced analysis methods are employed. An advanced analysis must include key factors of steel frames such as geometric nonlinearities (P-large delta and P-small delta effects), plasticity of material, nonlinear connections, geometric imperfections (out-ofstraightness and out-of-plumbness), and residual stress, simultaneously. There are two beam-column approaches for advanced analysis of steel frame structures: (i) the plastic hinge approach (concentrated plasticity) and (ii) the distributed plasticity approach (spread-of-plasticity). In the former approach, once yielding criteria is obtained, a plastic hinge will form at one of monitored points on the member (usually at the two ends). This method is a computationally efficient and simple way to consider the effect of inelastic material. However, the hinge methods overpredict the limit strength of structures [7–9], which can also lead to ⁎ Corresponding author. Tel.: +82 2 3408 3004; fax: +82 2 3408 3906. E-mail addresses: [email protected] (P.-C. Nguyen), [email protected] (S.-E. Kim).

http://dx.doi.org/10.1016/j.jcsr.2014.04.012 0143-974X/© 2014 Elsevier Ltd. All rights reserved.

unsafe designs. What's more, it may inadequately give information as to what is happening inside the member because the member is assumed to remain fully elastic between plastic hinges. On the other hand, by the distributed plasticity approach, yielding spreads throughout the whole length and depth of members. Therefore, the distributed plasticity method is more accurate than plastic hinge methods in capturing the inelastic behavior of frame structures under severe loadings. In the last two decades, there have not been many analytical studies about the second-order inelastic dynamic behavior of steel frames with nonlinear semi-rigid connections [10–14]. Gao and Haldar [10] presented an efficient and robust finite-element-based method for estimating nonlinear responses of space structures with partially restrained connections under dynamic and seismic loadings. Lui and Lopes [11] proposed an approach for dynamic analysis of semi-rigid frames using stability functions, the tangent modulus concept, and the bilinear model for capturing the effects of geometrical nonlinearities, inelastic behavior, and connection flexibility, respectively. In 1999, Awkar and Lui [12] developed the method of Lui and Lopes [11] for multi-story semi-rigid frames. Chan and Chui [13] published a book about static and dynamic analysis of semi-rigid steel frames, in which they proposed a spring-in-series model for simulating material plasticity and nonlinear connections; both plastic hinge and refined plastic-hinge methods are presented in detail. Recently, Sekulovic and Nefovska-Danilovic [14] applied the refined plastic hinge method and the spring-in-series concept proposed by Chan and Chui [13] for transient analysis of inelastic steel frames with nonlinear connections; however, their study ignored the P-small delta effects. All the above mentioned studies utilized the plastic hinge methods. Thus, analytical researches about the second-order

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

distributed plasticity analysis of semi-rigid steel frames under dynamic loadings are uncommon. In this paper, a sophisticated second-order spread-of-plasticity method proposed by Foley and Vinnakota [15–18] for static analysis is developed for nonlinear inelastic time-history analysis of plane semirigid steel frames. An elastic–perfectly plastic model with linear strain hardening is applied to establish a new nonlinear element tangent stiffness matrix based on the principle of stationary potential energy. Accurately, to capture the second-order effects and spread of plasticity, each frame member is divided into many sub-elements along the member length and the cross-section depth. The tangent stiffness matrix of the nonlinear beam-column element directly takes into account the effects of geometric nonlinearity, gradual yielding, and flexibility of nonlinear connections. Nonlinear connections are simulated by zero-length rotational springs. The moving of the strain-hardening and elastic neutral axis, which are due to gradual yielding of the cross-section, is directly included in the element tangent stiffness matrix, and this effect is updated during the analysis process. The bowing effect, geometrical imperfections, and residual stress are also considered in this study. Three major sources of damping are integrated in the same analysis. They are structural viscous damping, hysteretic damping due to nonlinear connections, and hysteretic damping due to material plasticity. A numerical procedure using the Hilber–Hughes–Taylor (HHT) method [19] and the well-known Newton–Raphson iterative algorithm is proposed to solve nonlinear equations of motion. Several numerical examples are performed to illustrate the accuracy, validity, and features of the proposed second-order inelastic dynamic analysis procedure for steel frames with nonlinear flexible connections.

37

y 3

1

4

5

…

n-1 n

2 J

I 1

3

4

5

Lsub

z

… …

…

…

i-1

i 2

x

L = n*Lsub

Fig. 2. Meshing of beam-column element into n sub-elements.

In the development of the second-order spread-of-plasticity beamcolumn element, the following assumptions are made: (1) the element is initially straight and prismatic; (2) plane cross-sections remain plane after deformation and normal to the deformed axis of the element; (3) out-of-plane deformations and the effect of Poisson are neglected; (4) shear strains are negligible; (5) member deformations are small, but overall structure displacements may be large; (6) residual stress is uniformly distributed along the member length; (7) yielding of the cross-section is governed by normal stress alone; (8) the material model is linearly strain-hardening elastic–perfectly plastic; and, (9) local buckling of the fiber elements does not occur. In this study, an elastic–perfectly plastic stress–strain relationship with linearly strain hardening used by Toma and Chen [21] is adopted as shown in Fig. 4. Strain hardening starts at the strain of εsh = 10εy, and its modulus Esh is assumed to be equal to 2% of the elastic modulus E. The total internal strain energy of a beam-column element can be expressed as follows: Z Z U¼

2. Nonlinear finite element formulation

σ dεdV: V

ð1Þ

ε

2.1. Beam-column element including the second-order effects and distributed plasticity

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

Investigation of a typical beam-column member subjected to loads is plotted in Fig. 1. In order to capture the distributed plasticity, the beamcolumn member is divided into n elements along the member length as illustrated in Fig. 2; each element is divided into m small fibers within its cross section as illustrated in Fig. 3; and, each fiber is represented by its material properties, geometric characteristic, area Aj, and its coordinate location (yj, zj) corresponding to its centroid. This way, residual stress is directly considered in assigning an initial stress value for each fiber. The second-order effects are included by the use of several sub-elements per member through updating of the element stiffness matrix and nodal coordinates at each iterative step. To reduce the computational time when assembling the structural stiffness matrix and solving the system of nonlinear equations, n subelements are condensed into a typical beam-column member with the six degrees of freedom at the two ends by using the static condensation algorithm derived by Wilson [20]. A reverse condensation algorithm is used to find the displacements along the member length for evaluating the effects of distributed plasticity and the second-order effects. The Appendix C presents the static condensation procedure in detail.

σ ¼ Eε σ ¼ Eε y ¼ σ y σ ¼ Eε y þ Esh ðε−ε sh Þ ¼ σ sh

 Z Z ε Z ε y EεdεdV e þ σ y dε− Eεdε dV p Ve 0 Vp 0 0 ) Z (Z ε Z ε Z ε y þ σ y dε− Eεdε þ Esh ðε−ε sh Þdε dV sh Z Z

V sh

ε

d 5 ,r5

w(x)

d1 ,r1 d 3 ,r3

d 4 ,r4 i

b

a

j

d6 ,r6

εsh

0

0

y

z

CG: Center of gravity

E

yj

zj

d CGsh

U¼

d CGe

d2 ,r2

ð2Þ

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

H

P

for elastic fibers for yielding fibers for hardening fibers:

Fiber states y

Elastic

y

Yielding

y

Hardening

Elastic core

L Fig. 1. Beam-column element modeling under arbitrary loads.

Fig. 3. Illustration of meshing of element cross-section and states of fibers.

