Francesca Passarella
email: passarella@diima.unisa.it

Vittorio Zampoli
email: zampoli@diima.unisa.it

Spatial estimates for transient and steady-state solutions in transversely isotropic plates of Mindlin-type

Keywords: Mindlin plates, Bending, Transverse isotropy, Strong elliptic elasticity tensor, Spatial behaviour

This paper concerns with a state of bending for a linear transversely isotropic plate model based on the Mindlin assumption on the displacement. By using appropriate families of lineintegral measures, we are able to establish results about the spatial behaviour of transient and steady-state solutions. All the results are obtained under relaxed hypotheses on the positive definiteness of the elasticity tensor.

Introduction

The theory concerning elastic plates is useful in a wide range of practical applications, from building materials to electronic production; for this reason, properties of such particular mechanical structure are investigated in several articles, see e.g. [START_REF] Naghdi | The Theory of Shells and Plates. Vol. VI a/2 of "Handbuch der Physik[END_REF], [START_REF] Green | The linear theory of an elastic Cosserat plate[END_REF], [START_REF] Lagnese | Modelling, analysis and control of thin plates[END_REF]. The classical linear theory based on Kirchhoff's elastic strain-displacement relations completely neglects the effects of transverse shear forces, cf. [START_REF] Lagnese | Modelling, analysis and control of thin plates[END_REF] and [START_REF] Nowinski | Theory of thermoelasticity with applications[END_REF]. For this reason, mathematical discrepancies due to such assumption have led studies toward increasingly refined models that take into account not only the deflection of plate's middle section, but also transverse shear deformations.

The following assumptions on the displacement field characterize the Mindlin-type thin elastic plate theory, cf. [START_REF] Mindlin | Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates[END_REF] and [START_REF] Reissner | On the theory of bending of elastic plates[END_REF] u α (x 1 , x 2 , x 3 , t) = u 0 α (x 1 , x 2 , t) + x 3 v α (x 1 , x 2 , t), u 3 (x 1 , x 2 , x 3 , t) = w(x 1 , x 2 , t),

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

with respect to rectangular coordinates x 1 , x 2 , x 3 such that the x 1 x 2 -plane represents the middle plane of the plate. Due to the linearity of the problem in concern, we can consider separately the extensional motion, characterized by u 0 1 and u 0 2 , and the state of bending, represented by v 1 , v 2 and w. In this context, we can remark that the theory based on the Mindlin-model is developed by [START_REF] Constanda | A mathematical analysis of bending of thin plates with transverse shear deformation[END_REF] for the elastostatic bending of a thin plate, including the effects of transverse shear deformation. Furthermore, a dynamic model for small deformations of a thin thermoelastic plate is described by [START_REF] Schiavone | Thermal effects in Mindlin-type plates[END_REF].

The Reissner-Mindlin and Kirchhoff-Love models are the two most common models of a thin linearly elastic plate. It is often remarked in the engineering literature that the Reissner-Mindlin model is more accurate, particularly for thin plates and when transverse shear plays a significant role, see [START_REF] Hughes | The finite element method: Linear static and dynamic finite element analysis[END_REF]. In fact, both [START_REF] Mindlin | Influence of rotatory inertia and shear on flexural motions of isotropic elastic plates[END_REF] and [START_REF] Reissner | The effect of transverse shear deformation on the bending of elastic plates[END_REF][START_REF] Reissner | On bending of elastic plates[END_REF] have independently proposed a plate theory that incorporates the effect of transverse shear deformation for analyzing thick plates.

Moreover, [START_REF] Arnold | On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models[END_REF] show that the Reissner-Mindlin plate bending model has a wider range of applicability than the Kirchhoff-Love model for the approximation of clamped linearly elastic plates. In fact, under the assumption that the body force density is constant in the transverse direction, they prove that the Reissner-Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads.

On the other hand, the theory for transversely isotropic materials shows a good applicability for plate mathematical models that take into account transverse shear deformations (see, for details, the paper by [START_REF] Paroni | The Reissner-Mindlin plate theory via Γ-convergence[END_REF]). Transversely isotropic materials, characterized by 5 elastic constants, have interesting applications in civil, mechanical and aerospace engineering. The main feature of such structures is represented by the existence of an axis of rotational symmetry, that is an axis for which any rotation about it does not change its properties; the plane perpendicular to this axis of rotational symmetry is called plane of isotropy. An example for this kind of materials is represented by laminates made of randomly oriented chopped fibers placed in a certain plane; mechanical properties of a bundled structure have no preferential direction in that plane. Each plane containing the axis of rotation is a plane of symmetry and therefore transversely isotropic materials admit an infinite number of elastic symmetries. Properties of such materials are widely investigated by [START_REF] Gurtin | Linear theory of elasticity[END_REF], [START_REF] Padovani | Strong ellipticity of transversely isotropic elasticity tensors[END_REF], [START_REF] Merodio | A note on strong ellipticity for transversely isotropic linearly elastic solids[END_REF][START_REF] Chirit ¸ȃ | On the strong ellipticity condition for transversely isotropic linearly elastic solids[END_REF][START_REF] Chirit ¸ȃ | On the strong ellipticity of the anisotropic linearly elastic materials[END_REF] and [START_REF] Zhang | Elasticity of transversely isotropic materials[END_REF]. In addition, [START_REF] Simmonds | Exact Lévy-type solution for the plate bending exist for transversely isotropic but not for general monoclinic materials[END_REF] obtains an exact 3-D Lévy-type solution for bending of an elastic slab taking into account transversal isotropy. In particular, [START_REF] Merodio | A note on strong ellipticity for transversely isotropic linearly elastic solids[END_REF]Ogden (2003), Chirit ¸ȃ (2006) In the present work, we take into account the linear theory characterizing a state of bending for (bounded and unbounded) plates of Mindlin-type, under the strong ellipticity condition on the elasticity tensor. In particular, using the time-weighted surface power function method (cf. Chirit ¸ȃ and [START_REF] Chirit ¸ȃ | Time-weighted surface power function method for the study of spatial behaviour in dynamics of continua[END_REF][START_REF] Chirit ¸ȃ | Some further growth and decay results in linear thermoelastodynamics[END_REF] and [START_REF] Ciarletta | Some results on the spatial behaviour in linear porous elasticity[END_REF]), we derive the first-order differential inequalities satisfied by a family of appropriate integral measures depending on a specific parameter. Then, we obtain a spatial decay estimate of Saint-Venant's type with time-independent decay rate and we prove the existence of a domain of influence.

Moreover, we consider a (bounded and semi-infinite) strip subjected to prescribed harmonic data on its end. Starting from an idea shown by [START_REF] Ciarletta | Some results on the spatial behaviour in linear porous elasticity[END_REF] and Chirit ¸ȃ (1995), we introduce a set of appropriate line-integral measures associated with the amplitude of time-harmonic vibrations. We derive a differential inequality describing the behaviour of steady-state solutions under the hypothesis that frequency of harmonic vibrations is lower than a certain critical value.

Formulation of the problem

Throughout this paper, we study the behaviour of a homogeneous and transversely isotropic elastic solid that occupies at time t = 0 the right cylinder B of length 2h with the (bounded and unbounded) cross-section Σ and the smooth lateral boundary Π. We call B and Σ be the interiors of B and Σ, we choose the rectangular Cartesian coordinate frame in such a way that Ox 1 x 2 is the middle plane and thus the faces of the plate are situated at x 3 = ±h. Moreover, we suppose that Σ is a simply connected region, L is the boundary of Σ and that h diam Σ.

We employ the usual summation and differentiation conventions. Latin subscripts (unless otherwise specified) are understood to range over the integers {1, 2, 3}, whereas Greek subscripts are confined to the range {1, 2}; summation over repeated subscripts is implied. Superposed dots or subscripts preceded by a comma mean partial derivative with respect to the time or the corresponding Cartesian variables. In this connection, we will disregard regularity questions, simply understanding a degree of smoothness sufficient to ensure analysis to be valid.

According to the linear theory of elastodynamics, the field equations are equations of motion stress-strain relation strain-displacement relation

t ji,j + f i = ρü i , on B × (0, ∞) t ij = C ijkl e kl e ij = 1 2 (u i,j + u j,i ) (1)
where u i are the components of the displacement vector, t ij are the components of the stress tensor, f i are the components of the body force vector, ρ is the reference mass density and C ijkl are the components of the symmetric elasticity tensor. We assume (see [START_REF] Gurtin | Linear theory of elasticity[END_REF]) that the direction of

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transversal isotropy is the one indicated by the x 3 coordinate axis and that

c ij = C iijj = c ji , i,j (not summed) ∈ {1, 2, 3}, c 11 = c 22 , c 13 = c 23 , c 44 = c 55 = C 2323 = C 1313 , c 66 = C 1212 = c 11 -c 12 2 .
(2)

Apart from the terms obtained using symmetries (2), these are the only non-zero components of C ijkl .

We consider the class of transversely isotropic materials having a strongly elliptic elasticity tensor, that is

C ijkl m i m k n j n l > 0 for every non-zero vector m i , n i .
The strong ellipticity condition becomes

c 66 (n 1 m 2 -n 2 m 1 ) 2 + c 11 (n 1 m 1 + n 2 m 2 ) 2 + 2 (c 13 + c 44 ) (n 1 m 1 + n 2 m 2 ) n 3 m 3 + +c 33 n 2 3 m 2 3 + c 44 n 2 3 m 2 1 + n 2 1 m 2 3 + n 2 3 m 2 2 + n 2 2 m 2 3 > 0
or equivalently (see e.g. [START_REF] Merodio | A note on strong ellipticity for transversely isotropic linearly elastic solids[END_REF] and Chirit ¸ȃ ( 2006))

c 11 > 0, c 11 > c 12 , c 33 > 0, c 44 > 0, |c 13 + c 44 | < c 44 + √ c 11 c 33 .
(3) [START_REF] Gurtin | Linear theory of elasticity[END_REF] proves that C is positive definite if

C ijkl φ ij φ kl > 0 for every non-zero symmetric tensor φ ij
and then it is strongly elliptic too.

