realistic modelling of composite and rc floor slabs under - spiral

1

Realistic Modelling of Composite and R/C Floor Slabs under

Extreme Loading – Part I: Analytical Method

B.A. Izzuddin1, X.Y. Tao

2 and A.Y. Elghazouli

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ABSTRACT

This paper presents a new flat shell element for composite and reinforced concrete (R/C)

floor slabs subject to extreme loading conditions, accounting for the effects of geometric as

well as material nonlinearities. A novel feature of the proposed element is its treatment of

problems associated with the geometric orthotropy of composite floor slabs, achieved through

a modification of the conventional Reissner-Mindlin hypothesis. The new element is

formulated in a local co-rotational framework, enabling the use of linear strain-displacement

relationships, with the influence of geometric nonlinearity addressed through transformations

between the local and global systems. In addition, a robust nonlinear material model is

proposed for concrete which captures the salient response characteristics under extreme

loading conditions, including the effects of elevated temperature due to fire. The proposed

element is implemented within the nonlinear structural analysis program ADAPTIC, which is

used in this paper to provide several verification examples, focusing principally on the

significance of the assumptions made in the element formulation. Extensive verification

against experiments on composite and R/C floor slabs is undertaken in the companion paper,

where favourable comparisons between the predictions of the proposed method and

experimental results are generally achieved.

CD Database subject headings: Composite structures; Concrete structures; Finite elements;

Fires; Nonlinear analysis; Slabs.

1 Reader in Computational Structural Mechanics, Department of Civil and Environmental Engineering,

Imperial College, London SW7 2AZ, United Kingdom. Member, ASCE.

2 Post-doctoral Research Associate, Department of Civil and Environmental Engineering,

Imperial College, London SW7 2AZ, United Kingdom.

3 Reader in Engineering Structures, Department of Civil and Environmental Engineering,

Imperial College, London SW7 2AZ, United Kingdom. Member, ASCE.

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INTRODUCTION

Recent findings from fire tests on a full-scale steel building (O‟Connor & Martin, 1998) have

demonstrated the important role played by the composite steel/concrete floor slab in

supporting gravity loading, particularly after the deterioration in the strength of the steel

beams at elevated temperature. While a considerable amount of analytical research has been

dedicated over the past few decades to the modelling of reinforced concrete slabs, research on

the modelling of ribbed steel-decked composite slabs received much less attention. This is

partly attributed to the difficulty of modelling the geometric orthotropy of composite slabs by

means of practical 2D shell elements.

Previous attempts at representing the response of composite floor slabs under fire conditions

can be classified under two main approaches. The first approach employs dimensional

simplification, where a grillage of 1D elements is used to approximate the 2D bending and

membrane response (Elghazouli & Izzuddin, 2001; Izzuddin & Moore, 2002). On the other

hand, the second approach employs geometric simplification, where uniform thickness shell

elements are used with equivalent orthotropic material properties to approximate the

influence of geometric orthotropy (Huang et al., 2000). Both approaches suffer from inherent

approximation which often leads to an inaccurate assessment of the fire resistance of

composite floor slabs. Clearly, in the context of finite element modelling, 3D elements could

be utilised to represent the geometric orthotropy of composite slabs; however, such an

approach is prohibitively expensive, from both the modelling and computational perspectives.

This paper proposes a new 2D shell element for ribbed composite slabs, which combines

computational efficiency with modelling accuracy, and which accounts for both geometric

and material nonlinearities. In contrast with a recent shell element for composite slabs

developed by Sebastian and McConnel (2000), the proposed element accounts for i) the effect

of the ribs on the membrane and bending actions transverse to the rib orientation, ii) the

influence of geometric nonlinearity due to large displacements and rotations, and iii) the

response of concrete under a non-monotonic strain variation and at elevated temperature.

These features enable the proposed element to be effectively used in the realistic modelling of

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composite floor slabs under extreme conditions, including those due to fire, blast and

earthquake loading. In addition, the element is equally applicable to the modelling of

reinforced concrete (R/C) slabs, which are clearly a special geometrically isotropic case of

composite slabs.

The paper proceeds with providing the formulation details of the new shell element, where a

modification of the conventional Reissner-Mindlin hypothesis is proposed for dealing with

geometric orthotropy and the material discontinuity between adjacent ribs. In addition, the

element is presented in two hierarchic linear/quadratic forms, enabling the selective

activation/deactivation of the higher-order quadratic approximation. The incorporation of the

proposed element within a recently developed large displacement co-rotational approach

(Izzuddin, 2002) is then discussed, achieved through transformations between the local co-

rotational and the global reference systems. Subsequently, a robust material model is

proposed for the nonlinear response of concrete, which accounts for the effects of tensile

cracking and softening, crack closure, compressive nonlinearity and elevated temperature.

The new shell element has been implemented within the nonlinear structural analysis

program ADAPTIC (Izzuddin, 1991), which is used in this paper to provide basic verification

of the element for uniform-thickness and ribbed plates in the elastic small and large

displacement ranges. The companion paper (Elghazouli & Izzuddin, 2004) provides further

extensive verification of the proposed nonlinear analysis method, focussing on the influence

of geometric and material nonlinearities for both ambient and elevated temperature

conditions. In addition, it discusses practical aspects of the response of R/C and composite

floor slabs subject to extreme loading, including the effects of compressive and tensile

membrane action.

COMPOSITE SLAB FORMULATION

As illustrated in Fig. 1, a steel-decked composite floor slab is characterised by geometric

orthotropy, a feature which renders its modelling by means of conventional shell elements

unrealistic. The proposed shell element addresses this issue in the context of small-strain

large-displacement analysis, where the following assumptions are made:

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1. The rib geometry is perfectly trapezoidal, but with an arbitrary aspect ratio

2. The concrete cover depth and the steel deck thickness are uniform

3. Steel reinforcement belonging to a particular layer and acting along a specific

direction lies at a constant distance from the concrete surface, and is smeared as an

equivalent uniform thickness plate acting uniaxially, several reinforcement layers

with different orientations being allowed

4. Perfect bond exists between concrete and both the steel reinforcement and deck

5. Direct out-of plane stresses in the concrete are negligible

6. Out-of-plane transverse shear stresses in the concrete rib are negligible

7. The inclined webs of the steel deck sustain only uniaxial stresses in the longitudinal

direction

The formulation of the new element in a local reference system is presented hereafter,

considering i) the proposed kinematic description, ii) discretisation using local element

freedoms, and iii) determination of the local element forces and tangent stiffness. The global

element response, including the treatment of geometric nonlinearity using a co-rotational

approach, is discussed in a later section.

Kinematics

The conventional Reissner-Mindlin hypothesis (Zienkiewicz & Taylor, 1991) is based on the

assumption that any through-thickness straight line initially normal to the slab surface

remains straight after deflection. Clearly, applying this hypothesis to ribbed composite-floor

slabs is unrealistic, since it can lead to considerable transverse direct stresses in the ribs when

the slab is subjected to transverse planar stretching or bending, stresses which cannot be

sustained in the real slab due to material discontinuity between two adjacent ribs (Fig. 1).

