Nonlinear Dynamic Analysis of Frames With Plastic Hinges

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NONLINEAR DYNAMIC ANALYSIS OF FRAMES WITH

PLASTIC HINGES AT ARBITRARY LOCATIONS

Z. H. YAN AND F. T. K. AU* Department of Civil Engineering, The University of Hong Kong, Hong Kong, PR China

SUMMARY

This paper presents a method for nonlinear dynamic analysis of frames subjected to distributed loads, which isbased on the semi-rigid technique and moving node strategy. The plastic hinge is modelled as a pseudo-semi-rigidconnection with nonlinear hysteretic moment–curvature characteristics at element ends. The stiffness matrix withmaterial and geometric nonlinearities is expressed as a sum of products of the standard and geometric stiffnessmatrices with their corresponding correction matrices based on the plasticity-factors developed from the sectionflexural stiffness at the plastic hinge locations. Each beam member is modelled by two elements. The moving

node strategy is applied to the intermediate node to track the exact location of any intermediate plastic hinge thatmay be formed. Equilibrium iterations and geometry updating are carried out in every time step. Stiffness deg-radation is adopted to describe the deterioration of plastic hinges, and the effects of various parameters in thedegradation model are evaluated. Examples are used to illustrate the applicability and excellent performance of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.

1. INTRODUCTION

The influence of material and geometric nonlinearities on the behaviour of frames is particularly

significant during extreme events such as earthquakes and typhoons. In static analysis when the exter-

nal loading increases monotonically, one often focuses on the yielding of members as an important

source of material nonlinearity. However, dynamic responses are often complicated by yielding,unloading, reloading, etc., especially around the plastic hinges, and the deterioration of members and

joints. Furthermore, in frame members subjected to distributed loads, plastic hinges may be formed

at intermediate locations in addition to those at the ends. On the other hand, modern structures also

suffer from more significant ‘P-Δ effects’ since they are often taller and more slender because of the

use of stronger materials and more advanced construction techniques. To better understand their

behaviour and to improve the design of modern buildings, it is necessary to develop methods for

dynamic analysis, taking into account such effects and nonlinearities.

The semi-rigid technique is a common method to analyse frames with semi-rigid connections

(Monforton and Wu, 1963). Geometric nonlinearity (Xu, 1992) and the plastification concept (Hasan

et al., 2002) were also introduced to the method, and a series of advanced analyses of steel frame

structures were developed (Xu et al., 2005; Gong et al., 2005; Gong et al., 2006; Gong, 2006;

Grierson et al., 2006). So far, the semi-rigid technique is mainly applied to static and pushoveranalyses. Although various researchers are trying to develop the pushover analysis as a means of

Copyright © 2009 John Wiley & Sons, Ltd.

* Correspondence to: Francis T. K. Au, Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, HongKong, PR China. E-mail: [email protected]

THE STRUCTURAL DESIGN OF TALL AND SPECIAL BUILDINGSStruct. Design Tall Spec. Build. 19, 778–801 (2010)Published online 2 April 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/tal.513



NONLINEAR DYNAMIC ANALYSIS OF FRAMES WITH PLASTIC HINGES 779

Copyright © 2009 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. 19, 778–801 (2010) DOI: 10.1002/tal

practical design, nonlinear dynamic analysis is still essential, especially to those structures that are

irregular and those of which the higher mode effects cannot be ignored. Besides, the strong-motion

peculiarity is also a problem that the pushover analysis needs to overcome (Elnashai, 2002). Therefore,

dynamic analysis is still important to frames with nonlinearities.

Under lateral loading, the frame members subjected to distributed load may have plastic hinges

formed at intermediate locations besides the member ends. To capture the behaviour of the intermedi-

ate plastic hinges, an efficient method for elasto-plastic large deflection analysis of steel frames using

an element with plastic hinges at mid-span and two ends was proposed (Chen and Chan, 1995). As

the intermediate plastic hinges can only be formed at mid-span, the results obtained are subject to

certain errors. Later, the moving node strategy was presented for the elasto-plastic analysis of frames

subjected to loads including linearly varying distributed load (Wong, 1996). In addition, certain appli-

cations of the moving node method in the second-order inelastic analysis of two- and three-dimensional

steel frames were reported (Kim et al., 2004; Kim and Choi, 2005). However, the above research on

the intermediate plastic hinge was only for the static analysis of frames. Such plastic hinges are often

formed when the frame is subjected to strong seismic excitations, and therefore, the corresponding

dynamic problem warrants further study.

The material and geometric nonlinearities in beam–column elements may be simulated by eitherthe plastic hinge element or the fibre element. Although the fibre element model can handle the

residual stresses and better simulate the yielding process, it is rather computation intensive. On the

other hand, the plastic hinge element model can simulate nonlinearities well and it is computationally

efficient. The two beam–column element models have been compared (Hall and Challa, 1995), and

good agreement has been obtained for frame problems, including the level of ground motion at which

collapses occur.

Dynamic analysis of frames with nonlinear semi-rigid connections and geometric nonlinearity has

been studied using Runge–Kutta integration and modified modal analysis (Lui and Lopes, 1997; Awkar

and Lui, 1999). However, the nonlinear connections can only be located in the beams but not in the

columns, since the stability functions to account for geometric nonlinearity are not compatible with

the nonlinear connections. It is, therefore, not easy to extend the method to inelastic analysis because

the bending moments in the columns of the lower floors are usually large enough to cause yielding.Another method to model the effect of connections is to adopt shape functions in the form of cubic

Hermitian functions that account for end-springs to simulate the connection flexibility and material

nonlinearity (Chan and Chui, 2000). Although this method is capable of predicting the dynamic response

of steel structures with geometric nonlinearities, hysteretic connection flexibility and hysteretic material

yielding, the element matrices are rather complicated and substantial modifications are necessary to

incorporate them into the existing linear dynamic analysis programmes. The force analogy method was

extended to the dynamic analysis of frames with material and geometric nonlinearities in time domain

(Zhao and Wong, 2006). The performance assessment and energy analysis for various structures were

recently carried out based on the method (Wong and Zhao, 2007; Zhang et al., 2007; Wong, 2008).

Recently, Au and Yan (2008) presented a method for nonlinear dynamic analysis of frames with

material and geometric nonlinearities based on the semi-rigid technique. This paper describes further

improvements, including the use of moving node strategy to track plastic hinges formed at arbitraryintermediate locations, as well as the degradation of members. The effects of distributed load on the

formation of intermediate plastic hinges and the performance of degradation model will be examined.