A state of bending for an elastic plate is characterized by the following relations

u α (x 1 , x 2 , x 3 , t) = -u α (x 1 , x 2 , -x 3 , t), u 3 (x 1 , x 2 , x 3 , t) = u 3 (x 1 , x 2 , -x 3 , t)
for every (x 1 , x 2 , x 3 ) ∈ B, t ∈ [0, ∞); moreover, we assume that the body loads obey to the relations

f α (x 1 , x 2 , x 3 , t) = -f α (x 1 , x 2 , -x 3 , t), f 3 (x 1 , x 2 , x 3 , t) = f 3 (x 1 , x 2 , -x 3 , t).
In the context of the theory of thin plates of uniform thickness such that u i vary smoothly with respect to x 3 , we consider as independent variables:

v α (x 1 , x 2 , t) = 3 2h 3 h -h x 3 u α (x 1 , x 2 , x 3 , t)dx 3 , w(x 1 , x 2 , t) = 1 2h h -h u 3 (x 1 , x 2 , x 3 , t)dx 3 .
If we use the following notations

M ij (x 1 , x 2 , t) = 3 2h 3 h -h x 3 t ij (x 1 , x 2 , x 3 , t)dx 3 , τ ij (x 1 , x 2 , t) = 1 2h h -h t ij (x 1 , x 2 , x 3 , t)dx 3 , F (x 1 , x 2 , t) = 1 2h h -h f 3 (x 1 , x 2 , x 3 , t) dx 3 + 1 h t 33 (x 1 , x 2 , h, t), H α (x 1 , x 2 , t) = 1 2h h -h x 3 f α (x 1 , x 2 , x 3 , t) dx 3 + t 3α (x 1 , x 2 , h, t); A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
we obtain M α3 = 0, τ αβ = 0, τ 33 = 0 since t αβ , t 33 are odd functions and t α3 is even with respect to

x 3 and we get the equations of motion

h 2 3 M βα,β -τ α3 + H α = ρ h 2 3 vα , τ β3,β + F = ρ ẅ on Σ × (0, ∞). ( 4
)
In the present paper, we restrict our attention to the state of bending characterized by

u α (x 1 , x 2 , x 3 , t) = x 3 v α (x 1 , x 2 , t), u 3 (x 1 , x 2 , x 3 , t) = w(x 1 , x 2 , t), ( 5
)
and we assume that

F (x 1 , x 2 , t) = 0, H α (x 1 , x 2 , t) = 0.
Now, we can write the following equations of motion

h 2 3 M βα,β -τ α3 = ρ h 2 3 vα , τ β3,β = ρ ẅ on Σ × (0, ∞), ( 6
)
the constitutive equations

M 11 = c 11 11 + c 12 22 , M 12 = M 21 = (c 11 -c 12 ) 12 , M 22 = c 12 11 + c 11 22 , τ α3 = τ 3α = 2c 44 ξ α , (7)
and the geometrical equations

αβ = 1 2 (v α,β + v β,α ), ξ α = 1 2 (w ,α + v α ) . ( 8
)
The system ( 6) -( 8) can be rewritten in terms of {v α , w}

h 2 3 c 11 -c 12 2 v α,ββ + h 2 3 c 11 + c 12 2 v β,βα -c 44 (w ,α + v α ) = ρ h 2 3 vα , c 44 w ,ββ + c 44 v β,β = ρ ẅ. ( 9
)
We consider the problem P defined by ( 9), the homogeneous initial conditions

v α (x 1 , x 2 , 0) = 0, w(x 1 , x 2 , 0) = 0, vα (x 1 , x 2 , 0) = 0, ẇ(x 1 , x 2 , 0) = 0, on Σ (10)
and the boundary conditions (Dirichlet problem)

v α = vα , w= w, on L 1 × [0, +∞), v α = 0, w= 0, on L 2 × [0, +∞), (11)
where In this section, we establish results describing the spatial behaviour of solutions of given data on the interval [0, T ] under the strong ellipticity condition on the elasticity tensor.

L
By using the equality v α,12 = v α,21 , we can rewrite the system (9) in the following form

h 2 3 M βα,β -τ α3 = ρ h 2 3 vα , τ β3,β = ρ ẅ on Σ × (0, ∞), ( 12
)
where M βα are given by

M 11 = c 11 v 1,1 + c 11 -κ 2 v 2,2 , M 12 = c 12 + κ 2 v 1,2 + c 11 -c 12 2 v 2,1 , M 21 = c 11 -c 12 2 v 1,2 + c 12 + κ 2 v 2,1 , M 22 = c 11 -κ 2 v 1,1 + c 11 v 2,2 . ( 13
)
Here the parameter κ is introduced for mathematical convenience, more precisely, in order to get an appropriate family of measures which can cover the whole class of strongly elliptic elastic materials with transverse isotropy. For example, if κ = -c 12 then we obtain

M βα = c 11 -c 12 2 v α,β + c 11 + c 12 2 δ αβ v τ,τ ,
and if κ = c 11 then we get

M βα = c 11 -c 12 2 v α,β + c 11 + c 12 2 v β,α . Moreover, if we choose κ = c 11 -2c 12 , then M αβ = M αβ .
We define the support D * T of external data D on the time interval [0, T ], that is

D * T = {x ∈ L : ∃s ∈ [0, T ] so that v1 (x, s) = 0 or v2 (x, s) = 0 or w (x, s) = 0}
and, for convenience, we assume that D * T is a bounded regular curve of L. We consider the following sets

D r = x ∈ Σ : D * T ∩ S(x, r) = ∅ , Σ r = Σ\D r , Σ(r 1 , r 2 ) = Σ r2 \Σ r1,
where r ≥ 0, r 2 ≤ r 1 and S(x, r) is the closed disk with radius r and center at x. Further, let L r be the subcurve of ∂Σ r contained inside Σ and whose unit normal vector n is forwarded to the exterior of D r . For any positive parameter λ, we define the following function

I κ (r, t) = - t 0 Lr e -λs τ β3 (s) ẇ(s) + h 2 3 M βα (s) vα (s) n β dl ds, r ≥ 0, t ∈ [0, T ]. (14)
Now, with the help of Eqs. ( 12), ( 13) we obtain

τ β3 ẇ + h 2 3 M βα vα ,β = ∂ ∂t [T + W] , ( 15
)
where

T = ρ 2 ẇ2 + h 2 3 vα vα , W = 1 2 W 0 + h 2 3 W 1 + h 2 3 W 2 , W 0 = 4c 44 (ξ 2 1 + ξ 2 2 ), W 1 = c 11 v 2 1,1 + v 2 2,2 + (c 11 -κ) v 1,1 v 2,2 , W 2 = c 11 -c 12 2 v 2 2,1 + v 2 1,2 + (c 12 + κ) v 2,1 v 1,2 . ( 16
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We use the definitions of Σ r and I κ (r, t), the divergence theorem and Eq. ( 15) in order to write, for

r 2 < r 1 , I κ (r 1 , t) -I κ (r 2 , t) = - t 0 Σ(r1,r2) e -λs ∂ ∂s [T (s) + W(s)] dσ ds = = - Σ(r1,r2) e -λt [T (t) + W(t)] dσ -λ t 0 Σ(r1,r2)
e -λs [T (s) + W(s)] dσ ds.

(

) 17
We can then prove that function I κ (r, t) satisfies some properties.

Theorem 1. Let ρ be greater than zero, the strong ellipticity condition (3) be valid and κ satisfy the relation

-c 11 < κ < min {3c 11 , c 11 -2c 12 } . ( 18
)
For a solution {v α , w} of initial-boundary value problem P and called D T the bounded support of external data D on the time interval [0, T ], it is possible to prove that 1) I κ (r, t) is a continuous differentiable function on r ≥ 0 and

∂I κ ∂t (r, t) = - Lr e -λt τ β3 (t) ẇ(t) + h 2 3 M βα (t) vα (t) n β dl, ( 19
)
∂I κ ∂r (r, t) = - Lr e -λt [T (t) + W(t)] dl -λ t 0 Lr e -λs [T (s) + W(s)] dl ds. ( 20
)
2) I κ (r, t) is a non-increasing function with respect to r, i.e.

I κ (r 1 , t) ≤ I κ (r 2 , t) r 2 < r 1 . ( 21
)
3) I κ (r, t) satisfies the following first-order differential inequalities

|I κ (r, t)| + c κ λ ∂I κ ∂r (r, t) ≤ 0 and ∂I κ ∂t (r, t) + c κ ∂I κ ∂r (r, t) ≤ 0, (22)
where

c κ = η κ ρ with η κ = max c 44 , c 11 + κ 2 , 3c 11 -κ 2 , c 11 -2c 12 -κ 2 . ( 23
)
4) I κ (r, t) is a non-negative function and

0 ≤ I κ (r, t) = Σr e -λt [T (t) + W(t)] dσ + λ t 0 Σr e -λs [T (s) + W(s)] dσ ds. ( 24
)
Proof. From a direct differentiation of Eq. ( 14) with respect to the variable t we get Eq. ( 19), while from Eq. ( 17) and through a differentiation with respect to r, we have Eq. ( 20).

The kinetic energy T and the quadratic form W are positive definite if ρ > 0 and hypotheses (3), (18) hold, thus Eq. ( 17) implies that I κ (r, t) is a non-increasing function with respect to r.

