Nonlinear analysis of reinforced concrete columns with holes · 3. Provide recommendations for the...

INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 3, No 3, 2013

© Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0

Research article ISSN 0976 – 4399

Received on February 2013 Published on March 2013 655

Nonlinear analysis of reinforced concrete columns with holes

Ehab M. Lotfy

Associate Professor, Civil Engineering Department,

Faculty of Engineering, Ismaelia, Suez Canal University, Egypt

[email protected]

doi:10.6088/ijcser.2 201203013060

ABSTRACT

The behavior of reinforced concrete columns with holes under axial load is not understood,

and researches in the subject are needed to help designers and structural code officials. Holes

drilled out to install additional services or equipment, such as for ducts through columns,

beams, or walls, can lead to loss of strength and possible structural failure. Until now little

work has been done on holes in columns and, hence, this study aims to examine the amount

of strength lost due to the presence of holes in columns. Nonlinear finite element analysis on

21-column specimens was achieved by using ANSYS software. The nonlinear finite element

analysis program ANSYS is utilized owing to its capabilities to predict either the response of

reinforced concrete columns in the post-elastic range or the ultimate strength of reinforced

concrete columns. An extensive set of parameters is investigated including different

parameters; dimensions of the holes with diameter 0.1, 0.15, 0.2 and 0.3 of column length,

their relative position in columns, and the shape of holes; circle and square. A comparison

between the experimental results and those predicted by the existing models are presented.

Results and conclusions may be useful for designers, have been raised, and represented.

Keyword: Inelastic finite element analysis, columns, holes, strength and testing of materials.

1. Introduction

In the construction of modern buildings, a network of pipes and ducts is necessary to

accommodate essential services like water supply, sewage, air-conditioning, electricity,

telephone, and computer network. Usually, these pipes and ducts are placed underneath the

beam soffit and, for aesthetic reasons, are covered by a suspended ceiling, thus creating a

dead space. Passing these ducts through transverse openings in the columns leads to a

reduction in the dead space and results in a more compact design. The provision of such

openings may result in the loss of strength, stiffness and ductility and, hence, significant

structural damage may be sustained, if the provision of the openings is not considered

adequately during the design or construction stages. This is especially true for un-braced

structures, since loss of stiffness leads to redistribution of internal forces and moments.

The mechanical behavior of concrete beams and slabs with openings has been examined in

several studies and design rules have been recommended (Ashouf A.F. et al., 1999), (Tayel

M. A. et al., 2004), (Simpson D., 2003), (Jiyang Wang et al., 2008) and (Mansur, M.A.,

1998). However, in the case of concrete columns and walls with transverse openings,

minimal research has been carried out and, currently, there is a lack of appropriate design

rules. Columns are critical elements, but in general only carry a fraction of their capacity at

normal service loads.

Nonlinear Analysis of Reinforced Concrete Columns with Holes

Ehab M. Lotfy

International Journal of Civil and Structural Engineering

Volume 3 Issue 3 2013 656

The research reported in this paper aims to investigate the compressive resistance-capacity of

concrete columns with transverse holes with diameters 0.1, 0.15, 0.2 and 0.3 of column

length, their relative position in columns; in middle third and edge third of tested columns ,

and the shape of holes; circle and square. Four columns with different holes were tested

experimentally to evaluate the effect of hole geometry and location. Analysis of the

experimental results is used to derive appropriate design recommendations.

2. Objective of the study

The main objectives of this study could be summarized in the following points

1. To investigate the reduction in load carrying capacity of the reinforced concrete

short columns having circle and square cross-sections with hole in different places.

2. To model the RC columns using three-dimensional non-linear finite element

analysis.

3. Provide recommendations for the design engineers and the structural codes for the

design of the reinforced concrete columns.

3. Experimental program

Four concrete columns with different holes in different position and control column without

holes were cast to evaluate the effect of section loss on the compressive resistance-capacity.

The parameters examined experimentally were the diameter, relative position; where column

is divided to three parts in the columns length and also in loading direction, middle third and

edge third, and the shape of holes; circle and square shape. Figure 1 shows the details of the

holes provided in each column.

All columns were 1600mm height, 300m length and 300mm wide and contained both

longitudinal and transverse reinforcement. The longitudinal reinforcement rebars comprised

4#16 mm in diameter, and the transverse reinforcement consisted of shear links, 8mm in

diameter @ 200 mm. A clear concrete cover of 25 mm was provided in all column specimens

and a strengthening jacket was provided at both ends of each column in order to minimize the

effect of local buckling of the longitudinal reinforcement, the test matrix is shown in table 1.