Accordingly, the Reissner-Mindlin hypothesis is modified in this work, effectively through

an enriched approximation for the rib displacement field beyond the requirements of the

conventional hypothesis.

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In discretising the composite floor slabs into shell elements, four variants of a generic flat

shell elements are employed, as illustrated in Fig. 2. The proposed kinematic description,

representing the modified Reissner-Mindlin hypothesis, is formulated with reference to a

local coordinates system, the origin located at the geometric centre of the cover region, as

depicted in Fig. 3. In addition, natural coordinates ),( , shown in Fig. 4, are employed for

planar reference so as to facilitate discretisation. Hereafter, the proposed kinematic

description is discussed for the cover and rib regions, considering the different requirements

of the four element variants.

Cover region

The conventional Reissner-Mindlin hypothesis is retained in the cover region for all four

element variants, where the displacements )w,v,u( of a material point located at )z,,( are

related to the reference planar displacements )w,v,u( ppp and rotations of the normal

),( yx by:

0

),(

),(

z

),(w

),(v

),(u

)z,,(w

)z,,(v

)z,,(u

y

x

p

p

p

(1)

with, referring to Figs. 3 and 4,

2

tz

2

t

2

Ly:11

2

Wx:11

(2)

The strain components considered for the concrete in the cover region are:

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y

w

z

vx

w

z

u

x

v

y

u

y

vx

u

yz

xz

xy

y

x

(3)

Considering (1) and (2), the concrete strains are therefore related to the natural derivatives of

the planar displacement and rotation fields by:

0

0

W

2

L

2

L

2

W

2

z

w

L

2

w

W

2

v

W

2u

L

2

v

L

2

u

W

2

yx

y

x

p

y

p

x

pp

p

p

yz

xz

xy

y

x

(4)

With the assumption of perfect bond between steel and concrete, the uniaxial strain within

reinforcement, acting at a distance )dz( s from the reference plane and making an angle

)( x with the local x-axis, is obtained as the component of the planar strains of (4) using:

)2sin()2cos(1)2cos(12

1xxyxyxxs (5)

In addition to strains in the concrete and reinforcement, element variant (II) requires the

strains in the steel deck located at )2tz( , which are obtained directly from (4) as the

planar components ),,( xyyx .

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Rib region

The rib region is considered only for element variants (I), (III) and (IV), where the

conventional Reissner-Mindlin hypothesis is modified through the incorporation of two

additional displacement fields )v,u( rr :

0

)z,,(v

)z,,(u

0

),(

),(

z

),(w

),(v

),(u

)z,,(w

)z,,(v

)z,,(u

r

r

y

x

p

p

p

(6)

The mapping between the real and natural coordinates accounts for the trapezoidal rib shape

(Fig. 2), and is accordingly specific to each of the three element variants:

02

tzh

2

Ly:11

h

)r1(

2

tz

2

Wx:11 r

(7)

with,

(IV)variant :1r

(III)variant :0

(I)variant :r1

r

r

r

(8)

where r is the trapezoidal rib ratio (Fig. 3).

The additional displacement fields )v,u( rr are introduced primarily to improve the

approximation of the planar direct strain )( x and shear strain )( xy within the rib.

Accordingly, the two additional displacement fields correspond to stretching and shear

modes, as demonstrated for element variant (III) in Fig. 5. More generally, the following

characteristics are proposed for the additional displacement fields, depending on the element

variant:

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)IV(Variant0)z,,1(v)z,,1(u

)III(Variant

)z,,(v)z,,(v

)z,,(u)z,,(u

0)z,,0(v)z,,0(u

)I(Variant0)z,,1(v)z,,1(u

rr

rr

rr

rr

rr

(9)

The above modes correspond to zero additional displacements )v,u( rr along the vertical line

of symmetry bisecting the rib, represented by values of 1,0,1 for variants (I), (III) and

(IV), respectively. Accordingly, deformations along the bisecting vertical lines are assumed

to comply with the Reissner-Mindlin hypothesis, but elsewhere within the rib this hypothesis

is clearly enhanced with stretching and shear modes, the discretisation of which is discussed

later.

Considering (3) and (6–9), the strain components ),,,,( yzxzxyyx are given by:

y

r

xr

y

z

x

y

x

z

prp

rr

y

p

z

rp

rr

x

rp

z

rp

rp

rp

z

yz

xz

xy

y

x

W

2

L

2

L

2

W

2

z

w

L

2vv

z

v

w

W

2uu

z

u

vv

W

2uu

L

2

vv

L

2

uu

W

2

(10)

where,

h

)r1(

2

tz1WWz (11)

and,

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)r1(2

tzh

)r1(rr

(12)

Although a compatible formulation is achieved with the above strains in (10), a better

response prediction is achieved if xz is neglected in the rib, as demonstrated in the

“Verification” section. This default assumption is considered to be realistic, since the

corresponding shear stress could not be sustained at significant levels in the rib region due to

material discontinuity between adjacent ribs.

Again, with the assumption of perfect bond between the steel and concrete, the strains in the

horizontal part of the deck located at )2thz( are obtained directly from (10) as the

planar components ),,( xyyx . For the inclined parts of the deck, on the other hand, only

the longitudinal strain component ( y ) is considered, as obtained directly from (10), the

stress components corresponding to the two remaining direct and shear strains assumed to be

relatively small.

Discretisation

The kinematic description proposed in the previous subsection is expressed in terms of

relationships between the various material strains on the one hand and the planar

displacements )w,v,u( ppp , the rotations of the normal ),( yx and the additional rib

displacements )v,u( rr on the other. These fields are discretised in terms of basic and

hierarchic element freedoms using polynomial shape functions as follows:

4 4L Q

p i i i i

i 1 i 1

u ( , ) ( , ) ( , )

N u N u (13.a)

4 4L Q

p i i i i

i 1 i 1

v ( , ) ( , ) ( , )

N v N v (13.b)

4 4 4L Q C

p i i i i i i

i 1 i 1 i 1

w ( , ) ( , ) ( , ) ( , )

N w N w N w (13.c)

4 4L Q

x i xi i xi

i 1 i 1

( , ) ( , ) ( , )

N N (13.d)

10

4 4L Q

y i yi i yi

i 1 i 1

( , ) ( , ) ( , )

N N (13.e)

2 3L Q

r i ri i ri

i 1 i 1

u ( , , z) ( , , z) ( , , z)

R u R u (13.f)

2 3L Q

r i ri i ri

i 1 i 1

v ( , , z) ( , , z) ( , , z)

R v R v (13.g)

where, as illustrated in Fig. 6,

]4,1i[yixiiii ),,,,( wvu are the basic translational and rotational local nodal

freedoms associated with bilinear shape functions )( LiN

]4,1i[yixiiii ),,,,( wvu are hierarchic translational and rotational side freedoms

associated with quadratic shape functions )( QiN

]4,1i[i )( w are hierarchic out-of-plane translational freedoms associated with cubic

shape functions )( CiN

]2,1i[riri ),( vu are rib freedoms along the two edges (1–2) and (3–4) associated with

linear approximation functions )( LiR

]2,1i[riri ),( vu are hierarchic rib freedoms along the two edges (1–2) and (3–4)

associated with quadratic approximation functions )( QiR

]3i[riri ),( vu are element-specific hierarchic rib freedoms associated with a quadratic

approximation function )( Q3R

and where all the shape functions are detailed in Appendix A.

