Hybrid finite element thermal modelling of fire protected structural elements strengthened with CFRP laminates

Composite Structures 113 (2014) 396–402

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Hybrid finite element thermal modelling of fire protected structural elements strengthened with CFRP laminates J.A. Teixeira de Freitas a,⇑, C. López a, P.T. Cuong a, Rui Faria b a b

Departamento de Engenharia Civil, Arquitectura e Georecursos, Instituto Superior Técnico, University of Lisbon, Portugal Departamento de Engenharia Civil, Faculty of Engineering, University of Porto, Portugal

a r t i c l e

i n f o

Article history: Available online 21 March 2014 Keywords: Hybrid finite elements Transient thermal analysis RC and CFRP materials Standard Fire

a b s t r a c t A hybrid formulation of the finite element method, based on the independent approximation of the temperature and heat flux fields, is used in the simulation of the thermal response of a reinforced concrete (RC) beam strengthened with carbon fibre reinforced polymer (CFRP) laminates subjected to the ISO-834 Standard Fire. Domain decomposition is strictly dictated by the adequate representation of geometry and thermo-physical properties, while the approximation of the state variables in each finite element is mainly constrained to adequately simulate the thermal response of the system. This uncoupling of the geometry mapping from the finite element approximation leads to a naturally p-adaptive formulation, well suited to parallelization, which can be implemented on unstructured, coarse meshes of high-degree elements to obtain accurate and numerically stable solutions. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The conventional formulation of the finite element method for heat transfer [1,2] leads, in general, to the use of highly refined meshes in the thermal analysis of systems involving components with dissimilar geometric and thermal properties, and subjected to markedly different heat flow fields. A typical case is the simulation of the fire performance of reinforced concrete (RC) structural elements strengthened with carbon fibre reinforced polymer (CFRP) laminates, the latter provided with fire protection calcium silicate (CS) layers [3]. Similar studies are reported in [4–9]. Most experimental studies on the thermal performance of concrete structural elements under fire conditions are supported by numerical simulations carried out with commercial finite element software or in-house codes, where usually refined meshes are employed [10–12]. This high level of mesh refinement is typically imposed by the option of coupling the mapping of geometry to the approximation of the temperature field and the use of structured meshes of isoparametric elements to ensure thermal continuity. In the application reported here this follows from the need to adequately represent the temperature field in a RC beam strengthened with thin layers of CFRP, a material highly sensitive to temperature, which is protected by boards of an insulating material (CS) ⇑ Corresponding author. Tel.: +351 218418236. E-mail address: [email protected] (J.A. Teixeira de Freitas). http://dx.doi.org/10.1016/j.compstruct.2014.03.021 0263-8223/Ó 2014 Elsevier Ltd. All rights reserved.

that is subjected to very high temperature gradients under the ISO-834 Standard Fire. An alternative approach is to address separately the needs posed by the two conflicting issues in terms of modelling, namely adequacy in the representation of geometry and thermal properties (constrained by domain decomposition) and adequacy in the modelling of the thermal field in each component of the system (constrained by the degree of the finite element approximation). This technique has been applied to the modelling of the thermochemical process of cement hydration in the staged construction of massive concrete structures [13], where it is often convenient to use elements with very high aspect ratios to reduce the computational effort. The research reported in [3] motivated the extension of this approach to the rather more demanding problem of simulating of heat transfer in fire protection systems of RC elements strengthened with CFRP laminates. Uncoupling the mapping of the geometry from the finite element approximation leads to a hybrid formulation, as both the temperature and heat flux fields must now be independently approximated. This approach entails an additional cost in the number of degrees-of-freedom, which is strongly counterbalanced by the possibility of implementing coarser meshes of elements with variable degrees of approximation. It is shown here that the shape of the finite element is basically dictated by geometry and material properties of the system, whilst the degree of approximation in each spatial direction can be freely selected towards fitting the

