An integrated framework for nonlinear analysis of plane frames exposed to fire using the direct stiffness method

Computers and Structures 190 (2017) 173–185

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An integrated framework for nonlinear analysis of plane frames exposed to fire using the direct stiffness method Gaurav Srivastava ⇑, P. Ravi Prakash Department of Civil Engineering, Indian Institute of Technology Gandhinagar, Gujarat 382355, India

a r t i c l e

i n f o

Article history: Received 19 February 2017 Accepted 28 May 2017

Keywords: Normal strength concrete (NSC) High strength concrete (HSC) Spalling Thermo-hydro-mechanical analysis Three-way coupling Steel structures

a b s t r a c t A novel coupled framework for analysis of reinforced concrete (RC) and steel planar frames subjected to fire is developed with three-way coupling between heat transfer, mechanical deformations and pore pressure build-up. Structural members are discretized in space using a two-level scheme where the mechanical solver utilizes 1D line elements, and the thermal and the pore pressure solvers work on 2D finite element (FE) meshes for each sub-span used by the mechanical solver. Such a strategy enables consideration of effects of large deformations, temperature-dependent material properties (thermal, moisture transport and mechanical), and spalling. None of the earlier developed frameworks considered a three-way coupling between mechanical, thermal and pore pressure solvers without employing a fullfledged 3D FE scheme. A matrix method type approach, developed herein, enables modeling of the three main physical processes taking place in RC members during fire without the need to consider full-fidelity 3D FEM. Several numerical examples are presented to demonstrate the accuracy and applicability of the developed framework in fire analysis of normal and high strength RC and steel structures. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Modern construction practice predominantly utilizes reinforced concrete (RC) and/or steel frames as main load-bearing mechanisms. Given that all structures carry the risk of being exposed to fire during their service lives, it is essential for them to be analyzed and designed for fire in addition to the regular service loads. The performance of structures under fire is usually described using fire resistance, which is the time required by structural member(s) to reach a pre-defined failure state. Fire resistance of structural members is typically prescribed by building codes and can also be evaluated using detailed engineering analysis of the system. Building codes [1–3], in general, specify fire ratings as a function of crosssectional dimensions and clear cover/insulation thickness. Apart from the codified procedures, several empirical relations have been proposed (e.g. [4]) to estimate temperature profile and subsequently the structural performance of the members. Almost all of these methods (either empirical or code-based) consider exposure of individual structural members (isolated beams or columns) to a standard fire (e.g. ISO 834 or ASTM E119) and are hence, unsuitable for assessing the performance of a structural system as a whole. ⇑ Corresponding author. E-mail addresses: [email protected] (G. Srivastava), patnayakuni.prakash@ iitgn.ac.in (P. Ravi Prakash). http://dx.doi.org/10.1016/j.compstruc.2017.05.013 0045-7949/Ó 2017 Elsevier Ltd. All rights reserved.

Moreover, all such relations are scenario specific, i.e., there are separate relations for steel and RC. In performance-based design procedures, where response of an entire structural system is to be computed, factors such as fire scenarios, geometric effects and consideration of appropriate support conditions become important. In this context, the importance of having an integrated computational framework that can perform such analyses for various scenarios in a unified manner is well acknowledged. Physical phenomena that must be considered by such models include (a) effects of material and geometric nonlinearity, (b) temperature-dependence of material properties, (c) effects of reversible and permanent thermal deformations, (d) consideration of permanent damage in material, and (e) effects of pore pressure in case of concrete with high moisture content or low permeability. One way to perform such analyses is through detailed finite element (FE) models that work at micro/meso scales. Several researchers have developed such models. For instance, Khennane and Baker [5] developed thermo-plasticity based 3D FE framework for analysis of isolated concrete members exposed to fire, Rigobello et al. [6] developed thermo-plasticity based 3D FE framework for coupled thermo-mechanical analysis of isolated steel members, Yu et al. [7] developed a 3D FE framework for analysis of steel/ composite structures exposed to fire and implemented it in the software VULCAN [8], Majorana et al. [9] develped a finite strain

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3D FE framework for coupled hydro-thermo-mechanical analysis of concrete structures subjcted to fire, Kumar and Srivastava [10] and Kumar [11] developed FE models for RC frames with and without infills. Such detailed FE models can be applied in a variety of situations but are computationally very intensive and hence, are applicable only to individual members or very small structural assemblies. Moreover, such models usually require a large number of parameters to be estimated/calibrated for proper functioning and face difficulties related to numerical convergence even after moderate fire exposures [11–13]. An alternative to detailed FE models is the macro-modeling approach that aims to reduce computational complexity through relevant physical idealizations. Kassimali and Garcilazo [14] developed a matrix analysis based framework for thermo-mechanical analysis of plane frame members considering linear elastic homogeneous material models and a linear temperature gradient across member cross-sections. Wang and Moore [15] developed a FE based macro model for thermo-mechanical analysis of steel frames. Kodur and Dwaikat [16] developed a moment-curvature based macro model for individual RC beams/columns incorporating the effects of spalling. Bionidi and Nero [17] developed a cellular FE based macro model for nonlinear analysis of RC structures subjected to elevated temperatures. Cai et al. [18] developed FE based 3D macro model for coupled thermo-mechanical analysis of steel/concrete structures and successfully implemented it in the software VULCAN [8]. Franssen [19] developed FE based 3D macro model for coupled thermo-mechanical analysis of composite steel-concrete structures and implemented it in the SAFIR program [20]. All of the aforementioned frameworks require discretization of the structural members into several subelements for reasonable level of accuracy. This results in higher computational intensiveness while modeling large scale structural systems subjected to fire. Furthermore, none of the available frameworks consider mutual coupling between heat transfer, moisture transport and mechanical deformations. No coupling implies that heat transfer analysis is performed first and mechanical deformation analysis is performed based on the pre-computed temperature field, thus, effects of cross-sectional changes due to spalling cannot be considered in thermal analysis. Prakash and Srivastava [21] recently developed a two-way coupled framework capable of performing coupled thermo-mechanical analysis. However, the effects of axial deformations were not considered and same constitutive behavior was considered for compression and tension. The present study develops a large deformation Euler-Bernoulli beam-column element comprising of several fibers along its length that can consider effects of material degradation at elevated temperatures and temperature-induced spalling. The developed element considers an updated Lagrangian co-rotational formulation and temperature-dependent stability and bowing functions. A two-level discretization strategy enables the typical 1D idealization in mechanical analysis while maintaining the cross-sectional details for heat transfer and pore pressure analysis through a 2D FE mesh. Integration of thermal effects into member stability and bowing functions yields higher accuracy with fewer number of sub-elements compared to previously discussed macro models. Use of the direct stiffness method approach enables much faster computation times when compared to full fidelity 3D FE models. Section 2 presents the detailed theoretical development of the model including discretization strategy, thermal, pore pressure and mechanical solvers, and the staggered solution strategy implemented to enable mutual coupling of the three solvers. Three-way coupling between the solvers is a unique feature of the developed framework. Section 3 discusses the relevant material models for concrete and steel. Section 5 presents several numerical examples