2. DEVELOPMENT OF SEMI-RIGID TECHNIQUE

Semi-rigid behaviour often exists in frame connections, since there is no perfectly rigid or pinned

connection in reality. Monforton and Wu (1963) modelled the semi-rigid connection as a zero-length



780 Z. H. YAN AND F. T. K. AU

Copyright © 2009 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. 19, 778–801 (2010)

DOI: 10.1002/tal

linear spring at each end of a member, as shown in Figure 1, and presented the fixity factor l i at end

i of the member as

λ α

α α

α

α i

ie

ie

is

ie

itotal

i EI r L

i=+

= =+

=( )1

1 3

1 2, (1)

where a ie is the elastic end-rotation at end i, a i

s is the spring rotation at end i, a itotal is the total end-

rotation at end i, r i is the rotational stiffness of the semi-rigid connection at end i, E is elastic modulus

of the member, I is moment of inertia of the member and L is the length of the member. The fixity

factor is defined as the ratio of the rotational stiffness of the semi-rigid connection to the total stiffness

comprising the connection and the member. It can also be interpreted as the ratio of the elastic end-

rotation of the member to the total end-rotation of the member. The fixity factor l i lies between 1 and

0, which corresponds to the rotational stiffness of the semi-rigid connection r i of infinity and zero,

respectively, namely the rigid connection and pinned connection.

Monforton and Wu (1963) also presented a correction matrix Se as a function of the fixity factors

to develop a first-order elastic analysis approach for semi-rigid frames. The stiffness matrix of an

element with semi-rigid connections is taken as the product of the standard elastic stiffness matrix Ke

and the correction matrix Se, i.e.

K K Ssemi rigid e e− = (2)

Much later, Xu (1992) presented another correction matrix Sg for geometric stiffness matrix as a func-

tion of the fixity factors to develop a second-order elastic analysis for semi-rigid frames. To account

for both first-order and second-order effects, the stiffness matrix of a member with semi-rigid con-

nection is written as the sum of product of the standard elastic stiffness matrix Ke with the correction

matrix Se, and the product of the geometric stiffness matrix Kg with the correction matrix Sg, i.e.

K K S K Ssemi rigid e e g g− = +

(3)

Hasan et al. (2002) later discovered that the model for post-elastic behaviour of a plastic hinge is

similar to that for the elastic behaviour of a semi-rigid connection. Therefore, regarding the spring

rotation a is at end i of the member as the plastic rotation a i

p of the plastic hinge at the corresponding

end of the member, and assuming that the rotational stiffness of the semi-rigid connection r i varies

under the external bending moment according to certain nonlinear moment–curvature relation, then a

potential plastic-hinge section can be regarded as a pseudo-semi-rigid connection. A plasticity factor

can then be introduced based on the fixity factor to describe the behaviour of a plastic hinge from the

initial elastic state to the plastic state under monotonic loading. Let the rotational stiffness r i of the

semi-rigid connection at end i be the section flexural stiffness dM i / d a i p at end i, namely

Figure 1. Concept of fixity factor





r dM

d i

i

i p

=α

(4)

where a i p is the plastic end-rotation of the member at end i under the external bending moment. The

plasticity factor (Hasan et al., 2002) can be defined as

p EI L dM d

i

i i p

=+ ( )

1

1 3 α (5)

The stiffness matrix with material and geometric nonlinearities can be expressed in the form of

plasticity factors by replacing the fixity factors l i (i = 1, 2) with the plasticity factors pi (i = 1, 2) in

Equation (3) as

K K S K S= +e e p

g g p (6)

where S pe and S p

g are the corresponding correction matrices. The standard elastic stiffness matrix Ke

and the geometric stiffness matrix Kg are, respectively

Ke

EA

L

EA

L EI

L

EI

L

EI

L

EI

L EI

L

EI

L

EI

L EA

L

=

−

−

−

0 0 0 0

12 60

12 6

40

6 2

0 0

3 2 3 2

2

ssym EI

L

EI

L EI

L

12 6

4

3 2−

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥

⎥⎥⎥⎥⎥

(7)

Kg

P

L

P P

L

P

PL P PL

symP

L

P

PL

=

−

− −

−

0 0 0 0 0 0

6

5 100

6

5 102

150

10 30

0 0 06

5 102

155

⎡

⎣

⎢⎢⎢⎢⎢

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

⎥⎥⎥⎥⎥⎥

(8)

Their corresponding correction matrices S pe and S p

g are expressed, respectively, as functions of plas-

ticity factors as (Hasan et al., 2002)





DOI: 10.1002/tal

Se p

p p

e

e e

e e

e

e e

=−( )

1

4

0 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0

0 0 0 01 2

11

22 23

32 33

44

55 566

65 660 0 0 0 e e

⎡

⎣

⎢⎢⎢⎢

⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎥⎥

(9a)

e e p p

e p p p p

e Lp p

e e p p

11 44 1 2

22 2 1 1 2

23 1 2

32 65 1

4

4 2

2 1

6

= = −= − += − −( )

= = − 22

33 1 2

55 1 2 1 2

56 2 1

66 2

3 2

4 2

2 1

3 2

( )

= −( )

= − += −( )

= −

L

e p p

e p p p p

e Lp p

e p p11( )

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

(9b)

Sg p

p p

g g g g

g g

=−( )

1

5 4

0 0 0 0 0 00 1 0 0 0 0

0 0

0 0 0 0 0 0

0 0 0 0 1 0

0

1 2

2

32 33 35 36

62 663 65 660 g g

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

(10a)

g g p p p p p p p p L

g p p

32 35 12

2 1 22

12

22

1 2

33 1

4 8 13 32 8 25 20

16

= − = − − − − + +( )= 22

21 2

21 2 1 2

36 2 12

12

2 1 2

25 96 128 28

4 16 5 9 2

+ − + −( )= − + −

p p p p p p

g p p p p p p 88 8

4 8 13 32 8 25 201 2

62 65 1 22

12

2 22

12

1 2

p p

g g p p p p p p p p

+( )= − = − − − − + +( ) L L

g p p p p p p p p

g p p p p

63 1 22

1 22

1 2 1 2

66 2 12

12

2

4 16 5 9 8 28

16 25= − + + −( )= + − 996 28 1281 2 1 2 p p p p− +( )

⎧

⎨

⎪⎪⎪

⎩⎪⎪⎪

(10b)

where P is the member axial force.

