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Eigenmode dominance and its application to structures with nonlinear materials
Compurers &Structures Vol.41,No.2,pp.281-291, Printedin Great Britain.
$3.00 + 0.00
co4.s7949/91 Pergamon Press
1991
EIGENMODE DOMINANCE AND ITS APPLICATION STRUCTURES WITH NONLINEAR MATERIALS
plc
TO
J. NAPOLEKO,Fo.,t A. E. ELWI$ and D. W. MURRAY% ?Departamento de Engenharia Civil, Pontificia Universidade Cathlica do Rio de Janeiro, R. Mq. S. Civente, 225, Rio de Janeiro, Brasil SDepartment of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 (Received 17 Augusi 1990) Abstract-Reduction methods have received considerable attention as solution strategies for geometrically nonlinear problems. In this work, the eigenmodes of the tangent stiffness matrix are investigated as a possible basis for a reduction strategy for problems with predominantly material nonlinearities. A number of parameters are developed for measuring the participation of the different eigenvectors in constructing an approximation of the incremental displacement vector. Certain mesh considerations and a series of problems including elastic strain hardening plasticity as well as softening materials are investigated. It is found that the incremental displacement vector is dominated by a few eigenvectors corresponding to the lowest eigenvalues. This domination becomes more evident as damage progresses.
1. INTRODUCTION The formulation of the incremental equilibrium equations for a nonlinear structure, within the framework of the finite element method, has traditionally employed the natural basis for the equilibrium description. The natural basis is formed by unit vectors which are associated with the nodal displacements in the direction of the global coordinate axes. A full set of coupled equilibrium equations, equal in number to the available degrees of freedom, is a direct consequence of this approach. Considerable computational effort is required in the solution of these equations because of the iterative repetition of the steps of factorization, reduction and back substitution necessary due to the material nonlinearities. In an attempt to develop more efficient solution techniques, a change of base vectors may be considered. Change of basis is common in some disciplines of structural analysis. The natural modes of vibration have been commonly adopted as an appropriate basis for the description of the dynamic equilibrium of linearly elastic structures in modal superposition techniques. Recently, Wilson and Bayo [l] have argued that special Ritz load dependent vectors would comprise a more effective basis for dynamic analysis. Almorth et al. [2] and Noor and Peters [3] introduced reduced basis techniques for geometrically nonlinear structures. Almroth et al. selected linear solution vectors, whereas Noor and Peters chose path derivatives to form the reduced basis. Nagy [4] adopted the buckling modes to form a reduced basis to investigate problems with mild changes in geometry. However, the majority of the base vector systems mentioned above have been applied to problems with linearly elastic materials. The intent, herein, is to address problems with nonlinear materials.
Any stable linear elastic structure responds to a given distribution of applied loads (the reference loads) with a unique pattern of displacements. The displacement pattern is generally attained by the solution of a complete set of equilibrium equations, one equation being associated with each global degree of freedom of the assembled structural stiffness matrix, which represents an integrated synthesis of the material and geometric properties of the structure. Consequently, the solution requires considerable computational effort. Once this has been determined, solutions for other load factors may be obtained by proportionality. As the material in the structure becomes nonlinear the incremental displacement pattern changes. The solution is usually obtained iteratively by solving for incremental displacements resulting from incremental loads and then correcting for unbalanced forces, which are computed as the difference between the applied forces and those required to equilibrate the current configuration. The incremental equilibrium equations, which are used to solve for incremental displacements, are formulated in terms of the tangent stiffness matrix which may become singular, but more generally simply ill-conditioned, in the vicinity of limit points, and indefinite in the softening region beyond a limit point. Generally this region is a difficult one to deal with in numerical solutions. This is particularly true in cases where the load-deflection curve is essentially flat over an extended region of deformation, such as in the case for elastic-perfectly plastic steel structures for which the displacement increments at the limit point reflect the amplification of a displacement pattern associated with a collapse mechanism. Singularity of the incremental stiffness matrix in a materially nonlinear structure reflects a condition in which the generalized incremental stiffness coefficient 281
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J. NAPOLE~O,Fo. et al.
associated with the incremental displacement pattern (the incremental collapse mode) becomes zero. Consequently the second-order energy absorption associated with the collapse mode is also zero. Assuming the collapse mode is unique there is only one zero energy displacement pattern. The eigenvectors of the stiffness matrix form an alternative basis which describes all possible deformed configurations of the structure. The eigenvalues represent the generalized stiffnesses associated with the generalized displacement coordinates of the eigenvectors. Hence the (unique) zero second-order energy incremental displacement pattern associated with the collapse mode must be the eigenvector associated with the zero eigenvalue. That is, the eigenvector for the zero eigenvalue should completely dominate the solution for displacement increments in the region of singularity of the tangent stiffness matrix. This implies that a solution for the generalized coordinate associated with the eigenvector of the lowest eigenvalue should capture the essential aspects of the behaviour of a materially nonlinear structure, regardless of its complexity, in the vicinity of its collapse. Furthermore, it can be argued that the eigenvectors associated with the lower energy modes of deformation of the structure can form a subspace within which the essential aspects of behavior of the structure can be captured at all stages of its load-deflection history. These eigenvectors can then be said to ‘dominate’ the solution. Domination of eigenvectors is well known in the dynamic analysis of linearly elastic structures. The fundamental mode of vibration, which usually corresponds to the lowest natural frequency, may dominate the dynamic response of standard buildings subjected to common load dist~butions, such as wind loads [S]. However, domination of eigenvectors for static equilibrium problems involving materially nonlinear structures has not been thoroughly investigated to date. Since a combination of a reduced eigenvector basis and the associated diagonalized incremental stiffness matrix, together with direct or indirect displacement control or arc-length constraint solution techniques, offers the potential of producing stable robus solutions for a variety of difficult materially nonlinear problems, an investigation of the dominance of eigenvectors in the solution of static problems is undertaken herein. The prime objective of the paper is to develop and apply analytical parameters in order to investigate and demonstrate whether certain eigenv~tors do dominate the displacement response of materially nonlinear structures. The incremental equilibrium equations are formulated in the basis of the eigenvectors of the tangent stiflness matrix for structures with material nonlin~~ties. The fo~ulation of incremental equilibrium in the basis of the eigenvectors is first presented followed by a discussion of the form of the resulting equilibrium equations. The eigenmodes of
the tangent stiffness matrices of different types of problems, with different types of nonlinear material models, are then investigated to determine the extent of domination of certain modes, and the ability of an approximate solution based on a small number of eigenvectors to represent the solution of the complete systems. 2.EQUILIBRIUMEQUATIONSINTHE NATURALBASIS
In the small displacement formulation of structural problems by the finite element method, the equilibrium equations for the assembled structure are generally expressed in matrix form as Kr=R
(1)
in which the elements of r are the displacements of the nodal points in the global coordinate directions, and the elements of R are the external forces at the nodes which are conjugate to the nodal displacements in the sense that 6W=6rTR
(2)
properly expresses the virtual work 6 W done on the structure for any set of virtual ~spla~ents, 6r. The element r, of the vector r may be identified as the measure number associated with a basis vector ei whose elements are all zero except for its ith element which has a value of unity. The set of all vectors ej form the ‘natural’ basis [6] for the configuration space for the structure. The same natural basis is the basis which must be used for the specification of external loads in order for (2) to be satisfied. The elements Ri of R may be identified as the measure numbers, associated with the natural basis vectors ei, required for this specification. In many solution procedures it is convenient to express the load vector in (1) as R=pR
(3)
in which R is a constant set of reference loads and p is a scalar load factor which controls the magnitude of the applied loads. When materials accumulate sufficient strain to become nonlinear the equilibrium equations (1) are no longer linear, and it is usual to apply some form of iterative technique to find incremental corrections to the displa~men~ in order to determine a conliguration which is closer to the ~~lib~~ conjuration. One form of such an incremental equilibrium equation is K,Ar = AQ
(4)
in which K, is the tangent stiffness matrix of the structure. The vector AQ in (4) represents the
Eigenmode dominance and its application unbalanced loads in the current configuration can be expressed as AQ=(p
+Ap)R-F
which
(5)
in which the first term is the set of applied external loads, and the second is the set of equilibrating internal forces evaluated by integration of the current stresses over the current configuration. 3. CHANGE OF BASIS
The general equations for change of bases are well known [6] but are reviewed here for completeness. Assume that N is the dimension of the configuration space, defined by the natural basis. There are then an infinite number of other sets of N basis vectors which can span the same space. Let vi be one of a set of such basis vectors. Then any set of displacements on the natural basis may be expressed as r=Vu
(6)
in which the columns of V are the basis vectors vi and the elements ui of u are the associated measure number or generalized displacement coordinates of these basis vectors. The generalized forces, U, which are conjugate to the generalized displacements of (6) are defined through the principle of contragradience [7] arising out of (2), as
283
The eigenvalues are assumed to be distinct and arranged in ascending order. The N equations (10) may then be assembled into the single matrix equation K,@ = 612
(11)
in which the columns of @ are the eigenvectors, and 1 is the diagonal matrix of eigenvalues in ascending order. It follows directly from (lo), and the assumption that the li are distinct, that the vectors of 8 are orthogonal and may be normalized so that are=1
(12)
in which I is the identity matrix. Since each element of any of the vectors +i is a particular value of the same element of Ar, and since the N vectors & are linearly independent, the columns of Q form an alternative basis for the configuration space of the structure. Hence, one may immediately write a coordinate transformation, similar to (6), for the incremental displacements of (4), as Ar=@Au
(13)
in which Aa are the generalized displacement coordinates associated with the respective eigenvectors. The generalized incremental forces, Ay, associated with the eigenvector basis then follow from (7) as Ay = aTAQ
U=VrR.
