Nonlinear analysis of reinforced concrete columns with holes

Transcription

1 INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume, No, 0 Copright b the authors - Licensee IPA- Under Creative Commons license.0 Research article ISSN Nonlinear analsis of reinforced concrete columns with holes Associate Professor, Civil Engineering Department, Facult of Engineering, Ismaelia, Suez Canal Universit, Egpt ehablotf000@ahoo.com doi:0.6088/ijcser 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 stud aims to examine the amount of strength lost due to the presence of holes in columns. Nonlinear finite element analsis on -column specimens was achieved b using ANSYS software. The nonlinear finite element analsis 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., 0.5, 0. and 0. 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 b the existing models are presented. Results and conclusions ma be useful for designers, have been raised, and represented. Keword: Inelastic finite element analsis, columns, holes, strength and testing of materials.. Introduction In the construction of modern buildings, a network of pipes and ducts is necessar to accommodate essential services like water suppl, sewage, air-conditioning, electricit, telephone, and computer network. Usuall, these pipes and ducts are placed underneath the beam soffit and, for aesthetic reasons, are covered b 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 ma result in the loss of strength, stiffness and ductilit and, hence, significant structural damage ma be sustained, if the provision of the openings is not considered adequatel during the design or construction stages. This is especiall 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., 999), (Tael M. A. et al., 004), (Simpson D., 00), (Jiang Wang et al., 008) and (Mansur, M.A., 998). However, in the case of concrete columns and walls with transverse openings, minimal research has been carried out and, currentl, there is a lack of appropriate design rules. Columns are critical elements, but in general onl carr a fraction of their capacit at normal service loads. Received on Februar 0 Published on March 0 655

2 The research reported in this paper aims to investigate the compressive resistance-capacit of concrete columns with transverse holes with diameters 0., 0.5, 0. and 0. 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 experimentall to evaluate the effect of hole geometr and location. Analsis of the experimental results is used to derive appropriate design recommendations.. Objective of the stud The main objectives of this stud could be summarized in the following points. To investigate the reduction in load carring capacit of the reinforced concrete short columns having circle and square cross-sections with hole in different places.. To model the RC columns using three-dimensional non-linear finite element analsis.. Provide recommendations for the design engineers and the structural codes for the design of the reinforced concrete columns.. 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-capacit. The parameters examined experimentall 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 shows the details of the holes provided in each column. All columns were 600mm height, 00m length and 00mm wide and contained both longitudinal and transverse reinforcement. The longitudinal reinforcement rebars comprised 4#6 mm in diameter, and the transverse reinforcement consisted of shear links, 8mm in 00 mm. A clear concrete cover of 5 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. No Col. No. C Dimension (mm) Table : Details of tested columns specimens f cu (N/mm ) Reinf. Shape of holes Dim. Of holes Position of holes Notes Control specimen C 4 no s 00X00 5 of circle D=60mm Case (a) C 6mm circle D=60mm Case (b) 4 C4 square L =60mm Case (c) 656

3 Figure : Details of reinforcement of tested columns 4. Numerical finite element The analsis is carried out on -RC columns; the parameters of stud were a holes dimensions with diameters 0., 0.5, 0. and 0. 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. 4.. Basic fundamentals of the FE method. The basic governing equations for two dimensions elastic plastic FEM have been well documented, and are briefl 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 geometr of the suggested element, it represents shape factor. The matrix [B] for a triangle element having nodal points, and is given b [ B ] = 0 x x 0 x x 0 x x x x 0 x x Where xi and i represent the coordinates of the node and represents the area of the triangular element, i.e. 0 x x 0 657