for validating the developed framework against available experimental data for RC and steel structural members and illustrates the use of the developed framework for fire analysis of a twostorey two-bay portal frame. 2. Theoretical development Mechanical analysis of systems of beams and columns can be performed in a computationally efficient manner by considering the so-called structural elements (e.g. Euler-Bernoulli beam element) as opposed to the usual continuum finite elements (e.g. the hexahederal element). Such an approach typically considers the behavior of the neutral axis of the member with certain assumptions with respect to its cross-section, hence, converting an otherwise 3D problem to an equivalent 1D idealization. While this allows computation of deformations under mechanical loads relatively accurately, considering the effects of additional physical phenomena (e.g. heat transfer and moisture transport) that demonstrate rich behavior across the cross-section of structural members becomes challenging. To facilitate consideration of heat and moisture transport along with mechanical deformations, necessary for appropriately modeling structures under fire, the present study considers a two-level discretization of the physical domain, as shown in Fig. 1. For mechanical analysis, the member is discretized along its length (shown by slices or nodes i) similar to a typical matrix analysis procedure. It is assumed that along the length of a sub-span (i.e. length between i and i þ 1, etc.), temperature and moisture conditions do not change with x. Variations of these conditions along the overall length of the member can be considered by increasing the total number of sub-spans. The variation in temperature and moisture across the crosssection is considered through a 2D finite element solution procedure using a separate yz mesh in each sub-span, as shown in Fig. 1. Thus, an exclusive 2D heat transfer and moisture transport analysis is performed for each sub-span of the member. In order to consider effects of variation of mechanical properties with changes in temperature and/or moisture, fibers are considered along the length of the member in accordance with the 2D mesh, as shown in Fig. 1, and an equivalent 1D fiber is constructed for mechanical analysis, as discussed later. 2.1. Heat transfer analysis Let a typical cross-section of the structural member be denoted by an area X enclosed by a boundary C having a unit normal n. Heat transfer analysis is performed by solving the transient heat equation in X given by

r  ðjrT Þ þ Q ¼ qc

dT ; dt

ð1Þ

where j; q and c are the temperature-dependent thermal conductivity, mass density and specific heat of the material, and Q represents the heat source; t represents time and T is the temperature. Convective and radiative boundary conditions can be enforced on C as

jrT  n ¼ hðT  T 1 Þ;

ð2Þ

where, T 1 is the ambient or fire temperature, and h is the combined convective and radiative heat transfer coefficient. Four node bilinear quadrilateral elements are used along with Galerkin approximation for spatial discretization of (1) which yields a first order nonlinear system of ODEs in time domain as

MT T_ þ jT T ¼ W;

ð3Þ

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175

Fig. 1. Discretization of structural member.

where MT is the heat capacity matrix, jT is the conductivity matrix and W is the heat flux vector. The generalized trapezoidal scheme is applied in time domain to (3) and a system of nonlinear algebraic equations is obtained as

ðMT þ sbjT ÞTnþ1 ¼ ðMT  sð1  bÞjT ÞTn þ sðbWnþ1  ð1  bÞWnþ1 Þ; ð4Þ

stant and Q p is the mass flow rate. Detailed expressions for some of the above mentioned physical variables can be found in [27]. Nonuniform permeability as observed in realistic structures is taken into account by using the constitutive relationship developed by Bary [29] as

vt ¼ vo 10Ad D ;

ð6Þ

where s is the time step, b is a constant between 0 and 1 (b P 0:5 for unconditional stability) and subscripts n and n þ 1 represent respective functions being evaluated at times n and n þ 1. Temperature-dependent material properties (density, specific heat and conductivity) and boundary conditions make (4) nonlinear. Thus, it is solved iteratively using the Newton-Raphson method until the L2 norm of the residual reaches a pre-defined tolerance.

where vo is the initial coefficient of intrinsic permeability and Ad is a constant equal to 4. D 2 ½0; 1 is the isotropic mechanical damage parameter defined in the usual sense such that D ¼ 0 and D ¼ 1 represent undamaged and fully damaged states, respectively, and the  ¼ ð1  DÞr. For a bilinear constitueffective stress is computed as r tive model for concrete, as discussed in Section 3.1, a functional form can be written for D as