ð3Þ

38

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

Substituting Eq. (5) into Eq. (4), the total internal strain energy of the partially strain-hardening elastic–plastic beam-column element is written as !  !2 ) Z L (  2 du d2 v du d2 v dx Ae −2Sze þ Ize dx dx dx2 dx2 0 !   4 ) Z L (   2 2 2 E du dv d v dv A dv dx Ae −Sze þ e þ 4 dx 2 0 dx dx dx dx2 ! ) Z L(   Z L P Ap dv2 du d2 v 1 þ dx− P Ap εy dx P Ap −M Ap þ 2 2 dx dx 2 dx 0 0 !  !2 ) Z L (  2 2 2 E du d v du d v ð6Þ dx Ash −2Szsh þ Izsh þ sh 2 0 dx dx dx2 dx2 !    ) Z (   2 Esh L du dv d2 v dv 2 Ash dv 4 dx Ash −Szsh þ þ 2 0 4 dx dx dx dx dx2   Z L du 1 dv2  dx þ σ y −Esh εsh Ash þ dx 2 dx 0 Z L 2 !     Z L d v 1 2 dx þ − σ y −Esh εsh Szsh Esh εsh −σ y εy Ash dx 2 2 dx 0 0

E U¼ 2

Fig. 4. Constitutive model is assumed for steel material.

where ε is the normal strain at any fiber within a cross section, σ is the normal stress at any fiber within a cross section, E is the elastic modulus for the material, Esh is the strain-hardening modulus of the material, V is the volume of fibers corresponding to their states within a cross section of an element, and subscripts e, p(y), sh stand for elastic, plastic, and strain-hardening states of fiber elements, respectively. Fig. 3 illustrates cross-section partitions with fiber states, in which dCGe and dCGsh are the shift of the center of the initial neutral axis and the distance from the initial neutral axis to the strain-hardening neutral axis created by fibers in the strain-hardening regime, respectively. Replacing the integrations over the volume of the element in Eq. (3) by integrating along the length and throughout the cross section of the element, Eq. (3) is expressed as

U¼

Z Z

Z Z 1 εdAp dx− σ y εy dAp dx 2 L Ae L Ap L Ap Z Z  Z Z E 2 ε dAsh dx þ σ y −Esh εsh εdAsh dx þ sh 2 L Ash L Ash Z Z 1 2 dAsh dx Esh εsh −σ y εy þ 2 L Ash E 2

2

where characteristics of the cross section illustrated in Fig. 3 as follows: Ae ¼

o   X Aj j¼1

Ap ¼

Z Z

p   X Aj j¼1

ε dAe dx þ σ y

ð4Þ

e

Ash ¼

p

Sze ¼ dCGe Ae

o   X 2 y j A j þ Iz; j

P Ap ¼

p   X σ y; j A j

Szsh ¼ dCGsh Ash

sh

M Ap ¼

p

j¼1

Izsh ¼

j¼1

0.000

Π ¼ U þ V:

ð9Þ

ð10Þ

ð11Þ

5 Beam-column member

3

sh

where w(x) is a function of the distributed load, P is the magnitude of the concentrated load, v(P) is the displacement at the position put by the concentrated load, {r} is the nodal load vector applied at two ends of the element, and {d} is the nodal displacement vector at the two ends of the element. The total potential energy of the element is written as follows:

where u is a function describing the longitudinal displacements along the element, v is a function describing the transverse displacements, and dx is an infinitesimal length of element. In this formulation, linear shape functions and cubic Hermite shape functions are employed for longitudinal displacements and transverse displacements, respectively.

7

q   X 2 y j A j þ Iz; j

ð8Þ

0

ð5Þ

1

p

j¼1

Z L T V ¼ − wðxÞvðxÞdx−PvðP Þ−fr g fdg

R k1 2

p   X y j σ y; j A j

where o, p, and q are the number of elastic, yielding, and strainhardening fibers, respectively. Iz,j is the z-axis moment of inertia of jth fiber around its centroid, and dCGe and dCGsh are the shift of the center of the initial neutral axis and the distance from the initial neutral axis to the new strain-hardening neutral axis created by fibers in the strain-hardening state, respectively. The potential energy of the element with loads indicated in Fig. 1 can be expressed as

where Ae is the remaining elastic area, Ap is the yielding area, Ash 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 [22])   du d2 v 1 dv 2 −y 2 þ ε¼ dx 2 dx dx

ð7Þ

e

j¼1

q   X Aj j¼1

Ize ¼

L

6 Rk2

8 4

0.000

Fig. 5. Beam-column member including nonlinear connections with eight degrees of freedom.

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49



Applying the principle of stationary potential energy, the change in the total potential energy for element vanishes for small variations in the generalized coordinates at an equilibrium configuration. In mathe¼ 0 with i ¼ 1; 2; …; 6. Taking matical terms, this can be written as: ∂Π ∂di the partial derivatives of Eq. (11), the set of equilibrium equations of the beam-column element can be given by n o fr g ¼ ½K Se fdg þ fEF g þ r p þ fr sh g

2 where [KSe] is the element secant stiffness matrix, {r} is the vector of element nodal forces, {EF} is the vector of fixed end-forces due to the superposition of both distributed and concentrated loads, {rp} is the nodal load vector resisted by the yielding area of the cross section and {rsh} is the nodal load vector resisted by 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:

ð13Þ

∂r i ∂2 Π ¼ with i; j ¼ 1; 2; –; 6: ∂d j ∂d j ∂di

ð14Þ

6 6 6 6 6 6 6 6 6 6 4

ðr i Þnew −ðr i Þold ¼

K T ði; jÞ ¼



h i h i ½K T  ¼ ½K 0  þ ½K 1  þ ½K 2  þ K p þ ½K sh0  þ ½K sh1  þ ½K sh2  þ K psh

ð15Þ

2.2. Nonlinear connections 2.2.1. Modified tangent stiffness matrix including nonlinear connections Neglecting axial and shear deformations in connections, semi-rigid connections are simulated by zero-length rotational springs attached at the two ends of the beam-column member developed above as illustrated in Fig. 5. The static condensation algorithm is again used to modify the beam-column member with eight degrees of freedom into the member with the conventional six degrees of freedom considering semi-rigid connections as shown in Fig. 6, and this also takes advantage of assembling the structural stiffness matrix. A similar procedure can be found in the ref. [23]. A modified process is presented as follows. Equilibrium equations of rotational spring elements are given by Rk1 −Rk1

−Rk1 Rk1



d3 d7



 ¼

r3 r7

1 7

0.000



 ¼

r6 r8



ðk33 þ Rk1 Þ k36 k13 k36 ðk66 þ Rk2 Þ k16 k13 k16 k11 k23 k26 k12 −Rk1 0 0 k34 k46 k14 k35 k56 k15 0 −R k2 8 9 8 9 8 09 r3 > M1 > > > > d3 > > > > > > > > > > > > > > > > r6 > M2 > d6 > > > > > > > > > > > > > > > > > > > > > > > > > r P d > > > > > > 1 1 1 > > < < > < > = = > = > r2 V1 d2  ¼ ¼ r > > 0 > d > > > > > > > 7> > 7> > > > > > > > > r4 > P2 > > > > > > > d4 > > > > > > > > > > > > >V > >d > >r > > > > > > > > > : 2> : 5> : 5> ; ; > ; > r8 0 d8 K aa K ba

K ab K bb



ð16bÞ

da db



 ¼

ra rb

−Rk1 0 0 0 Rk1 0 0 0

k23 k26 k12 k22 0 k24 k25 0

k34 k46 k14 k24 0 k44 k45 0

k35 k56 k15 k25 0 k45 k55 0

3 0 −Rk2 7 7 0 7 7 0 7 7 0 7 7 0 7 7 0 5 Rk2

ð17aÞ

 ð17bÞ

½K aa fda g þ ½K ab fdb g ¼ fr a g

ð18aÞ

½K ba fda g þ ½K bb fdb g ¼ fr b g:

ð18bÞ

fda g ¼ ½K aa 

−1

fr a g−½K aa 

−1

½K ab fdb g:

ð19Þ

Eq. (19) is used to solve the condensed displacements. Substituting Eq. (19) into Eq. (18b), equilibrium equations including the essential six degrees of freedom are written as

ð16aÞ

Rk1

d6 d8

From Eq. (18a), we have



2



where da is the condensed displacement vector including two degrees of freedom, db is the displacement vector of the modified beamcolumn member with the conventional six degrees of freedom, and P1, V1, M1 and P2, V2, M2 are the equivalent nodal forces of a beamcolumn member produced by distributed and concentrated forces applied between the member ends. Rewriting Eq. (17b) as algebraic equations

where [K0] is a linear stiffness matrix for elastic fibers, [K1] and [K2] are stiffness matrices considering the second-order and bowing effects for elastic fibers, respectively, [Kp] is a plastic stiffness matrix for yielding fibers, [Ksh0] is a linear stiffness matrix for strain-hardening fibers, [Ksh1] and [Ksh2] are stiffness matrices considering the second-order and bowing effects for strain-hardening fibers, respectively, and [Kpsh] is a plastic stiffness matrix for strain-hardening fibers. The detail of component matrices is presented in the Appendix A.