4 = det x x x 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]=[D e ]. For plastic elements, the Prandtl-Reuss stress-strain relations together with the differential form of the von Mises ield criterion as a plastic potential leads to [D] = [D p ]. The elastic matrix, [D e ], is given b the elastic properties of the material whereas the plastic matrix, [D e ], is a function of the material properties in the plastic regime and the stress-strain elevation. Obviousl, for two-dimensional analsis [D e ] and [D p ] depend on the stress-strain state, i.e. plane stress versus plane strain. The plastic matrix, [D p ], depends on the elastic-plastic properties of the material and the stress elevation. Comparing [D e ] and [D p ], it can be seen that the diagonal elements of [D p ] are definitel less than the corresponding diagonal elements in [D e ]. This amounts to an apparent (crease in stiffness or rigidit due to plastic ielding. Therefore, the plastic action reduces the strength of the material. III. Element stiffness matrix [K e ] [ K e ] = T [ B] [ D][ B]dv The transpose matrix of [B] is [B] T. In the case of the well-known triangular elements [k] is represented b; T [ K ] = [ B ] [ D ][ B ]V The element volume is V and for a two-dimensional bod equals the area of the element multiplied b its thickness t. IV. The overall stiffness matrix [K] The stiffness matrixes [K e ] 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] 658

5 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 boundar conditions, the nodal displacements are estimated, and consequentl the stress strain field for each element can be calculated. 4.. Material modeling A linear-elastic, isotropic constitutive relation is adopted to describe the behavior of uncracked concrete elements in tension or compression figure and figure. For steel reinforcement, elastic stress-strain behavior was assumed to obe the linear relation of Hook's law described as: e e [ D ]{ ε} = G( υ) [ D ]{ } { σ} = E + ε Where {σ} and {ε} are column matrices of stress σij and εij respectivel, G is the shear modulus; E is the modulus of elasticit and υ is the Poisson's ratio. In the plastic regime the stress-plastic strain; σ-εp, behavior of steel was assumed to obe a simple power law as shown in figure 4 with a strain hardening exponent of 0.0. Figure : Stress-strain relation for plain concrete in tension Figure : Stress-strain relation for plain concrete in compression Figure 4: Stress-Strain relation for steel reinforcement 659

6 4.. Resume about used program Nonlinear Analsis of Reinforced Concrete Columns with Holes The implementation of nonlinear material laws in finite element analsis codes is generall tackled b the software development industr in one of two was. In the first instance the material behaviour is programmed independentl of the elements to which it ma be specified. Using this approach the choice of element for a particular phsical sstem is not limited and best practice modelling techniques can be used in identifing an appropriate element tpe to which an, of a range, of nonlinear material properties are assigned. This is the most versatile approach and does not limit the analst to specific element tpes in configuring the problem of interest. Notwithstanding this however certain software developers provide specific specialised nonlinear material capabilities onl with dedicated element tpes. ANSYS (ANSYS Manual Set, 998) 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., 975) Finite element modeling 4.4. Geometr The details of tested columns were shown in Figure 5 and 6. Analses were carried out on - columns specimens, where all columns had square cross-section with a 00 mm side and 600 mm height, the longitudinal reinforcement rebars comprised 4#6 mm in diameter, and the transverse reinforcement consisted of shear links, 8mm in diameter@00mm, a clear concrete cover of 5 mm was provided in all column specimens 4.4. Element tpes Extensive inelastic finite element analses using the ANSYS program are carried out to stud the behavior of the tested columns. Two tpes of elements are emploed 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,, 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 polmer bar; two nodes are required for this element. Each node has three degrees of freedom, translation in the nodal x,, and z directions. The element is also capable of plastic deformation (ANSYS User s Manual) Material properties Normal weight concrete was used in the fabricated tested columns. The stress-strain curve is linearl elastic up to about 0% of the maximum compressive strength. Above this point, the stress increases graduall up to the maximum compressive strength fc \, after that the curve descends into softening region, and eventuall crushing failure occurs at an ultimate strain Loading and nonlinear solution 660