2.2. Pore pressure analysis

D¼

Presence and movement of moisture (in vapor phase) within concrete structures during fire causes pressure to build-up in the pores and can lead to spalling [22–27]. In the present study, transport of water vapor within concrete is modeled using Darcy’s law for the cross-section (X) of each sub-span. Assuming that vapor behaves as an ideal gas, the governing equations can be written directly for pore pressure as

where ET and em represent temperature-dependent elastic modulus and mechanical strain in concrete, respectively. In the aforementioned mathematical model, initial pore pressure Pv o is taken as

ap




dPv ¼ jp r2 Pv þ Q p ; dt

jp ¼ mv

vt ; l


  mv dml V v f ; ap ¼ 1 þ RT V v ql dPv

ð5Þ

where Pv is the pore pressure, mv is the mass of water vapor, ml is the mass of liquid water computed using Bazant’s isotherms [28], vt is the coefficient of intrinsic permeability, l is the coefficient of dynamic viscosity, V v is volume of water vapor, ql is the density of liquid water, f is the molar mass of water, R is the ideal gas con-



1

r ; ET em

Pv o ¼ Rh Pso ;

ð7Þ

ð8Þ

where Rh is the initial relative humidity of concrete and Pso is the initial saturation pressure of concrete. Along the boundaries, pore pressure is maintained at P v o . Upon spatial discretization of (5) using four node bilinear quadrilateral elements and Galerkin approximation, a first-order nonlinear system of ODEs is obtained as

AP P_ v þ jP Pv ¼ Q P :

ð9Þ

Next, a generalized trapezoidal scheme is applied to (9) which yields a system of nonlinear algebraic equations which is subsequently solved using NR method. Spalling is said to occur when the built-up pore pressure exceeds the temperature-dependent tensile strength of concrete.

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2.3. Mechanical analysis Mechanical analysis is performed through a beam-column element capable of considering the effects of thermal gradients, large deformations, mechanical damage and spalling. The developed element is based on the Euler-Bernoulli beam-column theory and employs an updated Lagrangian co-rotational formulation. For a realistic treatment of other physical phenomena (heat transfer and pore pressure development), several fibers are considered along the length of the member, as discussed earlier and shown in Fig. 1. The effect of all the fibers across the cross-section of a sub-span of the member are consolidated to model the overall behavior of the neutral axis of the member. The configurations, degrees-of-freedom and loading conditions of the neutral axis of the member in local and global coordinate systems are shown in Fig. 2. An additive decomposition of the total strain et in each fiber is considered as

et ¼ em þ eth þ etr þ ecr ;

ð10Þ

where em ; eth ; etr and ecr denote the mechanical, thermal, transient thermal and creep strains (computed independently for each fiber), respectively. Consideration of an additive decomposition of strain (instead of a multiplicative decomposition of deformation gradient) in a large deformation setting has been a matter of discussion since it was first introduced for finite strain elasto-plastic formulation by Green and Naghdi [30]. It is now recognized that this is an approximation which is valid when (a) small plastic deformations are accompanied by moderate elastic strains, or (b) small elastic strains are accompanied by moderate plastic deformations, or (c) small strains are accompanied by moderate rotations [31]. The maximum strain that concrete and steel can sustain before yielding at room temperature is approximately 0:35% and 0:45% (for steel with yield strength of 500 MPa), respectively. Concrete is incapable of undergoing large plastic deformations unless it is subjected to very high hydrostatic pressure that causes sufficient confinement. Such high pressures are usually observed in specialized loading scenarios such as blast or missile impact [32]. Steel, on the other hand, exhibits relatively small levels of elastic strain compared to plastic strain. Thus, it is reasonable to employ additive strain splitting for concrete and steel as they respectively follow conditions (a) and (b) mentioned previously. At high temperatures, both concrete and steel can withstand larger strains (2.5–5%) which are still relatively low and allow the use of additive strain decomposition. Additive strain decomposition has been used

(a)

successfully in previous studies (e.g. [6,15]). The Eurocodes [1,33] also suggest using additive strain decomposition for advanced analysis procedures pertaining to structures in fire. The total strain of the ith fiber can be computed as

eðiÞ t ¼ uyi þ

u3 ; L

ð11Þ

where u ¼ ðu2  u1 Þ=L is the curvature and yi the lever arm of ith fiber. u1 and u2 are the end rotations of the structural member while u3 is its axial displacement, as shown in Fig. 2a. Thermal strain is computed through the temperature-dependent coefficient of linear thermal expansion of the material, a, as

eth ¼ aDT:

ð12Þ

For concrete, a also depends on the type of aggregate being used. The transient thermal strain is considered only for concrete. Transient thermal and creep strains are computed through empirical relations that are discussed in detail in Section 3. While the total strain varies linearly across the cross-section (see (11)), admitting such a strain decomposition for each fiber allows consideration of nonlinear variations in individual straincomponents. The linear variation of total strain, although an approximation, is in line with the Euler-Bernoulli theory considered herein. The assumption of plane sections remaining plane after bending holds nearly exactly at ambient temperatures [34]. At elevated temperatures, it is an acceptable approximation and has been employed successfully in the past (e.g. [16,35,15]). Further, the Euler-Bernoulli theory neglects shear deformations, which is a reasonable assumption for slender members, typically utilized in framed construction. If a large number of stocky members (span to depth ratio less than 4) are present in a framed structure, the developed framework may not be able to give accurate predictions. Once various components of strain are computed for each fiber, the tangent modulus is computed at fiber level considering a temperature-dependent constitutive behavior of the fiber material (concrete or steel). The fiber level tangent moduli are then utilized to define the equivalent bending and axial rigidity of the beamcolumn element as

ðEIÞ ¼

n X Ei Ai y2i ;

ðAEÞ ¼

i¼1

n X Ai Ei ;

ð13Þ

i¼1

where Ei is the temperature-dependent tangent modulus of the ith fiber and Ai is the area associated with each fiber based on the underlying 2D FE mesh used for thermal and pore pressure solvers.