−Rk2 Rk2

where Rk1 and Rk2 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:

ð12Þ

  i ∂ri h  dj − dj new old ∂d j ∂r i Δd j Δr i ¼ ∂d j

Rk2 −Rk2

39

  T −1 T −1 ½K bb −½K ab  ½K aa  ½K ab  fdb g ¼ fr b g−½K ab  ½K aa  fr a g

ð20aÞ

0 0 K fdb g ¼ r :

ð20bÞ

Modified beam-column member

L

Rk2 5

8 4

0.000

Fig. 6. Modified beam-column member with conventional six degrees of freedom.

40

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

Substituting [Kaa]−1 into Eq. (20a), we obtain the modified tangent stiffness matrix [K′] including nonlinear connections and the modified load vector {r′} which are shown in the Appendix B. 2.2.2. Moment–rotation relationship of nonlinear connections In this study, nonlinear behavior of semi-rigid connections is represented by a nonlinear moment–rotation curve. It is expressed by a mathematical function in which the parameters are determined by the curve fitting of test results. The Richard–Abbott four-parameter model [24] and the Chen–Lui exponential model [25] are employed for tracing nonlinear moment–rotation behavior of semi-rigid connections. In 1975, Richard and Abbott proposed a four-parameter model [24]. The moment–rotation relationship of the connection is defined by   Rki −Rkp jθr j ð21Þ M¼   þ Rkp jθr j  ðR −R Þjθ jn 1n 1 þ  ki M kp r  0 where M and θr are the moment and the rotation of the connection, respectively, n is the shape parameter, Rki is the initial connection stiffness, and Rkp is the strain-hardening stiffness and M0 is the reference moment. In 1986, Lui and Chen proposed the following exponential model [25]: n X

  jθ j − r C j 1− exp 2jα þ Rkf jθr j

2) At point A, if the connection is unloaded, M ⋅ ΔM is negative and the M − θr curve goes back along line ABC with the initial stiffness Rki. 3) At point C, if the connection is continuously unloaded, M ⋅ ΔM is positive and the M − θr curve follows line CD with the initial stiffness Rki followed by the tangent stiffness Rkt. 4) At point D, if the connection is reloaded, M ⋅ ΔM is negative and the M − θr curve follows the straight line DE with the initial stiffness Rki. 5) At point E, if the connection is continuously reloaded, the M − θr 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). 3. Nonlinear solution procedure A nonlinear algorithm based on the Hilber–Hughes–Taylor (HHT) method [19] (also known as the alpha method) is developed for solving governing differential equations of motion described by Eq. (23) because the HHT method possesses unconditional stability and secondorder accuracy. In addition, it can induce numerical damping in the nonlinear solution which is impossible with the Newmark-beta method [27]. The incremental equation of motion of a structure can be modified as n ::tþΔt o n : o n o tþΔt tþΔt þ ð1 þ α Þ½C T  Δ D þ ð1 þ α Þ½K T  ΔD ¼… ½M  ΔD n o n : n o ð23Þ tþΔt t t þ α ½C T  Δ D g þ α ½K T  ΔD Δ F ext

ð22Þ

 where the dissipation of α∈ − 13 ; 0 for accuracy and numern ::o n coefficient :o ical stability; ΔD , Δ D , and {ΔD} are the vectors of incremental accel-

where M and |θr| are the moment and the absolute value of the rotational deformation of the connection, respectively, α is the scaling factor, Rkf is the strain-hardening stiffness of the connection, M0 is the initial moment, Cj is the curve-fitting coefficient, and n is the number of terms considered.

eration, velocity, and displacement, respectively; [M], [CT], and [KT] are mass, damping, and tangent stiffness matrices, respectively; {ΔFext} is the external incremental load vector; and, subscript t and t + Δt are used to distinguish the values at time t and t + Δt. The viscous damping matrix [CT] can be defined as Rayleigh damping [28]:

2.2.3. Cyclic behavior of nonlinear connections The independent hardening model shown in Fig. 7 is used to trace the cyclic behavior of nonlinear connections because of its simple application [26]. The virgin M − θr relationship is defined by the models shown in Eq. (21) or (22). The instantaneous tangent stiffness of connections is determined by taking the derivative M on the θr of Eq. (21) or (22). The hysteretic behavior of semi-rigid connections is described as follows:

½C T  ¼ α M ½M  þ βK ½K T 

M ¼ M0 þ

j¼1

1) If a connection is initially loaded, M ⋅ ΔM is positive and the M − θr curve follows line OA with the initial stiffness Rki shown in Fig. 7, the . instantaneous tangent stiffness will be Rkt ¼ ddM jθr j

ð24Þ

where αM and βK are mass- and stiffness-proportional damping factors, respectively. If both modes are assumed to have the same damping ratio ξ, then αM ¼ ξ

2ω1 ω2 ω1 þ ω2

;

βK ¼ ξ

2 ω1 þ ω2

ð25Þ

where ω1 and ω2 are the natural frequencies of the first and second modes of the considered frame, respectively. Using Newmark's approximate equations in standard 2form as shown Þ proposed by in [27] and using coefficients γ ¼ 12 −α and β ¼ ð1−α 4 Hughes [29], we have:  n o n o n : o 1 n o n o tþΔt t t 2 ::t 2 ::tþΔt ¼ D þ Δt D þ −β Δt D þ β  Δt D D 2 n: o n: o n ::t o n ::tþΔt o tþΔt t ¼ D þ ð1−γ ÞΔt D þ γ  Δt D : D

ð26Þ

ð27Þ

Transforming Eqs. (26) and (27), the incremental velocity and acceleration at the first iteration of each time step can be written as

Fig. 7. Independent hardening model.

n : o tþΔt ΔD ¼

  n o :: t γ n tþΔt o γ n : t o γ ΔD D þ 1− Δt D − β  Δt β 2β

ð28Þ

n ::tþΔt o ΔD ¼

1 n tþΔt o 1 n : t o 1 n :: t o ΔD D − D : − 2 β  Δt 2β β  Δt

ð29Þ

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

o n o h in ^ ΔDtþΔt ¼ Δ ^F K

ð30Þ

where the displacement vector and the velocity vector at the intermediate time are given by n o n o n o tþαΔt tþΔt t D ¼ ð1 þ α Þ D −α D

ð42Þ

n: o n: o n: o tþΔt tþαΔt t D ¼ ð1 þ α Þ D −α D :

ð43Þ

k

h i n o ^ and Δ ^F are the effective stiffness matrix and incrementally where K effective force vector, respectively, given as h i ^ ¼ ð1 þ α Þ½K  þ ð1 þ α Þ K T

γ 1 ½C  þ ½M  β  Δt T β  Δt 2

k

ð31Þ

  n o n o n o 1 n : to 1 n ::t o tþΔt t Δ ^F ¼ Δ F D þ D þ α ½K T  ΔD þ ½M  β  Δt 2β    n o n : o ::t γ n : to γ t þ ½C T  ð1 þ α Þ D −ð1 þ α Þ 1− Δt D þ α ΔD : β 2β

n o tþΔt ΔD

kþ1

Unbalanced forces in each time step can be eliminated by using the well-known Newton–Raphson method. At the first iteration of each time step, the total displacement, velocity and acceleration at the time t + Δt are updated based on the incremental displacement {ΔDt + Δt} as follows:

 n ::tþΔt o  1 n ::t o 1 n : to 1 n tþΔt o ΔD D − D þ D ¼ 1− : 2β β  Δt β  Δt 2

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 o h i n tþΔt ^ δΔD K k

kþ1

¼ fRgk

ð36Þ

h i ^ and the residual force vector where the effective stiffness matrix K k

{R}k are calculated at the unbalanced iterative step k, respectively, as follows h i ^ ¼ ð1 þ α Þ½K  þ ð1 þ α Þ K T k k

fRgk ¼

n

tþΔt

F ext

γ 1 ½C  þ ½M β  Δt T β  Δt 2

o −f F int gk −f F dam gk −f F ine gk

ð37Þ

ð38Þ

where {Ftext+ Δt} is the total external force vector. The inertial force vector {Fine}k, the damping force vector {Fdam}k and the updated internal force vector {Fint}k at the unbalanced iterative step k are respectively defined as:

Second-order inelastic dynamic analysis Form mass matrix 1. Form load vector, tangent stiffness matrix 2. Form viscous damping matrix 3. Form effective stiffness matrix, force vector

ts+1= ts+ t

"Solve equilibrium equations" Find incremental displacements Yes

Check [K]T < 0?

