adaptive space frame analysis : part a plastic hinge approach - spiral

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AADDAAPPTTIIVVEE SSPPAACCEE FFRRAAMMEE AANNAALLYYSSIISS:: PPAARRTT II,,

AA PPLLAASSTTIICC HHIINNGGEE AAPPPPRROOAACCHH

B.A. IZZUDDIN‡ AND A.S. ELNASHAI#

1. ABSTRACT

This is one of two companion papers presenting new procedures for the efficient large-

displacement analysis of steel frames in the elasto-plastic range. Emphasis is given in this paper to

the development and improvement of a plastic hinge approach utilizing the concept of adaptive

mesh refinement. In the companion paper, such a concept is discussed in the context of a more

accurate approach accounting for the spread of plasticity. The proposed plastic hinge approach is

formulated through the extension of an earlier 3D elastic quartic element into the inelastic domain,

where a general surface is suggested for representing plastic interaction between the axial force

and the biaxial moments. The numerical problems associated with the formation of adjacent plastic

hinges as well as the case of pure axial plasticity are highlighted, and methods for dealing with

such problems are discussed. The efficiency of the proposed approach derives partly from the

ability of the quartic formulation to represent beam-columns using only one element per member,

but more significantly from the utilization of adaptive mesh refinement. The latter consideration is

shown to have particular advantages in elasto-plastic analysis of braced structures. The

methodology presented in this paper and implemented in the nonlinear analysis program

ADAPTIC, is verified in terms of robustness, accuracy and efficiency using a number of examples

including geometric as well as material nonlinearity effects.

2. INTRODUCTION

Whilst the underlying methods for elasto-plastic analysis of framed structures, namely the plastic

hinge and the distributed plasticity idealizations, have been extensively used in the past, the present

‡ Lecturer in Computing, Civil Engineering Dept., Imperial College, London, UK. # Reader in Earthquake Engineering, Civil Engineering Dept., Imperial College, London, UK.

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papers (Part I: A Plastic Hinge Approach, and Part II: A Distributed Plasticity Approach) give a

rigorous and unified treatment in the context of the new concept of adaptive analysis. This is

proven, by the forwarded examples, to provide the desired levels of accuracy and economy that

render the use in design office practice feasible.

Previous work by the writers1 presented a new method for modelling geometric nonlinearities in the

analysis of three-dimensional framed structures, including those due to very large displacements

and beam-column effects. The two present papers describe the extension of that approach to model

material nonlinearity effects in steel frames, with particular emphasis placed on computational

efficiency.

In the first of these papers (Part I), adaptive analysis of steel frames based on the plastic hinge

approach is described. The paper first discusses the required extensions to the earlier elastic quartic

formulation2, and highlights potential numerical problems as well as methods for overcoming such

problems. Adaptive mesh refinement is then presented in the context of plastic hinge analysis, and

the significant computational and modelling advantages of such a process are pointed out. Finally,

verification examples using the nonlinear analysis program ADAPTIC are undertaken, and

comparisons are made where possible with other solutions to demonstrate the accuracy and extreme

efficiency of the proposed method.

3. THE PROPOSED PLASTIC HINGE APPROACH

In the elasto-plastic analysis of steel frames, two main approaches have been widely adopted; the

first employing lumped plastic hinge idealization3,4,5,6,7, and the second based on distributed

plasticity modelling8,9,10,11,12. Although the plastic hinge approach provides only an approximate

representation of steel frame behaviour, with its accuracy reducing as the spread of plasticity within

the section and along the member becomes important, it has a significant computational advantage

over the distributed plasticity approach.

In view of the considerable computational advantages of the plastic hinge approach, an earlier

elastic quartic formulation2 has been extended to include plastic hinges at the element ends.

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Contrary to the formulation proposed by Ueda et al6, the present formulation can be applied within

a general incremental-iterative procedure, and models the buckling behaviour through the inclusion

of geometric nonlinearities within the quartic formulation rather than the modification of the plastic

hinge interaction surface.

Although the inclusion of strain-hardening effects in the plastic hinges was contemplated, it was

eventually decided to ignore such effects for reasons reflected clearly in the previously referenced

works. Firstly, there is no guarantee of an improvement in accuracy commensurate to the significant

additional complexity in formulating plastic hinge behaviour with strain-hardening effects.

Secondly, the accuracy of a plastic hinge formulation is already questionable for cases where (i) the

spread of plasticity within the section depth is important, (ii) the spread of plasticity along the

member length is significant, and (iii) the material exhibits a response which cannot be represented

accurately by a bilinear curve, characteristic of high-strength steel or mild steel subjected to high

levels of cycling. Consequently, the plastic hinge formulation presented herein is based on elastic-

perfectly plastic modelling, and is therefore only intended for approximate yet efficient elasto-

plastic analysis of steel frames. Accurate modelling, including the effects of spread of plasticity,

strain-hardening, and general stress-strain relationships, is deferred to the companion paper (Part

II). The remainder of this paper is henceforth devoted to the discussion of the adaptive analysis

based on the plastic hinge approach.

4. PLASTIC HINGE QUARTIC FORMULATION

The new plastic hinge formulation is derived in a convected (Eulerian) system, in which the

element local displacements are always referred to the element chord in its deflected state. The

plastic hinge formulation is based on an elastic quartic formulation2 which has eight local degrees

of freedom, as shown in Figure 1, and which is capable of modelling elastic beam-columns using

only one element per member. Rigid-perfectly plastic hinges are added to the elastic quartic

formulation to provide a simple yet effective method for analysis involving material plasticity. The

resulting formulation is intended for preliminary investigations, since the effects of spread of

plasticity and strain-hardening are not accounted for.