One of the benefits of utilising hierarchic freedoms is that the element can be readily

employed in its basic linear form excluding the “[…]” terms of (13), or in its higher-order

quadratic form including these terms. An additional important benefit of hierarchic freedoms

is that the element may be used in either of its two forms within the same co-rotational large

displacement approach, as discussed in the next section. It should also be noted that the

approximation for the out-of-plane displacements )w( p is always one order higher than that

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for the rotations of the normal ),( yx , a feature which reduces inaccuracies associated with

the phenomenon of shear locking (Hughes & Tezduyar, 1981).

Element Response

For a given set of values for the element freedoms, the relevant strains in the concrete, the

steel reinforcement and the steel deck can be evaluated according to the kinematic

assumptions and the approximated displacement fields discussed in the two previous

subsections. This evaluation is undertaken over the element domain at several Gauss points

employed in the numerical integration of the element response. Corresponding to each set of

material strains at a given Gauss point, the material stresses can be established taking into

account the effects of material nonlinearity, as discussed in a later section. The element

resistance forces are then obtained as the weighted aggregate of the material stresses using

the principle of virtual work.

The following expressions demonstrate the contributions from various materials to the

element resistance vector (f), with subscripts c, d and s denoting the concrete, deck and

reinforcement, respectively:

)IV(&)III(),I(Variants

)II(Variant

s

s

ncovs

ribd

ribc

covc

ncovs

covd

covc

fffff

ffff

(14.a)

where,

)IV(Variant

)III(Variant

)I(Variant

ribmid,d

ribleft,d

ribd

ribright,d

ribmid,d

ribleft,d

ribd

ribright,d

ribmid,d

ribd

fff

ffff

fff

(14.b)

In the above expressions, sn refers to the total number of reinforcement layers, each layer

being associated with a unique combination of distance from the reference plane )dz( s

and orientation )( x , while subscripts left, mid and right refer to the three parts of the deck

attached to the rib.

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The contributions of the different materials to f are evaluated using the principle of virtual

work, where the stresses at every Gauss point for a given material are weighted by a B matrix

which relates the corresponding strain components at the Gauss point to the element

freedoms. With the terms of the various B matrices readily available from the expressions in

two previous subsections, the contributions to f are given by:

xyzgn

1i i

T

cyzxzxyyxTc

xyzi

covc w

8

tLWBf (15.a)

xygn

1i i

T

dxyyxTd

xyi

dcovd w

4

tLWBf (15.b)

xygn

1iis

Ts

xyi

scovs w

4

tLWBf (15.c)

xyzgn

1i i

T

cyzxzxyyxTc

xyzi

zribc w

2

W

4

hLBf (15.d)

xygn

1i i

T

dxyyxTd

xyi

dribmid,d w

4

tLrWBf (15.e)

)1at(00w

8

tLW)r1(h4yzgn

1i i

T

dyTd

yzi

d2

r2

ribleft,d

Bf (15.f)

)1at(00w

8

tLW)r1(h4yzgn

1i i

T

dyTd

yzi

d2

r2

ribright,d

Bf (15.g)

where,

)w,w,w(xyzi

yzi

xyi are Gauss weighting factors associated with integration

point (i) and used for 2D (x–y, y–z) and 3D (x–y–z) integration, respectively,

with )n,n,n( xyzg

yzg

xyg representing the corresponding number of Gauss points

),,,,,( yzxzxysyx are the material stresses corresponding to the strain

components ),,,,,( yzxzxysyx , respectively

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)t,t( sd represent the thickness of the deck and the smeared reinforcement

layer, respectively

)W,( zr are rib properties given by (8) and (11), respectively

with the remaining geometric parameters as depicted in Fig. 3.

In addition to the element resistance vector (f), an element tangent stiffness matrix (k) is

required for guiding the incremental iterative solution of the nonlinear equations of

equilibrium (Crisfield, 1991). This can be assembled from the various material contributions

in a similar way to f, where an example contribution from the concrete in the cover region is

given by:

xyzgn

1iictc

Tc

xyzi

covc w

8

tLWBDBk (16)

in which tcD represents the tangent modulus matrix of concrete.

It is worth noting that k accounts for material nonlinearity, which is discussed in more detail

later, with local geometric nonlinearity excluded as a result of the linear kinematic

relationship between material strains and the local displacement fields. It is important to

emphasise, however, that local geometric nonlinearity becomes negligible in relation to

global geometric nonlinearity when an element is used to represent small regions of the slab

domain, as is the intention of the present work. The following section discusses the

incorporation of the proposed shell element within a recently developed co-rotational

approach for the global geometrically nonlinear analysis of floor slabs.

GEOMETRIC NONLINEARITY

The proposed slab element is formulated in a local reference system where the corresponding

local deformations are assumed to be sufficiently small to justify linear strain-displacement

relationships. In the context of geometrically nonlinear analysis, this assumption becomes

exponentially accurate as the finite element mesh is refined, provided the local reference

system follows the deformed element configuration and excludes rigid body modes, in which

case it is referred to as a local co-rotational system (Crisfield, 1997; Belytschko et al., 2000).

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With such an approach, geometric nonlinearities in the local system may be ignored, but the

effect of rigid body modes are accounted for by means of geometrically nonlinear

transformations between the local and global reference systems.

Despite the diversity of approaches for geometrically nonlinear analysis, the co-rotational

approach offers exceptional benefits for structural problems with rotational degrees of

freedom, particularly when accounting for arbitrarily large rigid body rotations. Moreover,

this approach can be developed and implemented quite independently of the specific element

formulation, and can therefore act as a „harness‟ around different geometrically linear

elements (Crisfield, 1997), rendering them immediately applicable to the modelling of

geometric nonlinearities.

The proposed slab element is incorporated within a recently developed co-rotational approach

(Izzuddin, 2002), which offers several significant improvements over previous co-rotational

approaches (Crisfield, 1997), including i) the choice of global rotational freedoms, ii) the

definition of the local co-rotational reference system and iii) the determination of local

freedoms. Not only do these improvements lead to a much simpler, yet powerful, new co-

rotational method, but they additionally realise several important benefits including complete

insensitivity to the size of the load step as well as symmetry of the tangent stiffness matrix.