J.A. Teixeira de Freitas et al. / Composite Structures 113 (2014) 396–402

geometry of the element and the expected properties of the heat flow field. The present paper is organized in three parts. The mathematical model is defined first, with particular attention being paid to the different forms the boundary conditions may assume in the present application. The hybrid finite element formulation is summarized next. Emphasis is placed on the identification of the state variables subjected to direct approximation, on the (naturally hierarchical and orthogonal) bases used in the approximation and on the aspects that distinguish the hybrid and conventional formulations of the finite element method. As the main aspects of numerical implementation are presented with sufficient detail in [13], only the distinguishing features of the approach proposed here are highlighted and justified, namely those concerning the use of unstructured meshes and the suitability of the formulation to adaptive p-refinement and parallel processing. The paper closes with a detailed analysis of a transient, nonlinear testing problem with boundary conditions designed to simulate one-, two- and three-dimensional thermal responses of RC beams strengthened with CFRP laminates and protected to fire with CS boards.

on Cei

TD ¼ Tk

397

ð10Þ

T D ¼ T k þ hrs q on Cers

ð11Þ

Thus, the definitions of the generalized Neumann and Dirichlet boundaries are:

CeN ¼ Ceq [ Cec [ Cer [ Cecd

ð12Þ

CeD ¼ CeT [ Cei [ Cers

ð13Þ

The prescribed temperature, heat flux and ambient temperature fields are defined as functions of time and space in the supporting finite element code. The specific heat and the conductivity coefficients, as well as the convection, radiation, conductance and resistance coefficients, are defined in a similar way and may vary with temperature. The definition of the nonlinear, transient heat transfer problem is completed with the initial condition:

T ¼ T 0 at t ¼ t 0

in V e

ð14Þ

3. Finite element approximations 2. Heat transfer problem The heat transfer equations are written as follows for a typical finite element with domain Ve and boundary Ce referred to space and time systems x and t:

$T k$T ¼ qcT_ in V e nT k$T ¼ qN T ¼ TD

on CeN

on CeD

ð1Þ ð2Þ ð3Þ

In the thermal balance Eq. (1), $T and T_ represent the space gradient and the time rate of the temperature field, T(x, t), k is the local conductivity matrix and qc is the volumetric specific heat. The two latter material properties are assumed to vary with temperature. The finite element boundaries are uncoupled into complementary Neumann and Dirichlet parts,

Ce ¼ CeN [ CeD

ð4Þ

with CeN \ CeD ¼ ;, in order to implement alternative boundary conditions under the generalized notation used in Eqs. (2) and (3).  on The Neumann condition (2) is used to prescribe a heat flux q boundary Ceq ,

 on Ceq qN ¼ q

ð5Þ

as well as convection, radiation and inter-element conductance conditions,

Typical of the formulation of hybrid finite elements is the possibility to uncouple the geometric mapping of the element from the approximation of the state variables of the application problem. This feature is used here to implement coarse meshes of high-degree elements released from the usual constraints on geometry and connectivity. Although no conceptual constraints limit the topology of hybrid elements, e.g. [14], domain decomposition has been implemented in the tests reported here using the geometric mapping that typifies isoparametric master-elements,

x ¼ NðnÞc

in V e

ð15Þ

where vector c defines the global x-coordinates of the nodes of a typical element e, N is the shape function matrix and n = (n, g, f) represents the natural co-ordinate system. Separation of variables is used in the definition of the independent approximation of the temperature field,

Tðx; tÞ ¼ HðnÞTðtÞ in V e

ð16Þ

where vector T defines the amplitudes of the temperature modes listed in row-vector H. The temperature approximation basis is defined as the tensor product of (naturally hierarchical, orthogonal) Legendre polynomials expressed in the natural co-ordinate system of the element,