(b)

Deformed Configuration F6

y

L M1

u2

x L

F5

v6

J

u1

u3 M2

F1

v3

F3

L

v5

θ v4

F2 v2

F4

y v1 θ x

Fig. 2. Configuration of the beam-column element.

Un-deformed Configuration

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These equivalent rigidities are used to develop governing equations of the beam-column element to yield the member tangent stiffness matrix, which is discussed next. The mechanical behavior of the beam-column element shown in Fig. 2a, assuming a linear thermal gradient between top and bottom fibers, is governed by



 d v M1 þ M2 TB  TT ðEIÞ 2 þ J v ¼ M 1 þ x þ ðEIÞa ; dx L d

element displacement vector, DF is incremental external force vector and DFT is the incremental internal force vector due to temperature in global coordinate system. (24) is solved iteratively using the NR method. 3. Material models

2

ð14Þ

where M1 ; M2 are the end moments, J is the axial load, v is the neutral axis deflection and L is the length of the element. d is the depth of the section, and T T and T B are temperatures of top and bottom most fibers of the cross-section, respectively. Upon solving (14) one obtains


 ðEIÞ TB  TT ; ðe1 u1 þ e2 u2 Þ þ ðEIÞa L d
 ðEIÞ TB  TT M2 ¼ ðe2 u1 þ e1 u2 Þ  ðEIÞa ; L d

 u3 TB þ TT  cb þ a ; J ¼ ðAEÞ L 2

ð15Þ

M1 ¼

ð16Þ ð17Þ

where, e1 and e2 are the temperature-dependent stability functions and cb is the axial strain due to flexural bowing. Detailed mathematical expressions of these stability and bowing functions can be found in the literature [14,21]. (15)–(17) are differentiated with respect to u1 ; u2 ; u3 ; T B and T T and subsequently, incremental load displacement relationships are obtained in local coordinate system as

DS ¼ KS Du þ DST ;

ð18Þ

where, KS is the element tangent stiffness matrix in member coordinate system computed as

2

e1 þ

G21 p2 H

G1 G2 p2 H

G1 LH

e1 þ p22H

G2

G2 LH

G2 LH

L2 H

e2 þ

ðEIÞ 6 6 G1 G2 KS ¼ 6 L 4 e2 þ p2 H G1 LH

p2

3

DST ¼ ðEIÞa

G1 ð DT B 2LH 6 G2 6 ð DT B 4 2LH

7 7 7; 5

þ DT T Þ þ 1d ðDT B  DT T Þ

ð19Þ

3

7 þ DT T Þ  1d ðDT B  DT T Þ 7 5: p2

2L2 H

3.1. Concrete An additive strain decomposition is considered for concrete in accordance with (10). The total and the thermal strains are computed using (11) and (12). The incremental transient thermal strain for concrete Detr is computed using the incremental thermal strain Deth through the relationship proposed by Anderberg and Thelandersson [36] as

Detr ¼ sd

ð20Þ

ð DT B þ DT T Þ

e02 u1 2

G2 ¼ H¼

p

k2

þ

þ

ð21Þ

e01 u2 ;

0 b1 ðu1 0

þ u2 Þ þ

0 b2 ðu1

2

 u2 Þ ;

DF ¼

r¼

0

! 3 X T i B KS B þ Si KG Dv þ DFT

f c;T

0

ted ðT293Þ ;

ð26Þ

 8    em ep 2 > > ; f 1  > c;T ep <

em 6 ep

  3  > > > : f c;T 1  em3e ep ;

em > ep

;

ð27Þ

ep ¼ 0:0025 þ ð6T þ 0:04T 2 Þ  106 :

ð28Þ

ð29Þ

For HSC, similar bilinear behavior is considered through the temperature-dependent stress-strain relationships given by [39]

ð24Þ

where, is the element geometric matrix in global coordinate system corresponding to Si . S1 ; S2 represents end moments and S3 represents axial force in the member [14], Dv represents incremental

f c;T

8 f c;T ; 20  C 6 T 6 450  C > > <  T20 ¼ f c;T 2:011  2:353 1000 ; 450  C 6 T 6 874  C ; > > : 0; T P 874  C

i¼1

KiG

r pffiffi

p

ð23Þ

where, e01 ; e02 ; b1 ; b2 are the derivatives of stability and bowing function with respect to axial load factor. The element global forces F are linked to element member forces Q through the usual coordinate transformation matrix B. The incremental load displacement relationship in global coordinate system can be obtained by transforming (18) as

ð25Þ

0

ð22Þ 2

Deth ;

where, c1 ¼ 6:28  106 s0:5 , d ¼ 2:658  103 K1 ; f c;T is the yield strength of concrete at temperature T, and T is temperature in Kelvin. Once all the strain components are computed, the stressproducing component (mechanical strain) can be computed using (10). Several constitutive relations have been proposed for concrete (e.g. [38,1,36]). In the present study, the constitutive behavior of NSC in compression is assumed to be bilinear and is derived from the temperature-dependent stress-strain curves available in ASCE manual [39] given by

In (19) and (20), G1 ; G2 and H are defined as

G1 ¼ e01 u1 þ e02 u2 ;

r f c;20

where, r is the stress in concrete, f c;20 is the yield strength of concrete at room temperature, and sd is an empirical constant between 1.8 and 2.35. While (25) was proposed for NSC, the same relationship is considered for HSC due to the lack of availability of data. A parametric study has been performed with respect to etr to ascertain its appropriate level that can be used for HSC. A similar approach was used earlier by Cai et al. [18]. Creep strain in concrete is modeled using the relationship proposed by Harmathy [37] as

ecr ¼ c1

Du is the element displacement vector in member coordinate system, and ST is the element fixed-end force vector due to temperature which is given by

2

This section presents the material models for concrete and steel that have been implemented in the proposed framework in accordance with the additive strain decomposition introduced in Section 2.3.

r¼

 8    em ep Ha > > f ; 1  > c;T e p <

em 6 ep

  2  > > 30ðem ep Þ > : f c;T 1  ð130f ; Þe

em > ep

c;T

p

;

ð30Þ

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Fig. 3. Flowchart of the developed framework.