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The element forces and stiffness are considered in a local convected system, in line with the

derivation of the elastic element. The local element displacements and forces, after static

condensation of the two midside freedoms, are hence represented by the following vectors,

cu

1 y,

1 z,

2 y,

2 z, ,

T

T

cf M

1 y, M

1 z, M

2 y, M

2 z, F, M

T

T

(1)

The effects of geometric nonlinearity in the plastic hinge formulation are included in the same

manner as discussed by the writers1 for elastic formulations, where it was pointed out that the use

of element-based orientation vectors, as opposed to nodal triad vectors, permits the modelling of

large local displacements, which is an essential requirement for plastic hinge analysis.

4.1. Plastic hinge properties

Hinges of the rigid-plastic type are added at the two ends of the element, as shown in Figure 2. It is

assumed that the contribution of shear stresses to plasticity is negligible, consequently the effects of

the shear forces and the torsional moment on plastic behaviour are ignored. The formation of a

plastic hinge is hence governed by the interaction of the two principal moments and the axial force:

p1 (M

1 y, M

1 z, F)

1 hinge (1) plastic

p2 (M

2 y, M

2 z, F)

1 hinge (2 ) plastic

(2)

Plastic displacement increments are allowed at the plastic hinges, and are assumed to obey the

associated flow rule:

c

pu

1 y

p,

1 z

p,

2 y

p,

2 z

p,

p,

T

pT

c

pu

j N

j, hb

hh

NT

p1

M1 y

p1

M1 z

0 0p

1

F0

0 0p

2

M2 y

p2

M2 z

p2

F0

(3)

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N represents the components of the normals to the interaction surface, while b contains positive

scalars for the two hinges. Also, the summation range variable "h" indicates only the hinges which

are plastic; that is:

Only hinge (1) plastic h 1

Only hinge (2 ) plastic h 2

Both hinges plastic h 1, 2

(4)

A new representation has been developed for the plastic interaction surfaces of general symmetric

sections, which is based on the use of polynomial fitting to a selected number of interaction points.

Three curves determine the interaction surface, as shown in Figure 3:

Myp' f

1(F): reduced y axis plastic moment due to axial force

Mzp' f

2(F): reduced z axis plastic moment due to axial force

f3

My

Myp,

Mz

Mzp

0: biaxial moment interaction at zero axial force

(5)

It is assumed that the biaxial moment interaction in the presence of the axial force is identical to

that at zero axial force, but with reduced plastic moments. Hence, the equation of the interaction

surface can be expressed as:

(M y, Mz, F) Mzp

'

Mzp

f

3

M y

M yp'

,Mz

M zp'

1 1

(6)

Each of the interaction functions "f1" and "f2" is composed of three polynomial functions

established over three adjacent intervals, as depicted in Figure 4. Conditions of continuity of values

and slopes at the two intermediate interaction points are used to establish the constants of the

polynomial functions, with the slopes chosen to satisfy the curve convexity. On the other hand,

function "f3" defines the non-dimensional biaxial moment interaction, and is assumed to have the

form:

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f3

My

Myp'

,Mz

Mzp'

m c

b

m My

Myp'

2

Mz

Mzp'

2

& cb

M z

M zp'

My

Myp'

2

M z

M zp'

2

(7)

where "" is a function defined by three polynomials established over three adjacent intervals of

bending direction, as shown in Figure 4. A constant function "" (c

b) 1 corresponds to a

circular interaction curve between the biaxial moments.

4.2. Local forces

Since plasticity is lumped at the element ends, the local forces cf can be directly obtained from the

elastic local displacements ceu , but an incremental approach is necessary due to the path-

dependence of the problem. To ensure that the local forces remain within the boundaries of the

interaction surface, the plastic hinges undergo incremental plastic deformation c

pu so that only

part of the displacements increment cu is elastic; that is:

ceu cu c

pu (8)

Thus, for an increment of displacements cu , cf can be obtained using the elastic element

properties once c

pu is determined. If both hinges are rigid at the start of the current increment,

c

pu is taken as zero. If at least one hinge is plastic at the start of the current increment, c

pu is

determined in accordance with section 4.2.1.

4.2.1. Increment of plastic deformation

The calculation of the plastic deformation must ensure that the forces at the plastic hinges do not

exceed the interaction surface. This condition can be expressed infinitesimally using the following

equation:

Ni , g

cf ii 1

6

0 (9)

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where N is defined in eq. (3), and "g" is identical to "h" in eq. (4):

Only hinge(1) plastic g 1

Only hinge(2) plastic g 2

Both hinges plastic g 1, 2

(10)

Also, cf can be expressed infinitesimally as a function ceu :

cf i c

ek

i , j c

eu

jj 1

6

(11)

where cek

is the elastic local tangent stiffness of the element.

Hence, the combination of the flow rule in eq. (3) with eqs. (8), (9) and (11), results in the

following system of equations with the scaling factors b as unknowns:

Dg, h

bh

h

Ni , g c

ek

i , j cu j

j 1

6

i 1

6

(12.a)

where,

Dg, h

Ni , g c

ek

i , jN

j , hj 1

6

i 1

6

(12.b)

This represents one or two simultaneous equations, depending on the number of plastic hinges

which is reflected in the range variables "g" and "h". The solution to eq. (12.a) yields an estimate of

the scaling factors which can be expressed as:

bh D

h , g

1N

i , g cek

i , j cu j

j 1

6

i 1

6

g

(13)

in which D 1

is the inverse of the 1x1 or 2x2 part of the D matrix associated with plasticity.

If a scaling factor corresponding to a plastic hinge is negative, elastic unloading occurs. In this case,

the hinge is assumed rigid, and the scaling factor of the other plastic hinge, if present, is re-

calculated from eq. (13) after re-establishing the range variables "g" and "h".