3. MOVING NODE STRATEGY IN DYNAMIC ANALYSIS

When a member is under distributed loading, the location of intermediate plastic hinge varies accord-

ing to the member forces and loading. Certain errors will be introduced if the intermediate plastic

hinge is prescribed at mid-span, especially when the frame is under complicated load combination.

To capture the location and behaviour of the intermediate plastic hinge, Wong (1996) first presented

a moving node strategy for the elasto-plastic analysis of frames subjected to loads, including linearlyvarying distributed load. In particular, the location of maximum bending moment with zero shear force

is tracked by the moving node. Once the maximum bending moment reaches the yielding moment of

the member, an intermediate plastic hinge is formed there. The moving node then continues to track

the location of maximum bending moment until the next plastic hinge forms.

Each beam member is, therefore, modelled by two elements and three nodes. Consider the

portal frame in Figure 2 carrying distributed load and under earthquake excitation. At first, the

intermediate moving node is set at an arbitrary location such as mid-span of the beam member.





In each time step, the location of the maximum total bending moment where the total shear force

equals to 0 is obtained as

x T

w

total

= 1 (11)

where x is the distance from the left end of member to the location of maximum total bending moment,

w is the uniformly distributed load and T 1total is the total shear force at the left end of member. Then,

the maximum total bending moment of the member can be calculated using

M wx

T x M total total totalmax = − + −

2

1 12

(12)

where M mtot

ax

al

is the maximum total bending moment including the effects from the inertial force, thedamping force, the stiffness force, the earthquake excitation and the uniformly distributed load applied

on the beam member, and M 1total is the total bending moment at the left end of member. Then, if the

maximum total bending moment is larger than the yielding moment of the beam member, the moving

node initially at location 3 is shifted to 3′, where the total bending moment is the maximum and the

plastic hinge is formed. Afterwards, the dynamic analysis of the whole structure can proceed assum-

ing this new configuration, until another new intermediate plastic hinge forms at a different location

3″ . As seen from Figure 2, at any moment in time before the total collapse of structure, there is one

and only one possible intermediate plastic hinge within each beam and therefore there are at most two

plastic hinges formed in each beam. Using Figure 2 as an example, yielding can occur only in either

locations 1 and 3′ during loading in one direction, or locations 2 and 3″ during loading in the opposite

direction. The previous unloaded plastic hinge will become a damaged section if there is stiffness

degradation. In view of the finite size of plastic hinge, the intermediate plastic hinge cannot be tooclose to a member end as well.

4. STIFFNESS DEGRADATION MODEL

When the plastic hinge experiences yielding, unloading and reloading, its stiffness will deteriorate as

a result of damage. Degradation models to describe the deterioration of members include those of Park

and Ang (1985) and Rao et al. (1998) for concrete members, and models by Krawinkler and Zohrei

Figure 2. Illustration of the moving node strategy





DOI: 10.1002/tal

(1983), Ballio and Castiglioni (1994), and Azevedo and Calado (1994) for steel members. As the

plasticity factor in Equation (5) describes the plastic characteristics of a plastic hinge, the damage can

be simulated by the gradual decrease in plasticity factor. The plastic hinge may also move into and

out of the plastic region after the start of yielding. The idea of this stiffness degradation model comes

from Shin and Oh (2007) and Yoshida (2001). The illustration of the change of plasticity factor and

the corresponding change of plastic hinge stiffness ratio in the simplified stiffness degradation model

are shown in Figure 3. The plastic hinge stiffness ratio can be solved from

LdM d EI p

pi i

p i

i

α ( ) =−3

1(13)

Before a plastic hinge forms, the plasticity factor of the section is 1 and the corresponding stiffness

is infinite. The degradation of the stiffness starts when the section begins to yield, and finishes when

the plasticity factor reaches a certain residual value or when the intermediate plastic hinge moves to

another location. When the section yields, the plasticity factor becomes zero, and so does the stiffness

of the section at the plastic hinge, as shown in Figure 3.

Referring again to member 1–2 in Figure 2, when the previous plastic hinge 3 ′ unloads and a

new plastic hinge 3″ is formed, member 1–2 will then be modelled by elements 1–3″ and 3″ –2. In

particular, element 3″ –2 has damage associated with the previous plastic hinge at 3′ with stiffness

degradation. The stiffness matrix Kd of an element with damage at an intermediate position can be

expressed by the plasticity factor based on Equation (5) and the method adopted by Skrinar and

Plibersek (2007) as

Kd

h h

h h h h

h h h

h

sym h hh

=

⎡

⎣

⎢⎢⎢

11 14

22 23 25 26

33 35 36

44

55 56

66

0 0 0 0

0

0

0 0⎢⎢

⎢⎢

⎤

⎦

⎥⎥⎥⎥

⎥⎥

(14a)

Figure 3. Stiffness degradation model





h EA L h h h h

h EI p h h h hd

11 14 11 44 11

22 25 22 55 212 1 2

= = − == ( ) +( ) = − =

, ,

, ,Ω 22

23 1 35 23

26 1

6 3 2 1

6 2 2

h EI Lp L p h h

h EI L p Ld d

d

= ( ) + −( )[ ] = −= ( ) +( ) −

ΩΩ

,

11

12 1

6 2

56 26

33 1

2 2

36 1

−( )[ ] = −

=( )

−( )

+[ ]= ( )

p h h

h EI L p L p

h EI L

d

d d

,

ΩΩ L L L p L p

h EI L L L L p

L L

d d

d

−( ) −( ) +[ ]= ( ) − −( ) −( )[ ]

= −

12

662

1 1

2

1

12 2 1

4

ΩΩ p p L L L pd d ( ) − −( ) −( )[ ]

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪ 12 11 1

(14b)

where L1 is the distance from the damage to the left element end and pd is the plasticity factor denot-

ing the degree of degradation associated with the previous plastic hinge. Theoretically, there is no

limit to the number of such previous damaged locations that can be introduced to the element.

However, for practical cases of earthquake excitations, a beam member normally has an intermediate

plastic hinge either at a single location or alternating between two different locations. Therefore, a

member with an intermediate moving node to monitor the active plastic hinge is sufficient for most

practical cases.