We can estimate I κ (r, t) and ∂I κ ∂t (r, t) in terms of ∂I κ ∂r (r, t). At the beginning, we denote by A α the matrix associated to the quadratic form W α and we consider the functional

F [A α ; ϕ, γ] ≡ ϕ•A α γ
for the variables ϕ = {ϕ 1 , ϕ 2 } and γ = {γ 1 , γ 2 } . The hypotheses (3), (18) imply

k (α) m (γ 2 1 + γ 2 2 ) ≤ F[A α ; γ, γ] ≤ k (α) M (γ 2 1 + γ 2 2 ), (25)

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where

k (1) m = min c 11 + κ 2 , 3c 11 -κ 2 , k (2) m = min c 11 + κ 2 , c 11 -2c 12 -κ 2 , k (1) M = max c 11 + κ 2 , 3c 11 -κ 2 , k
(2)

M = max c 11 + κ 2 , c 11 -2c 12 -κ 2 . ( 26
)
Through Schwarz inequality and Eqs. ( 13), ( 16), we can see that

F [A α ; ϕ, γ] ≤ [F [A α ; ϕ, ϕ]] 1/2 [F [A α ; γ, γ]] 1/2 (27) and F A 1 ; M (1) , γ (1) = M 2 11 + M 2 22 , F A 1 ; γ (1) , γ (1) = W 1 , F A 2 ; M (2) , γ (2) = M 2 21 + M 2 12 , F A 2 ; γ (2) , γ (2) = W 2 , ( 28
)
where

M (1) = M 11 , M 22 , γ (1) = {v 1,1 , v 2,2 } , M (2) = M 21 , M 12 , γ (2) = {v 1,2 , v 2,1 } .
Collecting Eqs. ( 16), ( 25) -( 28), we deduce that

τ β3 τ β3 = c 44 W 0 , M 2 11 + M 2 22 ≤ F 1/2 A 1 ; M (1) , M (1) W 1/2 1 ≤ k (1) M M 2 11 + M 2 22 1/2 W 1/2 1 , M 2 21 + M 2 12 ≤ F 1/2 A 2 ; M (2) , M (2) W 1/2 2 ≤ k (2) M M 2 21 + M 2 12 1/2 W 1/2 2 (29)
and, consequently, we can conclude

τ β3 τ β3 + h 2 3 M βα M βα ≤ c 44 W 0 + h 2 k (1) M 3 W 1 + h 2 k (2) M 3 W 2 ≤ 2η κ W, ( 30
)
where η κ = max c 44 , k

(1)

M , k

(2) M is given by Eq. ( 23). Using Cauchy-Schwarz and arithmetic-geometric mean inequalities, it is

τ β3 ẇ + h 2 3 M βα vα n β ≤ ρ 2 ẇ2 + h 2 3 vα vα + 1 2 ρ τ β3 τ β3 + h 2 3 M βα M βα . ( 31
)
Furthermore, we use estimates ( 30), (31) into Eqs. ( 14) and ( 19) in order to obtain

|I κ (r, t)| ≤ t 0 Lr e -λs T (s) + η κ 2 ρ W(s) dl ds, ( 32
)
∂I κ ∂t (r, t) ≤ Lr e -λt T (t) + η κ 2 ρ W(t) dl . ( 33
)
Remembering Eq. ( 20) and setting = c κ = η κ ρ into (32), (33), we verify differential inequalities (22).

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If the plate is bounded, then r ranges in [0, ], with = max x∈ Σ min y∈D

* T (x 1 -y 1 ) 2 + (x 2 -y 2 ) 2 .
In view of the homogeneous initial conditions and by taking into consideration the definition for L r from ( 14) we deduce that

I κ (r, 0) = 0, I κ ( , t) = 0, r∈ [0, ], t∈ [0, T ].
Thus, by setting r 1 = , r 2 = r ∈ [0, ] into relations ( 17), ( 21), we arrive to Eq. ( 24).

On the other hand, if the plate is unbounded, then the variable r ranges in [0, ∞). Eq. ( 22) 2 can be rewritten as follows

∂I κ ∂t (r, t) + c κ ∂I κ ∂r (r, t) ≤ 0, - ∂I κ ∂t (r, t) + c κ ∂I κ ∂r (r, t) ≤ 0. ( 34
)
If we choose a pair (r 0 , t 0 ) such that t 0 ∈ [0, T ] and r 0 ≥ c κ t 0 and we put t = t 0 + rr 0 c κ into inequality (34) 1 , we obtain

d dr I κ r, t 0 + r -r 0 c κ ≤ 0 ⇒ I κ r 1 , t 0 + r 1 -r 0 c κ ≤ I κ r 2 , t 0 + r 2 -r 0 c κ , r 1 ≥ r 2 ,
so that, considering r 1 = r 0 and r 2 = r 0c κ t 0 , we deduce

I κ (r 0 , t 0 ) ≤ I κ (r 0 -c κ t 0 , 0) = 0. ( 35
)
Similarly, if we set t = t 0 -rr 0 c κ in inequality (34) 2 , then

d dr I κ r, t 0 - r -r 0 c κ ≤ 0 ⇒ I κ r 1 , t 0 - r 1 -r 0 c κ ≤ I κ r 2 , t 0 - r 2 -r 0 c κ , r 1 ≥ r 2
and thus, for r 1 = r 0 + c κ t 0 and r 2 = r 0 ,

I κ (r 0 , t 0 ) ≥ I κ (r 0 + c κ t 0 , 0) = 0. ( 36
)
We can notice from relations ( 35) and ( 36) that

I κ (∞, t) ≡ lim r0→∞ I κ (r 0 , t 0 ) = 0. ( 37
)
Finally, through Eqs. ( 17), ( 21), ( 37) we can arrive to inequality ( 24) and the proof is complete.

The results obtained up to now lead us to formulate the following theorem.

Theorem 2. Let hypotheses of previous theorem be still valid. Then, for each fixed t ∈ [0, T ], we have the following results.

1. Spatial decay:

I κ (r, t) ≤ exp - λ c κ r I κ (0, t) , 0 ≤ r < c κ t. ( 38
)
2. Domain of influence:

v α = w = 0, on Σ r × [0, T ], c κ t ≤ r. ( 39
) A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
Proof. Since I κ (r, t) is a non-negative function, then we can write from Eq. ( 22)

1 ∂ ∂r exp λ c κ r I κ (r, t) ≤ 0 = ⇒ I κ (r, t) ≤ exp - λ c κ r I κ (0, t).
Now, putting r = c κ t into ( 22) 2 , we have

d dt [I κ (c κ t, t)] ≤ 0 ⇒ I κ (c κ t, t) ≤ I κ (0, 0) = 0, 0 ≤ t. ( 40
)
On the other hand, we deduce from Eqs. ( 21), ( 24) that

0 ≤ I κ (r, t) ≤ I κ (c κ t, t), r≥ c κ t. ( 41
)
Thus, Eqs. ( 40), ( 41) imply

I κ (r, t) = 0, for r ≥ c κ t. ( 42
)
Obviously, Eq. ( 39) follows from Eqs. ( 24), ( 42) and from the positive definiteness of T and W.

Now, according to [START_REF] Gurtin | Linear theory of elasticity[END_REF], we depict the domain of influence of external given data at time

T

Spatial behaviour of steady-state solutions and estimates related to κ-measure

Throughout this section we study the problem of spatial behaviour of steady vibrations for transversely isotropic elastic plates having a strongly elliptic elasticity tensor. We assume that Σ is a rectangular strip and choose a Cartesian frame reference such that the middle surface of the plate Σ is defined by

Σ = {(x 1 , x 2 ) ∈ IR 2 : x 1 ∈ [0, 1 ], x 2 ∈ [0, 2 ]}, (43)
where 1 , 2 are some positive constants and 2 < ∞.

We discuss the problem of steady-state vibrations assuming that {v α , w} are separable with respect to space and time variables and that the time dependence is periodic, that is

v α = Re[ζ α (x 1 , x 2 ; ω)e iωt ], w= Re[ψ(x 1 , x 2 ; ω)e iωt ],
where Re[f ] represents the real part of f , ω ∈ IR + is the prescribed frequency of oscillation and ζ α , ψ are complex vector functions. The equations of motion (9) imply that the amplitude {ζ α , ψ} satisfies

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
the following system of differential equations

h 2 3 c 11 -c 12 2 ζ α,ββ + c 11 + c 12 2 ζ β,βα -c 44 (ψ ,α + ζ α ) + ρ h 2 ω 2 3 ζ α = 0, c 44 (ψ ,ββ + ζ β,β ) + ρω 2 ψ = 0. (44)
In what follows, we assume prescribed harmonic vibrations insisting on the end located at x 1 = 0, so that we consider the problem P 0 defined by Eqs. ( 44), homogeneous initial conditions and the following boundary conditions

ζ α (x 1 , 0) = 0, ψ(x 1 , 0) = 0, x 1 ∈ [0, 1 ], ζ α (x 1 , 2 ) = 0, ψ(x 1 , 2 ) = 0, x 1 ∈ [0, 1 ], ζ α ( 1 , x 2 ) = 0, ψ( 1 , x 2 ) = 0, x 2 ∈ [0, 2 ], ζ α (0, x 2 ) = ζα (x 2 ), ψ(0, x 2 ) = ψ(x 2 ), x 2 ∈ [0, 2 ] (45)
where ζα and ψ are prescribed continuous functions such that

ζα (0) = 0, ψ(0) = 0, ζα ( 2 ) = 0, ψ( 2 ) = 0.
The system (44) can be rewritten in the following form

h 2 3 Γ βα,β -χ α + ρ h 2 ω 2 3 ζ α = 0, χ β,β + ρω 2 ψ = 0 on Σ × (0, ∞) (46)
where

Γ 11 = c 11 ζ 1,1 + c 11 -κ 2 ζ 2,2 , Γ 12 = c 12 + κ 2 ζ 1,2 + c 11 -c 12 2 ζ 2,1 , Γ 21 = c 11 -c 12 2 ζ 1,2 + c 12 + κ 2 ζ 2,1 , Γ 22 = c 11 -κ 2 ζ 1,1 + c 11 ζ 2,2 (47)
and

χ α = c 44 (ψ ,α + ζ α ) . ( 48
)
We define the following function

J κ (x 1 ) = Lx 1 χ 1 ψ + χ 1 ψ + h 2 3 Γ 1α ζ α + Γ 1α ζα dx 2 , x 1 ∈ [0, 1 ], (49)
where superposed bars denote complex conjugates and

L x1 = {(x 1 , x 2 ): x 2 ∈ [0, 2 ]}. Considering
Eqs. ( 47) -( 49), we have

J κ (x 1 ) = Lx 1 c 44 ψ,1 + ζ1 ψ + (ψ ,1 + ζ 1 ) ψ + h 2 3 c 11 ζ1,1 ζ 1 + ζ 1,1 ζ1 + + c 11 -κ 2 ζ2,2 ζ 1 + ζ 2,2 ζ1 + c 12 + κ 2 ζ1,2 ζ 2 + ζ 1,2 ζ2 + c 11 -c 12 2 ζ2,1 ζ 2 + ζ 2,1 ζ2 dx 2 . ( 50
) A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
Arithmetic-geometric and Schwarz inequalities lead to

Lx 1 ζ1,1 ζ 1 + ζ 1,1 ζ1 dx 2 ≤ Lx 1 ζ 1,1 ζ1,1 dx 2 1/2 2 2 π 2 Lx 1 ζ 1,2 ζ1,2 dx 2 1/2 ≤ 2 2π Lx 1 ζ 1,α ζ1,α dx 2 , Lx 1 ζ2,2 ζ 1 + ζ 2,2 ζ1 dx 2 ≤ Lx 1 ζ 2,2 ζ2,2 dx 2 1/2 2 2 π 2 Lx 1 ζ 1,2 ζ1,2 dx 2 1/2 ≤ 2 2π Lx 1 ζ α,2 ζα,2 dx 2 , Lx 1 [( ζ1 + ψ,1 )ψ + (ζ 1 + ψ ,1 ) ψ]dx 2 ≤ Lx 1 ( ζ1 + ψ,1 )(ζ 1 + ψ ,1 )dx 2 1/2 2 2 π 2 Lx 1 ψ ,2 ψ,2 dx 2 1/2 ≤ 2 2π Lx 1 ( ζ1 + ψ,1 )(ζ 1 + ψ ,1 )dx 2 + 2 2π Lx 1 ψ ,2 ψ,2 dx 2 , ( 51
)
Lx 1 ζ1,2 ζ 2 + ζ 1,2 ζ2 dx 2 ≤ Lx 1 ζ 1,2 ζ1,2 dx 2 1/2 2 2 π 2 Lx 1 ζ 2,2 ζ2,2 dx 2 1/2 ≤ 2 2π Lx 1 ζ α,2 ζα,2 dx 2 , Lx 1 ζ2,1 ζ 2 + ζ 2,1 ζ2 dx 2 ≤ Lx 1 ζ 2,1 ζ2,1 dx 2 1/2 2 2 π 2 Lx 1 ζ 2,2 ζ2,2 dx 2 1/2 ≤ 2 2π Lx 1 ζ 2,α ζ2,α dx 2 ,
where we have used the well-known membrane problem and boundary conditions (45) 1,2 , i.e.