Proposed iterative algorithm

No Find nodal velocity and acceleration 1. Find displacements at two ends of members 2. Find displacements along member length 1. Calculate strain and stress of fibers 2. Calculate charateristic of sections 3. Calculate stiffness of connections 1. Find vector of total external force, inertial force, damping force, internal force 2. Form unbalanced force vector

Convergence?

No

ð39Þ

k

Next time step?

n: o tþαΔt f F dam gk ¼ ½C T  D

ð40Þ

k

f F int gk ¼

ð44Þ

kþ1

Yes

n :: tþΔt o f F ine gk ¼ ½M D

n

k

Second-order inelastic static analysis

ð34Þ

ð35Þ

n o n o tþΔt tþΔt ¼ ΔD þ δΔD

START: Input data

ð33Þ

 n o   n: o  ::t γ γ n : to γ n tþΔt o tþΔt D Δt D þ 1− D þ ΔD ¼ 1− 2β β β  Δt

k

At each iterative step, the state of each fiber and characteristic of a cross section of each element are updated for assembling the new structural stiffness matrix. Once the convergence criterion is satisfied, the structural response history is saved for the next time step as

ð32Þ

n o n o n o tþΔt t tþΔt ¼ D þ ΔD D

k

∇

Substituting Eqs. (28) and (29) into Eq. (23), the incremental displacement can be calculated from

41

n o o tþαΔt F int D k

No FINISH: Output data

ð41Þ

Fig. 8. Simplified analysis flowchart.

Yes

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

n o n o t tþΔt ¼ D þ ΔD

n: o n: o tþΔt tþΔt D ¼ D

¼

kþ1

þ

γ β  Δt

ð45Þ

kþ1

  n o   ::t γ γ n : to 1− Δt D þ 1− D 2β β o

n tþΔt ΔD

M

ð46Þ

M

W8x31

M = 10 Ns2/mm E = 200,000 MPa

kþ1

  1 n ::t o 1 n : to ¼ 1− D − D kþ1 2β β  Δt n o 1 tþΔt þ ΔD : kþ1 β  Δt 2

n ::tþΔt o n ::tþΔt o D ¼ D

ð47Þ

W8x31

kþ1

= 0.3 y

= 300 MPa

5m

n o n o tþΔt tþΔt ¼ D D

W8x31

42

Ground motion

a) Loma Prieta 1.2

Acceleration (g)

0.8

5m

0.4 Fig. 10. Portal steel frame.

0

0

5

10

15

20

25

30

35

40

-0.4

Fig. 8 shows a simplified step-by-step flowchart of the analysis procedure using the proposed iterative algorithm.

-0.8 -1.2

4. Numerical examples and discussions

Time (s)

b) San Fernando 1.2

Acceleration (g)

0.8 0.4 0

0

5

10

15

20

25

30

35

40

-0.4

A computer program written in the C++ programming language is developed based on the above-mentioned formulations to predict second-order spread-of-plasticity time-history responses of plane steel frames with nonlinear beam-to-column connections subjected to static loads, earthquakes, and dynamic loadings. It is verified for accuracy and efficiency by the comparison of the predictions with those generated by ABAQUS and previous studies in the literature. The isotropic hardening model for cyclic behavior is applied for steel material. All the frame members are divided into forty discrete elements. Each cross-section of elements is divided into sixty six fibers (twenty seven at each flange, twelve at the web) as shown in Fig. 3. In the dynamic analysis using the HHT method, the coefficient α of zero is adopted for these examples.

-0.8 4.1. Portal steel frame subjected to earthquakes

-1.2

Time (s) Fig. 10 illustrates the geometry and material properties of a portal frame with masses lumped at the frame nodes. In the numerical modeling, each frame member is modeled by using forty discrete elements in both the proposed program and ABAQUS (using B22 Timoshenko beam element) since the analysis can not accurately capture the second-order inelastic response of the frame if only a few elements per member are used in the modeling. All elements are divided into sixty-six fibers

c) El Centro 1.2

Acceleration (g)

0.8 0.4 0.0 0

5

10

15

-0.4 -0.8 -1.2

Time (s) Fig. 9. Earthquake records.

20

25

30

Table 1 Peak ground acceleration and its corresponding time steps of earthquake records [36]. Earthquakes

PGA (g)

Time step (s)

Loma Prieta (1989) (Capitola, 000, CDMG Station 47125) San Fernando (1971) (Pacoima Dam, 254, CDMG Station 279) El Centro (1940) (Array, #9, USGS Station 117)

0.529

0.005

1.160

0.010

0.319

0.020

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

ABAQUS

Present

Diff. (%)

1 2

0.8162 0.0290

0.8181 0.0291

0.23 0.34

(twenty seven at both flanges, twelve at the web) on the cross section in the proposed program. After performing the vibration analysis, the first two natural periods along the applied earthquake direction of the portal frame are obtained and compared in Table 2. It can be seen that a strong agreement of natural periods of the frame generated by ABAQUS and the proposed program is obtained. These two natural periods are used to estimate the Rayleigh damping matrix (Eq. (24)) by assuming the equivalent viscous damping ratio ξ of 5% in the next timehistory analysis step. The second-order elastic and inelastic displacement history of the frame under two different earthquakes of Loma Prieta and San Fernando (shown in Fig. 9 and Table 1) are compared in Fig. 11 and Fig. 12, respectively. A comparison of the peak displacements is given in Table 3 with the maximum difference of 4.98%. It can be observed that the proposed program and ABAQUS generate nearly identical results in all cases, including the permanent drifts of

120 90 Proposed

Displacement (mm)

Mode

a) Loma Prieta

60

ABAQUS

30 0 0

5

10

15

20

25

30

35

40

-30 -60 -90 -120

Time (s)

b) San Fernando 120 90 Proposed

Displacement (mm)

Table 2 Comparison of first two natural periods (s) along the applied earthquake direction of portal frame.

43

60

ABAQUS

30 0 0

5

10

15

20

25

30

35

40

-30 -60

a) Loma Prieta

-90

120

-120

90

Time (s)

Displacement (mm)

Proposed

60

Fig. 12. Second-order inelastic time-history responses of portal frame under earthquakes.

ABAQUS

30 0 0

5

10

15

20

25

30

35

40

-30 -60 -90 -120

Time (s)

b) San Fernando

displacement due to the gradual yielding behavior in the secondorder inelastic analysis cases. Using the same personal computer configuration (AMD Phenom II X4 955 Processor, 3.2 GHz; 4.00 GB RAM), the analysis time of the proposed program and ABAQUS for the second-order inelastic responses of the frame subjected to the San Fernando earthquake are 1 min 26 s and 25 min 52 s, respectively. The analysis time of ABAQUS is 18 times longer than the proposed program. This result demonstrates the higher computational efficiency of the proposed program.

4.2. Single-bay two-story steel frame with nonlinear connections

120 A single-bay two-story steel frame with flexible beam-to-column connections was studied by Chan and Chui [13]. The geometry and loading of the frame are given in Fig. 13. All the frame members are W8 × 48

90

Displacement (mm)

Proposed

60

ABAQUS

30 0 0

5

10

15

20

25

30

35

40

-30 -60

Table 3 Comparison of peak displacements (mm) of portal frame. Earthquakes

Max/min

Analysis type

ABAQUS

Present

Diff. (%)

Loma Prieta

Max

Elastic Inelastic Elastic Inelastic Elastic Inelastic Elastic Inelastic

104.47 104.76 −88.23 −84.46 119.52 122.50 −93.51 −79.07

104.42 105.23 −89.67 −83.12 116.07 119.27 −88.86 −78.25

−0.05 0.44 1.63 −1.59 −2.88 −2.64 −4.98 −1.03

Min

-90 San Fernando

-120

Max

Time (s) Min

Fig. 11. Second-order elastic time-history responses of portal frame under earthquakes.