H ¼ f   Li ðnÞLj ðgÞLk ðfÞ   g

ð17Þ

qN ¼ hc ðT  T a Þ on Cec

ð6Þ

with Lk = 1 (and Lj = 1) in two-dimensional (one-dimensional) problems. In a D-dimensional problem, the dimension of the temperature approximation basis is defined as follows,

qN ¼ hr ðT 4  T 4a Þ on Cer

ð7Þ

NV ðDÞ ¼

qN ¼ hcd ðT  T k Þ on Cecd

ð8Þ

ð18Þ

where Ta is the ambient temperature, Tk is the surface temperature of the connecting elements, and hc, hr and hcd denote the convection, radiation and conductance heat transfer coefficients, respectively. The Dirichlet condition (3) is used to prescribe temperature T on in boundary CeT ,

depending on whether different or the same degree of approximation, dj, is implemented in each natural co-ordinate direction. Thus, and consequent upon the uncoupling of the geometry mapping (15) and the temperature approximation (16), elements may have high aspect ratios, with adequate degrees of approximation in each direction. Besides the combination of different types of master-elements in the same mesh to obtain adequate representations of curved surfaces, this uncoupling of geometry and state variable

( D Y ðdj þ 1Þ j¼1

T eD ¼ T

on CeT

ð9Þ

and inter-element thermal continuity or resistance conditions 1 (hrs ¼ hcd ) on boundaries Cei and Cers ,

1 ðd D!

for different degrees in each dimension

þ jÞ for same degree d ¼ dj in all dimensions

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approximations simplifies the generalization of mapping (15). For instance, the surface of an element (or its side, in two-dimensional problems) can be subdivided in an arbitrary number of areas (or segments) to create unstructured meshes. It is shown in [13] that this technique is useful to reduce both the meshing effort and the number of degrees-of-freedom in the analysis of structures combining elements with distinct material and geometric profiles. It is known that the price that has to be paid to attain this higher level of modelling flexibility is the necessity to approximate also the heat flux on the Dirichlet boundaries of the element. This approximation is written in a form similar to Eq. (16),

As the approximation basis (17) is not defined on nodal (Lagrange-type) functions, the coefficients of vector T represent now the weights of the functions used to set up the approximation basis (16). As they no longer define nodal temperatures, the heat flux resultants (28) cannot be used to establish the equivalent nodal equilibrium conditions that typify conform elements. This is the root cause for the independent approximation of the heat flux on the Dirichlet boundaries of the element. Enforcement of approximation (19) in Eq. (28) yields the following definition for the heat flux resultant,

qðs; tÞ ¼ uðsÞqðtÞ on CeD

where the heat flux equilibrium matrix is expressed as:

ð19Þ

where vector q defines the amplitudes of the heat flux modes listed in row-vector u,

u ¼ f   Li ðuÞLj ðv Þ   g

ð20Þ

and s = (u, v) represents the surface coordinates of solid elements (s = (u) is the side coordinate of two-dimensional elements, with Lj = 1). Thus, the dimension of the boundary basis (20) for a Ddimensional problem, NC(D), is still defined by Eq. (18) under the following relation:

NC ðDÞ ¼ N V ðD  1Þ

ð21Þ

4. Finite element equations Under the temperature field approximation (16), the Galerkin weak form of the thermal balance condition (1) is,

Z

_ dV e ¼ 0 HT ð$T k$T  qc TÞ

ð22Þ

or, after integration by parts and using the heat flux definition (2):

Z

e HT qc T_ dV þ

Z

e

ð$HÞT k$T dV þ

Z

HT q dCe ¼ 0

ð23Þ

Implementation of approximation (16) and separation of the boundary in its Neumann and Dirichlet parts, Eq. (4), in order to enforce conditions (5)–(8), yields similar forms for the alternative hybrid and conform finite element formulations,

H T_ þ ðK þ CÞT þ Q D ¼ Q k  Q

ð24Þ

under the following definitions for the specific heat, conductivity, convection and conductance matrices:

H¼

K¼

C¼

Z

HT qcH dV

Z Z

e

ð$HÞT kð$HÞ dV T

e c

H hc H dC þ

Z

e

T

H hcd H dC

Z

Q¼

Z

Z

 dCeq  HT q

Z

ð29Þ HT hc T a dCec 

Z

Z

  HT hr T 4a  T 4 dCer

uT ðT  T D Þ dCeD ¼ 0

ð33Þ

and the weak form of thermal continuity is obtained enforcing in this equation the temperature approximation (16) under boundary conditions (9)–(11) and definition (32):