π 2 2 2 Lx 1 ζ α ζα dx 2 ≤ Lx 1 ζ α,2 ζα,2 dx 2 , π 2 2 2 Lx 1 ψ ψdx 2 ≤ Lx 1 ψ ,2 ψ,2 dx 2 . ( 52
)
By means of Eqs. ( 50), ( 51) we obtain

|J κ (x 1 )| ≤ Lx 1 2 c 44 2π ( ζ1 + ψ,1 )(ζ 1 + ψ ,1 ) + 2 c 44 2π ψ ,2 ψ,2 + h 2 2 c 11 6π ζ 1,α ζ1,α + + h 2 2 (c 11 -κ) 12π ζ α,2 ζα,2 + h 2 2 (c 12 + κ) 12π ζ α,2 ζα,2 + h 2 2 (c 11 -c 12 ) 12π ζ 2,α ζ2,α dx 2 . ( 53
)
This relation allows us to prove the following theorem about properties of function J κ .

Theorem 3. Let hypotheses of Theorem 1 be still valid and let frequency ω obey to

ω < ω m ( 54
)
where

ω m = π 2 2ρ μ κ 1 + 6 2 2 π 2 h 2 -1/2 , 0 < μ κ = min c 44 , k (1) m , k (2) m . ( 55
)
The function J κ is non-decreasing and it satisfies the following inequality

ν κ |J k (x 1 )| ≤ J k (x 1 ), with ν κ = min i=1,...,6 p i P i , ( 56
)
where P 1 = and

p 1 = 2μ κ , p 2 = 2μ κ h 2 π 2 2(6 2 2 + π 2 h 2 ) 1 - ω 2 ω 2 m , p 3 = 2μ κ , p 4 = 2μ κ , p 5 = 2μ κ 1 - ω 2 ω 2 m , p 6 = μ κ 1 - ω 2 ω 2 m . ( 58
)
Proof. Differentiating J κ we obtain

J κ (x 1 ) = Lx 1 χ 1 ψ ,1 + χ 1 ψ,1 + χ 1,1 ψ + χ 1,1 ψ+ + h 2 3 Γ 1α ζ α,1 + Γ 1α ζα,1 + Γ 1α,1 ζ α + Γ 1α,1 ζα dx 2 . ( 59
)
Eqs. ( 46) allow us to show that

h 2 3 Γ 1α,1 = - h 2 3 Γ 2α,2 + χ α -ρ h 2 ω 2 3 ζ α , χ 1,1 = -χ 2,2 -ρω 2 ψ; (60)
thus, with an integration by parts, boundary conditions (45) and Eq. ( 48), we arrive to

J κ (x 1 ) = 2 Lx 1 W -ρω 2 h 2 3 ζα ζ α + ψψ dx 2 , ( 61
)
where

W = W 0 + h 2 3 W 1 + h 2 3 W 2 , W 0 = c 44 ζα + ψ,α (ζ α + ψ ,α ) , W 1 = c 11 ζ1,1 ζ 1,1 + ζ2,2 ζ 2,2 + c 11 -κ 2 ζ1,1 ζ 2,2 + ζ 1,1 ζ2,2 , W 2 = c 11 -c 12 2 ζ 2,1 ζ2,1 + ζ 1,2 ζ1,2 + c 12 + κ 2 ζ2,1 ζ 1,2 + ζ 2,1 ζ1,2 . ( 62
)
The quadratic form W is positive definite if relations (3) and ( 18) are verified. It is then easy to see that

μ κ ζα + ψ,α (ζ α + ψ ,α ) + h 2 3 ζα,β ζ α,β ≤ W (63) and k (α)
m are defined into Eqs. (26). Considering Eqs. ( 52), ( 61), (63), we get

J κ (x 1 ) ≥ 2μ κ Lx 1 ζα + ψ,α (ζ α + ψ ,α ) + h 2 3 ζα,β ζ α,β - - ρω 2 2 2 π 2 μ κ h 2 3 ζ α,2 ζα,2 + ψ ,2 ψ,2 dx 2 . ( 64
)
Using the relations (52), we can observe that

6 2 2 π 2 h 2 Lx 1 (ζ 2 + ψ ,2 )( ζ2 + ψ,2 ) + h 2 6 ζ2,2 ζ 2,2 dx 2 ≥ 2 2 π 2 Lx 1 ζ 2,2 ζ2,2 dx 2 ≥ ≥ Lx 1 ζ 2 ζ2 dx 2 .
(65)

Further, using Schwarz and arithmetic-geometric mean inequalities, it follows

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT ζ2 + ψ,2 (ζ 2 + ψ ,2 ) ≥ (1 -) ζ 2 ζ2 + 1 - 1 ψ ,2 ψ,2 , for every > 0 and, for = 2, it is ζ2 + ψ,2 (ζ 2 + ψ ,2 ) ≥ -ζ 2 ζ2 + 1 2 ψ ,2 ψ,2 ; (66) thus Lx 1 (ζ 2 + ψ ,2 )( ζ2 + ψ,2 ) + h 2 6 ζ 2,2 ζ2,2 dx 2 ≥ Lx 1 (ζ 2 + ψ ,2 )( ζ2 + ψ,2 )dx 2 ≥ ≥ Lx 1 -ζ 2 ζ2 + 1 2 ψ ,2 ψ,2 dx 2 . ( 67
)
Combining Eqs. ( 65), ( 67) we obtain

Lx 1 (ζ 2 + ψ ,2 )( ζ2 + ψ,2 ) + h 2 6 ζ 2,2 ζ2,2 dx 2 ≥ h 2 π 2 2(6 2 2 + π 2 h 2 ) Lx 1 ψ ,2 ψ,2 dx 2 . ( 68
)
Moreover, Eqs. ( 64), (68) imply

J κ (x 1 ) ≥ μ κ Lx 1 2 ζ1 + ψ,1 (ζ 1 + ψ ,1 ) + h 2 π 2 6 2 2 + π 2 h 2 1 -ω 2 2ρ 2 2 π 2 μ κ 1 + 6 2 2 π 2 h 2 ψ,2 ψ ,2 + + 2h 2 3 ζ1,1 ζ 1,1 + 2h 2 3 ζ2,1 ζ 2,1 + 2h 2 3 1 -ω 2 ρ 2 2 π 2 μ κ ζ1,2 ζ 1,2 + + h 2 3 1 -ω 2 2ρ 2 2 π 2 μ κ ζ2,2 ζ 2,2 dx 2 . ( 69
)
If we note that the critical frequency ω m defined by Eq. ( 55) is such that

min 2 π 2ρ μ κ 1 + 6 2 2 π 2 h 2 , 2 π ρ μ κ , 2 π 2ρ μ κ = 1 ω m then we have J κ (x 1 ) ≥ Lx 1 p 1 ζ1 + ψ,1 (ζ 1 + ψ ,1 ) + p 2 ψ,2 ψ ,2 + h 2 3 p 3 ζ1,1 ζ 1,1 + + h 2 3 p 4 ζ2,1 ζ 2,1 + h 2 3 p 5 ζ1,2 ζ 1,2 + h 2 3 p 6 ζ2,2 ζ 2,2 dx 2 ≥ 0. (70)
This relation implies that J κ is a non-decreasing function. On the other hand, Eq. ( 53) implies

|J κ (x 1 )| ≤ Lx 1 P 1 ( ζ1 + ψ,1 )(ζ 1 + ψ ,1 ) + P 2 ψ ,2 ψ,2 + h 2 3 P 3 ζ 1,1 ζ1,1 + + h 2 3 P 4 ζ 2,1 ζ2,1 + h 2 3 P 5 ζ 1,2 ζ1,2 + h 2 3 P 6 ζ 2,2 ζ2,2 dx 2 (71)
and consequently

|J κ (x 1 )| ≤ 1 ν κ Lx 1 p 1 ( ζ1 + ψ,1 )(ζ 1 + ψ ,1 ) + p 2 ψ ,2 ψ,2 + h 2 3 p 3 ζ 1,1 ζ1,1 + + h 2 3 p 4 ζ 2,1 ζ2,1 + h 2 3 p 5 ζ 1,2 ζ1,2 + h 2 3 p 6 ζ 2,2 ζ2,2 dx 2 . (72) A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
Finally, Eqs. ( 70), (72) lead to Eq. ( 56).

We consider the class of steady-state vibrations {ζ α , ψ} for which

E(x 1 ) = Sx 1 P 1 ( ζ1 + ψ,1 )(ζ 1 + ψ ,1 ) + P 2 ψ ,2 ψ,2 + h 2 3 P 3 ζ 1,1 ζ1,1 + + h 2 3 P 4 ζ 2,1 ζ2,1 + h 2 3 P 5 ζ 1,2 ζ1,2 + h 2 3 P 6 ζ 2,2 ζ2,2 da < ∞, (73)
where

S x1 = Σ∩[x 1 , ∞[×[0, 2 ].
We can prove from Eqs. ( 50), ( 71), ( 73) that E(x 1 ) is a non-increasing function and -E (x 1 ) = (74)

When the strip Σ is bounded, then the hypothesis ( 73) is trivially verified and Eqs (45) 3 , (74) imply

E ( 1 ) = 0 ⇒ J κ ( 1 ) = 0. ( 75
)
Moreover, if the strip Σ is semi-infinite and the hypothesis ( 73) is satisfied, then Eq. ( 74) implies

lim x1→∞ E (x 1 ) = 0 ⇒ J κ (∞) ≡ lim x1→∞ J κ (x 1 ) = 0. (76)
Now, we can establish a theorem furnishing information about the spatial behaviour of the amplitude of vibrations, provided that the frequency of the harmonic vibrations is lower than the critical value ω m .