(7)
Substituting (6) into (l), and multiplying by VT yields the equilibrium equations in terms of the generalized forces and displacements as K,u=U
(8)
in which the stiffness matrix in (8), associated with the new basis, is K,=VrKV.
(9)
4. INCREMENTAL EQUILIBRIUM EQUATIONS IN THE EIGENVECTOR BASIS
It is well known that the equilibrium equations relative to eigenvector bases are particularly simple because the basis vectors are orthogonal with respect to the stiffness matrix. Since we are interested, herein, in the incremental equilibrium equations we perform the change of basis relative to the tangent stiffness matrix of (4). There are N eigenpairs (A,, 4,) for the matrix, where A, is the eigenvalue and 4, is the eigenvector, each of which satisfies the relation K,ti,=&d*
(10)
(14)
and the incremental stiffness follows from (9), with 0 replacing the transformation matrix V. However, multiplying (11) by 9 * and using the orthonormal relationship (12) it is apparent that the stiffness matrix in (9), associated with the eigenvalue basis, is the diagonal matrix of eigenvalues. Hence (4) has been transformed to 1Aa = Ay
(15)
and each of the equilibrium equations is uncoupled from the others. As a result the solution for each element in Aa arises directly from (15), and may be expressed as Aai = Ay,/li . The generalized unbalanced tracted from (14) as
(16)
force Ayi can be ex-
Ayi = +;AQ
(17)
and the portion of the global nodal displacement increment for the Aai determined in (16) follows from (13) as Ari = & Aai .
(18)
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J. NAPO&O, Fo. er al. 5. MEASURES OF PARTICIPATION
There is, of course, nothing new in the foregoing derivations. The primary objective of this paper is to investigate the degree to which a linear combination of a small number of eigenvectors can approximate the total displacement increment. It is therefore necessary to define some parameters by which the participation of each of the eignevectors in the total displacement increment can be measured. Equation (13) defines the total displacement increment as a linear combination of the eigenvectors and hence may be written more explicitly as
where the first equality follows from the definition, the second follows from the equation of the terms involving $7, and the third follows from (21). A constraint equation exists for the set of /Ii. This can be obtained by multiplying each side of (13) by its transpose and recognizing the orthonormality relation, (12), to yield.
(IIAr 11)’ = 5 (Aad*. Dividing
this equation
by its left-hand
side yields
1=&l:
N
Ar = 2 Aoc~~~.
(23)
j=l
(19)
(24)
j=l
j=l
Premultiplying each side of (19) by 47 and recognizing the orthogonality of the Q vectors, the generalized displacement associated with 4i may be obtained as Aai = Q TAr
It is proposed that fif be called the ‘participation parameter’ for the vector component Au,~~ and it will be designated, here, as pi. Hence we will write
(20)
and the component of the incremental displacement vector associated with this eigenvector may be extracted from (19) as A.ai4,. The relative magnitude of this displacement component, with respect to the displacement increment vector, is the ratio of the Euclidean norm of the component to that of the total. Hence a ‘relative size parameter’, /Ii, is defined as
(21)
and (24) becomes
/$,Pi=
l.
(26)
Note that pi can also be expressed as a percentage participation, denoted by Pi. Equation (26) may therefore be written in the alternative form f
P,=lOO%.
(264
j=l
in which t*t denotes the Euclidean norm. The second equality follows because of the orthonormality property of the eigenvectors. A ‘relative angle’, Bi, may also be defined for the vector component from the cosine interpretation of the scalar product as (22)
The ingredients of the analytical parameters may be interpreted geometrically in the schematic of the expansion of Ar in the N-dimensional space of Fig. l(a). It is noted that according to relation (22) the relative size parameter, /Ii, is the same as the (direction) cosine of the relative angle, Bi. This is a consequence of the orthogonality of the basis vectors, as may be seen for the orthogonal base vectors of
.
N
t
0
cf
i 2
Fig. 1. (a) The displacement increment vector in N-dimensional eigen basis. (b) The relative angles in three-dimensional eigen space.
Eigenmode dominance and its application the three-dimensional space in Fig. l(b). It is apparent from Fig. l(b) that the relative magnitudes of the vector components are equal to the direction cosines of the relative angles. It is also apparent in Fig. 7(a) that vector components orthogonal to Ar will not contribute to Ar and this will be reflected by a zero value of the associated /Ii (i.e. a 90” value of e,). If any set of M components is selected (A4 < N) for which (26) is satisfied, when the summation is over A4 rather than N (or for which Zr Pi= lOO%), the vector Ar is completely contained in the Mdimensional subspace spanned by the basis vectors of this subspace. The value of pi (or Pi) indicates the level of participation of the basis vector $i in the incremental displacement vector Ar, while the value of Zpi (or ZP,) when different from the constraint limit of 1 (or 100%) gives a measure of how good (or bad) is the approximation in the M-dimensional subspace.
6. APPROXIMATIONS BASED ON THE REDUCED EIGENVRCTOR BASIS
It is expected that the eigenvectors with the smallest eigenvalues will have the greater participation in the displacement response. This is not necessarily the case since the displacement response is load dependent and it may be that the generalized force associated with a small eigenvalue is zero, or nearly so. It can be seen from (16) and (17) that this situation arises if the vector AQ is orthogonal, or nearly so, with respect of c$~. In the following we describe the participation of each eigenvector in the total incremental displacement as orthogonal, quasi-orthogonal or participant, according to the value of its parameters, bi, Bi and Pi, with respect to the ranges shown in Table 1. The displacement increment Ar may be approximated in the M-dimensional space associated with the eigenmodes of the A4 smallest eigenvectors by truncating the sum in (19) to the first M terms. Using the subscript a to designate this approximation we may write Ar, = 5 aj~j.
285
The degree to which this approximation represents the total Ar vector may then be measured by
and
(29
P,=lOO/3f in which Aa, = Ar:Ar .