With reference to Fig. 7, the adopted co-rotational approach (Izzuddin, 2002) is developed for

4-noded quadrilateral elements, where five global degrees of freedoms ),( ii nd are employed

for each node (i), id representing the three global translations and in representing the two

smallest components of the initial normal to the element plane. With the local co-rotational

x–y axes defined as the bisectors of the element diagonals in the deformed configuration

(Fig. 7), the values of the local nodal freedoms ),,,,( yixiiii wvu can be readily determined

in terms of the global nodal freedoms (Izzuddin, 2002). Through the principle of virtual

work, the element forces and tangent stiffness in the global system can then be obtained as

transformations of the corresponding local element entities using first and second partial

derivatives of the local with respect to global freedoms (Izzuddin, 2002).

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Although in its basic form the adopted co-rotational approach is applicable to 4-noded

quadrilateral elements, its extension to higher-order quadrilateral elements is in fact straight

forward when the additional freedoms are hierarchic relative to the local element deformed

configuration (Izzuddin, 2002), as is the case with the proposed slab element. Such hierarchic

freedoms are then directly shared between adjacent elements, depending on side connectivity,

and are treated along with the global nodal freedoms as the primary unknowns sought by the

nonlinear solution procedure.

MATERIAL NONLINEARITY

The proposed slab element benefits from a wide range of nonlinear material models

previously developed for steel under ambient (Izzuddin, 1991; Izzuddin & Lloyd Smith,

1996) and elevated temperatures (Song et al., 2000), as well as under high strain-rates

(Izzuddin & Fang, 1997). Such models can be readily employed to represent the stress-strain

response of the reinforcement and the deck, as required by the local element formulation,

under a variety of normal/extreme static/dynamic conditions, including those due to fire,

explosion and earthquake loading.

In view of the above, detailed consideration is given here only to the modelling of concrete,

where the following features are sought in an appropriate nonlinear material model:

Simplicity and numerical robustness without compromising the main response

characteristics of concrete

Representation of tensile cracking and compressive nonlinearity, including

softening effects

Modelling of crack opening and closure, the latter being an important

requirement under dynamic loading and fire conditions

Consideration of the effects of elevated temperature, both in terms of the

resulting thermal strains and the change of material properties

A new material model is proposed for concrete which fulfils the above objectives, where

nonlinearity is considered for the biaxial planar stresses, with the out-of-plane shear response

16

assumed to be linear. The latter assumption is considered realistic for floor slabs that do not

exhibit transverse shear failure, including failure due to punching shear. Accordingly, the out-

of-plane shear response is governed by:

yz

xz

c

c

yz

xz

10

01

)1(2

E

6

5 (17)

where cE is the elastic Young‟s modulus, c is Poisson‟s ratio and 5/6 is a correction factor

relating to the fact that the distribution of transverse shear stresses over the slab depth is

closer to parabolic than to the constant form assumed by standard finite element discretisation

(Zienkiewicz & Taylor, 1991).

In view of the above, focus is given to the nonlinear biaxial response relating the concrete

strains ),,( xyyx to the stresses ),,( xyyx including the influence of compressive and

tensile nonlinearity. The most general form of the biaxial model, discussed hereafter,

corresponds to the post-cracking stage, where a fixed crack orientation is assumed, such

orientation being orthogonal to the direction of the maximum principal stress that first

exceeds the concrete tensile strength )f( t . In order to simplify the presentation of the model,

the biaxial response of concrete is considered hereafter only in the crack plane, where the first

crack is assumed to run orthogonal to the local element x-axis. However, in the more general

case, constant geometric transformations are simply employed to relate the biaxial

strains/stresses in the crack plane to those in the local element x-y reference system.

For a given biaxial strain state ),,( xyyx , the biaxial stresses ),,( xyyx are obtained

using the elastic constitutive properties as follows:

xyt

yt

xt

xyc

yc

xc

xyp

yp

xp

xy

y

x

cs

c

c

2c

c

xy

y

x

2

)1(00

01

01

1

E (18)

in which,

s is the shear retention factor, applied only after tensile cracking

17

),,( xypypxp are the plastic strains due to compressive nonlinearity

),,( xycycxc are the strains due to tensile cracking

),,( xytytxt are the thermal strains, related only to the material temperature,

where )( ytxt and )0( xyt for an isotropic material

The following subsections discuss in detail the treatment of compressive nonlinearity and

tensile cracking in the biaxial x-y plane.

Compressive Nonlinearity

Compressive nonlinearity in the biaxial response of concrete is accounted for using principles

of plasticity, where an evolving plastic interaction surface is employed (Fig. 8), as defined by

the constraint:

c1c2c21

cxy,yx IJ)b2(I

3

b12),(

C (19.a)

with,

yx1I (19.b)

2xy

2yyx

2x2 )(

3

1J (19.c)

In (19.a), c is the current compressive strength of concrete, and )6.0b( c is a compressive

interaction parameter which provides a best fit against experimental results (e.g. Kotsovos &

Pavlovic, 1995). It should be noted that the interaction curve proposed in (19), although

expressed in terms of stress invariants 1I and 2J , represents accurately the interaction of

biaxial stresses but is in fact inappropriate for triaxial compressive interaction, the latter being

outside the scope of the proposed material model.

The evolution of the plastic interaction surface is determined by the variation of the

compressive strength ( c ) with the cumulative equivalent plastic strain ( p ), where

compressive hardening and softening (Fig. 8) are experienced for ( cp ) and ( cp ),

respectively, according to the following parametric expression:

18

c2

c

p

c

c

c

p

c

c

c

p

cc

c f

r1

s11

r1

r2)s1(s

(20)

with,

c

cc

E

f (21)

In the above, cs and cr are non-dimensional material parameters used to reflect the start of

compressive nonlinearity and the residual post-crushing strength (Fig. 8).

Should the compressive constraint of (19.a) be exceeded, additional plastic strains are

introduced according to the associated flow rule:

xyc

xcyc

ycxc

c

oxyp

oyp

oxp

xy

y

x

oxyp

oyp

oxp

xyp

yp

xp

)b2(4

b24

b24

2

C

C

C

(22)

where, ),,( oxyp

oyp

oxp are the plastic strains at the start of the equilibrium step, and is a

positive scalar used to satisfy the interaction equation in (19.a). It is noted that the stresses

),,( xyyx and strength )( c in (22) are the current unknown entities, and hence the

correction to the interaction curve is based on the single-step backward Euler method

(Crisfield, 1991), which has the benefit of leading to a symmetric consistent tangent modulus

matrix (Izzuddin & Lloyd Smith, 1996).

Finally, the equivalent cumulative plastic strain ( p ), required for the evaluation of c in

(20), is based on the equivalence of incremental plastic work:

oxypxyp

oypyp

oxpxp

T

xy

y

x

oppc )( (23)

19

which in conjunction with (19) and (22) leads to:

c

1opp

2

I1 (24)

where, op is the equivalent cumulative plastic strain at the start of the equilibrium step, and

1I is the first stress invariant defined in (19.b). It is worth noting that p and c are

interdependent according to (20) and (24), and hence an iterative procedure is required on the

material level to determine these entities. However, this does not unduly complicate the

model application, since iterations are in any case required by the backward Euler correction

procedure and additionally for satisfying any simultaneous tensile cracking constraints, as

discussed in the following subsection.