Table 1: Details of tested columns specimens

No Col.

No.

Dimension

(mm)

fcu

(N/mm2) Reinf.

Shape

of

holes

Dim. Of

holes

Position

of holes Notes

1 C1

300X300 25

4 no’s

of

16mm

- - - Control

specimen

2 C2 circle D=60mm Case (a)

3 C3 circle D=60mm Case (b)

4 C4 square L

=60mm

Case (c)


Ehab M. Lotfy



Figure 1: Details of reinforcement of tested columns

4. Numerical finite element

The analysis is carried out on 21-RC columns; the parameters of study were a holes

dimensions with diameters 0.1, 0.15, 0.2 and 0.3 of column length, their relative position in

columns; in middle third and edge third of tested columns , and the shape of holes; circle and

square as shown in table 2.

4.1. Basic fundamentals of the FE method.

The basic governing equations for two dimensions elastic – plastic FEM have been well

documented, and are briefly reviewed here.

I. Strain - displacement of an element

[dε]=[B][dU]

Where: [B] is the strain - displacement transformation matrix. The matrix [B] is a function of

both the location and geometry of the suggested element, it represents shape factor. The

matrix [B] for a triangle element having nodal points 1, 2 and 3 is given by

[ ]

−−−−−−

−−−

−−−

∆=

211213313223

123123

211332

000

000

2

1

yyxxyyxxyyxx

xxxxxx

yyyyyy

B

Where xi and yi represent the coordinates of the node and ∆ represents the area of the

triangular element, i.e.


Ehab M. Lotfy



33

22

11

1

1

1

det2

yx

yx

yx

=∆

II. Stress - strain relation or field equation

[dσ] = [D] [dε]

Here, [D] is the stress- strain transformation matrix. For elastic elements the matrix from the

Hooke's law leads to [D]=[De]. For plastic elements, the Prandtl-Reuss stress-strain relations

together with the differential form of the von Mises yield criterion as a plastic potential leads

to [D] = [Dp

].

The elastic matrix, [De], is given by the elastic properties of the material whereas the plastic

matrix, [De], is a function of the material properties in the plastic regime and the stress-strain

elevation. Obviously, for two-dimensional analysis [De] and [D

p] depend on the stress-strain

state, i.e. plane stress versus plane strain.

The plastic matrix, [Dp

], depends on the elastic-plastic properties of the material and the

stress elevation. Comparing [De] and [D

p], it can be seen that the diagonal elements of [D

p]

are definitely less than the corresponding diagonal elements in [De]. This amounts to an

apparent (crease in stiffness or rigidity due to plastic yielding. Therefore, the plastic action

reduces the strength of the material.

III. Element stiffness matrix [Ke]

[ ] [ ][ ]dvBDBKT

e

∫∫∫=][

The transpose matrix of [B] is [B]T

. In the case of the well-known triangular elements [k] is

represented by;

[ ] [ ] [ ][ ]VBDBKT

=

The element volume is V and for a two-dimensional body equals the area of the element

∆ multiplied by its thickness t.

IV. The overall stiffness matrix [K]

The stiffness matrixes [Ke] of the elements are assembled to form the matrix [K] of the whole

domain. The overall stiffness matrix relates the nodal load increment [dP] to the nodal

displacement increment [du] and can be written as

[dP] = [K] [du]


Ehab M. Lotfy



This stiffness relation forms a set of simultaneous algebraic equations in terms of the nodal

displacement, nodal forces, and the stiffness of the whole domain. After imposing appropriate

boundary conditions, the nodal displacements are estimated, and consequently the stress

strain field for each element can be calculated.

4.2. Material modeling

A linear-elastic, isotropic constitutive relation is adopted to describe the behavior of

uncracked concrete elements in tension or compression figure 2 and figure 3.

For steel reinforcement, elastic stress-strain behavior was assumed to obey the linear relation

of Hook's law described as:

[ ] [ ] }{)1(2}{}{ ευεσee

DGDE +==

Where {σ} and {ε} are column matrices of stress σij and εij respectively, G is the shear

modulus; E is the modulus of elasticity and υ is the Poisson's ratio.

In the plastic regime the stress-plastic strain; σ-εp, behavior of steel was assumed to obey a

simple power law as shown in figure 4 with a strain hardening exponent of 0.02.