5. UPDATING OF GEOMETRY

To cope with the relatively large displacements, the geometry is updated according to the structural

response at the end of each time step as follows

L x x y y= −( ) + −( )1 2

2

1 2

2(15)

θ θ θ t = +0 Δ (16)

where L is the length of the member at any time, ( x 1, y1) and ( x 2, y2) are the updated global coordinates

at ends 1 and 2 of the member at the end of a time step, q 0 is the anticlockwise angle of rotation so

that the global X -axis becomes parallel to the initial element x -axis, q t is the anticlockwise angle of rotation so that the global X -axis becomes parallel to the updated element x ′-axis and Δq is the incre-

mental rotation from q 0 to q t .

6. PLASTIC ROTATION

The calculation of plastic rotation is only necessary when a potential plastic-hinge section becomes

active by moving into the plastic region. The process is similar to that adopted by Au and Yan (2008),

except that the effect of distributed load on the elastic rotation a –ie should also be included. There are

two ways to calculate the plastic rotation, depending on the nature of plastic hinge. If the plastic hinge

has strain-hardening or strain-softening property after the onset of yielding, the section flexural stiff-

ness is non-zero. From Equation (4), the plastic rotation a i p can be calculated simply as the summation

of plastic rotations in the previous time steps as

α i p i

i

M

r = ∑ Δ

(17)

where Δ M i is the incremental external bending moment at end i of the element from time t to t + Δt ,

and r i is the rotational stiffness of the section that is chosen according to the moment-rotation curve





DOI: 10.1002/tal

of the plastic hinge. Where necessary, an iteration process is carried out to ensure global force

equilibrium.

If the plastic hinge is perfectly elastic-plastic, which means that the section flexural stiffness after

the onset of yielding is zero (dM i / d a i p = 0), the plastic rotation can be solved indirectly. The deforma-

tion of a member under external bending moments and distributed load is shown in Figure 4. The

variable a total denotes the total rotation at end i of the member under external loading, a ie is the elastic

rotation at the end and a i p is the plastic rotation there. The orientation of the member before load

application is denoted by q 0, which is the anticlockwise angle of rotation so that the global X -axis

becomes parallel to the initial element x -axis. At the time considered, the member adopts a deformed

configuration with an updated member x ′-axis pointing from End 1 to End 2. The orientation of the

updated member x ′-axis is defined by q t , which is the anticlockwise angle of rotation so that the global

X -axis becomes parallel to the updated element x ′-axis. The incremental rotation from q 0 to q t is

then denoted by Δq . The plastic rotation at the plastic hinge a i p can be calculated by reference to

Figure 4 as

α α α θ i p

itotal

ie= − − Δ (18)

where the elastic rotation a ie can be evaluated from the end moments M i, M j, and the effect of distrib-

uted load a –ie by ignoring the geometric nonlinearity of element without intermediate damage as

α α ie i j

ie M L

EI

M L

EI i j i j= − + = = = =

3 61 2 2 1for or, , (19a)

Figure 4. Deformation of member with plastic hinge





For uniformly distributed load w acting normally on the whole beam, the effect on the elastic rotation

a –ie can be evaluated as

α ie i wL

EI i= − =( )( ) ,1

241 2

3

(19b)

For a beam element with intermediate damage as in Section 4, the elastic rotation a ie and the effect

of uniformly distributed load a –ie can be evaluated respectively as

α α

α

11 66 2 36

33 66 36

2 1

22 33 1 36

33 66 36

e e

e

M h M h

h h h

M h M h

h h h

=−− ( )

+

=−− ( ))

+

⎧

⎨⎪⎪

⎩⎪⎪ 2 2α e

(20a)

α

α

1

66 36

33 66 362

2

233 36

33 66 36

2

2

12

1

e

e

h h

h h h

wL

h h

h h h

wL= −

+

− ( )

=+− ( ) 22

⎧

⎨⎪⎪

⎩⎪⎪

(20b)

where hij (i, j = 3,6) are obtained from Equation (14b). The total rotation a itotal can be solved from the

global equilibrium equation in each time step, while the incremental rotation Δq can be calculated

from the updating of geometry in each time step. Note that Equations (17) or (18) applies only when

a potential plastic-hinge section becomes active. In Equations (19a) and (20a), the effects of shear

force and geometric nonlinearity on the local elastic rotation have been ignored for simplicity. These

effects are actually minimal, as elaborated in the subsequent sections.

7. YIELD CRITERION

The dynamic excitations mainly cause bending moments and shear forces in the frame members with

relatively minor variations in axial forces. The yield criterion (Duan and Chen, 1990) involving both

the bending moment M and the axial force P can be written as

1

1η

≤ + ( ) ≤ M

M

P

P p p

n

(21)

where the shape factor h depends on the type of cross section, M p is the moment capacity in the

absence of axial force, P p is the axial force capacity in the absence of bending moment and the yieldexponent n depends on the shape of cross section. As the transition from the fully elastic state to the

fully plastic state is assumed to take place over an extremely small range of curvature, only the full

yielding condition is used as the yield criterion, namely

M

M

P

P p p

n

+ ( ) ≤ 1 (22)





DOI: 10.1002/tal

8. ELEMENT FORCE RECOVERY PROCEDURE

In the dynamic analysis of nonlinear structures, the element force should be known for calculation of

the geometric stiffness matrix Kg and for determination of the unbalanced forces at nodal points. To

calculate the element force that eliminates any erroneous additional straining due to rigid body motion,

the external stiffness approach (Yang and Kuo, 1994) is adopted. The external stiffness matrix Keg for

a two-dimensional member is

Kge

M M

L

M M

LP

L

M M

L

P

L

M M

L

symP

L

=

+−

+

−+

−

+

0 0 0 0

0 0

0 0 0 0

0 0

0

0

1 2

2

1 2

2

1 2

2

1 2

2

⎡⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

(23)

where M i is the bending moment at End i (i = 1,2) of the element, P is the axial force of the element

and L is the length of the element, all of which are calculated for the last configuration.

Since the effects of geometric nonlinearity and rigid body motion in the beam members are relatively

small, the external stiffness approach is only applied to the column members in the present study.