Table 1 Specifications of simply supported NSC beam. Physical property

Value

Length (m) Cross-section dimensions (mm) Reinforcement Total load (kN) f ck (MPa) f y (MPa) Clear cover

6.1 300  355 2/19 mm at top, 4/19 mm at bottom 80 30 400 25 mm at bottom, 38 mm at sides

f c;T

8   > < f c;T ½1  0:003125ðT  20Þ; 20 C 6 T 6 100 C  100 C 6 T 6 400  C ; ¼ 0:75f c;T ; > : f c;T ½1:33  0:00145T ; T > 400  C

ep ¼ 0:0025 þ ð6T þ 0:04T 2 Þ  106 ; Ha ¼ 2:28  0:012f c;20 :

ð31Þ

ð32Þ

Here, r is stress in concrete and ep is the peak crushing strain in concrete at a given temperature. In tension, the constitutive behavior of concrete is assumed to be trilinear [40], characterized by its tensile strength, tensile cracking strain and tensile ultimate strain.

Fig. 4. Comparison of experimental and computed quantities for simply supported NSC beam.

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179

Fig. 5. Axially restrained HSC beam. Fig. 7. Simply supported HSC beam.

Table 2 Specifications of axially restrained HSC beam.

etc ¼

Physical property

Value

Length (m) Cross-section dimensions (mm) Reinforcement Total load (kN) Axial restraint stiffness (kN/mm) f ck (MPa) f y (MPa) Clear cover Cement(kg=m3 ) Moisture content (kg=m3 ) Relative humidity (%) vo (m2 )

3.962 252  406 2/13 mm at top, 3/19 mm at bottom 2-point load of 50 kN 13 93.3 420 38 mm at bottom, 38 mm at sides 513 65.5 92.5

f t;T

etu ¼ 11etc ;

ð33Þ

The temperature-dependent cracking and ultimate tensile strains in concrete are given by

ð34Þ

where etc is the tensile cracking strain of concrete, etu is the ultimate strain and ET is the modulus of elasticity of concrete at a given temperature. 3.2. Steel The total strain in steel is also assumed to follow the additive decomposition discussed in Section 2.3 except for the absence of the transient thermal component, i.e., for steel,

et ¼ em þ eth þ ecr ;

2  1018

The tensile strength f t;T of concrete at a given temperature is computed from its compressive strength f c;T as

qffiffiffiffiffiffiffi ¼ 0:33 f c;T ;

f t;T ; ET

ð35Þ

where the total and the thermal strains are computed using (11) and (12), respectively, and creep strain is computed using the relationship proposed by Harmathy [41] as 1

ecr ¼ ð3Z e2to Þ3 h3 þ Zh;

ð36Þ

8

< 6:755  1019 r=f 4:7 ; ðr=f Þ 6 0:42 y y Z¼ : 1:23  1016 e10:8ðr=f y Þ ; ðr=f Þ > 0:42 y

ð37Þ

1

Fig. 6. Axially restrained HSC beam subjected to parametric fire curve.

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without strain hardening is considered with temperaturedependent yield strength and modulus of elasticity taken from Eurocode 3 [33].

Table 3 Specifications of simply supported HSC beam. Physical property

Value

Length (m) Cross-section dimensions (mm) Reinforcement Total load (kN) f ck (MPa) f y (MPa) Clear cover Cement(kg=m3 ) Moisture content (kg=m3 ) Relative humidity (%) vo (m2 )

3.962 252  406 2/13 mm at top, 3/19 mm at bottom 2-point load of 50 kN 93.3 420 38 mm at bottom, 38 mm at sides 513 62 86 2  1018

R DH 1:75 where, h ¼ e RT dt; DRH ¼ 38900 K , eto ¼ 0:016ðr=f y Þ , and f y is the yield strength of steel. The constitutive behavior of steel is assumed to be same in tension and compression. A bilinear material model

4. Coupling of thermal, pressure, and mechanical solvers All the three solvers (thermal, pore pressure, and mechanical) are coupled with each other and are executed iteratively in a staggered scheme. Mechanical solver takes temperature as input from the thermal solver to determine relevant temperature-dependent mechanical properties, and vapor pressure build-up as input from the pore pressure solver to assess the extent of spalling. The spalled cross-sectional profiles obtained by the mechanical solver are communicated back to the thermal and the pore pressure solvers to appropriately consider new boundary conditions. Detailed computational implementation flowchart of the developed framework is shown in Fig. 3.

Fig. 8. Spalled cross-sections of simply-supported HSC beam (: unspalled, j: spalled) at t ¼ 0, 60 and 120 min (left to right).

Fig. 9. Comparison of experimental and computed values for HSC beam.

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181

Fig. 10. Fixed-fixed HSC column.

Table 4 Specifications of fixed-fixed HSC column. Physical property

Value

Length (m) Cross-section dimensions (mm) Reinforcement Total load (kN) f ck (MPa) f y (MPa) Clear cover Cement(kg=m3 ) Relative humidity vo (m2 )

3.81 406  406 8/25 mm 2406 86 414 38 mm at bottom, 38 mm at sides 296 86% 2  1018

Fig. 13. Steel frame exposed to fire.