Tensile Cracking

The tensile response of concrete is modelled by means of separate strength envelopes for the

biaxial stresses ),,( xyyx , as depicted in Figs. 9 and 10. Considering first the direct

stresses ),( yx , the envelope constraints can be expressed, depending on the respective

crack strains ),( ycxc , as follows:

0withotherwise

0for0

for0

xcxtx

xcx

oxcxcxtx

(25.a)

0withotherwise

0for0

for0

ycyty

ycy

oycycyty

(25.b)

where ),( oyc

oxc are the maximum crack strains accumulated up to the start of the current

equilibrium step.

The current tensile strengths, required in (25), are obtained from the nominal tensile strength

)f( t of the undamaged concrete, but accounting for the post-cracking softening effect:

20

txcoxcxcxt

txcoxcxcttxt

or0for0

foraf (26.a)

tycoycycyt

tycoycycttyt

or0for0

foraf (26.b)

with,

t

tt

a

f (27)

where ta is a material softening parameter. This parameter is typically chosen to represent

the tension stiffening effect in deflection evaluation for the serviceability limit state.

However, for the large displacement response under extreme loading, the tension stiffening

effect becomes less significant, and the sensitivity of the overall predictions to ta becomes

small, though ta may be chosen to reflect the cracking response of plain concrete.

It is clear that the above constraints in (25) account for crack opening )0,0( ycxc and

crack closure )0,0( ycxc , where in the latter case the stresses are allowed to be

compressive )0,0( yx .

Considering next the planar shear response, the elastic stiffness is reduced according to the

shear retention factor )( s , as indicated by (18). Additionally, however, a strength envelope

is also utilised for the shear response (Fig. 10), leading to the following constraints that are

similar to (25) but account for negative and positive shear:

0withotherwise)(

0withotherwise)(

for)()0(

0for)0()0(

for)0()(

xycxytxytxy

xycxytxytxy

oxycxycxytxytxyxyt

xycxytxyxyt

oxycxycxytxyxytxyt

(28)

where )( xytxyt is an effective current shear strength that accounts for interaction of the

shear stress with the direct stresses ),( yx :

))(()( ysxytxsxytxytxyt (29.a)

21

0for

0for

xxxs

xxsxs (29.b)

0for

0for

yyys

yysys

(29.c)

The interaction expression in (29.a) establishes a limiting shear stress )( xytxyt that

corresponds to a maximum tensile principal stress of xyt when the direct stresses ),( yx

are compressive. If either of the direct stresses ),( yx is tensile, then the interaction

expression utilises reduced direct stresses ),( ysxs , according to a shear interaction

parameter )1( s . This reflects the fact that direct tensile stresses for cracked concrete are

typically a smeared representation of the tension stiffening effect, and they do not necessarily

contribute in full to the resolution of the maximum principal stress in the presence of a

significant shear stress. In addition, a value of less than 1 for s leads to improved numerical

stability, particularly if tensile cracking has just occurred along both the x and y axes;

choosing a value for s close to 1 predicts, in such a case, shear cracking at a negligible

shear stress, since the maximum principal stress would already be very close to the tensile

strength ( tf ), thus leading to negligible overall material stiffness. Nevertheless, despite the

empirical nature of the proposed model in relation to the planar shear response, it is noted

that the shear stress at first crack formation is zero, and that the overall structural response is

subsequently much more sensitive to the nature of the direct material response in the crack

plane than to the ensuing shear characteristics.

Finally, the limiting maximum principal stress ( xyt ), required in (29.a), is related to the

tensile strength ( tf ), but allowing for softening according to a shear cracking strain ( xyc ):

txycoxycxyc

oxycxyt

oxycxyctxyc

tstxyt

txycoxycxyc

tstxyt

orfor0

for2

af

for2

af

(30)

with,

22

ts

tt

a

f2

(31)

where s is a parameter expressing the shear softening relative to direct tensile softening.

Considering (30) and Fig. 10, it is clear that two failure envelopes are effectively employed

for negative and positive shear, thus allowing, for example, considerable resistance to be

maintained for negative shear even after considerable cracking and loss of resistance has

occurred for positive shear.

Model Solution Procedure

An iterative solution procedure is typically required on the material level to determine the

biaxial concrete stresses ),,( xyyx corresponding to strains ),,( xyyx . If on the basis

of the elastic response in (18) any of the compressive or tensile constraints, (19), (25) or (28),

are exceeded, the plastic strains ),,( xypypxp and/or the cracking strains ),,( xycycxc are

modified, from their values at the start of the equilibrium step, in order to ensure that the

corresponding constraints are satisfied. A sophisticated procedure has been developed for this

purpose by the authors (Izzuddin & Elghazouli, 2003), which also enables the numerical

evaluation of the consistent tangent modulus matrix ( tcD ) without the need for a complex

analytical formulation.

After structural equilibrium is achieved, the plastic strains ),,( oxyp

oyp

oxp are updated to their

current respective values ),,( xypypxp . However, the maximum crack strains are updated

only if they are exceeded by their current respective values, that is:

xycoxycxyc

oxyc

xycoxycxyc

oxyc

ycoycyc

oyc

xcoxcxc

oxc

if

if

if

if

(32)

In addition, at the start of each equilibrium step, the effect of elevated temperature on the

thermal strains ),,( xytytxt , required in (18), and on the concrete material properties is

established. In the present model, the concrete elastic modulus )E( c , Poisson‟s ratio )( c ,

compressive strength )f( c , tensile strength )f( t , tensile softening parameter )a( t , and

23

thermal strain )( ytxt , are all assumed to vary with temperature according to different

trilinear curves.

Finally, it is again noted that the biaxial model details are presented above with reference to a

fixed crack plane, where it is assumed that the first crack runs orthogonal to the x axis.

Nevertheless, in the general case where the crack plane is not identical to the local element

x-y reference system, constant geometric transformations are employed to relate the strains,

stresses and tangent modulus matrix in the two reference planes.

VERIFICATION

The proposed 2D shell element for flat R/C and composite slabs has been implemented

within the nonlinear structural analysis program ADAPTIC (Izzuddin, 1991) v2.9.15. Two

verification examples are provided hereafter using ADAPTIC, focusing on the modelling

accuracy for uniform-thickness and ribbed plates, with specific reference to the elastic

response. Detailed verification of the formulation against experiments on R/C and composite

floor slabs, including the influence of steel and concrete material nonlinearity, is undertaken

in the companion paper (Elghazouli & Izzuddin, 2003).