44

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

100 kN

50 kN

50 kN

0.20

0.5F(t)

50 kN

Lateral displacement (m)

100 kN

3.0 m

All of members: W8x48 50 kN

0.16

E = 205 x 106 kN/m² = 7.8 T/m³ y = 235 MPa Lumped mass = 10.2 T = 5.1 T

F(t)

3.0 m

Connection "C"

F(t) 100 kN

= 1/438 0.5 time (s)

2 x 4.0 = 8.0 m

0.12 0.08 0.04 0.00 0.0

0.5

1.0

1.5

2.5

3.0

-0.08 Time (s)

-0.12 Proposed, rigid connection Proposed, linear con. Proposed, nonlinear con.

Fig. 13. Two-story steel frame.

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

Fig. 15. Second-order inelastic responses of two-story frame.

involving the connection flexibility, as shown in Fig. 17. An initial outof-plumbness ψ of 1/450 was assigned for all the column members. Young's modulus was 205 × 106 kN/m2, and viscous damping was ignored. The curve fitted parameters of the Chen–Lui exponential

a) Second-order elastic analysis 250

Moment (kN.m)

with Young's modulus E of 205 × 106 kN/m2, yielding stress σy of 235 MPa, and initial ECCS residual stress distribution [30]. An initial geometric imperfection of column ψ of 1/438 is considered. The vertical static loads are applied on the frame to consider the second-order effects, and then the horizontal forces are applied suddenly at each floor during 0.5 s, as shown in Fig. 13. The lumped masses of 5.1 and 10.2 ton are modeled at the top of columns and the middle of the beams, respectively. A time step Δt of 0.001 s is chosen and viscous damping of structure is ignored. The four parameters of the Richard– Abbott model for beam-to-column connections are: Rki = 23,000 kN ⋅ m/rad, Rkp = 70 kN ⋅ m/rad, Mo = 180 kN ⋅ m, and n = 1.6. The second-order elastic time-history responses predicted by the proposed program for the rigid, linear semi-rigid, and nonlinear semirigid frames match well with those of Chan and Chui [13] as shown in Figs. 14 and 16a. In the case of the second-order inelastic responses, as shown in Figs. 15 and 16b, it can be recognized that the differences are due to the modeling of plasticity. The proposed program captures spread of plasticity, whereas Chan and Chui's study uses the concentrated plastic hinge method. The moment–rotation relationships at connection C are also plotted in Fig. 16 for both second-order elastic and inelastic analyses.

2.0

-0.04

200 150 100 50 0 0.000

-0.005

Rotation (rad)

0.005

0.010

0.015

0.020

0.025

-50

4.3. Vogel six-story steel frame with nonlinear connections — a case study

-100 Vogel [31] presented a two-bay six-story steel frame as a calibration frame for second-order inelastic static analysis. Chui and Chan [32] built the semi-rigid beam-to-column joints to study the dynamic behavior

Proposed, rigid connection Proposed, linear con. Proposed, nonlinear con.

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

0.20

250

0.16

200

Moment (kN.m)

Lateral displacement (m)

b) Second-order inelastic analysis

0.12 0.08 0.04

150 100 50

0.00 0.0

0.5

1.0

1.5

2.0

2.5

-0.04

3.0 -0.005

0 0.000

Rotation (rad)

0.005

0.010

0.015

0.020

0.025

-50

-0.08

Time (s)

-0.12 Proposed, rigid connection Proposed, linear con. Proposed, nonlinear con.

-100 Chan and Chui, rigid con. Chan and Chui, linear con. Chan and Chui, nonlinear con.

Fig. 14. Second-order elastic responses of two-story frame.

Proposed, rigid connection Proposed, linear con. Proposed, nonlinear con.

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

Fig. 16. Hysteresis loops at connection C of two-story frame.

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

31.7 kN/m

6.0 m

F2(t) = 20.44 sin( t) kN = 7.8 T/m³ = 1/450

Fig. 17. Vogel six-story steel frame with semi-rigid connections.

3.75 m

3.0 2.0 1.0 0.0 0

5

10

15

20

-1.0 -2.0

3.75 m

Time (s)

-3.0

6 x 3.75 m = 22.5 m

3.75 m

6.0 m

: Semi-rigid connection. : Lumped mass due to vertical distributed static load. F1(t) = 10.23 sin( t) kN E = 205 x 10 6kN/m²

a) = 1.66 rad/sec

Proposed, rigid connection Proposed, linear con. Proposed, nonlinear con.

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

b) = 2.41 rad/sec 3.0

Lateral displacement (m)

= 1/450 Connection "C"

3.75 m

49.1 kN/m

3.75 m

49.1 kN/m

3.75 m

HEB200 HEB200 HEB240

IPE400

HEB240

IPE360

49.1 kN/m

HEB260

HEB160

IPE360

49.1 kN/m

49.1 kN/m

HEB260

F2(t)

HEB220

F2(t)

HEB220

F2(t)

IPE300

HEB220

F2(t)

IPE300

HEB220

F2(t)

IPE240

HEB160

F1(t)

nonlinear moment–rotation response at connection C generated by the proposed program also agrees with that of Chan and Chui, as plotted in Fig. 20. In this case study, the combined effect of inelastic hysteretic damping due to spread of plasticity, hysteresis loops of semi-rigid connections, residual stress, and initial geometric imperfections acting on overall structural responses is investigated. Yielding stress of 300 MPa and initial residual stress of ECCS [30] are used. Initial member out-ofstraightness is considered by employing the reduced tangent modulus method proposed by Kim and Chen [34] and Chen and Kim [35]. In the proposed program, Young's modulus of 0.85 × E is directly assigned for all steel members to consider initial member out-of-straightness. Before performing the second-order inelastic dynamic analysis, the frames are fully loaded by the distributed loadings on the beams. Viscous damping of the Rayleigh type is utilized, and its coefficients are presented in Table 4. The frame with various connections subjected to the El Centro earthquake (shown in Fig. 9 and Table 1) is analyzed for four cases of geometric imperfections (case 1 — without residual stress and initial member out-of-straightness, case 2 — considering only residual stress, case 3 — considering only initial member out-of-straightness, and case 4 — considering both residual stress and initial member out-ofstraightness). As shown in Fig. 21, the second-order inelastic dynamic responses of the frame with various connections are clearly

Lateral displacement (m)

model for a flush end plate connection were as follows: Rki = 12340.198 kN·m/rad, Rkf = 108.924 kN·m/rad, Mo = 0.0 kN·m, α = 0.00031783, C1 =− 28.286, C2 = 573.189, C3 = − 3433.98, C4 = 8511.3, C5 =−9362.567, and C6 = 3832.899 (unit of Ci is kN·m) [33]. The static loads of 31.7 and 49.1 kN/m2 uniformly distributed on beams and the self-weight density of 7.8 kN/m3 were converted to lumped masses at the frame joints. The fundamental natural frequencies for the cases of fully rigid and linear semi-rigid connections were found to be 2.41 rad/s and 1.66 rad/s, respectively [32]. The frame with various types of connections under different horizontal loads, F1(t) and F2(t), is investigated. Ignoring the bowing matrix [K2] in Eq. (15), the accuracy of the proposed program in the second-order elastic response analysis of the various frames under dynamic loadings with ω = 1.66 and 2.41 rad/s is illustrated in Fig. 18 by comparing with results of Chan and Chui [32] without bowing effects. It can be seen that the presented results are strongly identical with those of Chan and Chui. It is noted that resonances are observed for the linear semirigid and rigid frames shown in Fig. 18 when the frequency of the forced loadings is equal to the fundamental natural frequency of the frames. Resonance phenomenon does not occur with the frame including nonlinear connections because of existence of hysteretic damping through hysteresis loops in the connections. In case of including the bowing matrix, the second-order elastic displacement responses of the linear semi-rigid frame and the rigid frame under the dynamic loadings with ω = 1.66 and 2.41 rad/s, respectively, are different with those without bowing effects as shown in Fig. 19. It can be concluded that the bowing effect amplifies deflection of frames when their displacements are adequately large. The

45

2.0 1.0 0.0 0

5

10

15

20

-1.0 -2.0

Time (s)

-3.0 Proposed, rigid connection Proposed, linear con. Proposed, nonlinear con.