5. Numerical examples This section presents six numerical examples to demonstrate the accuracy, efficacy and applicability of the developed framework in the analysis of structural members/systems. The first five

Fig. 11. Spalled cross-sections of HSC column (: unspalled, j: spalled) at t ¼ 0, 60 and 120 min (left to right).

Fig. 12. Comparison of experimental and computed values for HSC column.

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G. Srivastava, P. Ravi Prakash / Computers and Structures 190 (2017) 173–185 Table 5 Specifications of steel frame. Physical property

Value

Section specification A (mm2 ) I (mm4 ) f y (MPa)

IPE80 (For all members) 764 801400 415

pressure solvers utilizes elements of size 10 mm for all the examples (this size was found through several mesh convergence studies). The mechanical solver utilizes a different number of sub-spans for each example, as mentioned later.

5.1. Simply supported NSC beam subjected to fire A simply supported RC beam with specifications as mentioned in Table 1 is considered. Lin et al. [42] experimentally characterized the response of this RC beam subjected to ASTM E119 [43] fire exposure on three sides and a total 80 kN uniformly distributed mechanical load. Utilizing the developed framework, the beam is analyzed with four sub-spans for the mechanical solver. A comparison of computed rebar temperature and the mid-span deflection of the beam against respective experimental counterparts is shown in Fig. 4. It can be observed that the computed values are in good agreement with the experimentally observed ones.

5.2. Spalling in axially restrained HSC beam

Fig. 14. Comparison of experimental and computed horizontal deflection at point C of the steel frame.

examples establish the predictive capabilities of the developed framework by comparing prediction of fire behavior of NSC, HSC and steel structural members/frames with respective experimentally observed behavior. The last example demonstrates the application of the proposed framework for simulation of a larger structural assembly. All the simulations were carried out on a desktop workstation with 8 GB RAM using MATLAB R2011a. A relative tolerance of 104 was used for NR iterations for all the three solvers (thermal, pore pressure, and mechanical). The 2D mesh for thermal and pore

An axially restrained HSC concrete beam (Fig. 5) with specifications as in Table 2 is considered. Dwaikat and Kodur [44] experimentally characterized the spalling response history of this beam subjected to a parametric fire as shown in Fig. 6a. The beam is analyzed with four sub-spans used by the mechanical solver. Volume of concrete spalled as a function of time is compared against experimental observations as shown in Fig. 6b. The volumetric spalling

Table 6 Specifications of two-storey two-bay frame. Physical property

Value

Cross-section dimensions (mm) f ck (MPa) f y (MPa) Clear cover Cement (kg=m3 ) Relative humidity vo (m2 )

300  300 for columns and 300  400 for beams 30 (NSC) and 93.3 (HSC) 420

Fig. 15. Two-storey two-bay frame subjected to local fire.

30 mm at bottom, 30 mm at sides 380 (NSC) and 513 (HSC) 86% 2  1018

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response history computed using developed framework shows good agreement with the experimental observations. 5.3. Simply supported HSC beam subjected to fire A simply supported HSC beam as shown in Fig. 7 with detailed specifications given in Table 3 is considered. Dwaikat and Kodur [45] experimentally characterized the response and the extent of spalling of this beam subjected to ASTM E119 [43] fire. The beam is analyzed with four sub-spans being used by the mechanical solver. Spalled concrete cross-sections at various times at is shown in Fig. 8. A comparison of computed temperature and mid-span deflection history against the respective experimental counterparts is shown in Fig. 9. The validation study indicates that the developed framework is reasonably accurate to predict the thermo-mechanical response with spalling effects. From the time vs. mid-span deflection plot (Fig. 9b) a steep increase in deflection at 140 min is observed due to spalling effects.

Fig. 16. Deformed geometry of NSC frame (10x scaling).

5.4. Fixed-fixed HSC column subjected to fire A fixed-fixed HSC column as shown in Fig. 10 with specifications mentioned in Table 4 is considered. Kodur et al. [46] experimentally characterized the response and spalling of this column subjected to ASTM E119 [43] fire. It has been analyzed with one sub-span being used by the mechanical solver. Spalled concrete cross-sections at various times are shown in Fig. 11. A comparison of temperature histories (Tc1 and Tc2) and axial deflection against respective experimental counterparts is shown in Fig. 12. Axial deflection plot (Fig. 12b) indicates failure of the column at 170 min due to instability whereas actual failure is observed at 210 min. The developed framework demonstrates reasonably good prediction capabilities for the first 2.5 h. 5.5. Steel frame exposed to fire

Fig. 17. Deformed geometry of HSC frame (10x scaling).

A steel frame as shown in Fig. 13 with specifications given in Table 5 is considered. Rubert et al. [47] experimentally characterized its response. The fire scenario had been designed in their experimental study such that the entire cross-section attains uniform temperature. Analysis has been performed using the developed framework with 10 sub-spans along each beam-column element. Horizontal deflection observed at point C in the experiment is compared against the corresponding simulated values, as shown in Fig. 14, and both are found to be in good agreement.

Fig. 18. Comparison of deflection at point D of two-storey two-bay frame.

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Fig. 19. Comparison of deflection at point E of two-storey two-bay frame.