Theorem 4. Let hypotheses of Theorem 3 be still true and let {ζ α , ψ} be the steady-state vibrations for which Eq. ( 73) holds. Then, the function -J κ is a measure of the amplitude {ζ α , ψ} of the harmonic vibration, that is -J κ ≥ 0 and -J κ = 0 implies {ζ α , ψ} = {0, 0}. Moreover, the following decay estimate of Saint-Venant type, with exponential decay factor 1 ν κ , is valid 0 ≤ -J κ (x 1 ) ≤ -J κ (0) exp -

x 1 ν κ . ( 77
)
Proof. In both bounded and semi-infinite cases, from hypothesis (54) about frequency ω and from the fact that J κ is non-decreasing, Eqs. ( 75), (76) imply J κ (x 1 ) ≤ 0, for all x 1 and inequality (56) becomes

J κ (x 1 ) + 1 ν κ J κ (x 1 ) ≥ 0. ( 78
)
Through an integration of Eq. ( 78), we get the decay estimate (77).

To conclude, we can remark that, in the context of steady-state vibrations for which E(x 1 ) is bounded and the excitation frequency is lower than the critical value ω m , Eq. ( 77) describes an exponential decay with the greatest decay factor 1 ν , where ν = inf κ ν κ .

A C C E P T E D M

A N U S C R I P T ACCEPTED MANUSCRIPT

Remark

The investigation performed is based on the assumption concerning the strong ellipticity of elasticity tensor. The results obtained under such hypothesis are thus valid also for classes of particular materials characterized by reversed properties, like negative Poisson's ratio and negative stiffness (auxetic or anti-rubber materials). These particular structures (see, for example, [START_REF] Park | Biomaterials. 3rd Ed[END_REF]) expand laterally when stretched, in contrast to ordinary materials. While optimal bounds for the spatial estimates are not so easy to obtain, in the present paper we offer the crude bounds in the spatial estimates of Theorem 2 and Theorem 4. Our results can be compared with other results of the literature (see e.g. Sanchez Palencia ( 1988), ( 1989)).

A C C E P T E D M

A N U S C R I P T

A

  and Chirit ¸ȃ et al. (2007) establish the necessary and sufficient conditions for strong ellipticity characterizing a transversely isotropic elastic solid. For a transversely isotropic plate described through the Mindlin-model and without positive definiteness assumptions on the elasticity tensor, Passarella and Zampoli (2007) have established a uniqueness result and a Galerkin representation of solution of the field equations. Furthermore, under the hypotheses of positive definiteness of the elasticity tensor, they prove a variational theorem of Gurtin

  1 and L 2 are disjoint and complementary nonempty subsets of L and vα and w are assigned fields representing external data D of the problem in concern. Throughout this paper we assume that a smooth solution of the problem P exists. The internal energy density of a transversely isotropic elastic plate (considering a state of bending) is a positive definite quadratic form if and only if c 44 > 0, c 11 > 0, |c 12 | < c 11 . From strong ellipticity conditions (3), we observe that it is c 12 < c 11 , thus there is a relaxation upon constitutive coefficients with respect to positiveness conditions.

  as the set of the points of Σ that can be reached by signals propagating from the support DT on the time interval [0, T ], with speeds equal to or less than the maximum speed of propagation κ ∈ ]-c 11 , min {3c 11 , c 11 -2c 12 }[; in particular, |c 12 | < c 11 ⇒ κ ∈ ]-c 11 , c 11 -2c 12 [ , while c 12 ≤ -c 11 ⇒ κ ∈ ]-c 11 , 3c 11 [.

Acknowledgment. The authors are very grateful to the referees for useful observations leading to improvement of paper.


O Barrera

A C F Cocks

A R S Ponter

Evaluation of the convergent properties of the Linear Matching Method for computing the collapse of structural components

Keywords: Limit analysis, Linear Matching Method, Convergence, Collapse mechanism. _____________________________________________________________________________________

The paper considers the application of the Linear Matching method to the limit analysis of perfectly plastic portal frames. This allows the display of the characteristic features of this method to a class of structural problems that have been studied by several other methods. The convergence of both upper and lower bounds is proven and a simple geometric interpretation displays the nature of the programming method. Through a sequence of examples the convergence properties of the method are displayed, showing that complete convergence can sometimes be delayed by the proximity of mechanisms with near equal limit loads.

INTRODUCTION

The development of numerical methods for the evaluation of limits in classical plasticity theory has a long history with a number of significant strands. The upper and lower bound theorems of limit and shakedown analysis naturally lend themselves to programming methods, both linear and non-linear, producing, today, efficient solution procedures [START_REF] Weichert | Inelastic Behaviour of Structures under Variable Repeated Loading -Direct Analysis Methods[END_REF]. Recently, the application of mathematical programming methods to the limit analysis of portal frames has been consolidated and summarised by Cocchetti and Maier (2003). These methods have also been extended to the behaviour of elastic-softening plastic portal frames, emphasing the importance of Mathematical Programming with Equilibrium Constraints (MPEC) methods by [START_REF] Ferris | Limit analysis of frictional block assemblies as a mathematical program with complementarity constraints[END_REF] and Tangaramvong andTin-Loi (2007, 2008). MPEC methods correspond to a class of methods that may be applied to structural mechanics problems and problems in business and finance. There has always existed a strong cross fertilization between mathematical programming methods in applied mechanics and in other applications. It is, therefore, beneficial to display new methods in applied mechanics in a form that allows a direct comparison with existing mathematical programming methods and the problem of the portal frame provides a useful vehicle in this respect, as well as being a significant problem in structural design.

In recent times, an alternative approach to classical plasticity problems has been developed, motivated by a desire to calculate classical plasticity limits using conventional finite element methods as the basis for iterative procedures. Marriot (1985) A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 2 has discussed the Modulus Variation Procedures (MVP) that generates lower bounds within some kinematic contexts by representing an elastic-plastic stress field as the solution of a linear problem with spatially varying moduli. Related design orientated methods have been discussed by [START_REF] Seshadri | Limit loads of framed structures and arches using the GLOSS R-node method[END_REF] and [START_REF] Mackenzie | A method of estimating limit loads by iterative elastic analysis.I simple example[END_REF]. These methods have been devised with structural design in mind.

Whereas programming methods have a well defined theoretical structure where convergence can be meaningfully discussed, generally MVP and related methods have been treated in a more ad-hoc manner and have proved to give a rather variable performance. Recently, [START_REF] Marin-Artieda | Approximate limit load evaluatio of structural frames using linear elastic analysis[END_REF] describe the application of these methods to portal frames and demonstrate both the strength (simplicity) and weakness (uncertainty over convergence) of this class of methods when applied to the simplest of the plasticity problems, limit analysis. They conclude that the original method of Marriot (1985) provides the most satisfactory lower bound solutions but rarely the exact limit state solution.

A further branch of these developments, the Linear Matching Methods (LMM) (Ponter et al, 2000 and[START_REF] Ponter | The Linear Matching Method for the Evaluation of Limit Loads, Shakedown Limits and Related Problems[END_REF]Engelhardt, 2000), arises from the same general approach of MVP methods, the use of standard linear finite element analysis with spatially varying moduli, but with the objective of defining programming methods with very well defined convergence properties. By this means, methods are derived with the sound theoretical foundation of programming methods and the simplicity of MVP. The primary motivation has been the development of methods that may be used as a computational tool for the UK Life Assessment Method R5 (Ainsworth, 1997) where thermomechanical loading for complex continuum structures is the primary concern. For this range of problems, methods have been derived for limit, shakedown and ratchet limits and a number of extensions allow for high temperature creep behaviour (Chen. et al, 2006 a,b). In all this work, the upper bound, kinematic, aspect of the method has been emphasised and, indeed, there is evidence that it may not be possible to derive a convergent method of this type that relies only on the lower bound theorems of plasticity [START_REF] Ponter | The Linear Matching Method for the Evaluation of Limit Loads, Shakedown Limits and Related Problems[END_REF]. However, amongst the published work on these methods, there is the lack of a straightforward description of the Linear Matching Method and its convergence properties within an easily understood structural context. The work of [START_REF] Marin-Artieda | Approximate limit load evaluatio of structural frames using linear elastic analysis[END_REF] provides an opportunity to fill this gap by displaying the essential elements of LMM within the simplest of contexts, where a comparison with other MVP methods already exists. Hence the purpose of this paper is two fold, to introduce the LMM into the context of structural design methods and, at the same time, demonstrate the methods strong convergence properties.

The outline of the paper is as follow. In section 2 some basic concepts of limit analysis are briefly summarized and the basic formulae for the evaluation of the upper and lower bound collapse multipliers are set up. Moreover, the fundamental assumptions of the LMM to rigid-plastic structures are given. In section 3 the Linear Matching Method is described and convergence is proven in Section 4. The iterative process results in a monotonically reducing upper bound which converges to the exact solution if the linear solutions are evaluated exactly; for finite element solutions the upper bound functional reduces to the least upper bound associated with the class of displacement fields allowed by the FE formulation. In section 5 the method is used to evaluate upper and lower bounds of collapse loads and collapse mechanisms in a single story portal frame structure and the method is given a graphical interpretation. This is followed by an extension of the procedure to multi-story portal frames. These solutions display a

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT
3 number of interesting convergence properties of the method. At convergence, upper and lower bounds become identical, verifying that the solution is exact. Convergence of the upper bound is generally geometric, the error reducing at each iteration by a constant factor. During the convergence process the upper bound monotonically reduces. The lower bound is observed to generally monotonically increase but, on occasions, can demonstrate a reduction. This is entirely consistent with theory. Final convergence of the upper and lower bounds to a common value can be delayed in cases where two distinctly differing mechanisms exist which give rise to near identical upper bounds. In this case the method can initially converge towards the mechanism with the non-optimal upper bound before then changing to the optimal mechanism. In such cases, lack of full convergence is clearly displayed through the difference between the upper and lower bound.