(30)
These parameters can now be used to test the hypothesis of domination and the quality of approximation in the reduced basis of eigenvectors used for the approximate solution. Three problems with different geometries and material behavior have been analyzed to determine the behavior of the eigenvalues and eigenmodes. The primary objective is to apply the analytical parameters formulated above in order to derive conclusions about domination of eigenvectors on the displacement response of structures. In the following some pertinent results are presented. A full description is presented by Napolego et al. [8]. 7. A PLANE STRUCTURE WITH VARYING GEOMETRIC PROPORTIONS
The geometric proportions of a structure form a contributing factor to the structural stiffness. Thus, the structural stiffness can be altered through variation of the geometric proportions. In turn, this implies a change in the structural behavior. For example, a shallow beam deforms fundamentally through bending, accompanied by negligible shear distortion of its cross sections. When the depth of the cross-section is substantially increased, shear distortion dominates the deformation of the beam. Figure 2 shows the discretization, type and intensity of the applied loads and the dimensions of the structure selected to form a parametric study in which the aspect ratio H/L is varied. The objective is to
(27)
j=l
Table 1. Definition of range of analytical parameters for eigenvector classification ifpe of Eigenvector Component Relative Size Parameter Angle Parameter (degrees) Participation Parameter (XI
Orthogonal
0
90
Quasi Orthogonal
(0
, 0.101
184,
90)
Participant
(0.10,
1.01
10, 84) t
0
(0,
II
(1,
1001
- 4H, ZH, H. H/2, H/4
w qy=lONla!m
Y 10 mm
Y
Fig. 2. Plane structure with varying geometric properties.
J. NAPOLE~O.Fo. et al.
286
investigate the influence of the geometric proportions on the participation of the eigenvector components in terms of level and number. The dimensions have been varied to give rise to values of the depth to span aspect ratio, H/L, of l/4, l/2, l/l, 2/l and 4/l. The material is linearly elastic with properties E = 400 MPa and v = 0.0. A null Poisson ratio has been adopted so that the eigenvectors can depict more distinguishable deformation patterns, without transverse effects such as contraction or expansion. The load intensity q,,has been kept constant, whereas qz varies according to the value of the aspect ratio. This results in a constant shear force at the fixed cross-section. The loads have been chosen so that a general deformation mode, with components such as extension, bending and shear distortion, can be present in the actual displacement vector of the structure. The level of contribution of these components depends on the value of the aspect ratio. A varying number of lowest eigenvalues and corresponding eigenvectors were extracted for each aspect ratio. This number has been chosen tentatively so that orthogonal, quasiorthogonal and participant eigenvector components are represented and the required accuracy in the approximate deflections
Fig. 3. The
is obtained. Figures 3 and 4 show the actual displacement vector, the normalized eigenvectors, the eigenvector components and the approximate displacement vector for two cases of the values of the aspect ratio, namely, for H/L of l/4 and 4/l, respectively. The approximate displacement vector is based on the participant eigenvectors as defined in Table 1. A constant plot scale has been used throughout the aforementioned figures. For eaGh value of the aspect ratio, the analytical parameters are computed for each eigenvector component. The values of the relative size and participation parameters for all the aspect ratios are plotted in the Figs 5 and 6, respectively. For H/L = l/4 (Fig. 3), Figs 5 and 6 indicate that only the first eigenvector component is participant according to the criteria of Table 1. The second and third are quasi-orthogonal, whereas the fourth is completely orthogonal. In addition, the normalized eigenvectors of Fig. 3 depict pure deformational patterns, such as simple bending for the first eigenvector, extension for the second and double curvature bending for the fourth. According to our definitions the approximate displacement vector is based only on the first eigenvector component and accurately
deformation modes for H/L = l/4.
Eigenmode dominance and its application c
Fig. 4. The deformation modes for H/L = 4. reproduces the actual displacement vector as deduced from the comparison between the approximate and the actual deflections at node 21 for which a ratio of 0.998 is shown in Table 2. For H/L = 4/l (Fig. 4) Figs 5 and 6 indicate that the fourth and fifth eigenvector components are quasi-orthogonal according to the criteria of Table 1. All components have lost the purity of deformation depicted for the case of H/L = l/4. The five remaining participant components included in the approximate displacement vector which provides a ratio of 0.936 with respect to the actual deflection at node 21, as shown in Table 2. Figure 7 illustrates the variation of the participation parameter computed for the first and second eigenvector components, with the value of the aspect ratio. For low aspect ratios, the first eigenvector component shows a predominant participation, whereas the first and second components share comparable participation levels when the aspect ratio is higher.
This parametric study demonstrates that a larger number of eigenvector components may be required while approximating the actual displacement vector of structures with high aspect ratios, such as deep beams and corbels than that required when the aspect ratio is low.
8. AN ELASTIC-PERFECTLY PLASTIC BEAM
CANTILEVER
Generally, a beam whose material is idealized as elastic-perfectly plastic exhibits three identifiable phases in its load-deflection curve. A linearly elastic phase is followed by an elasto-plastic phase where the structure experiences a gradually increasing deterioration due to yielding. The third and last phase, designated plastic, coincides with the onset of localized plastic hinges within the material. This, in turn, causes the formation of a failure mechanism which develops under approximately constant load. In the following an investigation of the domination of some
1.0 0.9 0.8
I
0.6 0.7
2
0.5
;
0.4
2
II:
0
wL=4/1
0.1 0.0
Fig. 5. The relative size parameter for different aspect ratios.
Fig. 6. The participation parameter for different aspect ratios.
288
J. NAPOLE;~O, Fo. et al.
Table 2. Actual and approximate deflections of the plane structure 21
Resultant Deflection at node 2 I
Actual (mm)
Approximate (mm)
Ratio: Approximate Actual
H/L-1/4
67.3 13
67.199
0.998
H/L-1/2
10.636
10.610
0.997
H/L-l/l
2.613
2.529
0.968
-.-
SECOND UGE-OR
-A-
SUM OF FIRST k SBCONTI
-I 0.0
0.5
1.0
1.5
2.5
2.0
3.0
3.5
4.0
4.5
5.0
AsPE!xRATM)n!L
Fig. 7. The participation parameter as a function of the aspect ratio.
eigenvector components on the incremental displacement response of an elastic perfectly plastic structure is presented. Figure 8 shows the discretization, dimensions and loads for a shallow cantilever steel beam. The beam material is modeled as elastic-perfectly plastic with the following properties: E = 200,000 MPa, v = 0.30 and F, = 300 MPa. A series of full eigenanalyses was performed throughout the load-deflection history. Only the first and second eigenvalues are of interest in this study, since domination is expected to be restricted to a small number of eigenvector components for the aspect ratio under consideration. The corresponding eigenvectors are plotted in Fig. 9, while Fig. 10 shows the variation of the first and second eigenvalues, normalized with respect to their initial values, with the tip deflection normalized with respect to its value at first yield. The initial values for the first and second eigenvalues, computed in the linearly elastic regime, are, respectively, 20.281 and 555.002 N/mm. The normalized eigenvalues remain constant in the linearly elastic phase and decrease gradually in the elasto-plastic phase. In the fully plastic phase, both normalized eigenvalues are approximately constant. It is noted that the first normalized eigenvalue decreases in the elasto-plastic phase with a higher gradient that the second normalized eigenvalue. Moreover, it reaches a very small non-negative value during the plastic phase. Since the eigenvalue is the stiffness associated with the corresponding eigenvector, it appears that the first eigen-
I-
vector component will give rise to the failure mechanism of the beam. However, this statement can only be substantiated by application of the analytical parameters throughout the load-deflection history. Figure 11 illustrates the variation of the relative size parameter for the first two eigenvectors with the deflection ratio. The size of the first eigenvector component approximates accurately the size of the actual displacement increment vector throughout the linearly elastic, elasto-plastic and plastic phases. The relative size of the second eigenvector component is very small for the linearly elastic and elasto-plastic phases and drops to zero in the plastic phase. Thevariationoftheparticipationparameterisshown in Fig. 12. It can be seen that the first eigenvector component dominates the incremental displacement response of the considered cantilever beam, from the beginning of the elastic phase to the onset of the failure mechanism. The domination becomes extreme in the plastic phase of the beam behaviour. It should, however, be noted that the first eigenvector does not represent a constant pattern of deformation throughout the load history because changes to K, are reflected by changes in the vector. However, the approximate displacement increment vector Ara, based on the first eigenvector component, incorporates all the essential characteristics of the known failure mechanism. 9. AN ELASTIC SOmENING
The beam-rod shown in Fig. 13 is subject to varying ratios of moment to axial force along its R-
supportat all nodes of this side.