Unlike the standard and geometric stiffness matrices, incorporating the plastic hinges into the element

has no effect on the external stiffness matrix Keg. Therefore, by subtracting the external stiffness matrix

Keg from the tangent stiffness matrix, a matrix that accounts duly for the effect of element deformation

can be obtained. The incremental element stiffness force Δf st +Δt from time t to t + Δt caused by defor-

mation increment of the element can be calculated in terms of the incremental displacement ΔXt +Δt

from time t to t + Δt as

Δ ΔΔ Δ Δf K S K S K Xt t s

e e p

g g p

ge

t t t t + + += + −( ) (24)

9. PROCEDURES FOR DYNAMIC ANALYSIS

In the present study that involves material and geometric nonlinearities, the Newmark method of direct

integration is adopted, together with an incremental iteration strategy. As usual, the inertial effect is

modelled by lumped masses at the beam–column joints. The incremental equilibrium equation of

motion of the whole structure can be written as

M X C X K X FΔ Δ Δ ΔΔ Δ Δ Δ

t t t t t t t t + + + ++ + = (25)

where M is the mass matrix, C is the damping matrix that may be formulated according to the damping

model chosen, K is stiffness matrix that can be obtained from Equation (6), ΔXt +Δt , ΔXt +Δt and ΔX ¨ t +Δt

are the incremental displacement, velocity and acceleration vectors relative to the ground, respectively,

from time t to t + Δt , in which the dot denotes differentiation with respect to time t , and ΔFt +Δt is the

incremental external excitation vector from time t to t + Δt . In the case of seismic loading, the incre-

mental external excitation vector ΔFt +Δt can be expressed as





Δ ΔΔF MHt t g x + = − (26)

where H is an index vector that assigns inertia forces to appropriate degrees of freedom, and Δ x g is

the incremental absolute ground acceleration from time t to t + Δt . In the present study, the stiffness

matrix K changes in the process of dynamic analysis, and straightforward application of the Rayleigh

damping model is not particularly rational. Therefore, for simplicity, mass proportional damping is

assumed and the damping matrix C is written in terms of the mass matrix M as

C M= 2ξ ω i i (27)

where x i and w i are, respectively, the critical damping ratio and the circular frequency for mode i. In

the subsequent simulations, the critical damping ratio and the circular frequency for the first mode are

chosen to calculate the damping matrix.

An incremental iteration strategy is used in each time step to ensure equilibrium in the broad sense

as elaborated below.

(1) Specify the material and geometric properties, and the initial conditions for dynamic analysis.Set the moving nodes at arbitrary positions such as the middle of each member. Initialize various

parameters and vectors, and evaluate the initial matrices M, C and K for analysis.

(2) Form the effective tangent stiffness Keff and the effective incremental load ΔFeff from time t to

t + Δt as

K K M Ceff t t

= + +1

2β

γ

β Δ Δ(28)

Δ ΔΔ

ΔΔF MH M C X M Ceff g t x

t

t t = − + +⎛

⎝ ⎜⎞ ⎠ ⎟ + + −⎛

⎝ ⎜⎞ ⎠ ⎟

⎡⎣⎢

⎤

1 1

2 2β

γ

β β

γ

β ⎦⎦⎥Xt (29)

where the stiffness matrix K is based on Equation (6), Xt is the displacement vector at time t ,

and the parameters for the Newmark method of direct integration are taken to be b = 0·25 and

g = 0·5.

(3) Solve for the incremental displacement ΔXt +Δt from time t to t + Δt from the equivalent equilib-

rium equation

K X Feff t t eff Δ ΔΔ+ = (30)

(4) Solve for the incremental velocity ΔXt +Δt and acceleration ΔX ¨ t +Δt from time t to t + Δt , respec-

tively, from

Δ Δ Δ

Δ

ΔΔ Δ

X X X Xt t t t t t t

t t

+ += − − −

⎛

⎝ ⎜

⎞

⎠ ⎟

γ

β

γ

β

γ

β 2 (31)

ΔΔ

ΔΔΔ Δ

X X X Xt t t t t t t t

+ += − −1 1 1

22β β β (32)

Then, calculate the displacement Xt +Δt , velocity Xt +Δt and acceleration X ¨ t +Δt at time t + Δt by

X X X X X X X X Xt t t t t t t t t t t t t t t + + + + + += + = + = +Δ Δ Δ Δ Δ ΔΔ Δ Δ, , (33)





DOI: 10.1002/tal

(5) Update the geometry of the frame according to Equations (15), (16) and (33).

(6) Calculate the incremental element stiffness forces of the elements Δf st +Δt corresponding to the

incremental displacements according to Equation (24), and then the element stiffness forces f st +Δt

at time t + Δt as

f f f t t s

t s

t t s

+ += +Δ ΔΔ (34)

Then, calculate the total element dynamic force f tot +Δt tal at time t + Δt as

f mx f mx mxMt t total

t t t t s

t t g t t + + + += + + + +Δ Δ Δ Δ Δ α ( ) (35)

where m is the element mass matrix, a M is the mass coefficient taken as a M = 2x iw i, xt +Δt and

xt +Δt are, respectively, the element velocity and acceleration vectors at time t + Δt with respect

to the local element axes, and x ˜ g(t + Δt) is the ground acceleration vector at time t + Δt with

respect to the local element axes. Actually, the four terms on the right-hand side of Equation

(35) account for the element inertial force, element stiffness forces, element damping force and

the element excitation force, respectively.(7) Determine the force state of each element.

(8) Update the plasticity factors according to the state of elements, the stiffness degradation model

and the simplified nonlinear hysteretic moment–curvature relation.

(9) Update the element and global stiffness matrices according to Equation (6).

(10) Calculate the incremental element stiffness forces corresponding to the incremental displace-

ments and the total element stiffness forces in the frame using the updated element stiffness

matrices, namely

Δ ΔΔ Δ Δf K S K S K Xt t s updated

e e p

g g p

ge

t t

updated

t t + + +( ) = + −( ) (36)

f f f t t s updated

t s

t t s updated

+ +( ) = + ( )Δ ΔΔ (37)

(11) Transform the total element stiffness forces to the nodal forces and assemble the internal nodal

force vector Nt +Δt in the global coordinate system

N T f t t k t t s

k

updated

k

ne

+−

+=

= ( )∑Δ Δ1

1

(38)

where Tk is the transformation matrix from the global to the element axis system for element k ,

ne is the number of elements in the frame, and the summation here implies the assembly

process.

