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A FINITE ELEMENT FOR REINFORCED CONCRETE SHELLS

Kitjapat Phuvoravan1

Lecturer, Department of Civil Engineering, Faculty of Engineering, Kasetsart University, Bangkok

fengkpp@ku.ac.th

ABSTRACT: Two techniques, discrete and layered modeling, exist for modeling reinforced concrete shells. In the
discrete modeling, concrete is modeled by three-dimensional solid elements while the reinforcing steel is modeled by
truss elements. The drawback of the discrete modeling is that a large number of degrees of freedom are required. In the
layered modeling, concrete is divided into a set of layers, while the reinforcing steel is smeared into a layer between
concrete layers. Layered modeling of RC shells is simple, but provides an unrealistic representation of the reinforcing
steel. This paper presents a new finite element for the analysis of reinforced concrete (RC) shells. The element
combines a four-node Kirchhoff shell element for concrete with two-node Euler beam elements for the steel
reinforcement bars. The connectivity between reinforcement beam elements and concrete shell element is achieved by
means of rigid links. By using the transformation method for rigid links, beam nodes are eliminated from the final
mesh of the structure. The stiffness from the reinforcement is implicitly included in the new element. In this manner,
this finite element is able to take into account the location of the reinforcement bars. This is in contrast to the smeared
method, which is often adopted in the layered modeling of RC shells. The new finite element is verified with
experimental results. The effect of different reinforcement spacing with the same reinforcement ratio is also evaluated.

KEYWORDS: Finite elements, Reinforced concrete, Shell structures, Slabs, Layered approach

1. INTRODUCTION Solid element

Reinforced concrete shell structures are used in a variety
of structural engineering applications. The behavior of
reinforced concrete (RC) structural systems is very
complex. Thus, in order to more accurately simulate
Truss element
them, there is a need for the development of efficient
sophisticated elements that are flexible enough to include
several important behaviors. The present work strives to Concrete node

develop a simple, yet accurate, finite element for

reinforced concrete shells. The goal is to provide a
practical approach for the modeling of these shell Steel node
structures.

1.1 Background & Motivation Figure 1 Discrete modeling of RC shells.

Two techniques exist for modeling reinforced concrete
shells: discrete and layered modeling. Figure 1 illustrates Figure 2 illustrates the layered approach (Hand [1])
the discrete modeling of a RC shell. In this case, for RC shells. In this formulation, a four-node or nine-
concrete is modeled by three-dimensional solid elements node shell element is adopted. Concrete is divided into a
while the reinforcing steel is modeled by truss elements. set of layers, while the reinforcing steel is smeared into a
The connectivity between a concrete node and a layer between concrete layers. The reinforcement layer
reinforcing steel node can be achieved by two methods. has stiffness only in the direction of the reinforcement.
In the first method, concrete and reinforcement share the In this manner, the variation of stress along the shell
same node; hence perfect bond is assumed. In the second thickness can be modeled. Furthermore, since the three-
method, a spring element is used to connect concrete and dimensional shell is simulated by a set of two-dimension
reinforcement nodes. The spring stiffness and the bond- layers, only a two-dimensional stress-strain relationship
slip relationship can be used to simulate the bond-slip is required. Layered modeling of RC shells is simple, but
behavior. Thus, discrete modeling of RC shells provides provides an unrealistic representation of the reinforcing
a realistic representation of steel. However, it is more steel. Since real reinforcement is discrete, only highly
expensive in terms of analysis time, since a large number reinforced shells can be appropriately modeled by the
of degrees of freedom are required. Furthermore, the layered approach. Finally, in this approach the
construction of the model is difficult and time incorporation of bond slip can only be achieved
consuming. artificially.
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 t
2 ⎛1 1
⎞
[k s ]20 x 20 = ∫ ⎜⎜ ∫ ∫ {B}sT ⋅ [G ]T ⋅ [D]s ⋅ [G ]⋅ {B}s ⋅ ab ⋅ dξdη⎟⎟dt
−
t ⎝ −1−1 ⎠
Slab Steel Layers 2
Thickness
(1)
Where, t is the shell thickness, a and b are half of shell
element length in the x̂ - and ŷ -direction, respectively,
Concrete Layers
and ξ and η are the natural coordinate axes. The
definition of other parameters can be obtained in Ref. [3].
Figure 2 Layered modeling of RC shells.