Once b is established, c

pu is obtained from eq. (3). Hence,

ceu

can be determined from eq. (8),

and cf can be calculated using the elastic element properties. However, since eqs. (9) and (11)

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apply only for infinitesimal increments, it is often necessary to correct the stress states at the plastic

hinges back to the interaction surface. This is performed by improving on the initial estimate of b

using an iterative procedure which accounts for the deviation p of the stress states from the

interaction surface, and which can be shown to have the form2:

bh b

h D

h , g

1p g 1

g

(14)

where again D 1

is the inverse of the 1x1 or 2x2 part of the D matrix.

4.2.2. Scaling to the interaction surface

As previously mentioned, hinges which are rigid at the start of an incremental step are not allowed

to exert plastic deformation. It is therefore possible that stress states of rigid hinges exceed the

interaction surface after the application of an increment of displacements cu . To remedy this

violation of hinge strength, cu is scaled down by a reduction factor 'r' until convergence to the

interaction surface is achieved.

Because of geometric and material nonlinearities within the element formulation, the relationship

between the interaction values p of rigid hinges and the reduction factor 'r' is nonlinear. Therefore,

the scaling procedure must be iterative, and proper allowance must be made for the case when both

element hinges are rigid and exceeding the interaction surface simultaneously.

In this work, an iterative procedure based on quadratic interpolation is employed, as demonstrated

in Figure 5 for hinge (1). For each iterative estimate of 'r', the local forces cf corresponding to

" r cu " are calculated in accordance with section 4.2.1, and are employed in the interaction

equation to obtain p. Convergence to the interaction surface is assumed when the values of p lie

within the interval 1, 1 10 6 .

Once convergence is achieved, the corresponding hinge is taken as plastic, before the rest of the

increment " 1 r cu "

is applied.

4.2.3. Sub-incrementation

The calculation of c

pu

according to section 4.2.1 is performed using the matrix of normals N at the

start of the incremental step. To allow for the continuous change in the normals due to the

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interaction surface curvature, a process of sub-incrementation is employed, with the direction of

normals assumed constant within a sub-increment.

In this work, cu is initially applied in one step, and the number of sub-increments is then

determined according to the relative position of the non-dimensional stress states of the plastic

hinges, as well as the relative orientation of the non-dimensional normals at the start and end of the

step. The mathematical expression for the number of sub-increments "n" is given in Appendix A.1.

4.2.4. Pure axial plasticity

Because the normal to the interaction surface is not uniquely defined at the point corresponding to

the full axial capacity (±Fp), numerical difficulties arise if a stress state of a plastic hinge crosses

this point. To avoid this problem, the interaction surface is assumed to extend smoothly beyond

(±Fp), and stress states are allowed to continue on the extended branch, as demonstrated in Figure 6

for bending in the x-y plane. Since this implies a violation of the hinge strength requirement, an

iterative scaling procedure, similar to that discussed in section 4.2.2, is employed to establish the

reduction factor 'r' needed to bring the stress states back to the point of full plastic axial capacity

(±Fp).

Once at the point (±Fp), a further increment of displacements cu will not cause any change in the

stress states if the components of plastic deformation lie within the boundary normals. This is

demonstrated in Figure 7 for stress states at (Fp), and assuming positive increment for the plastic

hinge rotations in the x-y plane:

If : p

N5, 1

N1, 1

1 y

p

N5, 2

N3, 2

2 y

p

then : No change in stress states

(15)

When biaxial hinge rotations are involved, a simple mathematical representation becomes more

difficult, since the boundary normals are now represented by a conical surface instead of two

vectors. However, if the boundary normal with components proportional to the hinge rotational

increments is established, the check for the change of stress states can be readily made. It is shown

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in Appendix A.2 that for stress states at (Fp) the condition of no change in stress states can be

expressed in terms of the increment of displacements cu as:

N5, 1

1 y

N1, 1

1 z

N2, 1

N

5, 2

2 y

N3, 2

2 z

N4, 2

c

pu

1 y,

1 z,

2 y,

2 z, , 0

T

ceu 0 , 0, 0 , 0, 0 ,

T

T

cf 0, 0, 0, 0, F p,GJL

T

T

(16)

where N is determined in Appendix A.2 for positive increments of rotations. Similar expressions

can be derived for different combinations of positive and negative increments of rotations, and for

the case of plasticity at (–Fp).

If the condition of eq. (16) is not satisfied for an increment cu , then the stress states at the plastic

hinges either undergo elastic unloading or follow a loading path on the interaction surface. In the

latter case, difficulties arise because the normals are not uniquely defined at (Fp), hence, c

pu

cannot be estimated. To avoid this problem, the element is partially unloaded from the condition of

axial plasticity before applying cu . Upon reloading, the scaling to the interaction surface brings

the stress states at the plastic hinges to points different from (Fp), and c

pu can then be determined

as usual.

4.3. Local tangent stiffness

The local tangent stiffness matrix ck must reflect the state of hinges at the element ends, whether

rigid or plastic. If both hinges are rigid, then ck is taken as equal to the elastic element local

tangent stiffness cek . If at least one hinge is plastic, then ck can be expressed as follows2:

ck i , j c

ek

i , kI

k, j N

k, hD

h , g

1N m, g c

ek

m, jm 1

6

h

g

k 1

6

(17)

where I is a 6x6 identity matrix.

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For the special case of axial plasticity at (±Fp), the condition of no change in the hinges stress states

is assumed, hence, the local tangent stiffness is taken as:

Full axial plasticity at Fp

ck i , j 0 for all (i, j) except (6, 6)

ck 6, 6

GJL

(18)

4.4. Global analysis

As previously pointed out, global structural analysis including geometric nonlinearity effects is

performed through the use of a new procedure developed by the writers1,2. The new procedure

involves three transformations on the element level between the local convected system and the

global reference system; namely, (i) a transformation from increment of global displacements to

increment of local displacements, (ii) a transformation from local forces to global forces, and (iii) a

transformation from local tangent stiffness to global tangent stiffness.