7 The analtical investigation carried out here is conducted on -RC columns; all columns are raised in vertical position with b 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 analsis, 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 analsis, 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 b straight lines. The curve starts at zero stress and strain. Point No., at 0. fc \ is calculated for the stress-strain relationship of the concrete in the linear range. Point Nos., and 4 are obtained from Equation (), in which is calculated from Equation (). Point No. 5 is at and f c. In this stud, an assumption was made of perfectl 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 stud..()..()..() Figure 5: Finite element mesh for a tpical column model 66

8 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 66

9 Table : Details of tested columns specimens No Col. Dim. f cu Shape of Dim of holes No. (mm) (N/mm Reinf. ) holes Notes C - - Control specimen C 0.L 00*00 5 4#6 mm C 0.5L Case (a) 4 C4 0.L 5 C5 0.L 6 C6 0.L 7 C7 0.5L 00*00 5 4#6 mm Case (b) 8 C8 0.L 9 C9 0.L Circle 0 C0 0.L holes C 0.5L 00*00 5 4#6 mm Case (c) C 0.L C 0.L 4 C4 0.L 5 C5 0.5L 00*00 5 4#6 mm Case (d) 6 C6 0.L 7 C7 0.L 8 C8 0.L 9 C9 0.5L Square 00*00 5 4#6 mm Case (e) 0 C0 0.L holes C 0.L 5. Inelastic analsis results and discussion The parametric studies included in this investigation are holes dimensions with diameters 0., 0.5, 0. and 0. of column length, their relative position in columns; case (a), (b), (c) and (d), and the shape of holes; case (a) and (d). Table shows the analticall results of the ultimate loads, deformations and compressive stress of concrete, respectivel. Table : Theoretical results of tested columns specimens No Col. Concrete stress Ultimate Def. Ultimate Load Notes No. (N/mm ) (mm) (KN) C Control specimen C C C C C C C C C C C C

10 4 C C C C C C C C Experimental validation The validit of the proposed analtical model is checked through extensive comparisons between analtical and experimental results of RC columns under compression load. Figure 8 shows the theoretical and experimental load-deformation curve of from C to C4 and control column. The theoretical results from finite element analsis showed in general a good agreement with the experimental values. Figure 8: The theoretical and experimental load-deformation curve of tested columns from C to C4 and control column. 5.. Holes dimensions Figures 9, 0, and show the theoretical load-deformation of columns (C, C, C, C4 and C5), (C, C6, C7, C8 and C9) and (C, C8, C9, C0, and C); which have hole 664

11 dimensions 0.00, 0.0, 0.5, 0., and 0. of columns length respectivel; increasing hole dimensions decrease the toughness and ductilit of tested columns. From Table, it can be seen that, ultimate loads, and ultimate strain of C, C, C4 and C5 to C are (99.6, 98.8, 9.6 and 80.5%), and (95., 95., 95. and 9.5%) respectivel. Ultimate loads, and ultimate strain of C6, C7, C8 and C9 to C are (97.4, 94.7, 88.5 and 74.5%), and (9.4, 88.4, 84.6 and 69.%) respectivel. Ultimate loads, and ultimate strain of C8, C9, C0 and C to C are (9.8, 9.9, 80. and 7.8%), and (86., 8.5, 79. and 75.4%) respectivel. Figure shows the effect of the increasing hole dimensions on the ultimate load of columns resists, where the increasing of hole dimensions more than 0.5 of tested columns length leads to reduction in ultimate loads of tested columns to 80%. The increasing of hole dimension more than 0.5 of column length decrease the toughness and ductilit 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 C, C, C, C and C5 Figure 0: The theoretical load-deformation of columns C, C6, C7, C8 and C9 Figure : The theoretical load-deformation of columns C, C8, C9, C0, and C Figure : Ultimate load of tested columns to control and hole dimensions/col. Length ratio 5.. Position of holes in columns Figures and 4 show the theoretical load-deformation of columns (C, C7, C, C5 and C) and (C4, C8, C, C6 and C) respectivel; which have position of holes case (a), case 665

