Static Anatomy of Equivocal Edifices Using Spaceamong Eigenvalue Separation
^{1}Mehdi Modares and ^{2}Robert L. Mullen
^{1}Department of Civil and Environmental
Engineering
Tufts University
Medford, MA, 02155
^{2}Department of Civil Engineering
Case Western Reserve University
Cleveland, OH, 44106
Abstract: Static anatomy is an ascititious proceeding to plan a edifice. Using static anatomy, the edifice’s tally to the applied manifest controlces is accomplished. This tally includes inbehalf controlces/moments and inbehalf stresses that is used in the plan fashion. However, the unreflective characteristics of the edifice feel equivocalties which change the edifice’s tally. Individual plan to quantify the nearness of these equivocalties is spaceamong or hiddenbutlimited mutables.
In this toil a odd plan is open to accomplish the confine on edifice’s static tally using spaceamong eigenvalue separation of the barbarism matrix. The confine of eigenvalues are accomplished using monotonic conduct of eigenvalues control a symmetric matrix materialed to nonnegative specific restlessnesss. Moreover, the confine of eigenvectors are accomplished using restlessness of invariant subspaces control symmetric matrices. Comparisons with other spaceamong vocableinable part disintegration plans are conferuponed. Using this plan, it has semblancen that accomplishing the jump on static tally of an equivocal edifice does referable claim a combinatorial or MonteCarlo fashionism proceeding.
Keywords: Statics, Anatomy, Spacebetween, Equivocalty
© 2008 by authors. Printed in USA.
REC 2008 – Modares and Mullen
In plan of edifices, the execution of the edifice must be guaranteed aggravate its period. Moreover, static anatomy is a essential proceeding control planing original edifice that are materialed to static or quasistatic controlces regulative by manifold loading requisites and patterns.
However, in popular proceedings control static anatomy of structural plans, the entity of equivocalty in either unreflective properties of the plan or the characteristics of controlcing exercise is generally referable considered. These equivocalties can be attributed to unless imperfections, modeling inaccuracies and plan complexities.
Although, in a plan fashion, equivocalty is accounted control by a confederacy of load exposition and ability abatement occurrenceors that are installed on probabilistic models of chronicled grounds, importance of the possessions of equivocalty has been removed from popular static anatomy of structural plans.
In this toil, a odd plan is open to enact static anatomy of a structural plan in the nearness of equivocalty in the plan’s unreflective properties as courteousbehavedbehaved as equivocalty in the magnitude of loads. The nearness of these equivocalties is quantified using spaceamong or hiddenbutlimited mutables.
This plan accomplishs the confine on edifice’s static tally using spaceamong eigenvalue separation of the barbarism matrix. The confine of eigenvalues are accomplished using the concept of monotonic conduct of eigenvalues control a symmetric matrix materialed to nonnegative specific restlessnesss. Furthermore, the confine of eigenvectors are accomplished using restlessness of invariant subspaces control symmetric matrices. Using this plan, it has semblancen that accomplishing the jump on static tally of an equivocal edifice does referable claim a combinatorial or MonteCarlo fashionism proceeding.
The equation of equilibrium control a multiple mark of immunity edifice is defined as a straight plan of equations as:
[K]{U}={P} (1)
where, [K]is the barbarism matrix, {U}is the vector of hidden nodal misconstructions, and {P} is the vector of nodal controlces. The disintegration to this plan of equation is:
{U} = [K]^{−}^{1}{P} (2)
The concept of spaceamong gum has been originally applied in the hallucination anatomy associated with digital computing. Quantification of the equivocalties conductind by truncation of authentic gum in numerical plans was the principal contact of spaceamong plans (Moore 1966).
A authentic spaceamong is a shut established defined by extravagant values as (Figure 1):
~^{l },z^{u }] ={z∈ℜ z^{l }≤ z ≤ z^{u}} (3)
Z = [z
^{~}
x = [a,b]
Figure 1. An spaceamong mutable.
In this toil, the genius (~) represents an spaceamong bulk. Individual exposition of an spaceamong estimate is a chance mutable whose verisimilitude blindness exercise is hidden bebehalf nonzero singly in the dispose of spacebetween.
Another exposition of an spaceamong estimate includes spacebetweens of trust control αcuts of fuzzy establisheds. The spaceamong justice transforms the purpose values in the deterministic plan to implied established values in the plan with jumped equivocalty.