1.2 Objective & Scope 4 3

The objective of this study is to develop a new finite
element for reinforced concrete shells that is simple, easy ẑ b
ŷ
to use, and efficient, yet can capture the effects of each
individual reinforcement bar, thus better capturing the w1 x̂
behavior of RC shells. The scope of the research is b
limited to flat shells. θx1
1 2
a a
1.3 Methodology θy1
Figure 3 Notation for the shell element.
First, a Kirchhoff shell element using thin shell
assumption is adopted for the concrete shells and Euler
For general layered shell element, the integration
beam elements with no shear deformation are adopted for
along the shell thickness is achieved by dividing the shell
the reinforcing steel bars. The selected shell and beam
into a set of layers. In each layer, it is assumed that
element are compatible, since neither includes transverse
parameters, such as stresses, strains, and elasticity
shear strains. The transformation method of rigid links is
moduli, are constant in the middle of the layer. In the
then used to combine reinforcement beam elements to the
present study, the integration along the shell thickness is
concrete shell element. The reinforcement nodes are
accomplished using the Simpson’s integration technique.
connected though the rigid links to the mid-surface of
A set of points is assigned with equal spacing along the
concrete shell element.
shell thickness. Normally, an odd number of points are
used to enable the application of the composite
2. PROPOSED FINITE ELEMENT
Simpson’s rule. The advantage of this integration
In this section, the formulation of the proposed finite
technique is that for the same level of accuracy, it uses a
element is presented. The formulations of the beam and
smaller number of points than the number of layers.
the modified layered shell element are presented. At the
Thus, the analysis time is reduced.
end, the new finite element is developed using the
The beam element used in the formulation of the
modified shell and beam elements and the transformation
proposed finite element is the Euler beam element. The
method for the rigid links
selected shell element and beam element are consistent
It is noted, in this study, that the finite element
since both possess no shear deformation. The stiffness of
software ABAQUS [2] is used to exercise the proposed
the Euler beam element, [kb]12x12, is a 12 by 12 matrix
element, which is implemented in the user-defined
determined by differentiating the strain energy of the
subroutine (UEL). The other finite element frameworks
beam element with respect to its degrees of freedom.
including model creators, engine solvers, and post-
processors are provided by ABAQUS program.
2.1.2 Material Modeling
In order to determine the element stiffness, the material
2.1 Finite Element Formulation
model must be provided. Since this study is limited to
2.1.1 Modified Shell Element and Beam Element
linear elastic analysis, only modulus of elasticity and
In this section, the determinations of the element stiffness
Poisson’s ratio for concrete shells and steel reinforcement
matrix the layered shell and the beam elements are
bars are required. This information can be obtained from
presented. A modified version of the layered shell
the experimental test.
approach is developed for the formulation of the shell
element.
2.1.3 Rigid Links
The general notation and the local coordinate axes
Rigid links or multipoint constraints are imaginary,
(xˆ, yˆ , zˆ ) of the shell element are shown in Figure 3. The weightless links used to constrain two nodes. This is
stiffness of the layered shell element with Kirchhoff achieved by means of transformations that make the
assumption is given by: degrees of freedom of one node slave to the degrees of
freedom of another node, which is called master node.
By using rigid links, only the degrees of freedom of the
master node appear in the assembled structure. An
example of the use of rigid links is that of a plate

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reinforced by a beam [4]. The nodes of the beam and the Rigid Link
nodes of the plate do not coincide. The beam helps the
plate resist the external load due to the connection 4* J 3*
between the plate and the beam. 6*
I z
b 1* 2*
2.2 RC Shell Element
The proposed RC shell element is formulated using the 5* y
x
concept of layered shell element for concrete combined Rigid Link
with the beam element for the reinforcement bar by System 1
means of rigid links. The reinforcement beam nodes
disappear in the combined element, however they
implicitly help resist external load through the shell 4
element. 3
To illustrate the proposed method, consider the
problem shown in Figure 4 where a single steel 1 2
reinforcement is placed in a concrete shell. In this figure,
the two systems used to derive the RC shell element are
System 2
shown. In the first system, all degrees of freedom from
the beam and the shell appear in the element. In the Figure 4 Proposed reinforced concrete shell element.
second system, only the degrees of freedom associated
with the shell are presented in the element. First, Given that point I and point J are on the mid-surface
consider nodes 5* and 6*, and points I and J only. The of the shell element, their displacements can be obtained
relationship between these nodes is given by: in terms of shell nodal displacements as:
{ }
u * = [T22 x10 ]⋅ {u IJ } (2) {u IJ } = [N IJ ]⋅ {u} (3)
Where, Where, {u} is shell displacement vector (20x1) at node 1,
[I ] [T ] 0 ⎤ , 2, 3, and 4, and [NIJ] is the matrix (10x20) of shell shape
[T22 x10 ] = ⎡⎢ 12 ⎤⎥ , [T12 x10 ] = ⎡⎢ 6 x5 functions at point I and point J.
⎣[T12 x10 ]⎦ ⎣ 0 [T6 x5 ]⎥⎦ From equation (2) and equation (3), the vector {u*}
⎡1 0 0 0 z ⎤ can be expressed in terms of the shell nodal displacement
⎢0 1 0 − z 0 ⎥ vector {u} as:
⎢
⎢0 0 1 0 0 ⎥
⎥ { }
u * = [T22 x10 ] ⋅ [N IJ ] ⋅ {u} (4)
[T6 x 5 ] = ⎢ ⎥, Next consider the whole shell and beam elements.
⎢0 0 0 1 0 ⎥ The relationship between degrees of freedom at nodes 1*,
⎢0 0 0 0 1 ⎥ 2*, 3*, 4*, 5*, and 6* of System 1 and degrees of freedom
⎢ ⎥ at nodes 1, 2, 3, and 4 of System 2 is given as:
⎣0 0 0 0 0 ⎦
[I12] is a 12x12 identity matrix, z is the z-coordinate from {u′} = [T32 x 20 ]⋅ {u} (6)
the reference mid-shell surface, Where,
{u } = {{u } {u } {u } {u }} [I ]
[T32 x 20 ] = ⎡⎢ 20 ⎤⎥ , [T12 x 20 ] = [T12 x10 ]⋅ [N IJ ] ,
* T * * * *
is displacement vector
⎣[T12 x 20 ]⎦
I J 5 6