(12) Calculate the unbalance force vector Γ t +Δt at time t + Δt by

G t t g t t t t t t d x t t + + + += − + − + +( )Δ Δ Δ ΔΔMH MX N F ( ) (39)

where Fdt +Δt is the damping force calculated according to the chosen damping model.

(13) Check the convergence parameters ε ε 1 2= = ⋅+ +G t t g t t x Δ ΔΔ ΔMH X and where, denotes

the norm of a vector. If e 1 > e 1max and e 2 > e 2max, go to Step (14); otherwise, go to Step (17).

(14) Solve for the incremental displacement ΔX′t +Δt for equilibrium iteration using unbalanced force

Γ t +Δt and effective tangent stiffness Keff evaluated with the updated stiffness matrices, namely





Δ Δ Δ′ =+−

+X Kt t eff t t 1G (40)

(15) Update the displacement, velocity and acceleration vectors at time t + Δt

X X Xt t

updated

t t t t + + +( ) = + ′Δ Δ ΔΔ (41)

X X Xt t

updated

t t t t t

+ + +( ) = + ′Δ Δ ΔΔΔ

γ

β (42)

X X Xt t

updated

t t t t t

+ + +( ) = + ′Δ Δ ΔΔΔ

12β

(43)

(16) Go to Step (5) and repeat the analysis until the convergence criteria are satisfied.

(17) If the maximum total bending moment within a beam element exceeds the yielding moment,

the moving node strategy is invoked and the intermediate plastic hinge is formed there. Any

previous intermediate plastic hinge unloaded will be recorded as damage if there is stiffness

degradation.(18) If time t + Δt < T , where T is the total time of excitation, set t = t + Δt and go to Step (2b);

otherwise, stop.

10. NUMERICAL EXAMPLES

After verification by a portal frame, a practical multi-storey frame is analysed to illustrate the appli-

cability and performance of the proposed method. Unless otherwise stated, the Young’s modulus is

2 × 1011 N/m2, while the yield stress is 2·48 × 108 N/m2. Mass-proportional damping with a 5%

critical damping ratio is assumed. For simplicity, the rotational inertial effect of the beam is ignored

and the lumped mass matrix is adopted. A lumped mass is assigned to each beam–column joint to

simulate the floor mass, and a joint load is applied to account for the gravity effect.

10.1 Seismic response of a portal frame

A portal frame made up of steel W-type I-beams, as shown in Figure 5, is used for verification. The

accurate locations of intermediate plastic hinges are determined from trial calculations beforehand.

Then, the results from the present method with moving node strategy are compared with those obtained

by confining the intermediate plastic hinges to the above predetermined location, namely the fixed

node strategy. As the bending capacity of the beam is lower than that of the column, the top of column

is unlikely to yield. The other properties of the steel W-type I-beams are given in Table 1. Each

beam–column joint is assigned a lumped mass of 50 000 kg and an applied joint load P0 of 500 kN.

To examine the basic performance, initially, the yield criterion adopted accounts for the bending

moment only and no stiffness degradation is considered. The moving node is initially set at the middle

of beam. The uniformly distributed load w on the beam is 50 kN/m. The assumed excitation is that of El Centro earthquake with magnified peak ground acceleration of 0·5 g, as shown in Figure 6, where

g is the acceleration due to gravity. A regular time step of 0·002 s is chosen. As the available El Centro

earthquake excitation has been sampled at 50 Hz with time steps of 0·02 s, piecewise cubic Hermite

interpolation is used to provide excitation data of time steps of 0·002 s. The above data apply below

unless otherwise stated.

Through accurate trial calculations, the intermediate plastic hinge of the beam is formed sequentially

as follows: 1·435 m from the left column at 2·210 s, and 5·296 m from the left column at 4·628 s.





DOI: 10.1002/tal

Therefore, to obtain the reference solution, the beam is divided into three elements of lengths of

1·435 m, 3·861 m and 1·404 m. With reference to Figure 5, the order of plastic hinge formations is asfollows: PPHL4, PPHL3, PPHL1, PPHL2 and the intermediate beam plastic hinge at various locations.

The horizontal beam displacement is shown in Figure 7, while the vertical displacement tracked by

the moving node of the beam is presented in Figure 8. The result in Figure 8 up to 2·210 s is actually

that at mid-span before the formation of intermediate beam plastic hinge. It is followed by the vertical

displacements of the first and second positions of intermediate beam plastic hinge, respectively, in the

periods 2·212 s to 4·628 s and beyond 4·628 s. The results also show that the two intermediate beam

plastic hinges are almost formed symmetrically, namely at 1·435 m from the left joint and 1·404 m

Table 1. Properties of steel members in portal frame

Member Area (m2) Moment of inertia (m4) Yield moment (Nm)

Column (W27 × 102) 1·9355 × 10−2 1·5068 × 10−3 1·2395 × 106

Beam (W12 × 58) 1·0968 × 10−2 1·9771 × 10−4 3·5113 × 105

Figure 5. A portal frame for verification

Figure 6. Magnified El Centro earthquake time history





from the right joint. Figures 7 and 8 show that the moving node strategy can be used effectively in

dynamic analyses.

Various factors affecting the formation of intermediate plastic hinges are then examined. Analyses

are repeated for uniformly distributed load w ranging from 0 to 50 kN/m at intervals of 10 kN/m while

keeping the applied joint load P0 = 500 kN unchanged. Figure 9 shows that the uniformly distributed

load has little effect on the horizontal displacements. When the uniformly distributed load is 30 kN/m

or below, no intermediate plastic hinge is formed within the beam. Figure 10 further shows the

variations of maximum span moment and its location in the beam with time when the uniformly

distributed load is increased to 40 and 50 kN/m. When the uniformly distributed load is 40 kN/m,

Figure 7. Horizontal beam displacement of portal frame under magnified El Centro earthquake (Δt = 0·002 s)

Figure 8. Vertical displacement of the moving node of beam in portal frame under magnified El Centroearthquake ((Δt = 0·002 s)





DOI: 10.1002/tal

Figure 10(a) shows that an intermediate plastic hinge forms briefly. When the uniformly distributed

load reaches 50 kN/m, Figure 10(b) confirms that the first intermediate plastic hinge forms briefly ata location close to the left column, and the second intermediate plastic hinge forms twice at a location

close to the right column.