Figure 2: Stress-strain relation for plain

concrete in tension

Figure 3: Stress-strain relation for plain

concrete in compression

Figure 4: Stress-Strain relation for steel reinforcement


Ehab M. Lotfy



4.3. Resume about used program

The implementation of nonlinear material laws in finite element analysis codes is generally

tackled by the software development industry in one of two ways. In the first instance the

material behaviour is programmed independently of the elements to which it may be

specified. Using this approach the choice of element for a particular physical system is not

limited and best practice modelling techniques can be used in identifying an appropriate

element type to which any, of a range, of nonlinear material properties are assigned. This is

the most versatile approach and does not limit the analyst to specific element types in

configuring the problem of interest. Notwithstanding this however certain software

developers provide specific specialised nonlinear material capabilities only with dedicated

element types.

ANSYS (ANSYS Manual Set, 1998) and (Installation Guide “ANYSYS) provides a

dedicated three-dimensional eight nodes solid isoparametric element, Solid65, to model the

nonlinear response of brittle materials based on a constitutive model for the triaxial behaviour

of concrete (William, K.J. et al., 1975).

4.4. Finite element modeling

4.4.1 Geometry

The details of tested columns were shown in Figure 5 and 6. Analyses were carried out on 21-

columns specimens, where all columns had square cross-section with a 300 mm side and

1600 mm height, the longitudinal reinforcement rebars comprised 4#16 mm in diameter, and

the transverse reinforcement consisted of shear links, 8mm in diameter@200mm, a clear

concrete cover of 25 mm was provided in all column specimens

4.4.2 Element types

Extensive inelastic finite element analyses using the ANSYS program are carried out to study

the behavior of the tested columns. Two types of elements are employed to model the

columns. An eight-node solid element, solid65, was used to model the concrete. The solid

element has eight nodes with three degrees of freedom at each node, translation in the nodal x,

y, and z directions. The used element is capable of plastic deformation, cracking in three

orthogonal directions, and crushing. A link8 element was used to model the reinforcement

polymer bar; two nodes are required for this element. Each node has three degrees of freedom,

translation in the nodal x, y, and z directions. The element is also capable of plastic

deformation (ANSYS User’s Manual).

4.4.3 Material properties

Normal weight concrete was used in the fabricated tested columns. The stress-strain curve is

linearly elastic up to about 30% of the maximum compressive strength. Above this point, the

stress increases gradually up to the maximum compressive strength fc\, after that the curve

descends into softening region, and eventually crushing failure occurs at an ultimate strain.

4.4.4 Loading and nonlinear solution


Ehab M. Lotfy



The analytical investigation carried out here is conducted on 21-RC columns; all columns are

raised in vertical position with by vertical load on top surface. At a plane of support location,

the degrees of freedom for all the nodes of the solid65 elements were held at zero. In

nonlinear analysis, the load applied to a finite element model is divided into a series of load

increments called load step. At the completion of each load increment, the stiffness matrix of

the model is adjusted to reflect the nonlinear changes in the structural stiffness before

proceeding to the next load increment. The ANSYS program uses Newton-Raphson

equilibrium iterations for updating the model stiffness. For the nonlinear analysis, automatic

stepping in ANSYS program predicts and controls load step size. The maximum and

minimum load step sizes are required for the automatic time stepping.

The simplified stress-strain curve for column model is constructed from six points connected

by straight lines. The curve starts at zero stress and strain. Point No.1, at 0.3 fc\ is calculated

for the stress-strain relationship of the concrete in the linear range. Point Nos.2, 3 and 4 are

obtained from Equation (1), in which is calculated from Equation (2). Point No. 5 is at

and f ′c. In this study, an assumption was made of perfectly plastic behavior after Point No. 5

as shown in figure 7, which shows the simplified compressive axial stress-strain relationship

that was used in this study

……………………………..(1)

…………………………………..(2)

……………………………………..(3)

Figure 5: Finite element mesh for a typical column model


Ehab M. Lotfy



Case (a) Case (b) Case (c) Case (d) Case (e)

Figure 6: Details of tested columns specimens

Figure 7: Simplified compressive axial stress-strain relationship


Ehab M. Lotfy



Table 2: Details of tested columns specimens

No Col.

No.

Dim.

(mm)

fcu

(N/mm2) Reinf.