(22x1) at point I, point J, node 5*, and node 6*, and

{u IJ }T = {{u I } {u J }} is displacement vector (10x1) at [I20] is the identity matrix of dimension 20 by 20, and
point I and point J. It should be noted that five degrees of {{ } { } { } { } { } { }}
{u '}T = u1* u 2* u 3* u 4* u 5* u 6* is the
freedom are considered at node I and node J, since there displacement vector (32x1) of system 1 at node 1 , 2 , 3*,
* *

is no z-rotational degree of freedom in the considered 4*, 5*, and 6*.

shell element. The stiffness matrix (20x20) of the reinforced
concrete element, [k], can then be written as:
[k ] = [T32 x 20 ]T ⋅ [k ′] ⋅ [T32 x 20 ] (7)
Where,
[k ] 0 ⎤ , [k ] is the shell stiffness matrix
[k ′] = ⎡⎢ s ⎥
⎣ 0 [k b ]⎦
s

(20x20), and [kb] is the beam stiffness matrix (12x12).

For general reinforcement, i.e., more than one bar
placed in arbitrary direction, the RC shell stiffness matrix
can be expressed more explicitly as:
[k ] = [kC ] + ∑ [k R ] (8)
nR
Where,
[kC ] = [k s ] is the stiffness from concrete component,
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 [k R ] = [N IJ ]T ⋅ [T12 x10 ]T ⋅ [Tb ]T ⋅ [kb ]⋅ [Tb ]⋅ [T12 x10 ]⋅ [N IJ ] is element analysis and the data within the linear elastic
the stiffness of the reinforcement, range from experimental test. The central deflection of
the slab is 0.843 mm. from finite element analysis and
⎡[λ ] 0 0 0⎤
0.838 mm. from experimental data. As can be seen, a
⎢ 0 [λ ] 0 ⎡ cosθ sin θ 0⎤
0 ⎥⎥ good agreement is observed between the results from the
[Tb ] = ⎢ , [λ ] = ⎢⎢− sin θ cosθ 0⎥⎥ ,
⎢0 0 [λ ] 0 ⎥ proposed finite element and those from the experiment.
⎢ ⎥ ⎣⎢ 0 0 1⎦⎥
⎣0 0 0 [λ ]⎦
4. EFFECT OF REINFORCEMENT SPACING
θ is the angle between the reinforcement and the x-axis, After the proposed RC shell element is verified, analyses
and nR is the total number of reinforcement bars in the are performed to evaluate the effect of the reinforcement
RC shell element. spacing. A problem with characteristics similar to those
of a RC building slab is developed. Figure 6 shows the
3. NUMERICAL VERIFICATION problem description. A 3.6 m. square reinforced concrete
The proposed RC shell element is implemented in the slab is modeled in quarter. The simply supported slab is
ABAQUS finite element software. Its formulation and subjected to a concentrated load at the slab center.
implementation are verified against the experimental
results provided by Taylor [5]. In this work, the RC slab Node Number
Symmetry
is a 1950 mm. square plate simply supported 75 mm.
21 22 23 24 25 y
from each edge resulting on a 1800 mm. span. The slab
thickness is 50 mm., which gives a span to depth ratio of
x
36. The slab is subjected to a uniform load. The 16 20
z

reinforcement in each direction is 4.8 mm. in diameter, t = 125 mm.

and is spaced 75 mm. in one direction and 62.5 mm. in (L/t = 28.8)
fc’ = 200 ksc.
1800 mm. 11
the orthogonal direction. The slab configuration and the (4 meshes) 15 Ec = 2.12x105 ksc.
vc = 0.18
reinforcement detail are shown in Figure 5. The Es = 2.04x106 ksc.
monitored degree of freedom is the central deflection of 6 vs = 0.3
10 fy = 5272 ksc.
the slab.
(a)
Symmetry
Roller supports 1 2 3 4 5 Symmetry
t = 50 mm.
75 mm.