The first transformation is utilized to obtain the element local displacements corresponding to the

current increment of structural global displacements. Once the local displacements are known, the

local element forces are determined according to section 4.2. The global element forces are

obtained from the local forces by applying the second transformation, and are assembled in a vector

representing the overall structural resistance. The global structural tangent stiffness, required for the

iterative solution procedure, is assembled from global element contributions, each obtained by

applying the third transformation to the local element tangent stiffness determined in accordance

with section 4.3.

5. ADAPTIVE MESH REFINEMENT

Most plastic hinge formulations are based on the assumption that plastic hinges occur only at the

element ends. This implies that one plastic hinge element can model a whole uniform structural

member in the elasto-plastic range, provided (i) the member is not loaded within its length, and (ii)

plasticity is mainly due to bending action. For braced structures, plastic hinges may occur within the

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lengths of a number of braces due to elasto-plastic buckling, and hence two plastic hinge elements

would be required for an adequate representation. In the context of conventional analysis, each

brace must be modelled using two plastic hinge elements, since the braces which undergo buckling

are not known a priori. Apart from the excessive computational requirements of such modelling,

since usually only a relatively small number of braces buckle during loading, the structural

idealization is complicated by the fact that the location of a plastic hinge within the brace length is

also not known a priori. The latter consideration is usually dealt with through the simplifying, but

potentially inaccurate, assumption that the plastic hinge occurs at the middle of the brace.

Ueda et al6 addressed the inefficiency of conventional methods by suggesting that analysis should

be started with one element per member, and automatic subdivision of an original element into two

equal-length elements is performed if a plastic hinge is detected at mid-length.

The present work adopts the suggestion of Ueda et al6, and further extends it to address the

inaccuracies associated with a plastic hinge occurring within the element length but not necessarily

at mid-length. Essentially, adaptive mesh refinement utilizes the accuracy of the quartic formulation

in the elastic range, and starts the analysis using only one element per member. In the course of

analysis, each element, already modelling plasticity effects at the ends, is checked for plasticity

anywhere within its length. If a plastic hinge is detected, the element is automatically subdivided

into two elements, after which the analysis is continued with a finer mesh. Consequently, the

suggested process of adaptive mesh refinement provides significant computational savings, and

deals with the uncertainty of plastic hinge location, as discussed in more detail in the following

sections.

5.1. Plasticity check

The check for plasticity within the element length is performed at each load step after global

equilibrium has been achieved. To establish the stress state within the element length, the

calculation of the biaxial bending moments must allow for the effect of the axial force in the

presence of transverse displacements. This effect can readily be accounted for if the convected

system is employed (Figure 9):

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My(x)

M1y

M2y

2

M1y

M2y

L

x F v(x) v i (x)

M z(x) M

1z M

2z

2

M1z M

2z

L

x F w(x) w i (x)

(19)

in which v(x) , w(x) , v i (x) and w i (x) are the transverse displacements and imperfections,

respectively.

To determine the plasticity condition at a section, the plastic interaction formula is used, expressed

as:

My (x), Mz(x), F 1 section at (x) is plastic (20)

where "" is the interaction formula given by (6).

The abscissa 'xd' along the element length with the highest interaction value "" is first established.

For the 2D formulation, this can be performed analytically, since the maximum value of ""

corresponds to the maximum value of bending moment which is a polynomial function of 'x'

according to equ. (19). For the 3D formulation, biaxial bending renders an analytical solution very

cumbersome. Therefore, a selected number of points along the element length are considered, with

'xd' chosen as the abscissa having the highest "".

If the interaction value "" corresponding to 'xd' satisfies the plasticity condition of equ. (34),

element sub-division is performed in accordance with section 5.2.

If none of the elements requires sub-division for the current load step, the solution proceeds to the

next step. Otherwise, the current load step is re-applied, so that global equilibrium corresponding to

the new mesh is established.

Plastic hinge elements which are the result of an earlier sub-division process are not allowed to

further sub-divide in the current load step, since the existence of more than one plastic hinge within

the member length leads to considerable numerical difficulties. Thus, the spread of plasticity within

the member length is neglected, and the buckling process is represented by two plastic hinge

elements only, where the location of the intermediate hinge is determined by the first occurrence of

plasticity.

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5.2. Element sub-division

The process of sub-division of a plastic hinge quartic element involves the addition of a new node

and two new quartic elements, as shown in Figure 10.

The only variables associated with the new node are global displacements. These are determined for

the last equilibrium configuration from the deflected shape of the original element and the global

displacements of its end nodes.

For each of the new elements, variables pertaining to the initial and last equilibrium configurations

must be established. These include initial direction cosines, initial imperfections, orientation of the

principal axes at both ends, local displacements, plastic hinge deformations, and local forces. The

determination of local displacements, hence local forces, must allow for the nonlinear distribution

of the axial displacement along the length of the original element, which is due to the nonlinear

effect of bending deformation on axial stretching. This proves to be an important factor for

convergence to be achieved when the current load step is re-applied.

6. VERIFICATION EXAMPLES

The methodology presented in this paper has been implemented in ADAPTIC13, a general purpose

computer program for the nonlinear static and dynamic analysis of space frames. In this section,

four examples are presented to demonstrate the accuracy and efficiency of the proposed plastic

hinge formulation and its use in the context of automatic mesh refinement. All reported CPU times

are for ADAPTIC v2.1.2, running on a Silicon Graphics workstation with 24 Mb of physical

memory and rated at 30 mips, 4.2 mflops and 26 Specmarks.