BEAM-ROD
(15KN) t
4
l-4 100 mm
3000 mm Fig. 8. An elastic-perfectly plastic cantilever beam.
10 mm
Eigenmode do~nan~
Fig. 9. The first and second modes at m~h~ism
length and was selected as a member type to study softening effects. The material is treated as elastic strain softening plastic with E = 2~,~ MPa, II’= - 5000 MPa, v = 0.30 and FY= 300 MPa. Pietruszczalc and Mroz [9] demonstrated that slope of the descending branch of the load-deflection curve of a strain softening structure is mesh dependent. Nevertheless, a coarse mesh has been adopted in the discretization of the beam-rod since the problem of uniqueness of the descending branch is outside the scope of the present study. The type and intensity of the applied reference loads have been designed to induce a combination of the flextural and extensional deformation modes. Figure 14 shows the relation between the load and displacement of node 11 of the beam-rod. The development of softening regions is ill~trated in Fig. 15 for the equilib~um states A-E marked on Fig. 14. After point E the spreading of
/j
1.0
g
0.8
289
and its appli~tion
formation stage.
softening stabilizes according to the pattern depicted in the last plot of Fig. 15. The load combination of uniformly distributed tension and pure bending induces the development of uneven softening through the height and span of the member. This condition influences the level of domination and the deformational patterns of the eigenvector components. Only the first three eigenvectors and eigenvalues are of interest as shown in Fig. 3. The fourth eigenvector component is orthogonal to the actual displacement pattern. Specifically, the bending and extensional modes are important while forming the approximate displacement increment vector, since these modes are expected to be activated by the uniformly distributed load and bending moment applied to the structure. Figure 16 illustrates the variation of the participation parameter with respect to the free end
FlRSTEIGE?WALUE
$
SECONDEKSENVAL 0.6
ii 0.4
0.2 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
D@FL&XlON RATlO (mm&in)
Fig. 10. The normalized eigenvalues as a function of deflection ratio.
0.0
0.s
1.0
1.5
2.0 2.5 3.0 3.5 4.0 DEFLECTION RAllO (mmimm)
4.5
Fig. 11. The relative size parameter as a function of the deflection ratio.
290
J.
NAPOLE;~O, Fo. et al.
d E
20.
2 _---. i -20 0.0
,
,
0.5
1.0
.
,
,
,
,
,
I
I
I
2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.5 0~0N
RATIO (mm/mm)
Fig. 12. The participation parameter variation with respect to the deflection ratio. displacement of node 11. During the linearly elastic stage, the bending and extensional eigenvector components compose the approximate displacement increment vector representing 12 and 88% participation, respectively. The greater participation of the extensional mode is intentional and has been achieved by specifying a reference value of 100 N/mm for the uniformly distributed load. The shear-flexure eigenvector component is quasi-orthogonal throughout this range of behavior. Upon initiation of uneven softening, the domination of the flexural eigenvector component starts to increase. At the limit point, all three modes participate. After the limit point, when the uneven softening stabilizes at state C (Fig. 14), the flexural eigenvector component dominates completely. In this range of behavior, the extensional eigenvector component is quasi-orthogonal to the actual displacement increment of the beam while the shear-flexure eigenvector component is nearly orthogonal. At the onset of even softening, represented by state D in Figs 14 and 15, the flexural and extensional eigenvector components contribute 84 and 16% participation, respectively. In this range, the shear-flexure eigenvector component is completely orthogonal to the actual displacement increment vector. This case study demonstrates that the level of domination of eigenvector components varies along the load-deflection history. In addition, the level of domination within a particular range of behavior
Fig. 14. Load-deflection curve for the elastic softening beam-rod. depends on the state of development of softening zones which characterizes that range. Furthermore, the pure bending and extensional deformation patterns of the modes extracted in the linearly elastic range disappear in the nonlinear range. Instead, modes with a combination of bending and axial deformations form in the nonlinear range. Nonetheless the number of participating eigenvectors is small relative to the number of degrees of freedom.
10. CONCLUSION In this paper the equations for a change of basis of the incremental equilibrium equations into a basis formed by the eigenvectors of the tangent stiffness matrix are reviewed. It is then suggested that a reduced basis of a small number of eigenvectors associated with the lowest eigenvalues should be able to model the deformational behavior of problems with material nonlinearities. In order to test this hypothesis, a number of analytical parameters were developed and applied to three different problems. The case studies included the linear elastic behavior of a beam-corbel with different aspect ratios, an elastic perfectly plastic beam, and an elastic strain softening beam-rod. The results of the investigation show that there are a small number of dominant modes that represent the deformation pattern. Clearly an approximation consisting of a linear combination of these modes can be used. Thus instead of
Z t
4
10
q = (loON/mm)
mm
a
400mm
Fig. 13. Discretization, dimensions and loads for the elastic softening beam-rod.
Eigenmode dominance and its application
291
at state B
at state A
at state D
at limit point and state C
at state E Fig. 15, ~elopment
of softening zones at the equilibrium states A-E for the elastic softening beam-rod. by the Comissao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES) and by the Natural Sciences and Engineering Research Council of Canada Operating Grant number A5877. REFERRNCES
3”
+
BENDING
*
ExTENS1GNAL
g40 F:
-
SlLEAR.BENDlNG
MODE
1. E. L. Wilson and E. P. Bayo, Use of special Ritz vectors
MODE MODE
2.
Ll E 2
0
3. -20 0
1
2
3
4
5
6
RESULTANT
7
8
DEFLRCTION
9
10
11
12
13
14
15
(NODE I I) (mm)
Fig. 16. The participation parameter variation with respect to the free end resultant deflection for the elastic softening beam-rod.
with a full set of N degrees of freedom, one can reduce the problem to a small number of generalized degrees of freedom. The concept of domination of the eigenmodes of the tangent stiffness matrix can be demonstrated for complex reinforced concrete structures. This is presented in a separate paper [lo]. A solution strategy based on this concept has been developed and is currently being submitted for publication.
4.
5.
working
Acknowledgements-us work was carried out in the course of research leading to partial fulfillment of the requirements of a Ph.D. degree of the first author at the University of Alberta. The work was funded in part
6. 7. 8.
9. 10.
in dynamic substructu~ analysis. J. Stnrct. Engng, ASCE 112, 1944-1954 (1988). B. 0. Almorth, P. Stern and F. A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA Jnl 16, 525-528 (1978). A. K. Noor and 3. M. Peters, Reduced basis technique for nonlinear analysis of structures, AZAA JnZ 118, 455-462 (1980). D. A. Nagy, Modal representation of geometrically nonlinear behavior by the finite element method. Trans. 4th Int. Conf. on Structural Mechanics in Reactor Technology, San Francisco, Vol. M, M4-M2 (1977). W. C. Hurty, J. D. Collins and G. C. Hart, Dynamic analysis of large structures by modal synthesis technique. Comput. Struct. 1, 553-563 (1971). G. A. Hawkins, M~tii~ear Andy& for Stua’ents in Engineering and Science, John Wiley, New York (1963). R. H. Gallagher, Finite Element Analysis, PrentiwHail, Englewood Cliffs, New Jersey (1975). J. Napoldo, Fo., A. E. Elwi and D. W. Murray, An eigenvector-based strategy for analysis of inelastic structures. Structural Engineering Report No. 166, &partment of Civil Engineering, University of Alberta, Canada (1990). S. T. Pietruszczak and Z. Mroz, Finite element analysis of deformation of strain-softening materials. ht. J. Numer. Meth. Engng 17, 327-334 (1981). J. Napoleiio, Fo., A. E. Elwi and D. W. Murray, An eigenvector-based strategy for analysis of inelastic structures. Comput. Struct. (submitted).