LIMIT ANALYSIS: PROBLEM STATEMENT

In common with all structural systems, portal frames can be analysed within a kinematic framework defined by a set of displacements, in this case the deflections c Let us consider structures composed of a rigid-plastic material which, therefore, has zero displacement up to the collapse load. Mechanism deformations are assumed to be infinitely small so that deflections have no effect on the equations of equilibrium. Reference, moreover, is made to a structure subjected to proportional loading, the intensity of loads being defined by the load factor λ, which is restricted to positive values.

Let the actual plastic collapse load factor of a structure under load be p λ , and let the collapse mechanism under loads

i p i F F λ = *
have small hinge rotations p j Φ at hinge positions j h with corresponding plastic moments p j M . The displacements corresponding to the loads p i F λ in the collapse mechanism are denoted as

p i Δ . ∑ ∑ Φ = Δ p j pj p i i p M F λ (2.1)
where since p λ and

pj j M Φ ∑ are positive, ∑ Δ p i i F is positive.
If the bending moment at position j h is denoted by j M , then,

pj j pj M M M ≤ ≤ - (2.2) A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 4
In the following we quote the limit theorems, without proof (see, for example [START_REF] Horne | Plastic theory of structures[END_REF].

Lower Bound

If, at any load factor LB λ , it is possible to find an equilibrium bending moment distribution * j M in equilibrium with the applied loads LB i F λ and everywhere satisfying the yield condition (2.2), then LB λ is either equal to or less than the load factor at failure.

LB p λ λ ≤ .
(2.3)

Upper Bound

If, for any assumed plastic mechanism, defined by a set c i Δ and c j Φ , the external work done by the loads at a positive load factor UB λ is equal to the internal work at the plastic hinges, then UB λ is either equal to or greater than the load factor at failure.

∑ ∑

Φ = Δ j c j pj c i i i UB M F λ , p UB λ λ ≥ (2.4)
It is well known that the limit load is uniquely defined although there may well be more than one corresponding mechanism. As the kinematics are defined by a limited class of deformation modes, the limit load so calculated is, strictly, an upper bound and the lower bound defines a lower bound to this upper bound

LINEAR MATCHING METHOD FOR LIMIT ANALYSIS

The simple rigid perfectly plastic moment-rotation relation is shown in Figure 1. Perfectly plastic deformation up to indefinitely high rotations is assumed to be possible at the yield moment pj M .

The Linear Matching Method attempts to construct, as the limit of an iterative procedure, linear solutions, j Φ ~ and j M ~, for the load i F λ by varying the set of linear moduli j R :

j j j M R 1 ~= Φ (3.1)
by a particular scaling factor, discussed below. Equation (3.1) describes an arbitrary sign consistent description of the relationship between the moments, in equilibrium, and compatible rotations which is capable of describing any type of holonomic constitutive assumption. Hence, for any limit state solution there exists a set of j R , for which the linear solution j M ~ is identical to the limit state solution.

The procedure described below provided an iterative procedure which seeks a sequence of values of j R denoted by k j R , so that each solution more closely approaches the correct solution. i.e.

1 ~+ = = Φ k j pj k j k j k j R M R M , hence k j k j pj k j R M M R 1 = + (3.2)
The quantity At each iteration a corresponding lower bound on the limit load can be found by scaling the moment distribution the scaled moments lie within yield,. Hence at a particular hinge point,

m j = pm k UB k LB k m M M = λ λ ~ (3.3) and for m j ≠ pj k UB k LB k j M M ≤ λ λ ~ (3.4)
Hence the lower bound is given by:

k m pm k UB k LB M M λ λ = (3.5)
Hence, at each iteration, both a lower and an upper bound is determined. In the following section we show that this procedure produces a monotonically reducing sequence of k UB λ that converges to the least upper bound. At the same time, at convergence, the lower bound equals the upper bound, demonstrating that the converged solution is the exact solution within the class of kinematic fields chosen.

CONVERGENCE PROOF

The rotations k j Φ ~ and displacements k i Δ at the k th iteration are used to determine the upper bound k UB λ via equation (2.4). The subsequent linear solution provides rotations

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 6 1 ~+ Φ k j , displacements 1 + Δ k i and a new upper bound 1 + k UB
λ . We now demonstrate that the upper bound load parameter always reduces during this process.

Consider Figure 1 and the following inequality:

( ) 0 ~1 1 ≥ Φ - Φ Φ ∫ ∫ + + Φ Φ Φ Φ k j k j k j k j j pj j j j d M d M (4.1) i.e. the area A 0 ≥ . Hence, ( ) ( ) ( ) 0 2 1 2 1 1 1 1 ≥ Φ - Φ - Φ Φ - Φ Φ + + + k j k j pj k j k j j k j k j j M M M (4.2)
where, according to equation (2.4)

k i i k UB k j pj F M Δ = Φ ∑ λ and ∑ + + + Δ = Φ 1 1 1 ~ki i k UB k j pj F M λ so inequality (4.2) becomes: ∑ ∑ ∑ ∑ ∑ ∑ + + + + + Δ - Δ ≥ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Δ - Δ - Φ ⋅ - Φ ⋅ i k i i k UB i k i i k UB i k i i k UB i k i i k UB k j j k j k j j k j F F F F R R 1 1 1 1 2 2 1 2 1 2 1 λ λ λ λ (4.3)
Now the potential energies for the consecutive linear solutions corresponding to stiffnesses k j R and

1 + k j
R are given by:

∑ ∑ ∑ ∑ Δ - Φ = Ω Δ - Φ = Ω + + + i k i i k UB k j i k j k i k j i k UB k j i k j k F R F R 2 1 2 1 2 1 2 1 1 λ λ (4.4)
Hence inequality (4.4) becomes:

∑ + + + Δ - ≥ Ω - Ω i k i i k UB k UB k k F 1 1 1 ) ( λ λ (4.5)
The solution of the linear problem with load . For this solution, at each hinge point two possibilities exist, depending on whether

k j k j Φ = Φ + 1
has a finite value (active hinges) or whether

k j k j Φ = Φ + 1
is infinitesimally small (inactive hinges). For active hinges equation (3.2) gives,

1 ~+ = = Φ k pj k k j k j R M R M and hence pj k j M M = ~ (4.7) A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 7 At inactive hinges pj k j M M ≤ ~.
Hence the conditions of both the upper and lower bound theorems are satisfied and the converged solution is the exact solution.

At each iteration the equilibrium moments provide a lower bound k LB λ which, we now see, becomes equal to the upper bound at convergence. However, it is not possible to prove that the lower bound forms a monotonically increasing sequence. In the examples that follow the lower bound does, generally, monotonically increase but in a few cases reductions do occur. However, the role of the lower bound in determining whether convergence has correctly occurred is essential, as will be demonstrated in the examples.

APPLICATIONS

In the following, we investigate the application of this method to the calculation of upper and lower bounds of the limit load for a range of portal framed structures. We begin with a single story frame to fix ideas. This is followed by the solution of multistory structures, with particular interest in the method's convergence properties.

Upper and Lower Bounds for a Single Story Frame.

Figure 2 shows a simple portal frame with a fixed base at 1 and 2. The vertical load

H V α =
and horizontal load λ = H remains proportional and defined in magnitude by the load factor λ . The frame is of uniform section with a plastic moment

Y pj M M = .
When two independent load systems can act simultaneously on a structure in any ratio, it is useful to study the load factors at collapse by means of an interaction diagram, Figure 3 [START_REF] Horne | Plastic theory of structures[END_REF]. Any line radiating from the origin represents proportional loading and in the following calculations three different values of α are considered: 0.25;1; 4 α = .

For problems where concentrated loads are applied, equilibrium requires that maximum and minimum bending moments occur at the ends of uniform beam sections, i.e. at nodes 1 to 2 in Figure 2. Hence plastic hinges may only occur at such nodes and the rotation of local plastic hinges are given by j Φ ,

1 = j to 5 = j
as shown in Figure 4. The exact limit load solution consists of three alternative mechanisms, as shown in Figure 4, giving rise to the interaction diagram of Figure 3.

According to the Linear Matching Method a sequence of linear analyses is undertaken in which the nodal stiffnesses are systematically changed to provide a sequence of upper and lower bounds to the limit load. The details of the procedure are given in Appendix A. , where u and v are the displacements in the directions of the applied loads as illustrated in Figure 4. The converged mechanism and limit load are consistent with the interaction diagram of Figure 3.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 8
The iteration process is terminated when

6 1 10 - -< -k UB k UB λ λ
and this is achieved in all cases in less than 15 iterations.

Graphical Representation of the Convergence Process

Convergence of the LMM may be shown through the following graphical representation of the iterative process. At the k th iteration the method chooses a stiffness matrix for the next iteration such that

: [ ] [ ][ ] k k p R M Φ = + 1
, and from the upper bound formula

[ ][ ] [ ] [ ] 0 ~F M T k k UB k p Δ = Φ λ
. Now consider the elastic analysis for the (k+1) th iteration, for which the magnitude of the applied load is k UB λ . The potential energy of the system for any [ ] Φ is given by:

[ ] [ ] [ ] [ ] [ ] 0 1 2 1 F R k UB T k T λ Δ - Φ Φ = Ω + (5.1) so when k Φ = Φ
~ , Ω is given by:

( ) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 0 0 0 0 2 1 2 1 2 1 ~F F F F M k UB T k UB T k UB T k UB T k T p k λ λ λ λ Δ - = Δ - Δ = Δ - Φ = Φ Ω (5.2)
Hence, a quantity Ω can be defined:

[ ] ( ) [ ] [ ] k UB T k F λ 2 3 ) ( 0 + Δ Φ Ω = Φ Ω (5.3) so that, when [ ] [ ] k Φ = Φ ~ [ ] k UB k λ = Φ Ω ) (
(5.4)

If we now consider the first variation of Ω about this point,

( ) [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ] 0 0 1 ~F M F R T k T p T k k T k k Δ Φ = Δ Φ Φ = Φ Ω + δ δ δ
(5.5)

The upper bound load parameter is given by;

[ ] ( ) [ ] [ ] [ ] [ ] 0 ~F M T k k T p k k UB Δ Φ = Φ λ (5.6) Hence for constant [ ] [ ] 0 ~F T k Δ A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 9 [ ] ( ) [ ] [ ] [ ] [ ] ( ) k T k T p k UB F M Φ Ω = Δ Φ = Φ 0 δ δ δλ (5.7)
This argument demonstrates that the normalized potential energy of the linear problem Ω and the upper bound load factor UB λ coincide at the kth iteration and their local slopes also coincide. . The plot also shows Ω for the (k+1) th =11 th iteration. It can be seen that the matching process chooses a linear material so the upper bound load parameter and the normalised potential energy coincide and are tangential at the k th solution at k β . The (k+1) th solution is obtained as the minimum of the potential energy and use of the resulting rotations provides a lower value of the upper bound. In terms of this graphical representation the convexity conditions (4.1) and (4.2) require that the potential energy curve lies everywhere above the upper bound curve.