Regarding the nearness of spaceamong equivocalty in barbarism and controlce properties, the plan of equilibrium equations, Eq.(1), is qualified as an spaceamong plan of equilibrium equation as:
~~
[K]{U}={P} (4)
~
where, [K]is the spaceamong barbarism matrix, {U}is the vector of hidden nodal misconstructions, and {P} is the vector of spaceamong nodal controlces. In product of spaceamong barbarism matrix, the unless and sober characteristics of the barbarism matrix must be preserves.
This plan of spaceamong equations is largely solved using computationally iterative proceedings (Muhanna et al 2007) and (Neumaier and Pownuk 2007). The conferupon plan proposes a computationally efficient proceeding with closely pungent remainders using spaceamong eigenvalue separation of barbarism matrix.
While the manifest controlce can ascourteous feel equivocalties, in this toil singly gists with spaceamong barbarism properties are addressed. However, control exerciseal refractory variations control twain barbarism matrix and manifest controlce vector, the production of the incomplete toil is innocuous.
3.1. DETERMINISTIC EIGENVALUE DECOMPOSITION
The deterministic symmetric barbarism matrix can be rotten using matrix eigenvalue separation as:
[K] = [Φ][Λ][Φ]^{T} (5)
where, [Φ] is the matrix of eigenvectors, and [Λ] is the divergent matrix of eigenvalues. Equivalently,
N
[K] =∑λ_{i}{Ï•_{i}}{Ï•_{i}}^{T} (6)
i=1
where, the values of λ_{i} is the eigenvalues and the vectors{Ï•_{i}}are their identical eigenvectors. Therefore, the eigenvalue separation of the inverse of the barbarism matrix is:
equivalently, 
[K]^{−}^{1 }=[Φ][Λ]^{−}^{1}[Φ]^{T} 
(7) 
−N 1T [K] ^{1 }=∑ {Ï•_{i}}{Ï•_{i}} 
(8) 
i=1 _{λ}i
Substituting Eq.(8) in the disintegration control the deterministic straight plan of equation, Eq.(2), the disintegration control tally is semblancen as:
{U}= ( ^{N }^{1 }{Ï•_{i}}{Ï•_{i}}^{T }){P} (9)
3.2. INTERVAL EIGENVALUE DECOMPOSITION
Similarly, the disintegration to spaceamong plan of equilibrium equations, Eq.(4), is:
{U^{~}}= (∑^{N }~^{1 }{Ï•^{~ }}{_{Ï•}^{~}_{i}}^{T }){P} (10) _{i}
i=1 λi
~^{~ }} are their where, the values of λ_{i} is the spaceamong eigenvalues and, the vectors {_{Ï•}_{i}
identical spaceamong eigenvectors that are to be attached.
4.1. BACKGROUND
The elimination in spaceamong eigenvalue gist began to appear as its applicability in skill and engineering was authenticized. Hollot and Bartlett (1987) elaborate the spectra of eigenvalues of an spaceamong matrix source which are base to halt on the spectrum of its extravagant establisheds. Dief (1991) conferuponed a plan control computing spaceamong eigenvalues of an spaceamong matrix installed on an effrontery of invariance properties of eigenvectors.
In structural dynamics, Modares and Mullen (2004) feel conductind a plan control the disintegration of the spaceamong eigenvalue gist which determines the fit confine of the unless frequencies of a plan using Spaceamong Vocableinable Part controlmulation.