$3.00 + 0.00
co4.s7949/91 Pergamon Press
1991
EIGENMODE DOMINANCE AND ITS APPLICATION STRUCTURES WITH NONLINEAR MATERIALS
plc
TO
J. NAPOLEKO,Fo.,t A. E. ELWI$ and D. W. MURRAY% ?Departamento de Engenharia Civil, Pontificia Universidade Cathlica do Rio de Janeiro, R. Mq. S. Civente, 225, Rio de Janeiro, Brasil SDepartment of Civil Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G7 (Received 17 Augusi 1990) Abstract-Reduction methods have received considerable attention as solution strategies for geometrically nonlinear problems. In this work, the eigenmodes of the tangent stiffness matrix are investigated as a possible basis for a reduction strategy for problems with predominantly material nonlinearities. A number of parameters are developed for measuring the participation of the different eigenvectors in constructing an approximation of the incremental displacement vector. Certain mesh considerations and a series of problems including elastic strain hardening plasticity as well as softening materials are investigated. It is found that the incremental displacement vector is dominated by a few eigenvectors corresponding to the lowest eigenvalues. This domination becomes more evident as damage progresses.
1. INTRODUCTION The formulation of the incremental equilibrium equations for a nonlinear structure, within the framework of the finite element method, has traditionally employed the natural basis for the equilibrium description. The natural basis is formed by unit vectors which are associated with the nodal displacements in the direction of the global coordinate axes. A full set of coupled equilibrium equations, equal in number to the available degrees of freedom, is a direct consequence of this approach. Considerable computational effort is required in the solution of these equations because of the iterative repetition of the steps of factorization, reduction and back substitution necessary due to the material nonlinearities. In an attempt to develop more efficient solution techniques, a change of base vectors may be considered. Change of basis is common in some disciplines of structural analysis. The natural modes of vibration have been commonly adopted as an appropriate basis for the description of the dynamic equilibrium of linearly elastic structures in modal superposition techniques. Recently, Wilson and Bayo [l] have argued that special Ritz load dependent vectors would comprise a more effective basis for dynamic analysis. Almorth et al. [2] and Noor and Peters [3] introduced reduced basis techniques for geometrically nonlinear structures. Almroth et al. selected linear solution vectors, whereas Noor and Peters chose path derivatives to form the reduced basis. Nagy [4] adopted the buckling modes to form a reduced basis to investigate problems with mild changes in geometry. However, the majority of the base vector systems mentioned above have been applied to problems with linearly elastic materials. The intent, herein, is to address problems with nonlinear materials.
Any stable linear elastic structure responds to a given distribution of applied loads (the reference loads) with a unique pattern of displacements. The displacement pattern is generally attained by the solution of a complete set of equilibrium equations, one equation being associated with each global degree of freedom of the assembled structural stiffness matrix, which represents an integrated synthesis of the material and geometric properties of the structure. Consequently, the solution requires considerable computational effort. Once this has been determined, solutions for other load factors may be obtained by proportionality. As the material in the structure becomes nonlinear the incremental displacement pattern changes. The solution is usually obtained iteratively by solving for incremental displacements resulting from incremental loads and then correcting for unbalanced forces, which are computed as the difference between the applied forces and those required to equilibrate the current configuration. The incremental equilibrium equations, which are used to solve for incremental displacements, are formulated in terms of the tangent stiffness matrix which may become singular, but more generally simply ill-conditioned, in the vicinity of limit points, and indefinite in the softening region beyond a limit point. Generally this region is a difficult one to deal with in numerical solutions. This is particularly true in cases where the load-deflection curve is essentially flat over an extended region of deformation, such as in the case for elastic-perfectly plastic steel structures for which the displacement increments at the limit point reflect the amplification of a displacement pattern associated with a collapse mechanism. Singularity of the incremental stiffness matrix in a materially nonlinear structure reflects a condition in which the generalized incremental stiffness coefficient 281
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J. NAPOLE~O,Fo. et al.
associated with the incremental displacement pattern (the incremental collapse mode) becomes zero. Consequently the second-order energy absorption associated with the collapse mode is also zero. Assuming the collapse mode is unique there is only one zero energy displacement pattern. The eigenvectors of the stiffness matrix form an alternative basis which describes all possible deformed configurations of the structure. The eigenvalues represent the generalized stiffnesses associated with the generalized displacement coordinates of the eigenvectors. Hence the (unique) zero second-order energy incremental displacement pattern associated with the collapse mode must be the eigenvector associated with the zero eigenvalue. That is, the eigenvector for the zero eigenvalue should completely dominate the solution for displacement increments in the region of singularity of the tangent stiffness matrix. This implies that a solution for the generalized coordinate associated with the eigenvector of the lowest eigenvalue should capture the essential aspects of the behaviour of a materially nonlinear structure, regardless of its complexity, in the vicinity of its collapse. Furthermore, it can be argued that the eigenvectors associated with the lower energy modes of deformation of the structure can form a subspace within which the essential aspects of behavior of the structure can be captured at all stages of its load-deflection history. These eigenvectors can then be said to ‘dominate’ the solution. Domination of eigenvectors is well known in the dynamic analysis of linearly elastic structures. The fundamental mode of vibration, which usually corresponds to the lowest natural frequency, may dominate the dynamic response of standard buildings subjected to common load dist~butions, such as wind loads [S]. However, domination of eigenvectors for static equilibrium problems involving materially nonlinear structures has not been thoroughly investigated to date. Since a combination of a reduced eigenvector basis and the associated diagonalized incremental stiffness matrix, together with direct or indirect displacement control or arc-length constraint solution techniques, offers the potential of producing stable robus solutions for a variety of difficult materially nonlinear problems, an investigation of the dominance of eigenvectors in the solution of static problems is undertaken herein. The prime objective of the paper is to develop and apply analytical parameters in order to investigate and demonstrate whether certain eigenv~tors do dominate the displacement response of materially nonlinear structures. The incremental equilibrium equations are formulated in the basis of the eigenvectors of the tangent stiflness matrix for structures with material nonlin~~ties. The fo~ulation of incremental equilibrium in the basis of the eigenvectors is first presented followed by a discussion of the form of the resulting equilibrium equations. The eigenmodes of
the tangent stiffness matrices of different types of problems, with different types of nonlinear material models, are then investigated to determine the extent of domination of certain modes, and the ability of an approximate solution based on a small number of eigenvectors to represent the solution of the complete systems. 2.EQUILIBRIUMEQUATIONSINTHE NATURALBASIS
In the small displacement formulation of structural problems by the finite element method, the equilibrium equations for the assembled structure are generally expressed in matrix form as Kr=R
(1)
in which the elements of r are the displacements of the nodal points in the global coordinate directions, and the elements of R are the external forces at the nodes which are conjugate to the nodal displacements in the sense that 6W=6rTR
(2)
properly expresses the virtual work 6 W done on the structure for any set of virtual ~spla~ents, 6r. The element r, of the vector r may be identified as the measure number associated with a basis vector ei whose elements are all zero except for its ith element which has a value of unity. The set of all vectors ej form the ‘natural’ basis [6] for the configuration space for the structure. The same natural basis is the basis which must be used for the specification of external loads in order for (2) to be satisfied. The elements Ri of R may be identified as the measure numbers, associated with the natural basis vectors ei, required for this specification. In many solution procedures it is convenient to express the load vector in (1) as R=pR
(3)
in which R is a constant set of reference loads and p is a scalar load factor which controls the magnitude of the applied loads. When materials accumulate sufficient strain to become nonlinear the equilibrium equations (1) are no longer linear, and it is usual to apply some form of iterative technique to find incremental corrections to the displa~men~ in order to determine a conliguration which is closer to the ~~lib~~ conjuration. One form of such an incremental equilibrium equation is K,Ar = AQ
(4)
in which K, is the tangent stiffness matrix of the structure. The vector AQ in (4) represents the
Eigenmode dominance and its application unbalanced loads in the current configuration can be expressed as AQ=(p
+Ap)R-F
which
(5)
in which the first term is the set of applied external loads, and the second is the set of equilibrating internal forces evaluated by integration of the current stresses over the current configuration. 3. CHANGE OF BASIS
The general equations for change of bases are well known [6] but are reviewed here for completeness. Assume that N is the dimension of the configuration space, defined by the natural basis. There are then an infinite number of other sets of N basis vectors which can span the same space. Let vi be one of a set of such basis vectors. Then any set of displacements on the natural basis may be expressed as r=Vu
(6)
in which the columns of V are the basis vectors vi and the elements ui of u are the associated measure number or generalized displacement coordinates of these basis vectors. The generalized forces, U, which are conjugate to the generalized displacements of (6) are defined through the principle of contragradience [7] arising out of (2), as
283
The eigenvalues are assumed to be distinct and arranged in ascending order. The N equations (10) may then be assembled into the single matrix equation K,@ = 612
(11)
in which the columns of @ are the eigenvectors, and 1 is the diagonal matrix of eigenvalues in ascending order. It follows directly from (lo), and the assumption that the li are distinct, that the vectors of 8 are orthogonal and may be normalized so that are=1
(12)
in which I is the identity matrix. Since each element of any of the vectors +i is a particular value of the same element of Ar, and since the N vectors & are linearly independent, the columns of Q form an alternative basis for the configuration space of the structure. Hence, one may immediately write a coordinate transformation, similar to (6), for the incremental displacements of (4), as Ar=@Au
(13)
in which Aa are the generalized displacement coordinates associated with the respective eigenvectors. The generalized incremental forces, Ay, associated with the eigenvector basis then follow from (7) as Ay = aTAQ
U=VrR.