Shape of

holes

Dim of holes Notes

1 C1

300*300 25 4#16 mm

- - Control

specimen

2 C2

Case (a)

0.1L

Circle

holes

3 C3 0.15L

4 C4 0.2L

5 C5 0.3L

6 C6

300*300 25 4#16 mm Case (b)

0.1L

7 C7 0.15L

8 C8 0.2L

9 C9 0.3L

10 C10

300*300 25 4#16 mm Case (c)

0.1L

11 C11 0.15L

12 C12 0.2L

13 C13 0.3L

14 C14

300*300 25 4#16 mm Case (d)

0.1L

15 C15 0.15L

16 C16 0.2L

17 C17 0.3L

18 C18

300*300 25 4#16 mm Case (e)

0.1L

Square

holes

19 C19 0.15L

20 C20 0.2L

21 C21 0.3L

5. Inelastic analysis results and discussion

The parametric studies included in this investigation are holes dimensions with diameters 0.1,

0.15, 0.2 and 0.3 of column length, their relative position in columns; case (a), (b), (c) and (d),

and the shape of holes; case (a) and (d). Table 3 shows the analytically results of the ultimate

loads, deformations and compressive stress of concrete, respectively.

Table 3: Theoretical results of tested columns specimens

No Col.

No.

Concrete stress

(N/mm2)

Ultimate Def.

(mm)

Ultimate Load

(KN)

Notes

1 C1 25 1.30 139.00 Control

specimen

2 C2 25 1.24 138.50

3 C3 25 1.24 137.40

4 C4 24.8 1.24 127.44

5 C5 24.6 1.19 112.32

6 C6 25 1.22 135.50

7 C7 25 1.15 131.76

8 C8 24.8 1.10 123.12

9 C9 24.6 0.90 103.68

10 C10 25 1.05 121.40

11 C11 25 0.89 107.60

12 C12 24.7 0.79 93.40

13 C13 24.6 0.58 65.70


Ehab M. Lotfy



14 C14 25 0.96 116.50

15 C15 25 0.89 106.00

16 C16 24.7 0.86 102.10

17 C17 24.6 0.82 98.00

18 C18 25 1.12 130.50

19 C19 25 1.06 119.50

20 C20 24.8 1.03 111.20

21 C21 24.6 0.98 101.52

5.1. Experimental validation

The validity of the proposed analytical model is checked through extensive comparisons

between analytical and experimental results of RC columns under compression load. Figure 8

shows the theoretical and experimental load-deformation curve of from C1 to C4 and control

column.

The theoretical results from finite element analysis showed in general a good agreement with

the experimental values.

Figure 8: The theoretical and experimental load-deformation curve of tested columns from

C1 to C4 and control column.

5.2. Holes dimensions

Figures 9, 10, and 11 show the theoretical load-deformation of columns (C1, C2, C3, C4 and

C5), (C1, C6, C7, C8 and C9) and (C1, C18, C19, C20, and C21); which have hole


Ehab M. Lotfy



dimensions 0.00, 0.10, 0.15, 0.2, and 0.3 of columns length respectively; increasing hole

dimensions decrease the toughness and ductility of tested columns.

From Table 3, it can be seen that, ultimate loads, and ultimate strain of C2, C3, C4 and C5 to

C1 are (99.6, 98.8, 91.6 and 80.5%), and (95.3, 95.3, 95.3 and 91.5%) respectively.

Ultimate loads, and ultimate strain of C6, C7, C8 and C9 to C1 are (97.4, 94.7, 88.5 and

74.5%), and (93.4, 88.4, 84.6 and 69.2%) respectively.

Ultimate loads, and ultimate strain of C18, C19, C20 and C21 to C1 are (93.8, 91.9, 80.2 and

72.8%), and (86.2, 81.5, 79.2 and 75.4%) respectively. Figure 12 shows the effect of the

increasing hole dimensions on the ultimate load of columns resists, where the increasing of

hole dimensions more than 0.15 of tested columns length leads to reduction in ultimate loads

of tested columns to 80%. The increasing of hole dimension more than 0.15 of column length

decrease the toughness and ductility of cross section, where it is increase the buckling effect

of tested column, so it has a significant effect on ultimate strain, and ultimate loads that the

columns resist.