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

Fig. 18. Second-order elastic displacement responses at roof floor of Vogel frame under forced loadings — without bowing effects.

46

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

a)

= 1.66 rad/sec

Table 4 Periods and Rayleigh damping coefficients of Vogel frame.

3.0

Lateral displacement (m)

2.0

Frame types

1st period (s)

2nd period (s)

ξ

αΜ

βΚ

Rigid Linear semi-rigid

2.6108 3.7662

1.0116 1.2846

0.05 0.05

0.1735 0.1244

0.0116 0.0152

1.0 0.0 0

5

10

15

20

a) Rigid connections 0.4

-1.0

Lateral displacement (m)

0.3

-2.0 Time (s)

-3.0

Proposed, rigid connection Proposed, linear con. Proposed, nonlinear con.

b)

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

= 2.41 rad/sec 3.0

0.1 0.0

0

5

10

15

20

25

30

-0.1 -0.2 -0.3

2.0

Lateral displacement (m)

0.2

Time (s)

-0.4

Case 1 - w/o RS+IMGI Case 2 - RS Case 3 - IMGI Case 4 - RS+IMGI

1.0

b) Linear connections

0.0 0

5

10

15

0.4

20

0.3

-2.0 Time (s)

-3.0 Proposed, rigid connection Proposed, linear con. Proposed, nonlinear con.

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

Fig. 19. Second-order elastic displacement responses at roof floor of Vogel frame under forced loadings — with bowing effects.

different. No significant differences in the second-order inelastic dynamic responses are observed between the frame models that include the initial residual stress and those without this effect, whereas the initial member out-of-straightness strongly acts on final behavior of the

Lateral displacement (m)

-1.0

0.2 0.1 0.0

0

5

10

15

20

25

30

-0.1 -0.2 -0.3 Time (s)

-0.4

Case 1 - w/o RS+IMGI Case 2 - RS Case 3 - IMGI Case 4 - RS+IMGI

c) Nonlinear connections 0.4

100 80

0.3

Lateral displacement (m)

Moment (kN.m)

60 40 20 0 -20 -40 -60 Proposed, nonlinear con.

-80 -100 -0.020 -0.015 -0.010 -0.005 0.000

Chui and Chan, nonlinear con.

0.005

0.010

0.015

0.2 0.1 0.0

0

5

10

15

0.020

Fig. 20. Moment–rotation responses at connection C of Vogel frame under forced loadings with ω = 1.66 rad/s in the second-order elastic analysis.

25

30

-0.2 -0.3

Rotation (rad)

20

-0.1

-0.4

Time (s)

Case 1 - w/o RS+IMGI Case 2 - RS Case 3 - IMGI Case 4 - RS+IMGI

Fig. 21. Second-order inelastic displacement responses at roof floor of Vogel frame under El Centro earthquake considering geometric imperfections.

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

100 80

Moment (kN.m)

60

47

• The presented numerical examples can be used to verify the validity and accuracy of simple practical advanced analysis methods as benchmarks. • The proposed algorithm can be also used to develop threedimensional advanced analysis programs for framed structures subjected to dynamic and seismic loadings.

Case 1 - w/o RS+IMGI Case 2 - RS Case 3 - IMGI Case 4 - RS+IMGI

40 20 0

Acknowledgments

-20 -40 -60 -80

-100 -0.030

-0.025

-0.020

-0.015

-0.010

-0.005

0.000

Rotation (rad) Fig. 22. Moment–rotation responses at connection C of Vogel frame under El Centro earthquake considering geometric imperfections in the second-order inelastic analysis.

This work was supported by a grant from the Human Resources Development program of the Korea Institute of Energy Technology Evaluation & Planning (KETEP) funded by the Korean government Ministry of Knowledge Economy (No. 20124030200050) and by the National Research Foundation (NRF) of Korea grant funded by the Korean government (MEST) (No. 2011-0030847). Appendix A. Element tangent stiffness matrix Linear shape functions and cubic Hermite shape functions used for the present element are respectively given by

frame from 5th to 30th second, as shown in Fig. 21. It can be concluded that the initial member out-of-straightness has much greater impact on the frame behavior than the effect of residual stress. Fig. 22 plots nonlinear moment–rotation responses at connection C corresponding to the four cases. It can be seen that hysteresis loops are unstable due to member-force redistribution caused by gradual yielding of framed members.

h x x iT fNgaxial ¼ 1− L L

 2 3 fNgbend ¼ 1− 3x þ 2x 2 3 L L

2

x−

3

2x x þ 2 L L

2

3

3

3x 2x − 3 L2 L

2

x x − L L2

T :

5. Conclusions An accurate numerical procedure is presented for the second-order spread-of-plasticity analysis of plane steel frames under dynamic and seismic loadings. By assuming that steel behaves as an elastic–perfectly plastic material with linear strain hardening, a new nonlinear beamcolumn stiffness matrix including flexibility of nonlinear connections is shown in this study. The effects of gradual yielding, geometric nonlinearity, connection flexibility, bowing, moving of the neutral axis, initial member out-of-straightness, and residual stress can be directly taken into account through the element tangent stiffness matrix. Three major sources of damping are considered. They are structural viscous damping, hysteretic damping due to gradual yielding of material, and hysteretic damping due to hysteresis loops of nonlinear connections. The accuracy and efficiency of the proposed procedure are proved by comparing the results with the commercial finite element package ABAQUS and previous studies. The following conclusions can be drawn from the present study: • The flexibility of nonlinear semi-rigid connections plays a major role in the overall structural responses during dynamic and seismic loadings. • Hysteretic damping created by energy dissipation of nonlinear semirigid connections helps eliminate resonance which amplifies timehistory responses of framed structures. • Residual stress might not cause significant influence on second-order inelastic time-history behavior of steel frames, whereas initial geometric imperfections clearly change the final response of steel frames. • It is necessary to include initial geometric imperfections and connection flexibility into advanced analysis methods to increase accuracy and safety for performance-based seismic designs of steel frames. • The accurate results obtained in a short analysis time prove that the proposed program can be effectively used in predicting secondorder inelastic time-history behavior of plane steel frames instead of using the time-consuming commercial finite element analysis software.

The component stiffness matrices of Eq. (15) are symmetric, and their non-zero terms are found as follows: Matrix [K0] EAe L 12EIze ¼ L3 4EIze ¼ L EAe ¼ L 6EI ¼ − 2ze L

ESze L 6EI ¼ 2ze L ES ¼ ze L ES ¼ − ze L 4EI ze ¼ L

EAe L 12EI ze ¼− L3 6EIze ¼− 2 L 12EI ze ¼ 3 L

ESze L 6EI ¼ 2ze L 2EIze ¼ L

K 0ð1;1Þ ¼

K 0ð1;3Þ ¼ −

K 0ð1;4Þ ¼ −

K 0ð1;6Þ ¼

K 0ð2;2Þ

K 0ð2;3Þ

K 0ð2;5Þ

K 0ð2;6Þ

K 0ð3;3Þ K 0ð4;4Þ K 0ð5;6Þ

K 0ð3;4Þ K 0ð4;6Þ K 0ð6;6Þ

K 0ð3;5Þ K 0ð5;5Þ

K 0ð3;6Þ

Matrix [K1]



6 1 ðd5 −d2 Þ− ðd3 þ d6 Þ 2 10L 5L 1 1 ¼ EAe ðd5 −d2 Þ þ ðd6 −4d3 Þ 30 10L 6 1 ¼ EAe − 2 ðd5 −d2 Þ þ ðd 3 þ d 6 Þ 10L  5L 6 1 ¼ EAe ðd5 −d2 Þ þ ðd3 −4d6 Þ 10L 30  6 1 ¼ EAe − 2 ðd5 −d2 Þ þ ðd 3 þ d 6 Þ 10L  5L 2 ¼ EAe ðd4 −d1 Þ þ ESze d3 15 1 1 ¼ EAe − ðd5 −d2 Þ þ ð4d3 −d6 Þ 30  10L 6 1 ¼ EAe ð d −d Þ− ðd3 þ d6 Þ 5 2 2 10L 5L 1 1 ðd5 −d2 Þ þ ð4d6 −d3 Þ ¼ EAe − 10L 30