Uniform-Thickness Plate

An elastic simply supported square plate, having a uniform thickness of 70 mm and an area

of 6×6 m2, is subjected to a uniformly distributed loading, as depicted in Fig. 11(a). The plate

is modelled with the proposed 2D shell formulation, using the basic linear as well as the

hierarchic quadratic forms, where three alternative meshes consisting of 6×6, 10×10 and

20×20 elements are employed in each case. For verification purposes, the plate is also

modelled using a conventional 20-noded quadratic 3D brick formulation already incorporated

within ADAPTIC (Izzuddin, 1991), where a mesh of 20×20×2 elements is utilised.

Considering a small out-of-plane load of 1 kN/m2, the predictions of the transverse

displacement at the centre of the plate, as obtained from the various element meshes, are

provided in Table 1. These results demonstrate the accuracy of the proposed 2D shell

formulation with reference to the prediction of the 3D brick formulation, where favourable

24

agreement is observed for the fine meshes of the 2D and 3D elements. As expected, the

quadratic form of the proposed shell formulation converges at a coarser mesh than required

by the linear form, although a good approximation is obtained by the linear form with a mesh

of 10×10 elements. Accordingly, the hierarchic quadratic freedoms are most beneficial for

coarse meshes, and they may be ignored as the mesh is sufficiently refined. Interestingly, the

prediction of the 2D quadratic shell formulation with a mesh of 20×20 elements is slightly

more flexible than that of the 3D brick formulation with a mesh of 20×20×2 element, which

is attributed to the different approximations presented by the two models of the out-of-plane

shear strains.

Ribbed Plate

A geometrically orthotropic ribbed plate is considered here, with the cross-section depicted in

Fig. 11(b), and with the remaining geometric and material properties identical to the previous

example. In order to establish the relative accuracy of the 2D shell element and the

significance of its underlying assumptions, three types of analysis are considered hereafter.

Central rib subject to basic deformation modes

One element type (III), representing a central rib of the considered plate with a length of

200mm, is subjected to the six basic deformation modes that induce planar stresses, as

indicated in Table 2. Four modelling cases (A to D) are considered, representing

combinations of i) whether a conforming or zero xz shear strain is assumed in the rib, and ii)

whether additional rib freedoms that represent the modified Reissner-Mindlin hypothesis are

employed. For comparison purposes, two alternative meshes (2×1×4 and 4×1×8) of 3D brick

elements are employed to model the same central rib, where the finest discretisation is over

the depth and the coarsest is in the longitudinal direction.

The results from the 2D shell element, considering both the linear and quadratic forms, and

from the 3D brick element are presented in Table 2, where the predicted strain energy is

normalised relative to that of a uniform thickness plate for which the rib region is ignored.

With the linear and quadratic shell elements providing identical results for cases (A) and (D),

25

it is clear that ignoring the rib freedoms over-estimates the normalised strain energy, or

relative response stiffness, by over 60% for three of the six modes. Overall, the most efficient

and effective representation of the six modes is achieved with the linear form of the shell

element based on case (B), for which additional rib freedoms are employed but the rib shear

strain xz is ignored. With this model, however, significant inaccuracy arises for the bending

mode in the transverse x direction, where the stiffness is underestimated by around 38%, and

for the twisting mode, where the stiffness is overestimated by around 20%. Interestingly, for

the bending mode, the underestimate is due to ignoring xz in the rib, where the true solution

is between the results of (B) and (C). Despite this shortcoming, the inaccuracies obtained

with the proposed ribbed element based on case (B) are only a fraction of those resulting

from a uniform thickness element that ignores the rib region, which underestimates the

stiffness for the two aforementioned modes by 63% and 82%, respectively. Furthermore,

these two modes do not feature strongly in the response of ribbed slabs subject to typical

boundary conditions, and hence the influence of any associated inaccuracy on the overall

response is small, as shown in the next subsection. Considering the remaining four modes,

which are more dominant in the large displacement analysis of realistic ribbed slabs, the

inaccuracy of the proposed element based is limited to a maximum of 7%, whereas the

uniform thickness element underestimates the response stiffness by as much as 88%.

Linear analysis of plate

The ribbed plate is considered here under a small out of-plane load of 1 kN/m2, where simply

supported conditions are assumed. Due to the ribbed configuration, the mesh utilising the

proposed 2D shell element is chosen to consist of 41 and 20 element divisions in the

transverse and longitudinal directions, respectively, leading to 20 elements type (I), 400

elements type (II), 380 elements type (III) and 20 elements type (IV), as depicted in

Fig. 12(a). For verification purposes, a mesh of 60×20 3D brick elements is employed, where

2 and 4 element layers are used in the cover and ribbed regions, respectively, as shown in

Fig. 12(b).

26

Comparison of the predictions of the proposed 2D shell element for the previously described

modelling cases (A to D) against the result from the fine mesh of 3D brick elements is

provided in Table 3. Considering first the predictions of the linear shell element, it is clear

that the most accurate prediction is based on case (B), where the rib freedoms are included

with xz ignored. This also shows that amongst the three cases, cases (A) and (D) are the

least accurate with an error of over 12 %, emphasising the significance of the proposed

modification of the Reissner-Mindlin hypothesis. Such an error becomes even larger as the

rib depth increases and/or as the plate attracts more of the load through bending in the

direction transverse to the ribs. Considering next the predictions of the quadratic shell

element, it is clear that any improvement over the linear form is rather marginal. This

highlights the adequacy of the linear form in the modelling of ribbed slabs, which typically

impose a fine mesh of elements due to geometric considerations. Interestingly, and as

observed for the uniform thickness plate, the prediction of the mesh of quadratic elements is

marginally more flexible than that of the 3D brick element, which is again attributed to the

different approximations presented by the two models of the out-of-plane shear strains. In

view of these findings, the linear form of the proposed 2D shell element based on case (B) is

used as the preferred modelling tool for ribbed composite floor slabs, and therefore forms the

basis of further verification and application studies undertaken in the companion paper

(Elghazouli & Izzuddin, 2003).

Large Displacement analysis of plate

Finally, the accuracy of the proposed 2D shell element in the large displacement range,

allowing for tensile membrane effects, is demonstrated by considering the elastic response

under a large out-of-plane load of 500 kN/m2. The deflected shapes obtained from the linear

form of the 2D shell element, based on case (B), and from the 3D brick element are depicted

in Fig. 12, where very good agreement is observed. Inspecting further the load deflection

response in Fig. 13, an excellent match is observed between the predictions of the proposed

shell element including rib freedoms, that is case (B), and the detailed 3D brick model.

27

Again, the importance of the additional rib freedoms is highlighted, where significant errors

are shown to arise in the large displacement range if these freedoms are suppressed.

With the accuracy of the proposed shell element verified against the most realistic model for

geometrically orthotropic plates based on a 3D brick element, its computational superiority

over such a model is clearly evident. Considering the large displacement analysis, the

analysis employing the 3D brick element consumed around 22hrs of CPU time on a 1 GHz

Pentium III PC, whereas the same analysis based on the proposed shell element required just

over 2mins of CPU time, thus representing a computational saving of over 99.8 %!