BT T  Rq ¼ T þ Rk

ð34Þ

The thermal resistance matrix and the vectors associated with prescribed temperature and the boundary temperature at connecting elements are defined as follows,

R¼

T¼

Z Z

Rk ¼

uT hrs u dCers

ð35Þ

uT T dCeT

ð36Þ

Z

uT T k dCei þ

Z

uT T k dCers

ð37Þ

As it has been done for the conductance condition, system (34) can be used to implement the inverse forms of the convection and radiation boundary conditions (6) and (7), which would then be assigned to the Dirichlet boundary of the element.

The governing system at element level is obtained combining the weak forms of thermal balance (24) and thermal continuity (34):

ð28Þ

HT hcd T k dCecd

ð32Þ

ð26Þ

ð27Þ

HT q dCeD

HT u dCeD

A second consequence of using a non-nodal temperature approximation is that it is no longer possible to enforce thermal continuity in strong form simply by equating the nodal temperatures of connecting elements. Instead, the approximation basis is used to enforce on (Galerkin) average the Dirichlet condition (3),



e cd

Z

5. Finite element solving system

and the heat flux terms associated with prescribed heat flux terms on the Neumann boundary (12) are:

Qk ¼

B¼

ð31Þ

ð25Þ

The vector associated with the unknown heat flux term on the Dirichlet boundary (13) is,

QD ¼

Q D ¼ Bq

ð30Þ

H

O

e ( _ )e T

O

O

q_

 þ

K þC

B

e  e T

BT

R

q

(

¼

Q T

)e

ð38Þ

The elementary systems (38) can no longer be assembled for the finite element mesh through incidence of nodal temperatures and summation of equivalent nodal heat fluxes. In fact, assemblage is simpler as it consists in listing in vectors T and q (and in their time rates) the degrees-of-freedom for each domain and boundary element, Te and qe, and organize accordingly the associated finite element matrices and vectors. Consequently, the structure of the assembled system is similar to the structure of the elementary system (38), where now matrices H, K and C block diagonal and the columns of the thermal resistance and heat flux matrices, R and B, are shared by at most two connecting elements (the same happens to thermal conductance term of matrix C).

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J.A. Teixeira de Freitas et al. / Composite Structures 113 (2014) 396–402

2b = 10cm

Assemblage and numerical implementation of the first-order nonlinear system (38) is discussed in [13]. The system is discretized in time using a standard trapezoidal rule,

T i ¼ T i1 þ c0 dt i T_ i1 þ cdt i T_ i

ð39Þ

h = 12

where dti is the time increment and c and c0 are time integration coefficients. The resulting system,

"

ðcdt Þ1 H þ K þ C

B

BT

R

( ¼

y

#  T i

q

x

i

ðcdti Þ1 H i ðT i1 þ c0 dt i T_ i1 Þ  Q i

h′ h

) ð40Þ

Ti

is then solved with the Newton–Raphson method. Besides being highly sparse, system (40) is well-suited to adaptive p-refinement and parallel processing. Suitability to adaptive refinement is a direct consequence of using the naturally hierarchical domain and boundary approximation bases (17) and (20): only the coefficients of the relevant matrices and vectors associated with the additional degrees-of-freedom are calculated anew. Suitability to parallelization follows from the fact that the temperature degrees-of-freedom, Te, are strictly element-dependent (not shared by elements connecting at the same node, as in conform elements), and the heat flux degrees-of-freedom, qe, are shared at most by two connecting elements.

2b′ = 5 Fig. 2. Cross-section dimensions.