(7)
Substituting (6) into (l), and multiplying by VT yields the equilibrium equations in terms of the generalized forces and displacements as K,u=U
(8)
in which the stiffness matrix in (8), associated with the new basis, is K,=VrKV.
(9)
4. INCREMENTAL EQUILIBRIUM EQUATIONS IN THE EIGENVECTOR BASIS
It is well known that the equilibrium equations relative to eigenvector bases are particularly simple because the basis vectors are orthogonal with respect to the stiffness matrix. Since we are interested, herein, in the incremental equilibrium equations we perform the change of basis relative to the tangent stiffness matrix of (4). There are N eigenpairs (A,, 4,) for the matrix, where A, is the eigenvalue and 4, is the eigenvector, each of which satisfies the relation K,ti,=&d*
(10)
(14)
and the incremental stiffness follows from (9), with 0 replacing the transformation matrix V. However, multiplying (11) by 9 * and using the orthonormal relationship (12) it is apparent that the stiffness matrix in (9), associated with the eigenvalue basis, is the diagonal matrix of eigenvalues. Hence (4) has been transformed to 1Aa = Ay
(15)
and each of the equilibrium equations is uncoupled from the others. As a result the solution for each element in Aa arises directly from (15), and may be expressed as Aai = Ay,/li . The generalized unbalanced tracted from (14) as
(16)
force Ayi can be ex-
Ayi = +;AQ
(17)
and the portion of the global nodal displacement increment for the Aai determined in (16) follows from (13) as Ari = & Aai .
(18)
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J. NAPO&O, Fo. er al. 5. MEASURES OF PARTICIPATION
There is, of course, nothing new in the foregoing derivations. The primary objective of this paper is to investigate the degree to which a linear combination of a small number of eigenvectors can approximate the total displacement increment. It is therefore necessary to define some parameters by which the participation of each of the eignevectors in the total displacement increment can be measured. Equation (13) defines the total displacement increment as a linear combination of the eigenvectors and hence may be written more explicitly as
where the first equality follows from the definition, the second follows from the equation of the terms involving $7, and the third follows from (21). A constraint equation exists for the set of /Ii. This can be obtained by multiplying each side of (13) by its transpose and recognizing the orthonormality relation, (12), to yield.
(IIAr 11)’ = 5 (Aad*. Dividing
this equation
by its left-hand
side yields
1=&l:
N
Ar = 2 Aoc~~~.
(23)
j=l
(19)
(24)
j=l
j=l
Premultiplying each side of (19) by 47 and recognizing the orthogonality of the Q vectors, the generalized displacement associated with 4i may be obtained as Aai = Q TAr
It is proposed that fif be called the ‘participation parameter’ for the vector component Au,~~ and it will be designated, here, as pi. Hence we will write
(20)
and the component of the incremental displacement vector associated with this eigenvector may be extracted from (19) as A.ai4,. The relative magnitude of this displacement component, with respect to the displacement increment vector, is the ratio of the Euclidean norm of the component to that of the total. Hence a ‘relative size parameter’, /Ii, is defined as
(21)
and (24) becomes
/$,Pi=
l.
(26)
Note that pi can also be expressed as a percentage participation, denoted by Pi. Equation (26) may therefore be written in the alternative form f
P,=lOO%.
(264
j=l
in which t*t denotes the Euclidean norm. The second equality follows because of the orthonormality property of the eigenvectors. A ‘relative angle’, Bi, may also be defined for the vector component from the cosine interpretation of the scalar product as (22)
The ingredients of the analytical parameters may be interpreted geometrically in the schematic of the expansion of Ar in the N-dimensional space of Fig. l(a). It is noted that according to relation (22) the relative size parameter, /Ii, is the same as the (direction) cosine of the relative angle, Bi. This is a consequence of the orthogonality of the basis vectors, as may be seen for the orthogonal base vectors of
.
N
t
0
cf
i 2
Fig. 1. (a) The displacement increment vector in N-dimensional eigen basis. (b) The relative angles in three-dimensional eigen space.
Eigenmode dominance and its application the three-dimensional space in Fig. l(b). It is apparent from Fig. l(b) that the relative magnitudes of the vector components are equal to the direction cosines of the relative angles. It is also apparent in Fig. 7(a) that vector components orthogonal to Ar will not contribute to Ar and this will be reflected by a zero value of the associated /Ii (i.e. a 90” value of e,). If any set of M components is selected (A4 < N) for which (26) is satisfied, when the summation is over A4 rather than N (or for which Zr Pi= lOO%), the vector Ar is completely contained in the Mdimensional subspace spanned by the basis vectors of this subspace. The value of pi (or Pi) indicates the level of participation of the basis vector $i in the incremental displacement vector Ar, while the value of Zpi (or ZP,) when different from the constraint limit of 1 (or 100%) gives a measure of how good (or bad) is the approximation in the M-dimensional subspace.
6. APPROXIMATIONS BASED ON THE REDUCED EIGENVRCTOR BASIS
It is expected that the eigenvectors with the smallest eigenvalues will have the greater participation in the displacement response. This is not necessarily the case since the displacement response is load dependent and it may be that the generalized force associated with a small eigenvalue is zero, or nearly so. It can be seen from (16) and (17) that this situation arises if the vector AQ is orthogonal, or nearly so, with respect of c$~. In the following we describe the participation of each eigenvector in the total incremental displacement as orthogonal, quasi-orthogonal or participant, according to the value of its parameters, bi, Bi and Pi, with respect to the ranges shown in Table 1. The displacement increment Ar may be approximated in the M-dimensional space associated with the eigenmodes of the A4 smallest eigenvectors by truncating the sum in (19) to the first M terms. Using the subscript a to designate this approximation we may write Ar, = 5 aj~j.
285
The degree to which this approximation represents the total Ar vector may then be measured by
and
(29
P,=lOO/3f in which Aa, = Ar:Ar .