CONCLUSION

This paper presents a new method for the nonlinear analysis of R/C and composite floor

slabs, accounting for geometric and material nonlinearity. In particular, the proposed method

deals with the geometric orthotropy of steel-decked composite slabs, which hitherto could

only be modelled using computationally demanding 3D elements, by means of an efficient

new shell element which is also applicable to R/C slabs as a special case.

The proposed shell element is formulated in a local reference system, where the conventional

Reissner-Mindlin hypothesis is modified to reflect the discontinuity in the material between

the ribs of a composite floor slab. This is achieved through the introduction of additional rib

freedoms, which are associated with hierarchic displacement fields specific to the ribs. In

addition, the element is presented in two hierarchic linear/quadratic forms, enabling the

selective activation/deactivation of the higher-order quadratic approximation. Full details of

the element formulation are presented, including the displacement and strain approximations

in the cover and rib regions for the concrete, steel reinforcement and steel decking.

The incorporation of the proposed element within a recently developed large displacement

co-rotational approach is then discussed, where the previously formulated local element

response is transformed to a global reference system common to all elements representing the

structure. Furthermore, a robust material model is proposed for the nonlinear response of

28

concrete, accounting for the effects of tensile cracking and softening, crack closure,

compressive nonlinearity and elevated temperature.

The new shell element has been implemented within ADAPTIC, which is used in this paper

to provide basic verification of the element for uniform-thickness and ribbed plates in the

elastic small and large displacement ranges. Comparisons against detailed 3D models

demonstrate excellent agreement for the uniform thickness plate, where very good

approximation is obtained with the linear form of the shell element using a relatively coarse

mesh of 10×10 elements. For the ribbed plate, investigations carried out on an isolated central

rib subject to basic deformation modes highlight the significance of additional rib freedoms

and, hence, the proposed modification of the Reissner-Mindlin hypothesis, where for three of

the six modes considered ignoring the rib freedoms can lead to more than 60% stiffer

response. The most efficient and effective approximation is achieved with the linear form of

the shell element utilising additional rib freedoms and ignoring the rib transverse shear strain,

where it is shown that the most significant deformation modes are modelled with very good

accuracy. For such modes, the inaccuracy of the proposed 2D shell element is limited to a

maximum of 7%, whereas that of a uniform thickness shell element based on the cover depth

can be as much as 88%. The least accurately represented mode by the proposed shell element

is the transverse bending mode, where the response stiffness can be underestimated by 38%,

though a uniform thickness element can underestimate the stiffness of the same mode by

63%. Notwithstanding this inaccuracy, when considering the overall response of a ribbed

plate, the proposed 2D shell element provides excellent agreement to within 1%, both for the

small and large displacement ranges. Significantly, the comparisons also highlight the

computational superiority of the proposed 2D shell modelling over 3D modelling, where

computational savings in excess of 99.8 % are achieved.

The companion paper provides further extensive verification of the proposed nonlinear

analysis method, focussing on the influence of geometric and material nonlinearities, where

favourable comparisons are made against a number of experiments under ambient and

elevated temperature conditions. In addition, it discusses practical aspects of the response of

29

R/C and composite floor slabs subject to extreme loading, including the effects of

compressive and tensile membrane action.

ACKNOWLEDGEMENT

The authors gratefully acknowledge the financial support provided for this work by the UK

EPSRC under grant GR/L96523.

30

REFERENCES

[1] Belytschko, T., Liu, W.K., and Moran, B. (2000), Nonlinear Finite Elements for

Continua and Structures, Wiley, Chichester, England.

[2] Crisfield, M.A. (1991), Non-linear Finite Element Analysis of Solids and Structures,

Vol. 1, Wiley, Chichester, UK.

[3] Crisfield, M.A. (1997), Non-linear Finite Element Analysis of Solids and Structures,

Vol. 2, Wiley, Chichester, UK.

[4] Elghazouli, A.Y., and Izzuddin, B.A. (2001), “Analytical Assessment of the Structural

Performance of Composite Floors Subject to Compartment Fires”, Fire Safety Journal,

36, 769-793.

[5] Elghazouli, A.Y., and Izzuddin, B.A. (2004), “Realistic Modelling of Composite and

R/C Floor Slabs under Extreme Loading – Part II: Verification and Application”,

Companion Paper.

[6] Huang, Z., Burgess, I.W., and Plank, R.J. (2000), “Effective Stiffness Modelling of

Composite Concrete Slabs in Fire”, Engineering Structures, 22, 1133-1144.

[7] Hughes, T.J.R., and Tezduyar, T.E. (1981), “Finite Elements Based upon Mindlin Plate

Theory with Particular Reference to the Four-Node Bilinear Isoparametric Element”,

Journal of Applied Mechanics, 48, 587-596.

[8] Izzuddin, B.A. (1991), Nonlinear Dynamic Analysis of Framed Structures, PhD Thesis,

Imperial College, University of London.

[9] Izzuddin, B.A. (2002), “Advanced Large Displacement Analysis of Composite Floor

Slabs”, Proc. 15th

ASCE Engineering Mechanics Conference, New York.

[10] Izzuddin, B.A., and Elghazouli, A.Y. (2003), “An Advanced Concrete Model for R/C

and Composite Floor Slabs Subject to Extreme Loading”, Proc. 9th

International

Conference on Civil and Structural Engineering Computing, The Netherlands.

31

[11] Izzuddin, B.A., and Fang, Q. (1997), “Rate-Sensitive Analysis of Framed Structures -

Part I: Model Formulation and Verification”, Structural Engineering and Mechanics,

5(3), 221-237.

[12] Izzuddin, B.A., and Lloyd Smith, D. (1996), “Large Displacement Analysis of Elasto-

Plastic Thin-Walled Frames - Part I: Formulation and Implementation”, Journal of

Structural Engineering, ASCE, 122(8), 905-914.

[13] Izzuddin, B.A., and Moore, D.B. (2002), “Lessons from a Full-Scale Fire Test”,

Structures and Buildings, Proc. Institution of Civil Engineers, 152(4), 319-329.

[14] Kotsovos, M.D. and Pavlovic, M.N. (1995), Structural Concrete: Finite Element

Analysis for Limit-State Design, Thomas Telford Publication Ltd., England, UK.

[15] O‟Connor, M.A., and Martin, D.M. (1998), “Behaviour of a Multi-Storey Steel Framed

Building Subjected to Fire Attack”, Journal of Constructional Steel Research, 46(1–3),

Paper No. 169.

[16] Sebastian, W.M., and McConnel, R.E. (2000), “Nonlinear FE Analysis of Steel-

Concrete Composite Structures”, Journal of Structural Engineering, ASCE, 126(6),

662-674.

[17] Song, L., Izzuddin, B.A., Elnashai, A.S., and Dowling, P.J. (2000), “An Integrated

Adaptive Environment for Fire and Explosion Analysis of Steel Frames - Part I:

Analytical Models”, Journal of Constructional Steel Research, 53(1), 63-85.