6.1. Elasto-plastic buckling of beam-column

The beam-column shown in Figure 11 is subjected to an eccentric axial force, and is analysed using

the plastic hinge formulation with automatic mesh refinement. This example is intended to

demonstrate the importance of allowing the plastic hinge induced by buckling to occur at locations

other than the element mid-length. For that purpose, the problem was analysed using the previous

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approach6 and the one suggested in this paper, and comparisons are made with the distributed

plasticity approach prsented in the compnion paper14.

The results depicted in Figure 12 indicate that the previous approach fails to detect a plastic hinge

within the element length, since the generalized stress-state does not exceed the interaction surface

at mid-length, hence no buckling behaviour is exhibited. On the other hand, using the approach

proposed in this paper, a plastic hinge is detected in the leftmost quarter of the beam-column, and

favourable comparison is demonstrated with the distributed plasticity approach. It is worth-noting,

however, that there is a case for assuming the plastic hinge to occur closer to midspan. This is

supported by Figure 13, where it is shown that the yielding region predicted by the distributed

plasticity approach migrates towards the mid-length as more deformation is accommodated. The

most accurate location for the plastic hinge lies in a region between the initial point of plasticity and

midspan; however, since this depends on several factors, a separate study would be required.

6.2. Four-storey frame

The frame shown in Figure 14 is subjected to the static action of vertical and sway forces, which are

increased proportionally up to plastic collapse. Three cases of sway to vertical load ratios (r=0.1,

0.24 & 0.5) were considered by Kassimali15, who employed a 2D plastic hinge formulation

neglecting plastic axial displacements, and assuming a bilinear interaction curve independent of the

section shape. The frame was later analysed by Kam16, who accounted for the spread of plasticity

across the section depth and along the member length.

The results given by ADAPTIC are based on the plastic hinge quartic formulation, where

favourable comparison is demonstrated in Figure 15 with the predictions of Kassimali for the three

load cases. The slight disagreement in the region of ultimate capacity is mainly attributed to the

difference in the interaction surface used, since that of Kassimali does not allow for any reduction

in the plastic moment capacity until the axial force exceeds 15% of the plastic axial capacity.

6.3. Elasto-plastic buckling of jacket

A 3D tubular jacket structure, with parabolic imperfections of (L/500) in three of its compression

members, is loaded asymmetrically as shown in Figure 16. The structure is loaded beyond its

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ultimate capacity, and the pre- and post-ultimate response is obtained using two approaches;

namely, the plastic hinge approach with automatic mesh refinement and the distributed plasticity

approach with an initially refined mesh14.

The load-deflection curves of Figure 17 demonstrate good agreement between the two adopted

approaches up to the point of ultimate capacity. In the post-ultimate range, the slight disagreement

is mainly due to the inability of the plastic hinge approach to account for the spread of plasticity to

the mid-length of the top buckled brace.

With the plastic hinge approach, the analysis is started using 28 quartic elements, and automatic

subdivision of members into two quartic elements is performed when a plastic hinge is detected

within the member length. At the end of analysis, 32 quartic elements are employed, as shown in

Figure 18. With the distributed plasticity approach, the analysis is started with a refined mesh for all

members of the structure, since the locations of plasticity are not known a priori. This consisted of

using 280 elasto-plastic cubic elements which account for the spread of plasticity within the section

depth and along the element length.

In this example, the plastic hinge approach requires only 9.5% of the CPU time needed by the

distributed plasticity approach (1min 56sec for plastic hinge, 20min 16sec for distributed plasticity),

which demonstrates the efficiency advantage of the plastic hinge approach. However, as discussed

in the companion paper14, the efficiency of the distributed plasticity approach can be significantly

improved, thus allowing the efficient and accurate elasto-plastic analysis of structures where the

spread of plasticity and the use of general stress-strain relationships are deemed important.

6.4. 3D jacket under earthquake loading

The 3D tubular jacket structure, depicted in Figure 19, is subjected at its supports to the transient

ground signal of Figure 20. The jacket supports a platform which is modelled as a superimposed

mass of 1000 tons, and is analysed using the plastic hinge approach with automatic mesh

refinement. In the dynamic analysis, the jacket mass has been included using distributed mass

elements2, and the weight of the platform has been applied as an initial static load. However, the

weight of the jacket has been ignored, although its inclusion in the analysis is straightforward.

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The analysis is started using one quartic plastic hinge element per member, facilitated by the

accuracy of the quartic formulation and its ability to model imperfect members with only one

element. During analysis, automatic mesh refinement is performed for six of the original elements,

which are subdivided into two elements each. The drift time-history for the jacket at the deck level

is shown in Figure 21, where a maximum lateral drift of 13.3 cm is predicted. Automatic

subdivision is performed first at t = 1.69 sec and last at t = 2.24 sec, with the deflected shapes and

subdivided members shown in Figure 22. With a total CPU time of 1hr 17min 43sec, this example

demonstrates the feasibility of nonlinear dynamic analysis of realistic structures using the proposed

plastic hinge approach combined with automatic mesh refinement.

7. CONCLUSIONS

This paper presented a treatment of adaptive space frame analysis based on the plastic hinge

approach. It was recognized that the efficiency of the plastic hinge approach can best be utilized

within an incremental-iterative solution procedure, since a purely incremental procedure would only

allow small load/time-steps. In that context, a new plastic hinge formulation based on an elastic

quartic element was proposed, and details of the interaction surface as well as the secant and

tangent stiffness were discussed. The numerical problems associated with adjacent plastic hinges

and the case of pure axial plasticity were pointed out, and remedial procedures were suggested.

It was also realized that the efficiency of the plastic hinge approach would be much enhanced by

adopting a process of automatic mesh refinement, comprising the subdivision of a plastic hinge

element into two elements if elasto-plastic buckling occurs during analysis. This process was

extended to provide a more realistic and accurate representation of member buckling by allowing

the plastic hinge to occur at any point within the member length. The proposed automatic mesh

refinement process not only provides an efficient solution where element subdivision is only

performed for the buckled members, but also relieves the analyst from having to assume a location

for the plastic hinge within the member length.