(30)
These parameters can now be used to test the hypothesis of domination and the quality of approximation in the reduced basis of eigenvectors used for the approximate solution. Three problems with different geometries and material behavior have been analyzed to determine the behavior of the eigenvalues and eigenmodes. The primary objective is to apply the analytical parameters formulated above in order to derive conclusions about domination of eigenvectors on the displacement response of structures. In the following some pertinent results are presented. A full description is presented by Napolego et al. [8]. 7. A PLANE STRUCTURE WITH VARYING GEOMETRIC PROPORTIONS
The geometric proportions of a structure form a contributing factor to the structural stiffness. Thus, the structural stiffness can be altered through variation of the geometric proportions. In turn, this implies a change in the structural behavior. For example, a shallow beam deforms fundamentally through bending, accompanied by negligible shear distortion of its cross sections. When the depth of the cross-section is substantially increased, shear distortion dominates the deformation of the beam. Figure 2 shows the discretization, type and intensity of the applied loads and the dimensions of the structure selected to form a parametric study in which the aspect ratio H/L is varied. The objective is to
(27)
j=l
Table 1. Definition of range of analytical parameters for eigenvector classification ifpe of Eigenvector Component Relative Size Parameter Angle Parameter (degrees) Participation Parameter (XI
Orthogonal
0
90
Quasi Orthogonal
(0
, 0.101
184,
90)
Participant
(0.10,
1.01
10, 84) t
0
(0,
II
(1,
1001
- 4H, ZH, H. H/2, H/4
w qy=lONla!m
Y 10 mm
Y
Fig. 2. Plane structure with varying geometric properties.
J. NAPOLE~O.Fo. et al.
286
investigate the influence of the geometric proportions on the participation of the eigenvector components in terms of level and number. The dimensions have been varied to give rise to values of the depth to span aspect ratio, H/L, of l/4, l/2, l/l, 2/l and 4/l. The material is linearly elastic with properties E = 400 MPa and v = 0.0. A null Poisson ratio has been adopted so that the eigenvectors can depict more distinguishable deformation patterns, without transverse effects such as contraction or expansion. The load intensity q,,has been kept constant, whereas qz varies according to the value of the aspect ratio. This results in a constant shear force at the fixed cross-section. The loads have been chosen so that a general deformation mode, with components such as extension, bending and shear distortion, can be present in the actual displacement vector of the structure. The level of contribution of these components depends on the value of the aspect ratio. A varying number of lowest eigenvalues and corresponding eigenvectors were extracted for each aspect ratio. This number has been chosen tentatively so that orthogonal, quasiorthogonal and participant eigenvector components are represented and the required accuracy in the approximate deflections
Fig. 3. The
is obtained. Figures 3 and 4 show the actual displacement vector, the normalized eigenvectors, the eigenvector components and the approximate displacement vector for two cases of the values of the aspect ratio, namely, for H/L of l/4 and 4/l, respectively. The approximate displacement vector is based on the participant eigenvectors as defined in Table 1. A constant plot scale has been used throughout the aforementioned figures. For eaGh value of the aspect ratio, the analytical parameters are computed for each eigenvector component. The values of the relative size and participation parameters for all the aspect ratios are plotted in the Figs 5 and 6, respectively. For H/L = l/4 (Fig. 3), Figs 5 and 6 indicate that only the first eigenvector component is participant according to the criteria of Table 1. The second and third are quasi-orthogonal, whereas the fourth is completely orthogonal. In addition, the normalized eigenvectors of Fig. 3 depict pure deformational patterns, such as simple bending for the first eigenvector, extension for the second and double curvature bending for the fourth. According to our definitions the approximate displacement vector is based only on the first eigenvector component and accurately
deformation modes for H/L = l/4.
Eigenmode dominance and its application c
Fig. 4. The deformation modes for H/L = 4. reproduces the actual displacement vector as deduced from the comparison between the approximate and the actual deflections at node 21 for which a ratio of 0.998 is shown in Table 2. For H/L = 4/l (Fig. 4) Figs 5 and 6 indicate that the fourth and fifth eigenvector components are quasi-orthogonal according to the criteria of Table 1. All components have lost the purity of deformation depicted for the case of H/L = l/4. The five remaining participant components included in the approximate displacement vector which provides a ratio of 0.936 with respect to the actual deflection at node 21, as shown in Table 2. Figure 7 illustrates the variation of the participation parameter computed for the first and second eigenvector components, with the value of the aspect ratio. For low aspect ratios, the first eigenvector component shows a predominant participation, whereas the first and second components share comparable participation levels when the aspect ratio is higher.
This parametric study demonstrates that a larger number of eigenvector components may be required while approximating the actual displacement vector of structures with high aspect ratios, such as deep beams and corbels than that required when the aspect ratio is low.
8. AN ELASTIC-PERFECTLY PLASTIC BEAM
CANTILEVER
Generally, a beam whose material is idealized as elastic-perfectly plastic exhibits three identifiable phases in its load-deflection curve. A linearly elastic phase is followed by an elasto-plastic phase where the structure experiences a gradually increasing deterioration due to yielding. The third and last phase, designated plastic, coincides with the onset of localized plastic hinges within the material. This, in turn, causes the formation of a failure mechanism which develops under approximately constant load. In the following an investigation of the domination of some
1.0 0.9 0.8
I
0.6 0.7
2
0.5
;
0.4
2
II:
0
wL=4/1
0.1 0.0
Fig. 5. The relative size parameter for different aspect ratios.
Fig. 6. The participation parameter for different aspect ratios.
288
J. NAPOLE;~O, Fo. et al.
Table 2. Actual and approximate deflections of the plane structure 21
Resultant Deflection at node 2 I
Actual (mm)
Approximate (mm)
Ratio: Approximate Actual
H/L-1/4
67.3 13
67.199
0.998
H/L-1/2
10.636
10.610
0.997
H/L-l/l
2.613
2.529
0.968
-.-
SECOND UGE-OR
-A-
SUM OF FIRST k SBCONTI
-I 0.0
0.5
1.0
1.5
2.5
2.0
3.0
3.5
4.0
4.5
5.0
AsPE!xRATM)n!L
Fig. 7. The participation parameter as a function of the aspect ratio.
eigenvector components on the incremental displacement response of an elastic perfectly plastic structure is presented. Figure 8 shows the discretization, dimensions and loads for a shallow cantilever steel beam. The beam material is modeled as elastic-perfectly plastic with the following properties: E = 200,000 MPa, v = 0.30 and F, = 300 MPa. A series of full eigenanalyses was performed throughout the load-deflection history. Only the first and second eigenvalues are of interest in this study, since domination is expected to be restricted to a small number of eigenvector components for the aspect ratio under consideration. The corresponding eigenvectors are plotted in Fig. 9, while Fig. 10 shows the variation of the first and second eigenvalues, normalized with respect to their initial values, with the tip deflection normalized with respect to its value at first yield. The initial values for the first and second eigenvalues, computed in the linearly elastic regime, are, respectively, 20.281 and 555.002 N/mm. The normalized eigenvalues remain constant in the linearly elastic phase and decrease gradually in the elasto-plastic phase. In the fully plastic phase, both normalized eigenvalues are approximately constant. It is noted that the first normalized eigenvalue decreases in the elasto-plastic phase with a higher gradient that the second normalized eigenvalue. Moreover, it reaches a very small non-negative value during the plastic phase. Since the eigenvalue is the stiffness associated with the corresponding eigenvector, it appears that the first eigen-
I-
vector component will give rise to the failure mechanism of the beam. However, this statement can only be substantiated by application of the analytical parameters throughout the load-deflection history. Figure 11 illustrates the variation of the relative size parameter for the first two eigenvectors with the deflection ratio. The size of the first eigenvector component approximates accurately the size of the actual displacement increment vector throughout the linearly elastic, elasto-plastic and plastic phases. The relative size of the second eigenvector component is very small for the linearly elastic and elasto-plastic phases and drops to zero in the plastic phase. Thevariationoftheparticipationparameterisshown in Fig. 12. It can be seen that the first eigenvector component dominates the incremental displacement response of the considered cantilever beam, from the beginning of the elastic phase to the onset of the failure mechanism. The domination becomes extreme in the plastic phase of the beam behaviour. It should, however, be noted that the first eigenvector does not represent a constant pattern of deformation throughout the load history because changes to K, are reflected by changes in the vector. However, the approximate displacement increment vector Ara, based on the first eigenvector component, incorporates all the essential characteristics of the known failure mechanism. 9. AN ELASTIC SOmENING
The beam-rod shown in Fig. 13 is subject to varying ratios of moment to axial force along its R-
supportat all nodes of this side.
