adaptive space frame analysis : part a plastic hinge approach - Spiral
Transcript of 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).
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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"
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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)
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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)
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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)
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
v i (x)
M2y
M1y
F
v (x)
y
21
x
F
Initial imperfection
M1z
M2z
F
w (x)
w i (x)x
z
21
F
Initial imperfection
Figure 9. Variables for plasticity check within the element length
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
Currentconfiguration
Initialconfiguration
Subdivisionpoint
Before subdivision After subdivision
Two newelements
Newnodex d
Figure 10. Subdivision of a plastic hinge quartic element
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
(a) Plastic hinge (b) Distributed plasticity
Yieldedregions
Figure 13. Deformation shapes for beam-column
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
Modelling
Elements Quartic
Initial
Final
28
32
Plastic hinges
Figure 18. Modelling of jacket structure using the plastic hinge approach
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
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 super- imposed mass
Figure 19. Geometric configuration of 3D jacket structure
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
-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
Izzuddin & Elnashai: Adaptive Space Frame Analysis, Part I
-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