(L/t = 36)
fc’ = 286 ksc Center point of slab 1800 mm.
fsy = 3831 ksc (4 meshes)
fsu = 4955 ksc
Spacing varies Ø Varies z
6 @150 mm. = 900 mm.

y
975 mm.

x
y
x
z
75 50
125 mm.

(b)
(a)
Reinforcement Ratio (ρ) = 0.218 %

Symmetry

7 @129 mm. = 900 mm. 75 mm. Figure 6 Problem Description Used to Evaluation the
975 mm.
Effect of Reinforcement Spacing; (a) Slab Configuration,
(b) Reinforcement Detail.
4.8 mm. @ 62.5 mm.
z Figure 7 shows the stiffness contribution of
reinforcement. The comparison is made among the
x
analyses using different reinforcement diameter and
y
50 mm. spacing but same reinforcement ratio. Contrary to the
smear method for reinforcement model, it can be seen
4.8 mm. (b)
4.8 mm.
that the proposed RC shell element can capture the effect
4.8 mm. of the reinforcement spacing and the effect of the
reinforcement diameter. The analysis results show that
Figure 5 Test of Pressure Load on Simply Supported RC the usage of larger reinforcement size and larger spacing
Slab by Taylor [3]; (a) Slab Configuration, (b) yields slightly weaker system. As the reinforcement size
Reinforcement Detail. and its spacing is decreased, the behavior of the slab
becomes closer to the analysis using smear reinforcement
The slab is analyzed using the proposed finite model. Figure 8 shows the effect of reinforcement
element. Due to its symmetry, a quarter of the slab is spacing to the stiffness of the RC slab. It can be seen that
modeled using an 8 by 7 mesh. The proposed RC shell the stiffness of the system decreases as the reinforcement
element is verified by comparing the result from finite spacing increases.
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 REFERENCES
[1] Hand, F.R., Pecknold, D.A., and Schnobrich, W.C.
2 Smear Reinforment
(1973), “Nonlinear layered analysis of RC plates and
Contribution of reinforcement

Proposed Elmt. (12mm Dia.)

1.8 Proposed Elmt. (6mm Dia.) shells,” Journal of Structural Engineering, ASCE,
99(ST7), pp. 1491-1505.
to deflection (%)

1.6 [2] ABAQUS/Standard User’s Manual (2001). Version

6.2. ABAQUS, Inc., Pawtucket, RI.
1.4 [3] Phuvoravan, K. and Sotelino, E.D. (2005), “Nonlinear
Finite Element for Reinforced Concrete Slabs,”
1.2 Journal of Structural Engineering, ASCE, Vol. 131,
No. 4, April 01, pp. 1-7.
1 [4] Cook, R.D., Malkus, D.S., and Plesha, M.E. (1989),
0 5 10 15 20 25 Concepts and applications of finite element analysis,
Node number Wiley, New York.
[5] Taylor, R., Maher, D. R. H., and Hayes, B. (1966),
Figure 7 Contribution of reinforcement to slab “Effect of the Arrangement of Reinforcement on the
deflection. Behavior of Reinforced Concrete Slabs,” Magazine of
Concrete Research, Vol. 18, No. 55, June, pp. 85-94.
1.60
Contribution of Reinforcement .
to Deflection (Node1)

1.55

1.50

1.45

1.40
0 100 200 300 400 500
Spacing (in)

Figure 8 Effect of Reinforcement Spacing to Stiffness of

RC Slab.

5. SUMMARY & CONCLUSIONS

A new finite element for the analysis of reinforced
concrete shell is presented. Having advantage over the
solid modeling, the proposed element requires
considerably less analysis time. Although not in the scope
of this paper, this is of particular concern in nonlinear
analysis. In addition, unlike the general layered shell
element, the proposed finite element takes into account
the actual location of the reinforcing steel bars. As a
result, the proposed finite element can better predict the
behavior of reinforced concrete shells than the general
layered shell element, especially for lightly reinforced
shells. Moreover, modeling reinforcement individually
makes it possible to include the effect of bond-slip
behavior, which is essential in the failure analysis of the
reinforced concrete structures. The proposed finite
element is verified with experimental results in linear
elastic range. It has been shown that the proposed finite
element yields a good representation of the behavior of
reinforced concrete shells.

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