A number of examples using the nonlinear analysis program ADAPTIC demonstrated the accuracy

and efficiency of the proposed plastic hinge approach. However, it was noted that the accuracy is

18

only reasonable for cases where the spread of plasticity is not significant, and where the material

stress-strain law is essentially elastic-plastic without strain-hardening. Otherwise, the distributed

plasticity approach must be used, which is outlined in the context of adaptive frame analysis in a

companion paper (Part II: A Distributed Plasticity Approach).

8. ACKNOWLEDGEMENTS

The authors would like to thank Professor Patrick J. Dowling, Head of Civil Engineering

Department, for his continuous technical and moral support of this work. The assistance provided

by the Edmund Davis fund of the University of London is also gratefully acknowledged.

9. REFERENCES

1. Izzuddin, B.A. and Elnashai, A.S. Eulerian Formulation for large displacement analysis of

space frames. J. Eng. Mech. Div., ASCE, 1993, Vol. 119, No. 3, pp. 549-569.

2. Izzuddin, B.A. Nonlinear dynamic analysis of framed structures. Thesis submitted for the

degree of Doctor of Philosophy in the University of London, Department of Civil Engineering,

Imperial College, London, 1991.

3. Wen R.K. and Farhoomand F. Dynamic analysis of inelastic space frames. J. Eng. Mech. Div.,

ASCE, 1970, Vol. 96, No. EM5, pp. 667-686.

4. Inoue K. and Ogawa K. Nonlinear analysis of strain-hardening frames subjected to variable

repeated loading, Technology Report of the Osaka University, 1974, Vol. 24, No. 1222, pp.

763-781.

5. Anagnostopoulos S.A. Inelastic beams for seismic analysis of structure. J. Struct. Div., ASCE,

1981, Vol. 107, No. ST7, pp. 1297-1311.

6. Ueda Y., Rashed S.M.H. and Nakacho K. New efficient and accurate method of nonlinear

analysis of offshore tubular frames (The idealized structural unit method). J. Energy Resources

Technonlogy, ASME, 1985, Vol. 107, pp. 204-211.

19

7. Powell G.H. and Chen P.F.S. 3D beam-column element with generalized plastic hinges. J. Eng.

Mech., ASCE, 1986, Vol. 112, No. 7, pp. 627-641.

8. Hobbs R.E. and Jowharzadeh A.M. An incremental analysis of beam-columns and frames

including finite deformations and bilinear elasticity. Comp. Struct., 1978, Vol. 9, pp. 323-330.

9. Yang T.Y. and Saigal S. A simple element for static and dynamic response of beams with

material and geometric nonlinearities. Int. J. Num. Meth. Eng., 1984, Vol. 20, pp. 851-867.

10. Corradi L. and Poggi C. A refined finite element model for the analysis of elastic-plastic

frames. Int. J. Num. Meth. Eng., 1984, Vol. 20, pp. 2155-2174.

11. Sugimoto H. and Chen W.F. Inelastic post-buckling behavior of tubular members. J. Struct.

Eng., ASCE, 1985, Vol. 111, No. 9, pp. 1965-1978.

12. Meek J.L. and Loganathan S. Geometric and material non-linear behaviour of beam-columns.

Comp. Struct., 1990, Vol. 34, No. 1, pp. 87-100.

13. Izzuddin, B.A. and Elnashai, A.S. ADAPTIC: A Program for the Adaptive Dynamic Analysis

of Space Frames. Report No. ESEE-89/7, 1989, Imperial College, London.

14. Izzuddin, B.A. and Elnashai, A.S.Adaptive space frame analysis: Part II, A distributed

plasticity approach. (companion paper), 1993.

15. Kassimali A. Large deformation analysis of elastic-plastic frames. J. Struct. Eng., ASCE, 1983,

Vol. 109, No. 8, pp. 1869-1886.

16. Kam T.Y. Large deflection analysis of inelastic plane frames. J. Struct. Eng., ASCE, 1988, Vol.

114, No. 1, pp. 184-197.

20

APPENDIX A

A.1 Sub-incrementation

The requirements of section 4.2.3 are represented mathematically by:

n1d

n1a

Integer 100 Dist .1d

o,

1d

c 1

Integer 100 Angle

1a

o,

1a

c

1

If hinge (1) plastic

1

1

If hinge (1) rigid

(21.a)

n2d

n2a

Integer 100 Dist .2d

o,

2d

c 1

Integer 100 Angle

2a

o,

2a

c

1

If hinge (2 ) plastic

1

1

If hinge (2 ) rigid

(21.b)

n Max. n1d , n

1a , n

2d , n

2a (21.c)

where, superscripts (o) and (c) denote start and end of step respectively,

1d

o

M1 y

o

M yp,

M1 z

o

M zp,

Fo

F p

T

1d

c

M1 y

c

M yp,

M1 z

c

M zp,

Fc

F p

T

(22.a)

1a

o Myp N

1, 1

o, M zp N

2, 1

o, F p N

5, 1

oT

1a

c Myp N

1, 1

c, M zp N

2, 1

c, F p N

5, 1

cT

(22.b)

21

2d

o

M2 y

o

M yp,

M2 z

o

M zp,

Fo

F p

T

2d

c

M2 y

c

M yp,

M2 z

c

M zp,

Fc

F p

T

(22.c)

2a

o Myp N

3, 2

o, M zp N

4, 2

o, F p N

5, 2

oT

2a

c Myp N

3, 2

c, M zp N

4, 2

c, F p N

5, 2

cT

(22.d)

and,

Dist .jd

o,

jd

c

j

di

c

jd

i

o

2

i 1

3

Angleja

o,

ja

c cos 1ja

i

c

ja

i

o i 1

3

/

ja

c

ja

o

(23)

A.2 Pure axial plasticity

The suggested representation for the interaction surface in section 4.1 has an advantage in respect of

determining normals to the interaction surface with a specific orientation. It can be shown from eqs.