5.6. Two-storey two-bay frame subjected to local fire A two storied two bay frame as shown in Fig. 15 with detailed specifications mentioned in Table 6 is considered. The frame is designed with NSC and HSC according to IS 456 [48] and corresponding reinforcement details are given in Appendix. The frame is analyzed with four sub-spans along each member for a local ASTM E119 exposure as shown in Fig. 15. Deformed geometry of frame under consideration is shown in Figs. 16 and 17 for NSC and HSC, respectively. Comparison of horizontal deflection histories of NSC and HSC frames at point D and E are shown in Figs. 18a and 19a, respectively. Increase in horizontal deflection with time is observed for both NSC and HSC frames. However, the increase is more predominant in HSC frame due to augmented stiffness degradation triggered by spalling. Similarly, vertical deflection histories of joints D and E are shown in Figs. 18b and 19b, respectively. Columns BE and AD elongate in case of NSC and the elongation increase with time. Whereas in case of HSC frame, columns BE and AD initially expand followed by rapid contraction. Increase in elongation in columns BE and AD with time is due to dominance of thermal dilation over material degradation. Whereas, in case of HSC frame, spalling results in reduction of cross-section followed by accelerated temperature increase in the rebars. This results in rapid reduction in stiffness and results in contraction in columns BE and AD.

Table 7 Reinforcement details of two-storied two-bay frame. Mix

Location

Reinforcement details

NSC

GH(0–2.5 m) GH(2.5–5 m) HI(0–2.5 m) HI(2.5–5 m) GD HE FI DE(0–2.5 m) DE(2.5–5 m) EF(0–2.5 m) EF(2.5–5 m) AD BE FC

3@12 mm 5@12 mm 5@12 mm 3@12 mm 8@16 mm 6@20 mm 8@16 mm 4@10 mm 7@10 mm 7@10 mm 4@10 mm 8@16 mm 8@20 mm 8@16 mm

GH(0–2.5 m) GH(2.5–5 m) HI(0–2.5 m) HI(2.5–5 m) GD HE FI DE(0–2.5 m) DE(2.5–5 m) EF(0–2.5 m) EF(2.5–5 m) AD BE FC

3@10 mm 7@10 mm 7@10 mm 3@10 mm 4@12 mm 4@12 mm 4@12 mm 4@10 mm 7@10 mm 7@10 mm 4@10 mm 4@12 mm 4@12 mm 4@12 mm

HSC

at at at at

top top top top

and and and and

4@10 mm 4@10 mm 4@10 mm 4@10 mm

at at at at

bottom bottom bottom bottom

at at at at

top top top top

and and and and

4@10 mm 4@10 mm 4@10 mm 4@10 mm

at at at at

bottom bottom bottom bottom

at at at at

top top top top

and and and and

4@10 mm 4@10 mm 4@10 mm 4@10 mm

at at at at

bottom bottom bottom bottom

at at at at

top top top top

and and and and

4@10 mm 4@10 mm 4@10 mm 4@10 mm

at at at at

bottom bottom bottom bottom

6. Conclusions An integrated framework has been developed for coupled thermo-hydro-mechanical analysis of RC or steel planar frames subjected to fire. The developed framework considers three-way coupling between the thermal, pore pressure, and mechanical solvers through a two-way spatial discretization strategy. Mechanical analysis is performed through 1D line elements based on the EulerBernoulli beam theory capable of considering effects of large deformations and spalling. The thermal and the pore pressure solvers utilize a 2D FE mesh for each sub-span of the mechanical solver and consider effects of changes in the cross-section through updated boundary conditions. While such a two-level spatial discretization allows high computational efficiency, the three-level physical coupling is a unique feature of the developed framework. The accuracy and applicability of the developed framework was demonstrated through six numerical examples considering various

structural members/assemblies of RC (both normal and high strength concrete) and steel. It was shown that the developed framework is capable of predicting temperatures, mechanical deformations and spalling profiles with reasonable levels of accuracy, when compared to experimental results available in the literature. It is expected that the developed framework will be useful for analyses required for design, optimization, progressive collapse, and uncertainty analyses of RC and steel structural assemblies subjected to fire. Appendix A Details of reinforcements used for NSC and HSC frame of sixth example are presented in Table 7.

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References [1] Eurocode 2 – design of concrete structures: part 1-2 – Structural fire design, BSI, London; 2004. [2] ACI216.1-07: standard method for determining fire resistance of concrete and masonry construction assemblies. American Concrete Institute, Detroit; 2007. [3] IS-1641 – Indian code of practice for fire safety of buildings(general): details of construction-code of practice. Bureau of Indian Standards, New Delhi; 1989. [4] Wickström U. Temperature analysis of heavily-insulated steel structures exposed to fire. Fire Safety J 1985(9):281–5. [5] Khennane A, Baker G. Thermoplasticity model for concrete under transient temperature and biaxial stress. Proc. royal society of London a: mathematical, physical and engineering sciences, vol. 439. p. 59–80. [6] Rigobello R, Coda HB, Neto JM. A 3D solid-like frame finite element applied to steel structures under high temperatures. Finite Elem Anal Des 2014;91:68–83. [7] Yu C, Huang Z, Burgess IW, Plank RJ. Development and validation of 3D composite structural elements at elevated temperatures. J Struct Eng 2010;136 (3):275–84. [8] Vulcan solutions. [accessed: 2016-09-1]. [9] Majorana C, Salomoni V, Mazzucco G, Khoury G. An approach for modelling concrete spalling in finite strains. Math Comput Simul 2010;80(8):1694–712. [10] Kumar P, Srivastava G. Thermo-mechanical modeling of reinforced concrete masonry infill panels exposed to fire. In: Proc. ASCE engineering mechanics institute international conference. [11] Kumar P. Characterization of in-plane and out-of-plane behavior of infill panels subjected to thermal exposure Master’s thesis. Indian Institute of Technology Gandhinagar; 2015. [12] Kumar P, Srivastava G. Numerical modeling of structural frames with infills subjected to thermal exposure: state-of-the-art review. J Struct Fire Eng, in press. [13] Mohyeddin A, Goldsworthy HM, Gad EF. FE modelling of RC frames with masonry infill panels under in-plane and out-of-plane loading. Eng Struct 2013;51:73–87. [14] Kassimali A, Garcilazo JJ. Geometrically nonlinear analysis of plane frames subjected to temperature changes. J Struct Eng 2010;136(11):1342–9. [15] Wang Y, Moore D. Steel frames in fire: analysis. Eng Struct 1995;17(6):462–72. [16] Kodur V, Dwaikat M. A numerical model for predicting the fire resistance of reinforced concrete beams. Cem Concr Compos 2008;30(5):431–43. [17] Biondini F, Nero A. Cellular finite beam element for nonlinear analysis of concrete structures under fire. J Struct Eng 2010;137(5):543–58. [18] Cai J, Burgess I, Plank R. A generalised steel/reinforced concrete beam-column element model for fire conditions. Eng Struct 2003;25(6):817–33. [19] Franssen JM. Analysis of the fire behaviour of composite steel-concrete structures Ph.D. thesis. Belgium: University of Liege; 1987. [20] Franssen J. SAFIR: a thermal/structural program for modeling structures under fire. Eng J-Am Inst Steel Construct 2005;42(3):143. [21] Prakash P, Srivastava G. Development of matrix method based frame work for thermo-mechanical analysis of RCC frames. In: Response of structures under extreme loading. p. 972–80. [22] Diederichs U, Jumppanen U, Schneider U. High temperature properties and spalling behaviour of high strength concrete. In: Proceedings of fourth Weimar workshop on high performance concrete. Germany: HAB Weimar; 1995. p. 219–35. [23] Kodur V. Spalling in high strength concrete exposed to fire concerns, causes, critical parameters and cures. In: Proceedings, ASCE structures congress, Philadelphia, PA. p. 1–8.