Figure 8 shows the same representation as Figure 7 except over a wider range of β and for iterations where the current solution is far from convergence, B, and close to convergence, C. Note that the shape of the potential energy curve depends on the distance from the converged solution. There is a vertex in the limit load plot at the minimum. As a result, an Ω curve which is tangential to the limit load curve close to the vertex must be narrow in order to ensure that it does not cross the upper bound curve. At the vertex, it must collapse onto a vertical line. This is indicative of the fact that there is no rotation at node 2 of Figure 2 for the mechanism associated with the vertex (ie mechanism III of Figure 4). The corresponding linear solution requires that the rotational stiffness of this node is ∞. Any variation of β away from this point requires rotation of this node resulting in a sharp increase in Ω . Away from the vertex, where there is rotation and finite stiffnesses at all the nodes, there is a more gradual variation of Ω with β. The mechanism corresponding to point A also has zero rotation at one of the nodes (node 3). For the same reason as discussed above, the Ω curve becomes narrow again as this point is approached.

An estimate of the Rate of Convergence

The rate of convergence of the method depends on both the nature of the particular problem and the choice of the initial set of 0 j R . Note that the value of 1 UB λ is independent of the initial choice of λ and also the absolute magnitude of R . For problems where all the possible hinges are active convergence tends to be rapid. But in the problem just described and those discussed later, there are a significant number of hinges which become inactive in the converged solution. For such hinges, j R begins with the initial this argument would imply that

1 1 - = k k C δ δ
where C>1 is a constant, an average of the constant values associated with the inactive hinges. Hence

k N N k C δ δ 1 = + or k N k C N δ δ ln ln ln + - = + (5.9)
Figure 9 shows the logarithm of k δ versus the iteration numbers for the cases

0.25 α = , 1 = α , 4 = α
as shown in Figure 6 (a,b,c). It can be seen that convergence closely follows equation (5.9) for 0.25 α = and

4 = α with, in this case 2 = C
, except when the error is extremely small, where numerical rounding errors begin to play a part. For 1 = α , the initial mechanism for j R constant is very close to the optimal mechanism and convergence is more rapid.

EXTENTION TO MULTI-STORY FRAME STRUCTURES

In this section the extension to multi-story frame structures is discussed to verify the effectiveness of the proposed approach. The procedure is described in detail in Appendix B.

Three Story Portal Frames

Figure 10 presents the geometry and the sections at which it is necessary to allow for the possible occurrence of plastic hinges of a three story portal frame for a model with 15 elements.

Figure 11 show the exact collapse mechanism, namely a sway in the first two stories and Figure 12 shows the convergence of UB λ and LB λ to the limit load corresponding to this mechanism. Note that the lower bound does not always monotonically increase, but eventually converges to the optimal upper bound. In Figure 13 the rate of convergence is shown where, again, geometric convergence with a slope of ln 2 occurs over most of the iterations. It is interesting to note that the rate of geometric convergence is the same as that shown in Figure 9 for the structure of Figure 2, although there is no theoretical reason why this should be the case.

A Ten Story Frame

Figure 14 presents the geometry of a ten-story plane frame studied previously by [START_REF] Marin-Artieda | Approximate limit load evaluatio of structural frames using linear elastic analysis[END_REF] where a number of alternative methods were applied. Figure 15 shows the optimal collapse mechanism in the form of an overall sway In the same paper, an ABAQUS step by step calculation is reported with a significantly lower collapse load. The reported collection of active hinges at collapse does not form a mechanism and hence no meaningful comparison can be made with this result. The authors place greatest emphasis on the Reduced Elastic Modulus (REM) method of [START_REF] Marriott | Evaluation of deformation on load control of stresses under inelastic conditions using elastic finite element stress analysis[END_REF], which gives a lower bound which tends to converge to a limit load of 30.1kN, 4% less than the exact solution. The REM method gives no indication of the collapse mechanism. Note that final convergence of the lower bound to the upper bound in Figure 16 takes a significant number of iterations. The rate of convergence is shown in Figure 17 where it can be seen that convergence is initially rapid and geometric so that by 50 iterations the error has reduced to 8 10 -. There is then about 200 iterations during which the error reduces slowly until, finally, geometric convergence is re-established at the same rate as initially. This behaviour arises from the very close proximity of two mechanisms with near identical collapse loads. The method initially converges towards the mechanism with the higher collapse load before moving over to the optimal mechanism. This phenomenon is best understood in the following example where the two mechanisms are moved further apart by a small change in a plastic modulus.

Consider the case when the plastic moment in beams labelled f in Figure 14 is changed by 8% from ( ) 0.216

P Y M f M = to ( ) 0.2 P Y M f M =
. In this case the structure has two very distinct collapse mechanisms I and II, shown in Figure 18. Mechanism I consists of a sway of the entire structure with = 1 λ 0.8832, and for mechanism II only the last seven stories sway, giving = 2 λ 0.8785. Figure 19 shows the evolution of the two bounds with increasing number of iterations. The upper bound shows a rapid convergence to mechanism I, followed by a very slow convergence to the optimal mechanism II where the upper and lower bounds coincide. The initial apparent convergence to mechanism I is shown to be false by the remaining large difference between the upper and lower bounds. Full convergence requires a large number of iterations; in excess of 500. It is evident from Figure 20 that the convergence is geometric in the early stages as the deformed shape converges towards mechanism I and again in the latter stages as mechanism II is approached, but it is much slower in-between as the solution process seeks out the new mechanism.

It is clear that the existence of distinct mechanisms with near equal collapse load limits can delay eventual convergence to the optimal mechanism. The fact that the upper and lower bound do not coincide after the initial convergence process clearly indicates that this phenomenon has occurred and the lower bound will certainly be less at this stage than the eventual converged upper bound. In the following we ask the question: how close do collapse loads for adjacent mechanisms have to be for this phenomenon to occur. Consider again the structure and plastic moments shown in Figure 14 . Figure 23a shows the number of iterations to convergence over the entire range of α , which is assumed to occur when

6 1 10 - -< -k UB k UB λ λ
For this very precise criterion of convergence, it can be seen that large numbers of iterations are only required when there are two mechanisms with upper bounds that are very close to each other. The load factors for mechanism 1 and mechanism 2 coincide when

1 = α . For 1 < α and 05 . 1 > α
convergence occurs in the order of 100 iterations. For 05 . 1 1 < < α the load factors for the two competing mechanism differ by, at most 2%.

CONCLUSIONS

The Linear Matching Method provides a programming method for the evaluation of limits in classical plasticity that differs significantly from other programming methods. Although it is best described as a convergent upper bound method, each iteration provides an equilibrium set of moments as well as a compatible set of kinematic variables. Hence both upper and lower bounds are generated which, at convergence, coincide and verifying that the exact solution has been obtained. This property is unique to the Linear Matching Method. This paper has concentrated on the convergence properties of the method applied to rigid plastic portal frames. We find convergence tends to be geometric and reasonably rapid. We also find that the close proximity of upper bounds to the optimal mechanism can delay convergence, by the method first seeking out the mechanism associated with the non-optimal solution before then slowly converging to the optimal solution. It would appear that this phenomenon will only occur if the distributions of hinges in the two mechanisms are very distinct and the upper bounds very close together. The existence of the phenomenon in a particular problem is clearly displayed through the difference between the upper and lower bound.

The inherent simplicity of the method by using, as its basis, linear analysis, together with its strong convergence properties, encourages the view that it is a method that may usefully be seen alongside classical programming approaches to structural design.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

13 Appendix A-Linear matching method for single story frame-Matrix Formulation

The Linear Matching Method involves performing a sequence of linear analyses by systematically varying the stiffness, to provide a sequence of upper and lower bounds to the limit load. The details of the procedure are reported in this appendix. Let us assign a rotational stiffness i R at each plastic hinge j Φ ~and the related moments are given by:

[ ] [ ][ ] Φ = R M
where [ ] M ~is a bending moment vector, [ ] R is the diagonal matrix of rotational stiffnesses, where

[ ] ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 5 4 3 2 1 M M M M M M , [ ] ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = 5 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 R R R R R R and [ ] [ ][ ] Δ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ Φ Φ Φ Φ Φ = Φ 54 3 2 1 B (A.1) Here [ ] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = Δ v u
is the vector of independent degrees of freedom and [B] is the shape

function matrix: [ ] [ ] 1 0 1 1 2 0 1 0 L u L L and v L L ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ - ⎡ ⎤ ⎢ ⎥ = Δ= ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ B (A.2) With the load vector ; [ ] [ ] 0 0 0 F V H F λ λ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = (A.3)
the total potential energy for the (k+1) th linear solution is given by:

[ ] [ ][ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] 0 1 1 1 0 1 1 1 2 1 2 1 F K F R T k k UB k k T k T k k UB k k T k + + + + + + Δ - Δ Δ = Δ - Φ Φ = Ω λ λ (A.4) where [ ] [ ] [ ][ ] B R B K k T k 1 1 + + =
denote the stiffness matrix. The linear problem is solved by minimizing the potential energy, i.e.

[ ][ ] [ ]

0 1 1 ~F K k UB k k λ = Δ + +
At each independent node the rotations are related to the independent displacement vector [ ] Δ by (A.1).