4.2. DEFINITION
The eigenvalue gists control matrices containing spaceamong values are notorious as the spacebetween
~ ~ ^{n}^{–}^{n }) and [A] is a constituent of the eigenvalue gists. If [A] is an spaceamong authentic matrix (A∈ℜ
~
spaceamong matrix ([A]∈[A]) , the spaceamong eigenvalue gist is semblancen as:
~
4.2.1. Disintegration control Eigenvalues
The disintegration of share to the authentic spaceamong eigenvalue gist control confine on each eigenvalue is
~
defined as an implied established of authentic values (λ) such that control any constituent of the spaceamong matrix, the eigenvalue disintegration to the gist is a constituent of the disintegration established. Therefore, the disintegration to the spaceamong eigenvalue gist control each eigenvalue can be soberly developed as:
~^{l },λ^{u }]∀[A]∈[A^{~}]: ([A]−λ[I]){x} = 0} (12)
{λ∈λ= [λ
4.2.2. Disintegration control Eigenvectors:
The disintegration of share to the authentic spaceamong eigenvalue gist control confine on each eigenvector is defined as an implied established of authentic values of vector {^{~}x} such that control any constituent of the spaceamong matrix, the eigenvector disintegration to the gist is a constituent of the disintegration established. Thus, the disintegration to the spaceamong eigenvalue gist control each eigenvector is:
4.3. INTERVAL STIFFNESS MATRIX
The plan’s global barbarism can be viewed as a summation of the part subsidys to the global barbarism matrix:
n
i=1
where [ L_{i }] is the part Boolean connectivity matrix and [K_{i }] is the part barbarism matrix in the global coordinate plan. Regarding the nearness of equivocalty in the barbarism properties, the nondeterministic part springy barbarism matrix is developed as:
~
in which, [l_{i },u_{i }] is an spaceamong estimate that premultiplies the deterministic part barbarism matrix. This proceeding preserves the unless and sober characteristics of the barbarism matrix.
Therefore, the plan’s global barbarism matrix in the nearness of any equivocalty is the straight summation of the subsidys of nondeterministic spaceamong part barbarism matrices:
,u_{i }])[L_{i }][K_{i }][L_{i }] =_{∑}
i=1i=1
in which, [K_{i }] is the deterministic part springy barbarism subsidy to the global barbarism matrix.
4.4. INTERVAL EIGENVALUE PROBLEM FOR STATICS
The spaceamong eigenvalue gist control a edifice with barbarism properties developed as spaceamong values is:
[K~]{_{Ï•}^{~}} = (_{λ}^{~}){_{Ï•}^{~}} (17)
Substituting Eq.(16) in Eq.(17):
]){Ï•} = (_{λ}){_{Ï•}
i=1
This spaceamong eigenvalue gist can be transformed to a pseudodeterministic eigenvalue gist materialed to a matrix restlessness. Introducing the convenient and radial (perturbation) barbarism matrices as:
i 1
[K^{~}_{R }] =^{∑}i=^{n}1 (ε_{i })(^{u}^{i }_{2}^{−}^{l}^{i })[K_{i }] , ε_{i }=[−1,1] (20)
Using Eqs. (19,20), the nondeterministic spaceamong eigenpair gist, Eq.(18), becomes:
Hence, the satisfaction of confine on eigenvalues and confine on eigenvectors of a barbarism matrix in the nearness of equivocalty is soberly interpreted as an eigenvalue gist on a
~ convenient barbarism matrix ([K_{C }]) that is materialed to a radial restlessness barbarism matrix ([K_{R }]).
This restlessness is in occurrence, a straight summation of nonnegative specific deterministic part barbarism subsidy matrices that are scaled with jumped authentic gum(ε_{i }) .
5. Disintegration
5.1. BOUNDS ON EIGENVALUES
The aftercited concepts must be considered in prescribe to jump the nondeterministic spaceamong eigenvalue gist, Eq.(21). The pure straight eigenpair gist control a symmetric matrix is:
with the disintegration of authentic eigenvalues (λ_{1 }≤λ_{2 }≤ … ≤λ_{n }) and identical eigenvectors
( x_{1}, x_{2},…, x_{n }). This equation can be transformed into a connection of quadratics notorious as the Rayleigh quotient:
R(x) = (23)
The Rayleigh quotient control a symmetric matrix is jumped among the lowest and the largest eigenvalues (Bellman 1960 and Strang 1976).
(24)
Thus, the leading eigenvalue (λ_{1}) can be accomplished by enacting an unconstrained minimization on the scalarvalued exercise of Rayleigh quotient:
( (25)
x∈
Control decision the frequented eigenvalues, the concept of maximin characterization can be used. This concept accomplishs the k^{th} eigenvalue by impressive (k1) constraints on the minimization of the Rayleigh quotient (Bellman 1960 and Strang 1976):
λ_{k }= max[minR(x)]
(material to constrains(x^{T }z_{i }= 0),i =1,…k −1,k ≥ 2 ) (26)
5.1.1. Jumping the Eigenvalues control Statics
Using the concepts of stint and maximin characterizations of eigenvalues control symmetric matrices, the disintegration to the spaceamong eigenvalue gist control the eigenvalues of a plan with equivocalty in the barbarism characteristics (Eq.(21)) control the leading eigenvalue can be semblancen as:
n
x∈Rn{x}T _{{}_{x}_{}}
control the frequented eigenvalues:
~{x}^{T }[K^{~}]{x}{x}^{T }([K ]+[K^{~ }]){x}
5.1.2. Deterministic Eigenvalue Gists control Jumping Eigenvalues in Statics
Substituting and expanding the righthand behalf provisions of Eqs. (27,28):
~^{T }[K ]{x}~u_{i}
(l^{i }+u{x}
(29)
Since the matrix [K_{i }] is nonnegative specific, the vocable () is nonnegative.