Thermal data: Concrete 1.0

ρc/ρcmax

0.8

0.6

k/kmax 0.4

6. Numerical application

0.2

The test on the simulation of the thermal response of a RC beam strengthened with a CFRP laminate insulated from fire by CS boards is taken from [3]. The dimensions of the simplified model of the beam are given in Figs. 1 and 2, and the temperature-dependent thermo-physical properties of the different materials [3,15] are defined in Figs. 3 and 4. In the present thermal model the steel reinforcement bars are not modelled, as they will not influence significantly the temperature field of the beam [15] due to their high thermal conductivity. The peaks in the variation of the specific heat of concrete and of the CFRP laminate are associated with evaporation of moisture and resin decomposition, respectively. Different values are defined for  and differthe thickness of the CFRP laminate, h0 , and CS boards, h, ent boundary conditions are applied on the lateral surfaces, x = ±b, and on the end sections, z = ±L, to illustrate alternative modelling situations, namely 1D, 2D and 3D thermal fields, as well as the transition in time from one- to two-dimensional solutions. Convection and radiation conditions are applied on the top and bottom surfaces of the beam in all tests. The air temperature on the upper surface is constant, Ta = 20 °C, and the ISO-834 Standard Fire (°C) is prescribed on the lower surface, Ta = TISO = T0 + 345log(8t + 1), with t in minutes [16]. The heat flux is null on the (insulated) lateral faces. The radiation coefficient is constant, hr = 0.7r, where r is the Stefan–Boltzmann constant (W/m2 K4), and the convection coefficient (W/m2 K) varies linearly with the surface tem9 perature (°C) of the material [17]: hc ¼ 5 þ 196 ðT  20Þ. The initial temperature is T0 = 20 °C in all components and the period of the

0.0

kmax ≈ 1.6422 W/m°C ρcmax=3.5280 MJ/m3°C

T (°C) 0

200

400

600

800

1000

Fig. 3. Thermal properties of concrete.

analysis is 5000 s in all tests, to reach a peak fire temperature close to 1000 °C. 6.1. One-dimensional behaviour In order to induce a one-dimensional thermal field, adiabatic conditions are assumed on boundaries x = ±b and the laminate and the insulator are removed in the first analysis. Furthermore, the Standard Fire is prescribed on side y = 0 of the beam crosssection. As the temperature field spans most of the range of the concrete thermal properties, the main objective of this test is to assess the performance of the formulation under nonlinear material properties and boundary conditions. The problem is solved with the two-element mesh shown in 0  ¼ 0). The temperature Fig. 2 (using symmetry and setting h ¼ h profile along the height y = h  g (with 0 6 g 6 1) and the estimates at control points are presented in Fig. 5(a) and (b), respectively. They are determined with a fifth-degree approximation, d = 5, in bases (17) and (20), for a solving system (40) with a total of N = 80 degrees-of-freedom. As it can be seen, they recover the ‘exact’ values obtained with a (over-) refined mesh of 50  120 conform elements (bilinear, 4-node elements, to yield N = 5880). 6.2. Two-dimensional behaviour

Concrete CFRP

y

h = 12

z

h′ h

CS 30

L′ = 70

L = 100 cm

Fig. 1. RC beam strengthened with a CFRP laminate subjected to fire.

In order to illustrate the two-dimensional effect of the asymmetry caused by the CFRP laminate and the insulation layer, the latter being now directly exposed to the Standard Fire, the cross-section is still analysed assuming adiabatic conditions on boundaries x = ±b. The recommended values for the thickness of the laminate and  ¼ 2:5 cm (see Fig. 2), are for the protection layer, h0 = 0.42 cm and h

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J.A. Teixeira de Freitas et al. / Composite Structures 113 (2014) 396–402

Thermal data: CFRP

Thermal data: CS

1.0

1.0

ρc/ρcmax

kmax ≈ 1.3641 ρcmax ≈ 7.3342

0.8

0.8

k/kmax

0.6

0.6

ρc/ρcmax

0.4 0.2

0.4

T (°C)

0.0 0

200

kmax = 0.1820 W/m°C ρcmax ≈ 0.4297 MJ/m3 °C

0.2

k/k max

400

600

800

1000

T (°C)

0.0 0

200

400

600

800

1000

Fig. 4. Conductivity and specific heat of CFRP reinforcement and CS insulation.