To study the effects of the plastic hinge deterioration on the response, the stiffness degradation

model, as shown in Figure 3, is adopted in the subsequent analysis in order to examine the effects of

residual value and rate of reduction of plasticity factor. In accordance with the model, when the plas-

ticity factor decreases from 1 to the residual value, the plastic hinge stiffness ratio also decreases from

positive infinity to the corresponding residual stiffness. When a plastic hinge moves to another

Figure 9. Horizontal beam displacements of portal frame under magnified El Centro earthquake and differentuniformly distributed loads

Figure 10. Variations of maximum span moment and its location in the beam of portal frame under magnifiedEl Centro earthquake





location, there will be stiffness degradation associated with the previous location, and hence the stiff-

ness matrix Kd of an element with intermediate damage, as shown in Equation (14), should be applied

as appropriate.

Analyses are repeated for uniformly distributed load of 50 kN/m, covering cases in which the plas-

ticity factor begins to decrease from 1 upon yielding at a rate of 0·05/s until it reaches a residual value

ranging from 1 (no degradation) to 0·8. The horizontal beam displacements shown in Figure 11 indi-

cate that the response with stiffness degradation associated with plastic hinges obviously results in a

lower frequency and larger amplitudes. For a residual plasticity factor of 0·95, the intermediate plastic

hinge forms at two different locations of the beam. However, for a residual plasticity factor of 0·90

or below, the intermediate plastic hinge forms only at one single location of the beam. The variation

Figure 11. Horizontal beam displacements of portal frame under magnified El Centro earthquake anduniformly distributed load of 50 kN/m with rate of reduction of plasticity factor of 0·05/s

Figure 12. Variation of plasticity factor of active intermediate beam plastic hinge of portal frame under magnifiedEl Centro earthquake and uniformly distributed load of 50 kN/m with rate of reduction of plasticity

factor of 0·05/s





DOI: 10.1002/tal

of plasticity factor of the active intermediate beam plastic hinge shown in Figure 12(a) verifies that

the intermediate plastic hinge forms at two different locations, while that in Figure 12(b) indicates

formation of intermediate plastic hinge at one single location.

Analyses are repeated for uniformly distributed load of 50 kN/m, covering cases in which the plas-

ticity factor begins to decrease from 1 upon yielding to 0·95 at a rate of reduction ranging from 0·05

to 0·2/s. Figure 13 shows that the horizontal beam displacements are not as sensitive to the rate of

reduction of plasticity factor. The variations of plasticity factor of the active intermediate beam plastic

hinge shown in Figures 12(a), 14(a) and 14(b) for the reduction rates of 0·05, 0·1 and 0·2/s, respec-

tively, verify that the intermediate plastic hinge forms at two different locations in the first case only,

but at one single location in the last two cases. Generally, a higher rate of reduction causes the frame

to degrade faster, so that the subsequent intermediate maximum moment becomes smaller. By proper

Figure 13. Horizontal beam displacements of portal frame under magnified El Centro earthquake anduniformly distributed load of 50 kN/m with residual plasticity factor of 0·95

Figure 14. Variation of plasticity factor of active intermediate beam plastic hinge of portal frame under magnifiedEl Centro earthquake and uniformly distributed load of 50 kN/m with residual plasticity factor of 0·95





adjustment of the rate of reduction and residual value of plasticity factor of the stiffness degradation

model, the deterioration characteristics of a plastic hinge can be simulated approximately.

10.2 Seismic response of a multi-storey frame

A more realistic three-storey frame made up of steel W-type I-beams, as shown in Figure 15, is then

studied to illustrate the capability of the proposed method. The member used and their properties are

shown in Table 2. At the roof level, the lumped masses on exterior and interior columns are 10 000 kg

and 20 000 kg, respectively. On each floor below the roof, the lumped masses on exterior and interior

columns are 20 000 kg and 40 000 kg, respectively. The applied joint loads to account for gravity

effects are as follows: P1 = 500 kN, P′1 = 1000 kN, P2 = 300 kN, P′2 = 600 kN, P3 = 100 kN and P′3 =

200 kN. The assumed excitation is that of the El Centro earthquake with magnified peak ground

acceleration of 0·7 g. The multi-parameter yield criterion that takes into account bending moment M

and axial force P is adopted with a yield exponent n = 1·3. The stiffness degradation model adopted

has a residual plasticity factor of 0·9 and a rate of reduction of 0·1/s. Dynamic analysis is carried out

with rather fine time steps of 0·002 s. The horizontal displacements at various floors of the frame are

shown in Figure 16, indicating that the response generally increases with height. The formation of intermediate beam plastic hinge differs from floor to floor. In each beam of the first floor, the inter-

mediate plastic hinge forms at two different locations, as observed in the plasticity factor of MN1 in

Figure 17(a). However, Figure 17(b) shows that the intermediate plastic hinge (i.e., MN3) forms at

one single location in each span of the second floor. Incidentally, no intermediate plastic hinge is

formed in any span of the third floor. The locations of plastic hinges shown in Figure 18 indicate that,

Table 2. Properties of steel members in three-storey frame

Member Area (m2) Moment of inertia (m4) Yield moment (Nm)

Exterior column (W24 × 103) 1·9548 × 10−2 1·2487 × 10−3 1·1379 × 106

Interior column (W24 × 117) 2·2194 × 10−2 1·4735 × 10−3 1·3289 × 106

Beam (W10 × 77) 1·4581 × 10−2 1·8939 × 10−4 3·9665 × 105

Figure 15. A three-storey steel frame with moving nodes (MN) for tracking formation of intermediate plastic

hinges in beams





DOI: 10.1002/tal

besides the beam ends and column ends at ground level, intermediate beam plastic hinges can also be

formed at various locations at certain instants. To evaluate the dynamic response of frames accurately,

it is important that such phenomena are taken into account. The time histories of the plastic rotationsat selected plastic hinges are further shown in Figure 19. The proposed method is also capable of

efficiently analysing complicated multi-storey frames commonly encountered in practice.