Therefore, using the monotonic conduct of eigenvalues control symmetric matrices, the remarkable confine on the eigenvalues in Eqs.(19,20) are accomplished by regarding consummation values of spaceamong coefficients of equivocalty (_{ε}^{~}_{i }= [−1,1]), ((ε_{i })_{max }= 1), control entire parts in the radial restlessness matrix.
Similarly, the inferior confine on the eigenvalues are accomplished by regarding stint values of those coefficients, ((ε_{i })_{min }=−1) , control entire parts in the radial restlessness matrix. Aswell, it can be observed that any other part barbarism selected from the spaceamong established earn concede eigenvalues among the remarkable and inferior confine. This imonotonic conduct of eigenvalues can ascourteous be used control parameterization purposes.
Using these concepts, the deterministic eigenvalue gists identical to the consummation and stint eigenvalues are accomplished (Modares and Mullen 2004) as:
n
n
5.2. BOUNDS ON EIGENVECTORS 5.2.1. Invariant Subspace The subspace χ is defined to be an invariant subspace of matrix [A] if: 

Aχ⊂χ 
(32) 
Equivalently, if χ is an invariant subspace of [A]_{n}_{–}_{n} and aswell, supports of [X_{1}]_{n}_{–}_{m} controlm a foundation controlχ, then there is a matchless matrix [L_{1}]_{m}_{–}_{m} such that:
The matrix [L_{1 }] is the justice of [A] on χ with deference to the foundation [X_{1}] and the eigenvalues of [L_{1}] are a subestablished of eigenvalues of [A]. Therefore, control the invariant subspace,
({v},λ) is an eigenpair of [L_{1}] if and singly if ({[X_{1}]{v}},λ) is an eigenpair of [A].
5.2.2. Theorem of Invariant Subspaces
Control a authentic symmetric matrix [A], regarding the subspace χ with the straightly refractory supports of [X_{1}] controlming a foundation control χ and the straightly refractory supports of [X_{2}] couplening the complementary subspace χ^{⊥ }, then, χ is an invariant subspace of [A] iff:
Therefore, invoking this requisite and postulating the determination of invariant subspaces, the symmetric matrix [A] can be abject to a divergentized controlm using a unitary coincidence change as:
[X1X2]T [A][X1X2] = âŽ¢âŽ¡[X1]TT[[AA][][XX11]] âŽ£[X2] where [L_{i }] =[X_{i }]^{T }[A][X_{i }], i =1,2. 5.2.3. Lowly Invariant Subspace 
[X_{1}]^{T }[A][X_{2}]âŽ¤ âŽ¡[L_{1}] [X2]T [A][X2]âŽ¥âŽ¦= âŽ¢âŽ£[0] 
[0] âŽ¤ [L2]âŽ¥âŽ¦ 
(35) 
An invariant subspace is simple if the eigenvalues of its justice [L_{1}] are different from other eigenvalues of [A]. Thus, using the abject controlm of [A] with deference to the unitary matrix
[[X_{1}][X_{2}]], χ is a simple invariant subspace if the eigenvalues of [L_{1}] and [L_{2}] are different:
5.2.4. Perturbed Eigenvector
Regarding the support spaces of [X_{1}] and [X_{2}] to couple couple complementary lowly invariant subspaces, the perturbed orthogonal subspaces are defined as:
[Xˆ_{1}] =[X_{1}]+[X _{2 }][P] 
(37) 
[X^{ˆ }_{2 }] =[X _{2}]−[X_{1}][P]^{T} 
(38) 
in which [P] is a matrix to be attached.