T (°C)

T (°C)

b ≤ x ≤ +b

1000

b ≤ x ≤ +b

1000

y=0

t = 5000 s

800

1920

h = 12 (cm) h′ = h = 0

600

960

y = h⋅η

400

240 120

η

0

0.0

-0.2

0.2

0.4

0.6

600

Hybrid

480

200

800

TISO

0.8

2.5

Conform

400

4.5

200

7.0 12.0

t (s)

0

1.0

0

(a) Temperature profiles

1000

2000

3000

4000

5000

(b) Temperature at control points

Fig. 5. Concrete cross-section (without CFRP and CS layer).

used in the analysis. The temperature profiles along the height and breadth of the cross-section presented in Fig. 6(a) and (b) show that the two-dimensional effect is increasingly visible as time evolves. It is caused by the different levels of CS protection on the lower edge of the RC section. The analysis is implemented on the six-element discretization of the cross-section shown in Fig. 2, with two ‘concrete elements’, one ‘CFRP element’ and three ‘CS elements’. A seventh-degree approximation is used in the concrete elements, (dn; dg) = (7; 7) in Eq (18) for D = 2, and basis (dn; dg) = (7; 3)is applied to both CFRP and CS elements (n and g correspond to the x and y directions). The boundary approximation is dimensioned to enforce thermal continuity in the y-direction, by letting du = dn on horizontal sides. This condition is relaxed on vertical sides of the mesh, setting du = dg  1. The resulting solving system (40) has 245 degrees-of-freedom and recovers the reference solution obtained with 4-node

1000

isoparametric elements. The reference solution shown in Fig. 6 is determined using a highly refined mesh of 7450 elements (6000 elements in the concrete section, 100 in the CFRP and 1350 in the CS layer), for a total of 7650 degrees-of-freedom. 6.3. Transition from two-dimensional into one-dimensional behaviour The third cross-section analysis is implemented to confirm that the hybrid finite element formulation can adequately model qualitative changes in the thermal response as time evolves. To this effect, the same thickness is used for the laminate and the insulator, 0  ¼ 0:42 cm, providing a light insulation for the CFRP, with the h ¼h purpose of reaching temperatures in the laminate that exhaust its heat capacity and thus inducing the two-dimensional thermal field to evolve towards a one-dimensional one. The problem is solved with the six-element mesh and the approximations bases used in the previous test. As shown in

Section x = 0 (y= h⋅η)

T (°C)

200

Section y = 0 (x= b⋅ξ)

T (°C)

t = 5000 (s)

t = 5000 (s)

800 1920

160

960

600

CFRP

CFRP

480 240

400 120

h = 12 (cm) h′ = 0.42

120

h = 2.5

80 960

(o) Conform solutions

200 0

0 -0.4

1920

40

η

-0.2

0.0

0.2

0.4

0.6

0.8

(a) Temperature profiles in height

1.0

480

t=0

0 0.0

0.2

0.4

0.6

0.8

(b) Temperature profiles in breadth

Fig. 6. Temperature profiles in the RC cross-section with CFRP and high insulation.

ξ 1.0

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J.A. Teixeira de Freitas et al. / Composite Structures 113 (2014) 396–402

1000

Section x = 0 (y = h⋅η)

T (°C) 5000

Section y = 0 (x = b⋅ξ)

T (°C)

600

t = 5000 (s)

800

480

1920

CFRP

960

600 400

360

480

h = 12 (cm) h′ = 0.42

240

h = 0.42

240

1920 960

120

120

200 t= 0 (s)

0 -0.4

-0.2

η 0.0

0.2

0.4

0.6

0.8

1.0

480

t = 240

ξ

0

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 7. Temperature profiles in the RC cross-section with low CS insulation.

t = 240s

480s

960s

1920s

5000s

Fig. 8. Temperature distribution in the beam middle-plane (x ¼ 0; 0 6 z 6 L).