6 ðd4 −d1 Þ 2 5L 1 ¼ EAe ðd4 −d1 Þ 10L 6 ¼ EAe − 2 ðd4 −d1 Þ  5L 1 ¼ EAe ðd4 −d1 Þ 10L  1 ¼ EAe − ðd4 −d1 Þ  10L 1 ¼ EAe − ðd4 −d1 Þ  30 6 ¼ EAe ð d −d Þ 4 1 2 5L 1 ¼ EAe − ðd4 −d1 Þ 10L  2 ðd4 −d1 Þ −ESze d6 ¼ EAe 15

K 1ð1;2Þ ¼ EAe

K 1ð2;2Þ ¼ EAe

K 1ð1;3Þ

K 1ð2;3Þ

K 1ð1;5Þ K 1ð1;6Þ K 1ð2;4Þ K 1ð3;3Þ K 1ð3;4Þ K 1ð4;5Þ K 1ð4;6Þ

K 1ð2;5Þ K 1ð2;6Þ K 1ð3;5Þ K 1ð3;6Þ K 1ð5;5Þ K 1ð5;6Þ K 1ð6;6Þ

48

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

Matrix [K2]

K 2ð2;2Þ K 2ð2;3Þ K 2ð2;5Þ K 2ð2;6Þ K 2ð3;3Þ K 2ð3;5Þ K 2ð3;6Þ K 2ð5;5Þ K 2ð5;6Þ K 2ð6;6Þ

Matrix [Ksh2]

  432 EAe 18  2 108 2 2 ¼ d3 þ d6 þ 3 ðd5 −d2 Þ − 2 ðd5 −d2 Þðd3 þ d6 Þ 140  L L L  EAe  2 72 108 2 2 ¼ 3 d6 −d3 þ 6d3 d6 − d3 ðd5 −d2 Þ þ 2 ðd5 −d2 Þ 280  L L  432 EAe 18  2 108 2 2 ¼ d3 þ d6 − 3 ðd5 −d2 Þ þ 2 ðd5 −d2 Þðd3 þ d6 Þ − 140  L L L  EAe  2 72 108 2 2 ¼ 3 d3 −d6 þ 6d3 d6 − d6 ðd5 −d2 Þ þ 2 ðd5 −d2 Þ L 280  L EAe 18 2 2 2 ¼ ðd5 −d2 Þ −3ðd5 −d2 Þðd6 −d3 Þ 12Ld3 þ Ld6 −3Ld3 d6 þ L 140    EAe 72 108 2 2 2 ¼ d3 ðd5 −d2 Þ− 2 ðd5 −d2 Þ −3 d6 −d3 −6d3 d6 þ L 280 L i   EAe h 2 2 ¼ 4Ld3 d6 −3L d3 þ d6 −6ðd5 −d2 Þðd3 þ d6 Þ 280    432 EAe 18 2 108 2 2 ¼ d3 þ d6 þ 3 ðd5 −d2 Þ − 2 ðd5 −d2 Þðd3 þ d6 Þ 140  L L L  EAe  2 72 108 2 2 3 d6 −d3 −6d3 d6 þ ¼ d6 ðd5 −d2 Þ− 2 ðd5 −d2 Þ 280  L L EAe 18 2 2 2 ¼ ðd5 −d2 Þ þ Ld3 þ 12Ld6 −3Ld3 d6 þ 3ðd5 −d2 Þðd6 −d3 Þ 140 L

  432 Esh Ash 18  2 108 2 2 d3 þ d6 þ 3 ðd5 −d2 Þ − 2 ðd5 −d2 Þðd3 þ d6 Þ 140  L L L   E A 72 108 2 2 2 ¼ sh sh 3 d6 −d3 þ 6d3 d6 − d3 ðd5 −d2 Þ þ 2 ðd5 −d2 Þ L 280  L  432 E A 18  2 108 2 2 d3 þ d6 − 3 ðd5 −d2 Þ þ 2 ðd5 −d2 Þðd3 þ d6 Þ ¼ sh sh − 140  L L L  Esh Ash  2 72 108 2 2 3 d3 −d6 þ 6d3 d6 − d6 ðd5 −d2 Þ þ 2 ðd5 −d2 Þ ¼ L 280  L E A 18 2 2 2 ¼ sh sh 12Ld3 þ Ld6 −3Ld3 d6 þ ðd5 −d2 Þ −3ðd5 −d2 Þðd6 −d3 Þ L 140    E A 72 108 2 2 2 d3 ðd5 −d2 Þ− 2 ðd5 −d2 Þ ¼ sh sh −3 d6 −d3 −6d3 d6 þ L 280 L i   Esh Ash h 2 2 4Ld3 d6 −3L d3 þ d6 −6ðd5 −d2 Þðd3 þ d6 Þ ¼ 280    432 E A 18 2 108 2 2 ¼ sh sh d3 þ d6 þ 3 ðd5 −d2 Þ − 2 ðd5 −d2 Þðd3 þ d6 Þ 140  L L L  Esh Ash  2 72 108 2 2 ¼ 3 d6 −d3 −6d3 d6 þ d6 ðd5 −d2 Þ− 2 ðd5 −d2 Þ 280  L L E A 18 2 2 2 ¼ sh sh ðd5 −d2 Þ þ Ld3 þ 12Ld6 −3Ld3 d6 þ 3ðd5 −d2 Þðd6 −d3 Þ 140 L

K sh2ð2;2Þ ¼ K sh2ð2;3Þ K sh2ð2;5Þ K sh2ð2;6Þ K sh2ð3;3Þ K sh2ð3;5Þ K sh2ð3;6Þ K sh2ð5;5Þ K sh2ð5;6Þ K sh2ð6;6Þ

Matrix [Kpsh] Matrix [Kp] 6 P 5L Ap 2L ¼ P 15 Ap 6 ¼ P Ap 5L

1 P 10 Ap 1 ¼ − P Ap 10 1 ¼ − P Ap 10

6 P 5L Ap L ¼ − P Ap 30 2L ¼ P 15 Ap

K 3ð2;2Þ ¼

K 3ð2;3Þ ¼

K 3ð2;5Þ ¼ −

K 3ð3;3Þ

K 3ð3;5Þ

K 3ð3;6Þ

K 3ð5;5Þ

K 3ð5;6Þ

K 3ð6;6Þ

K 3ð2;6Þ ¼

1 P 10 Ap

Esh Ash L E S ¼ sh zsh L 12Esh Izsh ¼− L3 E S ¼ sh zsh L E A ¼ sh sh L 6E I ¼ − sh2 zsh L

Esh Szsh L 12Esh Izsh ¼ L3 6Esh I zsh ¼ L2 6E I ¼ − sh2 zsh L E S ¼ − sh zsh L 4Esh I zsh ¼ L

Esh Ash L 6E I ¼ sh2 zsh L 4E I ¼ sh zsh L 2E I ¼ sh zsh L 12Esh Izsh ¼ L3

K sh0ð1;3Þ ¼ −

K sh0ð1;4Þ ¼ −

K sh0ð1;6Þ

K sh0ð2;2Þ

K sh0ð2;3Þ

K sh0ð3;4Þ K sh0ð4;4Þ K sh0ð5;6Þ

K sh0ð3;5Þ K sh0ð4;6Þ K sh0ð6;6Þ

K sh0ð3;3Þ K sh0ð3;6Þ K sh0ð5;5Þ



6 1 ðd5 −d2 Þ− ðd3 þ d6 Þ 2 10L 5L 1 1 ¼ Esh Ash ðd5 −d2 Þ þ ðd6 −4d3 Þ 10L 30  6 1 ¼ Esh Ash − 2 ðd5 −d2 Þ þ ðd3 þ d6 Þ 10L 5L  6 1 ¼ Esh Ash ðd5 −d2 Þ þ ðd3 −4d6 Þ 10L 30  6 1 ¼ Esh Ash − 2 ðd5 −d2 Þ þ ðd3 þ d6 Þ 10L  5L 2 ¼ Esh Ash ðd4 −d1 Þ þ Esh Szsh d3 15 1 1 ¼ Esh Ash − ðd5 −d2 Þ− ðd6 −4d3 Þ 30  10L 6 1 ¼ Esh Ash ð d −d Þ− ðd3 þ d6 Þ 5 2 2 10L 5L  1 1 ¼ Esh Ash − ðd5 −d2 Þ− ðd3 −4d6 Þ 10L 30  2 ¼ Esh Ash ðd −d1 Þ −Esh Szsh d6 15 4