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 14

Upper Bound Evaluation

For a particular load ratio α the vector of external loads can be written as follow:

[ ] [ ] 0 F H H V H F λ α λ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = (A.5)
For any compatible mechanism from the upper bound formula eq. ( 2.3):

[ ] [ ] j j Y T UB M F Φ = Δ ∑ = 51 0 λ (A.6)
For this particular case i Φ 's can defined so that they are always positive, see Figure 4; and hence

[ ] [ ] [ ] [ ][ ] ∑ = Δ ≤ Δ 5 1 0 j T Y T UB B I M F λ (A.7)
where the vector [ ] I is a unit matrix. The upper bound load multiplier from equation (A.6) is then given by :

[ ] [ ][ ] [ ] [ ] 0 5 1 ~F B I M T n T Y UB Δ Δ = ∑ = λ (A.8)
Briefly, the main steps for determinig the upper bound using the LMM are:

1) The first linear analysis is made with R i=1,5 =1 and 1

UB λ = .
2) The rotational stiffness R i at each node is adjusted according to the rules of LMM so that at iteration k+1 the new distribution of rotational stiffness is :

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = + k j Y k i k j M M R R 1
3) A new linear analysis is performed with the new stiffness matrix

1 + k j R so a new displacement vector [ ] Δ ~and [ ] [ ][ ] Δ = Φ B and k UB λ can be evaluated and [ ] [ ] α λ k UB F = is the load for the next iteration.
The process is iterated until the lowest upper bound is reached. The lower bound is given by eq. (3.5):

k j Y k UB k LB M j M max λ λ = (A.9) A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 15 Appendix B

-Matrix formulation for multistory frame

Let us consider a single beam element (Fig. B1a) with hinge positions at its ends adjacent but distinct from nodes i and j where the beam is connected to other beams. As before i Φ denotes the hinge rotation and the rotation of the nodes is separately denoted by i ϑ . The displacements of the nodes parallel to and perpendicular to the beam length are denoted by ,

i i u v respectively.
For rotation of hinges with no rotation of nodes, Figure B1b :

L v v L v v i j j i j i - = Φ - = Φ ; (B.1)
and Fig. B1c shows the rotation of hinges due to rotations of nodes :

j j i i ϑ ϑ - = Φ - = Φ ; (B.2)
Hence the total rotation of the hinges is the sum of these (B.1) and (B.2):

j i j j i i j i L v v L v v ϑ ϑ - - = Φ - - = Φ ; (B.3)
The element is considered inextensible and hence the beam extension:

j i u u e - =

= constant

Hence for a typical element the vector [ ] Φ of the local hinge rotations and beam extension is related to the element displacement vector [ ] u of nodal displacements and rotations as follow:

[ ] [ ] [ ] [ ][ ] u C v u v u L L L L L L e u j j j i i i C j i = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ - - - - = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Φ Φ = Φ ϑ ϑ 0 0 0 0 1 1 0 0 1 0 0 1 0 1 1 0 (B.4)
where [ ] C is the compatibility matrix. In turn the element displacement vector [ ] u is related to a global matrix of global degrees of freedom [ ]

U by the matrix [ ] Ψ , [ ] [ ][ ] U u Ψ = Hence [ ] [ ][ ] U B = Φ where [ ] [ ][ ] Ψ = C B A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 16
The stiffness matrix for each element is now:

[ ] [ ] [ ][ ] B R B K T el ' = where [ ] ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = l j i E R R R
Where L E is the elongation stiffness, given a suitable large value; in our calculations

6 10 . 1 = E .
The solution of the linear problem for a particular set of [ ] R is, therefore, given by,

[ ][ ] [ ] 0 ~F U K UB λ = where [ ] [ ] ∑ = el el K K (B.5)
The rotation at plastic hinges and the upper bound load parameter can be evaluated from the following relations:

[ ] [ ][ ] U B = Φ (B.6) [ ] [ ] ∑ Φ = hinges j pj T UB M F U λ (B.
Y L M λ ϑ ϑ ⋅ ⋅ = + + + ⋅ + ⋅ ie. 2 Y L M λ =0.87857 (C.2)
For the third mechanism :

Work done:

( ) Figure 7 Graphical Representation of the convergence process for the k=10 th iteration.

3 3 3 7 4 0.85 2 2 8 1 2 Y L M λ ϑ ϑ α ⎛ ⎞ ⋅ ⋅ + ⋅ = ⋅ + ⋅ + ⋅ ⎡ ⎤ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠
Figure 8 The variation of Ω at different stages of the convergence process. A

C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT M k j M ~kj ~1 ~ k j k j R 1 A M k j M ~kj ~1 ~ k j k j R 1 A M k j M ~kj ~1 ~ k j k j R 1 k j R k j R k j R k j k j R k j R k j R k j A pj M Fig.1 A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

1

  Evaluation of the Convergent Properties of the Linear Matching Method for Computing the Collapse of Structural Components. O. Barrera * , A.C.F. Cocks and A.R.S. Ponter ⊕

iF

  Δ at the intersection of beams at ix that are compatible with a set of plastic hinge rotations c j Φ at positions j h . Hence the deformation of the structure is subjected to a severe subclass of all the possible modes of behaviour, those defined by c i is then defined by the Galerkin criterion that equilibrium is satisfied if the following virtual work relationships holds for all possible sets

R

  can be found so that for fixed k j Φ ~ the moment can be brought to the yield surface, as shown in Figure (1).

  for this (k+1) th solution is chosen by computing the upper bound load parameter k UB

  Figures (5 a,b,c) show the evolution of the upper and lower bound values for three values of α, corresponding to different regions of the activation diagram shown in Fig 3. The evolution of the shape of the collapse mechanism as the solution converges is given in Figures (6 a,b,c), expressed as a function of the parameter v u = β

Figure 7

 7 Figure 7 shows UB λ versus the shape of mechanism parameter u v / = β for the problem of Figure 2 for the case of 1 = α. The plot also shows Ω for the (k+1) th =11 th iteration. It can be seen that the matching process chooses a linear material so the upper bound load parameter and the normalised potential energy coincide and are tangential at the k th solution at k β . The (k+1) th solution is obtained as the minimum of the potential energy and use of the resulting rotations provides a lower value of the upper bound. In terms of this graphical representation the convexity conditions (4.1) and (4.2) require that the potential energy curve lies everywhere above the upper bound curve.

M~

  assigned value and increases to much larger values as the solution converges towards the exact value. At each iteration, the value of the corresponding k j is dominated by equilibrium considerations and remains relatively constant. Hence from (3.1) the corresponding rotation may be expected to reduce by a constant factor and hence the contribution to the upper bound will also reduce by a constant factor. If the convergence of the method were dominated by the reduction of the contribution to the upper bound inactive hinges, the rate of convergence would be expected to be geometric, i.e. the error reduces by a constant factor over each iteration. If the error in the upper bound at the kth iteration is k δ ,

  convergence process is shown in Figure16in terms of values of UB λ , LB λ . The converged values of the bounds and the collapse mechanism are consistent with the exact solution reported by[START_REF] Marin-Artieda | Approximate limit load evaluatio of structural frames using linear elastic analysis[END_REF] where they give a collapse load of 31.3kN for L= 6m and y M =210kN

  . To answer this question, the values of the plastic moments for elements a(*), b(*), c(*) in Figure 21 are changed by an amount α, such that NEW p P possible optimal mechanisms occur, as shown in Figure 22. The converged values compared in Figure 23b with the upper bounds associated with the three mechanisms (see Appendix C), showing that a change in the optimal mechanism occurs at 5

KFor

  7)Now according to the procedure we can adjust iteration by iteration the rotational stiffness at each hinge k j matrix can be reconstructed. We solve (B.5) and can determine the new rotations from (B.6) and the new upper bound from (B.7), so: can be evaluated from the upper bound from equation (3.5) Appendix C-Upper Bound for frame of Figure21In Fig.22three different values of the collapse load multiplier can be determined assuming the 3 collapse mechanisms.

Figure. 1

 1 Figure CaptionsFigure.1 Graphical interpretation of matching procedure at increment k+1.

Figure 2 A

 2 Figure 2 A single story portal frame.

Figure 3

 3 Figure 3 Interaction diagram for the structure of Figure 1[START_REF] Horne | Plastic theory of structures[END_REF].

Figure 4

 4 Figure 4 Possible collapse mechanisms for the structure of Fig 2. The number of each node is given in Fig.2.

Figure 5 Figure 6

 56 Figure 5 Evolution of the upper and lower bounds for (a) 25 . 0 = α , (b) 1 = α and (c) 4 = α . Figure 6 Evolution of the shape of the collapse mechanism for (a) 25 . 0 = α , (b) 1 = α and (c) 4 = α .

Figure

  Figure.9 Rate of convergence for 25 . 0 = α , 1 = α and 4 = α . The plot shows geometric convergence ( 25 . 0 = α , 1 = α) except when the error is extremely small.

Figure. 10

 10 Figure.10 Three story portal frame analysed in section 6.1. The plastic moment of each beam element is indicated on the figure.Figure 11 The collapse mechanism observed in the analysis of the structure in Fig 10.

Figure 11

 11 Figure.10 Three story portal frame analysed in section 6.1. The plastic moment of each beam element is indicated on the figure.Figure 11 The collapse mechanism observed in the analysis of the structure in Fig 10.

Figure 12

 12 Figure 12 Evolution of the upper and lower bound for the structure in Fig 10.

Figure. 13

 13 Figure.13 Rate of convergence shows approximately geometric convergence.

Figure. 14

 14 Figure.14 Ten-story frame. The plastic moments assumed in the initial analysis of section 6.2 are indicated adjacent to each beam element, Figure.15 Collapse mechanism for the ten-story frame of Fig 14.

Figure. 16

 16 Figure.16 Evolution of the upper and lower bound related to Fig.15.

Figure. 17

 17 Figure.17 Rate of convergence initially rapid and geometric (the error has reduced to 8 10 -) then the error reduces slowly until, finally, geometric convergence Figure 18 The collapse mechanisms observed in the analysis of the structure shown in Fig 14.

Figure 19

 19 Figure19Evolution of upper and lower bounds for the structure of Fig14, showing the initial rapid convergence towards the limit load corresponding to mechanism I followed by a slower convergence to the exact limit load corresponding to mechanism II Figure.20 Rate of convergence shows a rapid and geometric convergence to the first upper bound and then a slower convergence to the second upper bound Figure. 21 Structure in Fig14 with a range of plastic moment values given by p

Figure

  Figure A1 Nodal rotations.

Figure

  Figure B1 (a) Beam element from nodes i and j , (b) Beam with no rotation of nodes, (c) beams with the rotation of hinges due to rotations of nodes.

  Fig.2

  Fig.5a, b, c

Fig. 12

 12 Fig.12

  Fig.13

  Fig.14

Fig. 16

 16 Fig.16

  Fig.17

  Fig.18

  Fig.20

  Fig.22

ACKNOWLEDGEMENTS

O.B. would like to acknowledge support from the University 'Mediterranea' of Reggio Calabria to visit the University of Oxford, during which period this research was conducted.