Figure 9: The theoretical load-deformation

of columns C1, C2, C3, C3 and C5




of columns C1, C18, C19, C20, and C21

Figure 12: Ultimate load of tested columns to

control and hole dimensions/col. Length ratio

5.3. Position of holes in columns

Figures 13 and 14 show the theoretical load-deformation of columns (C3, C7, C11, C15 and

C1) and (C4, C8, C12, C16 and C1) respectively; which have position of holes case (a), case


Ehab M. Lotfy



(b), case (c), case (d), and control specimen; holes in the edge third has significant effect on

the ultimate loads and deformations of tested columns, hence affect the toughness of tested

specimens, but holes in middle third has limited effect on the ultimate loads and deformations

of tested columns

From Table 3, it can be seen that, ultimate loads, and ultimate strain of C3, C7, C11, C15 and

C1 are (98.8, 94.8, 77.4 and 76.2%), and (95.3, 88.4, 68.4 and 68.9%) respectively. Ultimate

loads, and ultimate strain of C4, C8, C12, C16 and C1 are (91.6, 88.5, 77.4 and 73.4%), and

(95.3, 84.6, 68.4 and 66.15%) respectively

Figure 15 shows that; hole with case (c ) and (d) has a significant effect on the ultimate load

of tested columns with hole dimensions 0.15 and 0.2 of column length

Figure 16 shows that; hole with case (b), case (c ) and (d) has a significant effect on the

deformation of tested columns with hole dimensions 0.15 and 0.2 of column length





Figure 15: Position of holes and Pu/P

control for hole Dim. (0.15L and 0.2L)

Figure 16: Position of holes and Def./Def.

control for hole Dim. (0.15L and 0.2L)

5.4. Shape of holes

Figures 17 and 18 show the theoretical load-deformation of tested columns (C3 and C19 to

C1) and (C4 and C20 to C1); which confirm that using square hole in tested column has a

significant effect on the ultimate loads and deformation so it decreased the toughness and

ductility of tested columns.


Ehab M. Lotfy



From Table 3, it can be seen that, ultimate loads, and ultimate strain of C3 and C19 to C1 are

(98.8, and 85.9%), and (95.3 and 81.53%) respectively, ultimate loads, and ultimate strain of

C4 and C20 to C1 are (91.6, and 80%), and (97.6 and 79.2%) respectively.

Using square hole in tested column has a significant effect on the behavior of tested columns;

where it reduced the ductility, toughness, ultimate load and increased deformation


of columns C3, C19, and C1


of columns C4, C20, and C1

5.5 Conclusion

The inelastic behavior of 21 columns are investigated in the current study under the effect of

increasing loading employing the inelastic FE analysis program ANSYS. Several parameters

are investigated including the parameters of study were a holes dimensions with diameters

0.1, 0.15, 0.2 and 0.3 of column length, their relative position in columns; in middle third and

edge third , and the shape of holes; circle and square. The study focuses on the consequences

of the investigated parameters on the deformation and ultimate resisting load. The

conclusions made from this investigation are:

1. The theoretical results from Finite Element Analysis showed in general a good

agreement with the experimental values.

2. The hole with diameter more than 0.15 of columns length has significant effect of the

column behavior; reducing the ductility and toughness of tested columns.

3. The increasing of hole dimensions to more than 0.15 of columns length leads to

reduction in ultimate loads of tested columns to 80%.

4. Using square hole in tested column has a significant effect on the behavior of tested

columns

5. Holes can be made in middle third of columns with diameter up to 0.15 column length.

6. References

1. Ashouf A.F. and Rishi G., (1999), Tests of reinforced concrete continuous deep

beams with web openings, ACI structural journal, 97(3), pp 418-426.

2. Tayel M. A., Soliman M. H. and Ibrahim K. A., (2004), Experimental behavior of flat

slabs with openings under the effect of concentrated loads, Alexandria engineering

journal, 43(2), pp 203-214.


Ehab M. Lotfy



3. Simpson D., (2003), The provision of holes in reinforced concrete beams, Concrete

(London), 37(3), pp 24-25.

4. Jiyang Wang, Masanobu SAKASHITA, Susumu Kono, Hitoshi Tanaka, Makoto

Warashina., (2008), A macro model for reinforced concrete structural walls having

various opening ratios, 14 th world conference on earthquake engineering, October 12-

17, Beijing, China

5. Mansur, M.A., (1998), Effect of openings on the behavior and strength of R/C beams

in shear, Cement and concrete composites, Elsevier science Ltd., 20(6), pp 477-486.

6. ANSYS Manual Set, (1998), ANSYS Inc., Southpoint, 275 Technology Drive,

Canonsburg, PA 15317, USA.

7. Installation Guide (2010), ANYSYS – VERSION 10, Computer software for

structural engineering.

8. William, K.J. and Warnke, E.D., (1975), Constitutive model for the Triaxial behavior

of concrete, Proceedings of the international association for bridge and structural

engineering, 19, p 174, ISMES, Bergamo, Italy

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