12 (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, it can be seen that, ultimate loads, and ultimate strain of C, C7, C, C5 and C are (98.8, 94.8, 77.4 and 76.%), and (95., 88.4, 68.4 and 68.9%) respectivel. Ultimate loads, and ultimate strain of C4, C8, C, C6 and C are (9.6, 88.5, 77.4 and 7.4%), and (95., 84.6, 68.4 and 66.5%) respectivel Figure 5 shows that; hole with case (c ) and (d) has a significant effect on the ultimate load of tested columns with hole dimensions 0.5 and 0. of column length Figure 6 shows that; hole with case (b), case (c ) and (d) has a significant effect on the deformation of tested columns with hole dimensions 0.5 and 0. of column length Figure : The theoretical load-deformation of columns C, C7, C, C5 and C Figure 4: The theoretical load-deformation of columns C4, C8, C, C6 and C Figure 5: Position of holes and Pu/P control for hole Dim. (0.5L and 0.L) Figure 6: Position of holes and Def./Def. control for hole Dim. (0.5L and 0.L) 5.4. Shape of holes Figures 7 and 8 show the theoretical load-deformation of tested columns (C and C9 to C) and (C4 and C0 to C); 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 ductilit of tested columns. 666

13 From Table, it can be seen that, ultimate loads, and ultimate strain of C and C9 to C are (98.8, and 85.9%), and (95. and 8.5%) respectivel, ultimate loads, and ultimate strain of C4 and C0 to C are (9.6, and 80%), and (97.6 and 79.%) respectivel. Using square hole in tested column has a significant effect on the behavior of tested columns; where it reduced the ductilit, toughness, ultimate load and increased deformation Figure 7: The theoretical load-deformation of columns C, C9, and C Figure 8: The theoretical load-deformation of columns C4, C0, and C 5.5 Conclusion The inelastic behavior of columns are investigated in the current stud under the effect of increasing loading emploing the inelastic FE analsis program ANSYS. Several parameters are investigated including the parameters of stud were a holes dimensions with diameters 0., 0.5, 0. and 0. of column length, their relative position in columns; in middle third and edge third, and the shape of holes; circle and square. The stud focuses on the consequences of the investigated parameters on the deformation and ultimate resisting load. The conclusions made from this investigation are:. The theoretical results from Finite Element Analsis showed in general a good agreement with the experimental values.. The hole with diameter more than 0.5 of columns length has significant effect of the column behavior; reducing the ductilit and toughness of tested columns.. The increasing of hole dimensions to more than 0.5 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 0.5 column length. 6. References. Ashouf A.F. and Rishi G., (999), Tests of reinforced concrete continuous deep beams with web openings, ACI structural journal, 97(), pp Tael M. A., Soliman M. H. and Ibrahim K. A., (004), Experimental behavior of flat slabs with openings under the effect of concentrated loads, Alexandria engineering journal, 4(), pp

14 . Simpson D., (00), The provision of holes in reinforced concrete beams, Concrete (London), 7(), pp Jiang Wang, Masanobu SAKASHITA, Susumu Kono, Hitoshi Tanaka, Makoto Warashina., (008), A macro model for reinforced concrete structural walls having various opening ratios, 4 th world conference on earthquake engineering, October - 7, Beijing, China 5. Mansur, M.A., (998), Effect of openings on the behavior and strength of R/C beams in shear, Cement and concrete composites, Elsevier science Ltd., 0(6), pp ANSYS Manual Set, (998), ANSYS Inc., Southpoint, 75 Technolog Drive, Canonsburg, PA 57, USA. 7. Installation Guide (00), ANYSYS VERSION, 0Computer software for structural engineering. 8. William, K.J. and Warnke, E.D., (975), Constitutive model for the Triaxial behavior of concrete, Proceedings of the international association for bridge and structural engineering, 9, p 74, ISMES, Bergamo, Ital 668