185

[24] Kalifa P, Menneteau F-D, Quenard D. Spalling and pore pressure in HPC at high temperatures. Cem Concr Res 2000;30(12):1915–27. [25] Gawin D, Majorana C, Schrefler B. Numerical analysis of hygro-thermal behaviour and damage of concrete at high temperature. Mech Cohesivefrictional Mater 1999;4(1):37–74. [26] Gawin D, Pesavento F, Schrefler B. Towards prediction of the thermal spalling risk through a multi-phase porous media model of concrete. Comput Methods Appl Mech Eng 2006;195(41):5707–29. [27] Dwaikat MB, Kodur V. Hydrothermal model for predicting fire-induced spalling in concrete structural systems. Fire Safety J 2009;44(3):425–34. [28] Bazˇant ZP, Thonguthai W. Pore pressure and drying of concrete at high temperature. J Eng Mech Div 1978;104(5):1059–79. [29] Bary B. Etude du couplage hydraulique-mecanique dans le beton endommage. Ph.D. thesis, Cachan, Ecole normale superieure; 1996. [30] Green A, Naghdi P. A general theory of an elastic-plastic continuum. Arch Rational Mech Anal 1965;16(4):251–81. [31] Lubliner J. Plasticity theory. Pearson Education Inc.; 2006. [32] Bazˇant ZP, Adley MD, Carol I, Jirásek M, Akers SA, Rohani B, Cargile JD, Caner FC. Large-strain generalization of microplane model for concrete and application. J Eng Mech 2000;126(9):971–80. [33] Eurocode-3: design of steel structures–Part 1.2: General rules–structural fire design. BSI, London; 1993. [34] Park R, Paulay T. Reinforced concrete structures. John Wiley & Sons; 1975. [35] Caldas RB, Fakury RH, Sousa Jr JBM. Finite element implementation for the analysis of 3D steel and composite frames subjected to fire. Latin Am J Solids Struct 2014;11(1):1–18. [36] Anderberg Y, Thelandersson S. Stress and deformation characteristics of concrete at high temperatures. Tech. rep., Lund Institute of Technology, Division of Structural Mechanics and Concrete Construction; 1976. [37] Harmathy TZ. Fire safety design and concrete. Tech. rep., Concrete Design and Construction Series, UK: Longman Scientific and Technical; 1993. [38] Lie T. Structural fire protection, manuals and reports on engineering practice, no. 78. ASCE, New York, NY; 1992. [39] Kodur V, Wang T, Cheng F. Predicting the fire resistance behaviour of high strength concrete columns. Cem Concr Compos 2004;26(2):141–53. [40] Rots J, Kusters G, Blaauwendraad J. The need for fracture mechanics options in finite element models for concrete structures. Proc. int. conf. on computer aided analysis and design of concrete structures part, vol. 1. p. 19–32. [41] Harmathy T. A comprehensive creep model. J Basic Eng 1967;89(3):496–502. [42] Lin T, Gustaferro A, Abrams MS. Fire endurance of continuous reinforced concrete beams. PCA R & D Bull; 1981. [43] ASTM-E119. Standard methods of fire tests of building construction and materials. American Society for Testing and Materials, Philadelphia, USA; 2016. [44] Dwaikat MB, Kodur VKR. Fire induced spalling in high strength concrete beams. Fire Technol 2009;46(1):251–74. [45] Dwaikat M, Kodur V. Response of restrained concrete beams under design fire exposure. J Struct Eng 2009;135(11):1408–17. [46] Kodur V, Mcgrath R. Fire endurance of high strength concrete columns. Fire Technol 2003;39(1):73–87. [47] Rubert A, Schaumann P. Structural steel and plane frame assemblies under fire action. Fire Safety J 1986;10(3):173–84. [48] IS456. Indian standard plain and reinforced concrete-code of practice (fourth revision). Bureau of Indian Standards, New Delhi; 2000.