Fig. 7, when the temperature in the CFRP reaches the 600–800 °C range the heat transfer into the concrete section strongly increases and consequently the one-dimensional response is gradually recovered in time.

of dimension ratios ‘x:‘y:‘z, are 1.0:4.8:28.0 in concrete elements and 6.0:1.0:166.7 in CFRP and CS elements. The same bases are used in all domain and boundary approximations, irrespectively of the aspect ratios of the elements. The first test was run for bases dn = dg = df = dV = 5 and du = dv = dC = 4, for a total of N = 882 degrees-of-freedom. Convergence was confirmed increasing the degree of the approximation to dV = 7 and dC = 6 (N = 1382). The solution obtained with the latter approximation is shown in Figs. 8 and 9. The representation is complicated by the ranges that must be encompassed both in temperature (from 20 °C to 1000 °C) and in dimension (from 4.2 mm to 1000 mm). The option has been to present the variations in the beam middle-plane, x = 0 in Fig. 2, and in the lower part, y < 0.5 h, of the mid-span cross-section, z = 0 in Fig. 1. The temperature distributions presented in Figs. 8 and 9 are close but do not match the results shown in Fig. 7, as different boundary and inter-element continuity conditions are applied. Thermal continuity is adequately modelled, both between the CFRP element and the connecting concrete element and on the interfaces of CS elements. The results presented in Figs. 8 and 9 also capture the delaying effect on heat transfer caused by the conductance condition applied to the interface between the concrete and the CS elements.

6.4. Three-dimensional behaviour 7. Closure Symmetry is used to analyse one quarter of the beam with the twelve element mesh resulting from the combination of the discretizations shown in Figs. 1 and 2. The initial temperature and the convection and radiation conditions defined above still hold. 0  The Standard Fire is applied to the bottom surface, y ¼ ðh þ hÞ, and the air temperature on the complementary part of the boundary (x = ±b, y = h, z = ±L) is 20 °C. In addition, the conductance condition (8) is applied with hcd = 50 W/m2 K on the interface of the CS elements with the concrete and CFRP elements to model thermal bridging. 0  ¼ 0:42 cm) is The more demanding low insulation case (h ¼ h used to assess the performance of the formulation implemented with high aspect ratio elements. Extreme cases, in the sequence

Coupling of the approximation of state variables with the domain decomposition of fire protected structural systems, typical of isoparametric finite elements, implies the use of structured meshes that often lead to unnecessarily high levels of refinement. This excessive refinement is caused by the combination of structural elements (e.g. RC beams and slabs) with components that are either subjected to high temperature gradients (the fire protection layers) or have geometries with high aspect ratios (the reinforcing CFRP laminates). The option followed here is to uncouple the representation of the geometry from the approximation of the thermal response. It leads to a hybrid finite element formulation implemented on

800ºC

100ºC

t = 240s

480s

960s

1920s

Fig. 9. Temperature distribution in the lower part of the mid-span section (z = 0, y < 0.5 h).

5000s

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coarse meshes of elements with varying degrees of approximation of the temperature and heat flux fields. Domain decomposition is strictly dictated by the adequate representation of geometry and material properties, whilst the approximation of the state variables in each element is mainly constrained to adequately simulate the thermal response of the system. To enhance numerical stability under high-order p-refinement, naturally hierarchical, orthogonal functions are used to set up the temperature and heat flux approximations. Modelling is very flexible because these approximations are independent and different degrees can be used in each spatial direction of both temperature and heat flux fields. Moreover, it leads to solving systems that are well suited to adaptive refinement and parallel processing. The numerical application presented here justified neither the use of those properties nor the implementation of unstructured meshes, illustrated in [13]. However, the results obtained in the simulation of the thermal response of a RC beam strengthened with CFRP laminates subject to the Standard Fire are useful to show that the hybrid finite element formulation yields accurate and stable solutions using coarse meshes of higher-degree elements with widely varying aspect ratios and with distinct and strongly nonlinear thermo-physical properties. Acknowledgements This research work has been supported by Fundação para a Ciência e Tecnologia (FCT) through research units ICIST (IST-id) and LABEST (University of Porto), and contracts SFRH/BD/76632/ 2011 and PTDC/ECM/100779/2008. The authors gratefully acknowledge the advice and support given by colleague C. Tiago. References [1] Zienkiewicz OC, Cheung YK. Finite elements in the solution of field problems. Engineer 1965;220:507–10.

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