11. CONCLUSIONS

The paper describes a method for nonlinear dynamic analysis of frames carrying distributed

loads, which is based on the semi-rigid technique and moving node strategy. The method uses

Figure 16. Horizontal floor displacements of three-storey frame under magnified El Centro earthquake withmulti-parameter yield criterion

Figure 17. Variation of plasticity factor of active intermediate beam plastic hinge at moving nodes MN1 andMN3 of three-storey frame under magnified El Centro earthquake





pseudo-semi-rigid connections to simulate the plastic hinges that may form in a frame. The moving

node strategy is applied in the dynamic analysis to track the formation of intermediate plastic hinges

accurately. Stiffness degradation is also included in the plastic hinge model to simulate its deteriora-tion. With the increase of distributed loads, the intermediate beam plastic hinge may shift from one

location to another. Once the frame starts yielding, the lower the residual plasticity factor is, the weaker

the frame becomes. The apparent frequency of the frame after yielding normally drops. The rate of

reduction of plasticity factor also affects the formation of intermediate beam plastic hinge. The

numerical examples presented demonstrate that the method is efficient in dealing with multi-storey

frames that may have intermediate plastic hinges.

Figure 18. Locations of plastic hinges in three-storey frame under magnified El Centro earthquake with time

shown in brackets (PH, plastic hinge; •, end plastic hinge; , intermediate plastic hinge)

Figure 19. Plastic rotations at selected plastic hinges of three-storey frame under magnified El Centroearthquake with multi-parameter yield criterion





DOI: 10.1002/tal

REFERENCES

Au FTK, Yan ZH. 2008. Dynamic analysis of frames with material and geometric nonlinearities based on semi-rigid technique. International Journal of Structural Stability and Dynamics 8(3): 415–438.

Awkar JC, Lui EM. 1999. Seismic analysis and response of multistory semirigid frames. Engineering Structures 21: 425–441.

Azevedo J, Calado L. 1994. Hysteretic behaviour of steel members: Analytical models and experimental tests. Journal of Constructional Steel Research 29: 71–94.

Ballio G, Castiglioni CA. 1994. Seismic behaviour of steel sections. Journal of Constructional Steel Research 29: 21–54.

Chan SL, Chui PPT. 2000. Non-Linear Static and Cyclic Analysis of Steel Frames with Semi-Rigid Connections.Elsevier: Kidlington.

Chen WF, Chan SL. 1995. Second-order inelastic analysis of steel frames using element with midspan and endsprings. Journal of Structural Engineering ASCE 121(3): 530–541.

Duan L, Chen WF. 1990. A yield surface equation for doubly symmetrical sections. Engineering Structures 12(2):114–119.

Elnashai AS. 2002. Do we really need inelastic dynamic analysis? Journal of Earthquake Engineering 6(Specialissue 1): 123–130.

Gong Y. 2006. Adaptive gradual plastic hinge model for nonlinear analysis of steel frameworks. Canadian Journal of Civil Engineering 33(9): 1125–1139.

Gong Y, Xu L, Grierson DE. 2005. Performance-based design sensitivity analysis of steel moment framesunder earthquake loading. International Journal for Numerical Method in Engineering 63: 1229–1249.

Gong Y, Xu L, Grierson DE. 2006. Sensitivity analysis of steel moment frames accounting for geometric andmaterial nonlinearity. Computers and Structures 84: 462–475.

Grierson DE, Gong Y, Xu L. 2006. Optimal performance-based seismic design using modal pushover analysis. Journal of Earthquake Engineering 10(1): 73–96.

Hall JF, Challa VRM. 1995. Beam-column modelling. Journal of Engineering Mechanics ASCE 121(12): 1284–1291.

Hasan R, Xu L, Grierson DE. 2002. Push-over analysis for performance-based seismic design. Computers and Structures 80: 2483–2493.

Kim SE, Lee JS, Choi SH, Kim CS. 2004. Practical second-order inelastic analysis for steel frames subjected todistributed load. Engineering Structures 26: 51–61.

Kim SE, Choi SH. 2005. Practical second-order inelastic analysis for three-dimensional steel frames subjected

to distributed load. Thin-Wall Structures 43: 125–160.Krawinkler H, Zohrei M. 1983. Cumulative damage in steel structures subjected to earthquake ground motions.Computers and Structures 16(1–4): 531–541.

Lui EM, Lopes A. 1997. Dynamic analysis and response of semirigid frames. Engineering Structures 19(8):644–654.

Monforton GR, Wu TS. 1963. Matrix analysis of semi-rigidly connected frames. Journal of the Structural Divi-

sion ASCE 89(6): 13–42.Park YJ, Ang AHS. 1985. Mechanistic seismic damage model for reinforced concrete. Journal of Structural

Engineering, ASCE 111(4): 722–739.Rao PS, Sarma BS, Lakshmanan N, Stangenberg F. 1998. Damage model for reinforced concrete elements under

cyclic loading. ACI Material Journal 95(6): 682–690.Shin S, Oh SH. 2007. Damage assessment of shear buildings by synchronous estimation of stiffness and damping

using measured acceleration. Smart Structures and Systems 3(3): 245–261.Skrinar M, Plibersek T. 2007. New finite element for transversely cracked slender beams subjected to transverse

loads. Computational Materials Science 39: 250–260.Wong KKF. 2008. Seismic energy dissipation of inelastic structures with tuned mass dampers. Journal of Engi-

neering Mechanics ASCE 134(2):163–172.Wong KKF, Zhao D. 2007. Uncoupling of potential energy in nonlinear seismic analysis of framed structures.

Journal of Engineering Mechanics ASCE 133(10):1061–1071.Wong MB. 1996. Effects of linearly varying distributed load on the collapse behaviour of frames. Computers

and Structures 61(5): 909–914.Xu L. 1992. Geometrical stiffness and sensitivity matrices for optimization of semi-rigid steel frameworks.

Structural Optimization 5: 95–99.





Xu L, Liu Y, Grierson DE. 2005. Nonlinear analysis of steel frameworks through direct modification of memberstiffness properties. Advances in Engineering Software 36: 312–324.

Yang YB, Kuo SR. 1994. Theory and Analysis of Nonlinear Framed Structures. Prentice-Hall: Englewood Cliffs,NJ.

Yoshida I. 2001. Damage detection using Monte Carlo filter based on non-Gaussian noise. Proceedings of Non-

linear System Identification and Structural Health Monitoring, DPRI, Kyoto University, 281–292.Zhao DF, Wong KKF. 2006. New approach for seismic nonlinear analysis of inelastic framed structures. Journal

of Engineering Mechanics ASCE 132(9): 959–966.Zhang X, Wong KKF, Wang Y. 2007. Performance assessment of moment resisting frames during earthquakes

based on the force analogy method. Engineering Structures 29(10): 2792–2802.

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Nonlinear Dynamic Analysis of Frames With Plastic Hinges

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