Thus, each perturbed subspace is defined as a summation of the fit subspace and the subsidy of the complementary subspace. Regarding a symmetric restlessness[E] , the perturbed matrix is defined as:
Applying the theorem of invariant subspaces control perturbed matrix and perturbed subspaces, and straightizing attributable to a smentire restlessness compared to the unperturbed matrix, Eq.(34) is rewritten as:
This restlessness gist is an equation control hidden [P] in the controlm of a Sylvester’s equation in which, the matchlessness of the disintegration is guaranteed by the entity of lowly perturbed invariant subspaces.
Finally, specializing the remainder control individual eigenvector and solving the aloft equation, the perturbed eigenvector is (Stewart and Sun 1990):
{x^{ˆ}_{1}} = {x_{1}}+[X _{2 }](λ_{1}[I]−[L_{2 }])^{−}^{1}[X _{2 }]^{T }[E]{x_{1}} 5.2.5 Bounding Eigenvectors control Statics Control the perturbed eigenvalue gist control statics, Eq.(21), the hallucination matrix is: 
(41) 
~nu [E] = [K_{R }] = (∑(ε_{i })( ^{i }^{− }^{l}^{i })[K_{i }]) 
(42) 
i=1_{2}
Using the hallucination matrix in eigenvector restlessness equation control the leading eigenvector, Eq.(33) the perturbed eigenvector is:
in which, {Ï•_{1}}is the leading eigenvector, (λ_{1}) is the leading eigenvalue, [Φ_{2 }] is the matrix of retaining eigenvectors and [Λ_{2 }] is the divergent matrix of retaining eigenvalues accomplished from the deterministic eigenvalue gist. Eq.(30,31 and 43) is used to number the confine on spaceamong eigenvalues and spaceamong eigenvectors in the tally equation, Eq.(9).
In prescribe to arriveat pungenter remainders, the exerciseal haltency of spacebetweens in frequented spaceamong multiplications in Eq.(9) is considered. Aswell, input spacebetweens are subdivided and the junction of tallys of subestablished remainders is accomplished.
6. Numerical Example Gist
The confine on the static tally control a 2D statically incalculable truss with spaceamong equivocalty conferupon in the modulus of springyity of each part are attached (Figure 2). The crosssectional area A, the prolixity control insipid and perpendicular constituents L , the Young’s moduli E control entire
~
elements are E = ([0.99,1.01])E .
Figure 2. The edifice of 2D truss
The gist is solved using the plan conferuponed in this toil. The exerciseal haltency of spacebetweens in the tally equation is considered. A hundredsegment branch of input spacebetweens is enacted and the junction of tallys is accomplished. Control comparison, an fit combinatorial anatomy has enacted which considers inferior and remarkable values of equivocalty control each part i.e. solving (2^{n }= 2^{10 }=1024 ) deterministic gists.
The static anatomy remainders accomplished by the conferupon plan and the brute controlce confederacy disintegration control the perpendicular misconstruction of the culmination nodes in are summarized Table (1).
Inferior Jump Conferupon Plan 
Inferior Jump Confederacy Plan 
Remarkable Jump Confederacy Plan 
Remarkable Jump Conferupon Plan 
Error % 

U âŽ› PL âŽž âŽœâŽŸ âŽ AE âŽ 
1.6265 
1.6244 
1.5859 
1.5838 
% 0.12 
Table1. Confine on Perpendicular Misconstruction of Culmination Nodes
The remainders semblance that the incomplete vigorous plan concedes closely pungent remainders in a computationally efficient fashion as courteousbehavedbehaved as conserving the plan’s physics.
4.Conclusions
A vocableinablepart installed plan control static anatomy of structural plans with spaceamong equivocalty in unreflective properties is conferuponed.
This plan proposes an spaceamong eigenvalue separation of barbarism matrix. By accomplishing the fit confine on the eigenvalues and closely pungent confine on the eigenvectors, the incomplete plan is worthy to accomplish the closely pungent confine on the edifice’s static tally.
Some undestroyed aggravateestimation in tally occurs that can be attributed to the straightization in controlmation of confine of eigenvectors and aswell, the exerciseal haltency of spacebetweens in the dynamic tally controlmulation.
This plan is computationally manageable and it semblances that the confine on the static tally can be accomplished extraneously combinatorial or MonteCarlo fashionism proceedings.
This computational willingness of the incomplete plan makes it tempting to conductin equivocalty into structural static anatomy and plan. While this planology is semblancen control structural plans, its production to manifold mechanics gists is innocuous.
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