[18] Zienkiewicz, O.C., and Taylor, R.L. (1991), The Finite Element Method: Vol. 2 – Solid

and Fluid Mechanics, Dynamics and Nonlinearity, McGraw Hill.

32

TABLES

2D shell 3D brick

Type Linear Quadratic -

Mesh 6×6 10×10 20×20 6×6 10×10 20×20 20×20×2

Displacement (mm) 7.49 8.62 8.82 8.79 8.85 8.87 8.83

Table 1. Central transverse displacement of uniform-thickness plate

2D shell 3D brick

Type Linear Quadratic

Mesh:

2×1×4

Mesh:

4×1×8

Case (A) (B) (C) (D) (B) (C)

xzrib No No Yes Yes No Yes

rib freedoms No Yes Yes No Yes Yes

Extension (x) 2.00 1.23 1.38 2.02 1.10 1.34 1.29 1.25

Extension (y) 1.74 1.72 1.73 1.74 1.71 1.72 1.72 1.72

Shear (xy) 1.88 1.58 1.58 1.88 1.53 1.53 1.50 1.47

Bending (x) 12.64 1.69 4.42 12.68 1.30 4.09 3.06 2.72

Bending (y) 8.81 8.53 8.65 8.81 8.50 8.63 8.58 8.57

Twisting (xy) 10.78 6.74 6.78 10.83 6.46 6.51 5.78 5.58

Table 2. Normalised strain energy of one central rib for various deformation modes

2D shell 3D brick

Type Linear Quadratic

Mesh:

60×20×(2:4)

Case (A) (B) (C) (D) (B) (C)

xzrib No No Yes Yes No Yes

rib freedoms No Yes Yes No Yes Yes

Displacement (mm) 2.68 3.05 2.82 2.65 3.15 2.87 3.04

Table 3. Central transverse displacement of ribbed plate

33

APPENDIX A: SHAPE FUNCTIONS

The linear, quadratic and cubic shape functions are given by the following respective

expressions:

)1)(1(

)1)(1(

)1)(1(

)1)(1(

4

1),(L

N (33)

)1)(1(

)1)(1(

)1)(1(

)1)(1(

2

1),(

2

2

2

2

QN (34)

)1()1(

)1)(1(

)1()1(

)1)(1(

2

1),(

2

2

2

2

CN (35)

The linear and quadratic approximation functions, utilised for the additional rib displacement

fields, depend on the element variant (Figure 2), and are respectively given by:

)1)(()1(

)1)()(1(

8

1)z,,(L

R Variant (I) (36.a)

)1)((

)1)((

4

1)z,,(L

R Variant (III) (36.b)

)1)(()1(

)1)()(1(

8

1)z,,(L

R Variant (IV) (36.c)

)1)()(1(

)1)(()1(

)1)()(1(

4

1)z,,(

2

2

2

QR Variant (I) (37.a)

)1)((

)1)((

)1)((

2

1)z,,(

2

2

2

QR Variant (III) (37.b)

34

)1)()(1(

)1)(()1(

)1)()(1(

2

1)z,,(

2

2

2

QR Variant (IV) (37.c)

where,

h

tz21

(38)

35

LIST OF FIGURES

Figure 1. Geometric configuration of composite floor slab

Figure 2. Variants of composite shell element

Figure 3. Geometric configuration and local reference system

Figure 4. Local reference systems: real and natural coordinates

Figure 5. Modified Reissner-Mindlin hypothesis: additional rib deformations

Figure 6. Local freedoms of slab element

Figure 7. Global nodal freedoms and local co-rotational system (Izzuddin, 2002)

Figure 8. Plasticity-based compressive response of concrete

Figure 9. Tensile envelopes for direct stresses

Figure 10. Tensile envelope for shear stress

Figure 11. Ribbed plate configuration

Figure 12. Element meshes and deflected shapes: (UDL = 500 kN/m2):

(a) 2D shell formulation, (b) 3D brick formulation

Figure 13. Large displacement response of ribbed slab

Izzuddin et al.: Realistic Modelling of Composite and R/C Floor Slabs …: Part I

Transverse

Longit

udina

l

Out

-of-

plan

e

Concrete

Steel deck

Steel reinforcement

Rib

Cover

Figure 1. Geometric configuration of composite floor slab


(I) (II) (III) (IV)

Figure 2. Variants of composite shell element


L

t

h

r W

W

2

34

1

x,u

y,vz,w

Figure 3. Geometric configuration and local reference system


��

2

34

1

x

yz

Real coordinates Natural coordinates

��

2

34

1

ξ

ηz

Figure 4. Local reference systems: real and natural coordinates


z

ru,ξ

rv,η

1 2

34

z

ru,ξ

rv,η

1 2

34

Transverse deformations Longitudinal deformations

Figure 5. Modified Reissner-Mindlin hypothesis: additional rib deformations


1y1x111 ,,,, θθθθθθθθwvu 2y2x222 ,,,, θθθθθθθθwvu

3y3x333 ,,,, θθθθθθθθwvu4y4x444 ,,,, θθθθθθθθwvu

1 2

34

2y2x222

2

,,,, θθθθθθθθwvu

w

4y4x444

4

,,,, θθθθθθθθwvu

w

3r3r , vu ξ

η

1r1r

1y1x111

1r1r1

,

,,,,

,,

vu

wvuvuw

θθθθθθθθ

2r2r

3y3x333

2r2r3

,

,,,,

,,

vu

wvuvuw

θθθθθθθθ

Figure 6. Local freedoms of slab element


X,U

Y,V

oxc

oyc

ozc

o13v

o1313vα

o24v

o2424vα

1

2 3

4

Initial configuration

Z,W

3

24v

13v

xc

yc

zc

x

y

z

1

2

4

1d

1nCurrent configuration

Figure 7. Global nodal freedoms and local co-rotational system (Izzuddin, 2002)


cσ

cf

cc fs

cc fr

pεcε

1σ

2σ

cσ−

cσ−

cσ=C

Figure 8. Plasticity-based compressive response of concrete


tf ta

xcεoxcε

xtσ

tε

tf ta

ycεoycε

ytσ

tε

Figure 9. Tensile envelopes for direct stresses


tf

tγ− tγ xycγ−γoxyc

+γoxyc

xytσ

2

atsα2

atsα

Figure 10. Tensile envelope for shear stress


60mm200mm

144.6mm70mm

(b) Ribbed plate cross-section

(a) Uniform-thickness plate

2.0

mm/N1020E 23

=ν×=

6 m

6 m

70 mm

Uniformly distributedloading = 1 kPa

Figure 11. Ribbed plate configuration


(a) 2D shell formulation (b) 3D brick formulation

Figure 12. Element meshes and deflected shapes: (UDL = 500 kN/m2)


0

100

200

300

400

500

600

0 100 200 300 400 500

Displacement (mm)

Loa

d (k

Pa)

2D shell (with rib freedoms)

2D shell (without rib freedoms)

3D brick

Figure 13. Large displacement response of ribbed slab

realistic modelling of composite and rc floor slabs under - spiral

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