6 ðd4 −d1 Þ 2 5L 1 ¼ Esh Ash ðd4 −d1 Þ 10L  6 ¼ Esh Ash − 2 ðd4 −d1 Þ 5L  1 ¼ Esh Ash ðd4 −d1 Þ 10L  1 ¼ Esh Ash − ðd4 −d1 Þ 10L

K sh1ð2;2Þ ¼ Esh Ash

K sh1ð1;3Þ

K sh1ð2;3Þ

K sh1ð2;4Þ K sh1ð3;3Þ K sh1ð3;4Þ K sh1ð4;5Þ K sh1ð4;6Þ K sh1ð6;6Þ

The modified tangent stiffness matrix including semi-rigid connections is symmetric, and its non-zero terms are found as follows: 2

Let α 1 ¼ ðk33 þ Rk1 Þ; α 2 ¼ ðk66 þ Rk2 Þ; β ¼ α 1 α 2 −k36 2

Matrix ½K aa 

−1

α2 6β ¼6 4 −k36

3 −k36 7 β 7 α1 5 β

Matrix [K′]

K sh1ð1;2Þ ¼ Esh Ash

K sh1ð1;6Þ

  1 Ash σ y −Esh εsh 10   1 ¼ A σ −Esh εsh 10 sh y  1 ¼ − Ash σ y −Esh εsh 10   6 ¼ Ash σ y −Esh εsh 5L   2L ¼ Ash σ y −Esh εsh 15 ¼

Appendix B. Modified tangent stiffness matrix [K′] and modified load vector {r′} for beam-column element including nonlinear connections

β

Matrix [Ksh1]

K sh1ð1;5Þ

K sh3ð3;3Þ

K sh3ð5;6Þ

K sh0ð1;1Þ ¼

K sh0ð2;6Þ

K sh3ð2;5Þ

K sh3ð3;6Þ

Matrix [Ksh0]

K sh0ð2;5Þ

  6 K sh3ð2;3Þ Ash σ y −Esh εsh 5L   6 K sh3ð2;6Þ ¼ − Ash σ y −Esh εsh 5L   2L ¼ A σ −Esh εsh K sh3ð3;5Þ 15 sh y  L ¼ − Ash σ y −Esh εsh K sh3ð5;5Þ 30   1 K sh3ð6;6Þ ¼ − Ash σ y −Esh εsh 10

K sh3ð2;2Þ ¼

K sh1ð2;5Þ K sh1ð2;6Þ K sh1ð3;5Þ



K sh1ð3;6Þ K sh1ð5;5Þ K sh1ð5;6Þ

1 ¼ Esh Ash − ðd4 −d1 Þ  30 6 ¼ Esh Ash ðd4 −d1 Þ 2 5L  1 ¼ Esh Ash − ðd −d1 Þ 10L 4

0

k13 ðα 2 k13 −k16 k36 Þ k16 ðα 1 k16 −k13 k36 Þ R ðα k −k16 k36 Þ 0 − K ð1;3Þ ¼ k1 2 13 β β β k ðα k −k16 k36 Þ k26 ðα 1 k16 −k13 k36 Þ R ðα k −k13 k36 Þ 0 − K ð1;6Þ ¼ k2 1 16 ¼ k12 − 23 2 13 β β β k ðα k −k16 k36 Þ k46 ðα 1 k16 −k13 k36 Þ R ðα k −k26 k36 Þ 0 ¼ k14 − 34 2 13 − K ð2;3Þ ¼ k1 2 23 β β β k ðα k −k16 k36 Þ k56 ðα 1 k16 −k13 k36 Þ R ðα k −k23 k36 Þ 0 − K ð2;6Þ ¼ k2 1 26 ¼ k15 − 35 2 13 β β β k ðα k −k26 k36 Þ k26 ðα 1 k26 −k23 k36 Þ − ¼ k22 − 23 2 23 β β k ðα k −k26 k36 Þ k46 ðα 1 k26 −k23 k36 Þ R ðα k −k46 k36 Þ 0 ¼ k24 − 34 2 23 − K ð3;4Þ ¼ k1 2 34 β β β k ðα k −k26 k36 Þ k56 ðα 1 k26 −k23 k36 Þ R ðα k −k56 k36 Þ 0 − K ð3;5Þ ¼ k1 2 35 ¼ k25 − 35 2 23 β β β k ðα k −k46 k36 Þ k46 ðα 1 k46 −k34 k36 Þ − ¼ k44 − 34 2 34 β β k ðα k −k46 k36 Þ k56 ðα 1 k46 −k34 k36 Þ R ðα k −k34 k36 Þ 0 ¼ k45 − 35 2 34 − K ð4;6Þ ¼ k2 1 46 β β β k ðα k −k56 k36 Þ k56 ðα 1 k56 −k35 k36 Þ R ðα k −k35 k36 Þ 0 − K ð5;6Þ ¼ k2 1 56 ¼ k55 − 35 2 35 β β β R ðβ−Rk1 α 2 Þ R R R ðβ−Rk2 α 1 Þ 0 0 ¼ k1 K ð3;6Þ ¼ k36 k1 k2 K ð6;6Þ ¼ k2 β β β

K ð1;1Þ ¼ k11 − 0

K ð1;2Þ 0

K ð1;4Þ 0

K ð1;5Þ 0

K ð2;2Þ 0

K ð2;4Þ 0

K ð2;5Þ 0

K ð4;4Þ 0

K ð4;5Þ 0

K ð5;5Þ 0

K ð3;3Þ

P.-C. Nguyen, S.-E. Kim / Journal of Constructional Steel Research 100 (2014) 36–49

References

Vector {r′} M ðα k −k16 k36 Þ þ M 2 ðα 1 k16 −k13 k36 Þ ¼ P 1 − 1 2 13 β M 1 ðα 2 k23 −k26 k36 Þ þ M 2 ðα 1 k26 −k23 k36 Þ 0 ðM 1 α 2 −M2 k36 Þ 0 r 3 ¼ Rk1 r2 ¼ V 1 − β β M ðα k −k46 k36 Þ þ M 2 ðα 1 k46 −k34 k36 Þ 0 r4 ¼ P 2 − 1 2 34 β M ðα k −k56 k36 Þ þ M 2 ðα 1 k56 −k35 k36 Þ 0 ðM 2 α 1 −M1 k36 Þ 0 r5 ¼ V 2 − 1 2 35 r 6 ¼ Rk2 β β 0 r1

Appendix C. Static condensation procedure for beam-column member Considering a beam-column member divided into n sub-elements as shown in Fig. 2. The static condensation procedure can be illustrated by the use of matrix notation. The element equilibrium equations can be written in partitioned matrix forms as follows: 

K aa K ba

K ab K bb

49



da db



 ¼

ra rb

 ð48Þ

where da is the displacement vector of nodes along the member length (from node 3 to node i), db is the condensed displacement vector of the typical beam-column member with the conventional six degrees of freedom at two ends (I and J). Rewriting Eq. (48) as algebraic equations ½K aa fda g þ ½K ab fdb g ¼ fr a g

ð49Þ

½K ba fda g þ ½K bb fdb g ¼ fr b g:

ð50Þ

From Eq. (49), we have fda g ¼ ½K aa 

−1

fr a g−½K aa 

−1

½K ab fdb g:

ð51Þ

Substituting Eq. (51) into Eq. (50), equilibrium equations including only the conventional six degrees of freedom are written as   T −1 T −1 ½K bb −½K ab  ½K aa  ½K ab  fdb g ¼ fr b g−½K ab  ½K aa  fr a g

ð52aÞ

0 0 K fdb g ¼ r

ð52bÞ

where [K′] is the condensed stiffness matrix associated with the ‘one element per member’ concept of analysis, and {r′} is the condensed load vector. To include the effects of nonlinear connections, Eq. (52b) is condensed again as shown in Section 2.2.1. Eq. (52b) is used to assemble the structural equilibrium equation. After solving the structural equilibrium equation, nodal displacements {db} at two ends of the member can be extracted to find displacements {da} along the member length by using Eq. (51).

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