(5)-(7) that the normals for a stress state at (Fp) are expressed in the positive rotations quadrant as:

N1, 1

f

2' F p

f1' Fp

1 cb 12

M zp

, N3, 2

f

2' Fp

f1' F p

1 cb 22

M zp

N2, 1

c

b 1

Mzp, N

4, 2

cb 2

M zp

N5, 1

c

b 1 f2' F p

M zp, N

5, 2 c

b 2 f2' Fp

M zp

(24)

where,

22

f1' F p : first derivative of f

1(F) at F F p

f2' F p : first derivative of f

2(F) at F F p

cb1

& cb2

: non dimensional bending direction cosines for hinges (1) & (2 )

The direction cosines " c

b1" and

" cb2

" are chosen such that the components of the corresponding

normals are proportional to the increments of hinge rotations at both ends. Hence,

N1, 1

N2, 1

1 y

p

1 z

p

cb 1

f2' F p

f1' F p

f2' F p

f1' F p

2

1 y

p

1 z

p

2

(25.a)

Similarly,

cb 2

f2' F p

f1' F p

f2' F p

f1' F p

2

2 y

p

2 z

p

2

(25.b)

Once " c

b1" and

" cb2

" are determined, N can be established from (24), and a check similar to eq.

(15) can be performed. Thus,

If : p N

5, 1

1 y

p

N1, 1

1 z

p

N2, 1

N

5, 2

2 y

p

N3, 2

2 z

p

N4, 2

then : No change in stress states

(26)

However, since the condition of no change in stress states implies a zero increment of elastic

rotations and axial displacement, this condition can be expressed in terms of the increment of

displacements cu as:

23

N5, 1

1 y

N1, 1

1 z

N2, 1

N

5, 2

2 y

N3, 2

2 z

N4, 2

c

pu

1 y,

1 z,

2 y,

2 z, , 0

T

ceu 0 , 0, 0 , 0, 0 ,

T

T

cf 0, 0, 0, 0, F p,GJL

T

T

(27)

with similar expressions for different combinations of positive and negative increments of rotations,

and for the case of plasticity at (–Fp).

24

NOTATION

- Generic symbols of matrices and vectors are represented by bold font-type with left side

subscripts or superscripts (e.g. sG

, qau

). This rule also applies to three-dimensional matrices.

- Subscripts and superscripts to the right side of the generic symbol indicate the term of the vector

or matrix under consideration (e.g. sG

i , j , k , qau

i ).

Operators

c : right-side superscript, denotes current values during an incremental step.

o : right-side superscript, denotes initial values during an incremental step.

: right-side superscript, transpose sign.

: incremental operator for variables, vectors and matrices.

: partial differentiation.

i

: summation over range variable (i).

: encloses terms of a matrix.

: encloses terms of a row vector.

a : magnitude of vector a .

ai : absolute value of term

ai .

Symbols

b : vector of plastic hinge scalars.

cb : cosine of the angle formed by the vector representing the non-dimensional

bending moments in the biaxial plastic interaction space.

D : 1x1 or 2x2 matrix defined in equation (12.b).

f1 : plastic moment in the local y direction function of axial force.

f2 : plastic moment in the local z direction function of axial force.

f3 : plastic interaction function between the local y and z direction moments.

cf : element basic local forces

M

1 y, M

1 z, M

2 y, M

2 z, F, M

T

F p : plastic axial force capacity.

25

ck : element local tangent stiffness matrix.

cek : element elastic local tangent stiffness

matrix

.

L : element length before deformation.

m : magnitude of the vector representing the non-dimensional bending moments in the

biaxial plastic interaction space.

M y : section moment in the local y direction.

M yp : plastic moment capacity in the local y direction.

M yp'

: reduced plastic moment capacity in the local y direction due to axial force.

M z : section moment in the local z direction.

M zp : plastic moment capacity in the local z direction.

M zp'

: reduced plastic moment capacity in the local z direction due to axial force.

N : matrix of normals to the interaction surface.

p : plastic hinge interaction values at the two ends of quartic element.

r : step reduction factor.

cu : element basic local displacements

1 y,

1 z,

2 y,

2 z, ,

T

ceu : basic elastic local displacements of plastic hinge element.

c

pu : plastic hinge displacements

1y

p,

1z

p,

2y

p,

2z

p,

p,

T

p.

v(x) : centroidal displacement in the local y direction.

v i (x) : imperfection shape in local y direction for quartic element.

w(x) : centroidal displacement in the local z direction.

wi (x) : imperfection shape in local y direction for quartic element.

x : reference abscissa along the element chord.

: plastic interaction surface function of (My , Mz , F)

1 function at end (1)

2 function at end (2).

26

: function of (cb) used for plastic interaction between the biaxial moments.

Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I

t yt yi

2y

i

1y

i

Imperfect configuration

1y

2y

y (0 )y L2

y

L2

x221

L/2 L/2

z (0 )

t zi

1z

t z

Imperfect configuration

1z

i

2z

2z

i x221

L/2 L/2

z L2

z

L2

T

zL2

z (0 )

y (0 )

yL2

y

L2

z L2

T

12

Figure 1. Local freedoms of the elastic quartic formulation


2z

p

2y

e

1z

e1y

e1y

p

2y

p

2z

e

Rigid-plastic hinge

Initial imperfection

Elastic deformed shape

1z

p

y z

x x

1

2 2

1

L e

L e

p

L e

L e

p

Figure 2. Plastic hinge configuration in the convected system


Myp

MzpMz

F p

My

F

Curve(3): f3

My

Myp,

Mz

Mzp

0

Curve(1): Myp' f

1(F)

Curve(2): Mzp' f

2(F)

My , M z, F M zp

'

M zp

f

3

My

Myp'

,Mz

M zp'

1 1

Figure 3. Interaction surface idealization


parabolic

cubic

parabolicf

1(F)

Myp

F p F

parabolic

cubic

parabolicf

2(F)

M zp

F p F

f3(m y , m z) m y

2 m z2

m z

m y2 m z

2

0

cubic

parabolic

cubic

m y

m z1

1

Figure 4. Idealization of interaction functions "f1", "f2" and "f3"


10

1

Iteration (1): Linear

Iteration (2): Quadratic

r(1)

r(2)

Convergence : r(n) Full increment

Reduction factor (r)

p1

Interactionvalue Iteration (3): Quadratic

Figure 5. Iterative scaling to interaction surface of hinge(1)


Fo

Myp

Fc

FF p

My

Myp

Original state at hinge(1)

Original state at hinge(2)

Current state at hinge(2)

Current state at hinge(1)

M1y

o

M1y

c

M2y

o

M2y

c

Figure 6. Extension of the interaction surface at (Fp)


F

Fp

N 5, 1

, N 1, 1

N 5, 2

, N 3, 2

N 5, 1 , N 1, 1

N 5, 2

, N 3, 2

1

p,

1y

p

2

p,

2y

p

p

1

p

2

p

Figure 7. Condition of no change in stress states at (Fp)


P

Very smallrotational stiffness

Rigid Rigid

PlasticPlastic

a. Formation of two adjacent plastic hinges b. Suppression of plastic hinge

P+P

Active hinge

Suppressed hinge

Figure 8. Example on plastic hinge suppression


v i (x)

M2y

M1y

F

v (x)

y

21

x

F


M1z

M2z

F

w (x)

w i (x)x

z

21

F


Figure 9. Variables for plasticity check within the element length


Currentconfiguration

Initialconfiguration

Subdivisionpoint

Before subdivision After subdivision

Two newelements

Newnodex d

Figure 10. Subdivision of a plastic hinge quartic element


L

eP = λ P0

Section: Φ150×7.5 mm2

(chs)

E = 210,000 N/mm2

σy = 300 N/mm2

e = 25 mmL = 3,000 mm

P 0 = 1×106

N

Figure 11. Geometric and loading configuration of beam-column


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.01 0.02 0.03 0.04 0.05

Loa

d fa

ctor

Displacement (m)

Subdivision at any point

Subdivision only at midlength

Distributed plasticity approach

Figure 12. Load-deflection curves for beam-column


(a) Plastic hinge (b) Distributed plasticity

Yieldedregions

Figure 13. Deformation shapes for beam-column


Top storey

Other storeys

P/2 P P/2

rP/2

P/2 P P/2

rP

Lc

Lg

/ 2Lg

/ 2

Columns:

W12x79 (Bottom storey)

W10x60 (Other storeys)

Girders: W16x40

Lc 12 ft

Lg 30 ft

E 13000 tsi

y 15. 25tsi

N: Node

QP: Quartic plastic-

hinge elementN1 N2

N3 N4 N5

N6 N7 N8

N9 N10 N11

N12 N13 N14

QP1 QP2

QP3 QP4

QP5 QP6

QP7 QP8

QP9 QP10

QP11 QP12

QP13 QP14

QP15 QP16

Meshing configuration

Figure 14. Geometry and loading of four-storey frame


1086420

0

5

10

15

20

25

L.F._QP_r0.1

L.F._QP_r0.24

Quartic plastic-hinge elements

L.F._Kas_r0.1

L.F._Kas_r0.24Kassimali (1982)

Horizontal de fle ction at top right joint (in)

Load P

(T

ons)

r=0.1

r=0.24

r=0.5

Figure 15. Load-deflection curves of four-storey frame


P

i2

3m

2.4m

2.4m

Vertical legs:

Tubular 270 6 mm2

y 300 N / mm2

Other members:

Tubular 90 3 mm2

y 350 N / mm2

E 210 103

N / mm2

i1

i2

: (L/500) imperfection in horizontal plane

i1

: (L/500) imperfection in vertical plane

i2

Figure 16. Geometric and loading configurations of jacket structure


0.050.040.030.020.010.00

0

100

200

300

400

500

600

Plastic hinge

Distributed plasticity

Displaceme nt (m)

Loa

d (

KN

)

Figure 17. Load-deflection curve of jacket structure


Modelling

Elements Quartic

Initial

Final

28

32

Plastic hinges

Figure 18. Modelling of jacket structure using the plastic hinge approach


6m

10m

4 @

5m

5m

3 @ 5m

Ground motionSide view Front view

3 @ 3m

Top viewΦ175×10 mm

2

Φ350×20mm2

Φ450×25 mm2

chords

legs

deck

E = 210×103

N/mm2

σy = 300 N/mm2

ρ = 7800 kg/m3

Pinned supports at all legs

Parabolic imperfection of 1cm at midspan for all diagonal braces

1000 tons superimposed mass

Figure 19. Geometric configuration of 3D jacket structure


-1

-0.5

0

0.5

1

0 1 2 3 4 5

Acc

./g

Time (sec)

Figure 20. Transient signal applied to 3D jacket structure


-0.1

-0.05

0

0.05

0.1

0.15

0 1 2 3 4 5

Lat

eral

dri

ft (

m)

Time (sec)

Figure 21. Response of 3D jacket to transient signal


(a) t=1.83 sec(b) t=5.0 sec

Scale = 10

(a)(b)

Figure 22. Deflected shapes of 3D jacket

adaptive space frame analysis : part a plastic hinge approach - spiral

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