BEAM-ROD
(15KN) t
4
l-4 100 mm
3000 mm Fig. 8. An elastic-perfectly plastic cantilever beam.
10 mm
Eigenmode do~nan~
Fig. 9. The first and second modes at m~h~ism
length and was selected as a member type to study softening effects. The material is treated as elastic strain softening plastic with E = 2~,~ MPa, II’= - 5000 MPa, v = 0.30 and FY= 300 MPa. Pietruszczalc and Mroz [9] demonstrated that slope of the descending branch of the load-deflection curve of a strain softening structure is mesh dependent. Nevertheless, a coarse mesh has been adopted in the discretization of the beam-rod since the problem of uniqueness of the descending branch is outside the scope of the present study. The type and intensity of the applied reference loads have been designed to induce a combination of the flextural and extensional deformation modes. Figure 14 shows the relation between the load and displacement of node 11 of the beam-rod. The development of softening regions is ill~trated in Fig. 15 for the equilib~um states A-E marked on Fig. 14. After point E the spreading of
/j
1.0
g
0.8
289
and its appli~tion
formation stage.
softening stabilizes according to the pattern depicted in the last plot of Fig. 15. The load combination of uniformly distributed tension and pure bending induces the development of uneven softening through the height and span of the member. This condition influences the level of domination and the deformational patterns of the eigenvector components. Only the first three eigenvectors and eigenvalues are of interest as shown in Fig. 3. The fourth eigenvector component is orthogonal to the actual displacement pattern. Specifically, the bending and extensional modes are important while forming the approximate displacement increment vector, since these modes are expected to be activated by the uniformly distributed load and bending moment applied to the structure. Figure 16 illustrates the variation of the participation parameter with respect to the free end
FlRSTEIGE?WALUE
$
SECONDEKSENVAL 0.6
ii 0.4
0.2 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
D@FL&XlON RATlO (mm&in)
Fig. 10. The normalized eigenvalues as a function of deflection ratio.
0.0
0.s
1.0
1.5
2.0 2.5 3.0 3.5 4.0 DEFLECTION RAllO (mmimm)
4.5
Fig. 11. The relative size parameter as a function of the deflection ratio.
290
J.
NAPOLE;~O, Fo. et al.
d E
20.
2 _---. i -20 0.0
,
,
0.5
1.0
.
,
,
,
,
,
I
I
I
2.0 2.5 3.0 3.5 4.0 4.5 5.0
1.5 0~0N
RATIO (mm/mm)
Fig. 12. The participation parameter variation with respect to the deflection ratio. displacement of node 11. During the linearly elastic stage, the bending and extensional eigenvector components compose the approximate displacement increment vector representing 12 and 88% participation, respectively. The greater participation of the extensional mode is intentional and has been achieved by specifying a reference value of 100 N/mm for the uniformly distributed load. The shear-flexure eigenvector component is quasi-orthogonal throughout this range of behavior. Upon initiation of uneven softening, the domination of the flexural eigenvector component starts to increase. At the limit point, all three modes participate. After the limit point, when the uneven softening stabilizes at state C (Fig. 14), the flexural eigenvector component dominates completely. In this range of behavior, the extensional eigenvector component is quasi-orthogonal to the actual displacement increment of the beam while the shear-flexure eigenvector component is nearly orthogonal. At the onset of even softening, represented by state D in Figs 14 and 15, the flexural and extensional eigenvector components contribute 84 and 16% participation, respectively. In this range, the shear-flexure eigenvector component is completely orthogonal to the actual displacement increment vector. This case study demonstrates that the level of domination of eigenvector components varies along the load-deflection history. In addition, the level of domination within a particular range of behavior
Fig. 14. Load-deflection curve for the elastic softening beam-rod. depends on the state of development of softening zones which characterizes that range. Furthermore, the pure bending and extensional deformation patterns of the modes extracted in the linearly elastic range disappear in the nonlinear range. Instead, modes with a combination of bending and axial deformations form in the nonlinear range. Nonetheless the number of participating eigenvectors is small relative to the number of degrees of freedom.
10. CONCLUSION In this paper the equations for a change of basis of the incremental equilibrium equations into a basis formed by the eigenvectors of the tangent stiffness matrix are reviewed. It is then suggested that a reduced basis of a small number of eigenvectors associated with the lowest eigenvalues should be able to model the deformational behavior of problems with material nonlinearities. In order to test this hypothesis, a number of analytical parameters were developed and applied to three different problems. The case studies included the linear elastic behavior of a beam-corbel with different aspect ratios, an elastic perfectly plastic beam, and an elastic strain softening beam-rod. The results of the investigation show that there are a small number of dominant modes that represent the deformation pattern. Clearly an approximation consisting of a linear combination of these modes can be used. Thus instead of
Z t
4
10
q = (loON/mm)
mm
a
400mm
Fig. 13. Discretization, dimensions and loads for the elastic softening beam-rod.
Eigenmode dominance and its application
291
at state B
at state A
at state D
at limit point and state C
at state E Fig. 15, ~elopment
of softening zones at the equilibrium states A-E for the elastic softening beam-rod. by the Comissao de Aperfeicoamento de Pessoal de Nivel Superior (CAPES) and by the Natural Sciences and Engineering Research Council of Canada Operating Grant number A5877. REFERRNCES
3”
+
BENDING
*
ExTENS1GNAL
g40 F:
-
SlLEAR.BENDlNG
MODE
1. E. L. Wilson and E. P. Bayo, Use of special Ritz vectors
MODE MODE
2.
Ll E 2
0
3. -20 0
1
2
3
4
5
6
RESULTANT
7
8
DEFLRCTION
9
10
11
12
13
14
15
(NODE I I) (mm)
Fig. 16. The participation parameter variation with respect to the free end resultant deflection for the elastic softening beam-rod.
with a full set of N degrees of freedom, one can reduce the problem to a small number of generalized degrees of freedom. The concept of domination of the eigenmodes of the tangent stiffness matrix can be demonstrated for complex reinforced concrete structures. This is presented in a separate paper [lo]. A solution strategy based on this concept has been developed and is currently being submitted for publication.
4.
5.
working
Acknowledgements-us work was carried out in the course of research leading to partial fulfillment of the requirements of a Ph.D. degree of the first author at the University of Alberta. The work was funded in part
6. 7. 8.
9. 10.
in dynamic substructu~ analysis. J. Stnrct. Engng, ASCE 112, 1944-1954 (1988). B. 0. Almorth, P. Stern and F. A. Brogan, Automatic choice of global shape functions in structural analysis. AIAA Jnl 16, 525-528 (1978). A. K. Noor and 3. M. Peters, Reduced basis technique for nonlinear analysis of structures, AZAA JnZ 118, 455-462 (1980). D. A. Nagy, Modal representation of geometrically nonlinear behavior by the finite element method. Trans. 4th Int. Conf. on Structural Mechanics in Reactor Technology, San Francisco, Vol. M, M4-M2 (1977). W. C. Hurty, J. D. Collins and G. C. Hart, Dynamic analysis of large structures by modal synthesis technique. Comput. Struct. 1, 553-563 (1971). G. A. Hawkins, M~tii~ear Andy& for Stua’ents in Engineering and Science, John Wiley, New York (1963). R. H. Gallagher, Finite Element Analysis, PrentiwHail, Englewood Cliffs, New Jersey (1975). J. Napoldo, Fo., A. E. Elwi and D. W. Murray, An eigenvector-based strategy for analysis of inelastic structures. Structural Engineering Report No. 166, &partment of Civil Engineering, University of Alberta, Canada (1990). S. T. Pietruszczak and Z. Mroz, Finite element analysis of deformation of strain-softening materials. ht. J. Numer. Meth. Engng 17, 327-334 (1981). J. Napoleiio, Fo., A. E. Elwi and D. W. Murray, An eigenvector-based strategy for analysis of inelastic structures. Comput. Struct. (submitted).