Sfakianakis, M.G. Biaxial bending 2001.pdf

Advances in Engineering Software 33 (2002) 227±242

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Biaxial bending with axial force of reinforced, composite and repaired concrete sections of arbitrary shape by ®ber model and computer graphics M.G. Sfakianakis* Structural Engineering Division, Department of Civil Engineering, University of Patras, GR 265 00 Patras, Greece Received 20 August 2001; accepted 28 November 2001

Abstract A new method is proposed for the study of the failure mechanism of reinforced concrete sections of arbitrary shape in biaxial bending with axial force. The procedure is an alternative ®ber model which employs computer graphics as a computational tool for the integration of normal stresses over the section area. In addition to the cases of classic reinforced concrete sections with longitudinal reinforcement steel bars, the method is extended to sections of reinforced concrete structural members repaired by jackets as well as to members with composite steel±concrete sections. Such a computational tool satis®es the needs of nonlinear analysis of reinforced concrete structures as well as the needs of the daily design practice. The method does not include any iterative procedure within its steps and thus it does not have the disadvantage of possible nonconvergence. On the other hand, it is fast and gives accurate results. Four representative numerical applications of the method are presented for the clari®cation of its validity and advantages. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Biaxial bending; Fiber model; Computer graphics; Failure surface; Bounding surface; Design charts; Reinforced concrete jacket; Composite section

1. Introduction The failure of any reinforced concrete cross-section under bending, for a given value of a compressive or tensile axial load, corresponds to the top of the curve of the moment± curvature diagrams. Thus, in the triaxial space which is de®ned by the axial load and the two components of the bending moment, the resulting failure moments for various values of the axial load form a close surface, known as failure or bounding surface of the cross-section. An inner surface, almost similar in shape to the failure surface, is the conventional failure surface, which corresponds to points lower than those on top of the moment±curvature diagrams. These points are de®ned by the design codes by means of prede®ned maximum allowable strains of the most compressive and of the most tensile vertex of the crosssection. Usually, these strains correspond to the yield strain of the two materials (steel and concrete). Meridians and isoload equators of the conventional failure surfaces form the well-known design charts used in daily practice. Such design charts have been constructed by the use of numerical algorithms for the majority of the usual cross-sectional shapes (orthogonal, circular, etc.). For cross-section shapes other than these classical ones, simpli®ed assumptions are * Tel.: 130-61-997-748; fax: 130-61-996-154. E-mail address: [email protected] (M.G. Sfakianakis).

made for the design. On the other hand, for the needs of nonlinear analysis, the detailed knowledge of the failure surface is extremely important since the plastic deformations of a structural element are functions of its load history and of the distance of its load vector from this surface. The `bounding surface' concept was originally developed for metals [6], and then was appropriately applied to soils [5] and concrete [9,10]. Until now, most of the existing models for the nonlinear behavior of reinforced concrete structural elements under normal actions (biaxial bending with axial load) assume the shape of the bounding surface known and described by closed form relationships (i.e. ellipsoid). An exception to this trend, is the model of Sfakianakis and Fardis [20] for orthogonal cross-sections, in which the size and shape of the bounding surface is computed by means of closed form relationships. Further research [11± 15] showed that the shape and size of these surfaces solely depend on section geometry, longitudinal reinforcement amount and the way it is placed in the section. On the other hand, it is quite dif®cult and time consuming to produce closed form relationships for these surfaces, as it was done earlier for the orthogonal section. Thus, numerical procedures are necessary. In this direction, several algorithms have been proposed for the study of a crosssection under biaxial bending with combined normal force [1,2,7,13,19,23]. Alternatively, design equations have been proposed by other researchers. The parameter calibration of

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M.G. Sfakianakis / Advances in Engineering Software 33 (2002) 227±242

Fig. 1. (a) Typical failure surface, (b) neutral axis positions.

these equations has been based on experimental results or results obtained by the use of ®ber models [8,11,12,15,18]. In this paper a new general method, based on computer graphics, is presented for the construction of the full shape and size of the failure surface of reinforced concrete sections of arbitrary shape. The general case is a polygonal (convex or not) cross-sectional shape, which includes openings and reinforced with longitudinal reinforcement steel bars and/or structural steel (composite section). Moreover, the case of repaired structural elements by concrete jackets (full or partial) is included. The main idea of the method is the use of computer graphics capabilities as an indirect computational tool in the classical ®ber method. Thus, any previous numerical problems or large computer storage demands are fully eliminated. 2. Geometrical de®nition of the failure surface The failure surface of an arbitrary cross-section in the N± My ±Mz space, can be de®ned as the geometrical locus of points …N; My ; Mz † which correspond to the ultimate strength of the section. The result is a closed surface which cannot be described by simple relationships of closed form. This surface fully depends on the detailed section geometry, on its reinforcement amount and on the way it is placed inside it. Fig. 1(a) shows meridians of a typical failure surface which correspond to speci®c angle locations u n of the neutral axis in the section plane, with respect to the My axis (Fig. 1(b)). In general, the angles a i and a j of two meridional points which belong to two equators of distance DN between them, are not equal to u n but usually have a small divergence from this value. In other words, a ˆ tan21 …Mz =My † ± un : This means that the meridians are not always plane. This is due to secondary moments that may occur about the axis j ± j

perpendicular to the neutral axis n±n; which passes through the origin of the section Cartesian system Y±Z (Fig. 1(b)). These secondary moments about the axis j ± j may happen because of possible variations of concrete and steel stresses from both sides of this axis. This variation depends on the unsymmetry of the cross-section and on any unsymmetric distribution of the longitudinal reinforcement bars too. For example, if the cross-section is circular and the longitudinal reinforcement bars are distributed uniformly around its perimeter, the failure surface is axisymmetric, and all its meridians are identical. For polygonal cross-sections with 458 rotational symmetry and a uniform distribution of the reinforcement bars, all eight meridians of the failure surface at 458 intervals are also identical, and the assumption of axisymmetry gives a good approximation to the shape of the failure surface. If the cross-section is rectangular and its reinforcement is equally divided among its four sides, the failure surface exhibits symmetry with respect to the four meridional planes M y ˆ 0; Mz ˆ 0; and My ˆ ^Mz : Its four meridians in the former two planes have identical shape, and the same holds for the four others in the latter. A clear example of the effect of unsymmetry (partial or not) is the U-shaped cross-section, of Fig. 2. Almost all the meridians of the failure surface in this section are plane. For example, consider a uniaxial imposed curvature w ˆ wy for un ˆ 08: Although the only expected moment is that of 2My, a secondary moment ^Mz (very small in value) is produced due to the fact that the application point of the resultant force of the material internal forces may not lie on the Z axis. The sign of this secondary moment mainly depends on the compressive zone depth. Thus, even though axis Y is an axis of symmetry, secondary moments about axis Z are inevitable. In general, these secondary moments rotate a little the moment vector M and make it nonparallel to the cross-section neutral axis n±n: Herein, a ± un ˆ 08:

M.G. Sfakianakis / Advances in Engineering Software 33 (2002) 227±242

229

Fig. 2. Secondary moments due to section unsymmetry.

3. Construction of the failure surface Consider the general case of an arbitrary cross-section shape with a reinforced concrete jacket (Fig. 3). The initial cross-section has longitudinal reinforcement steel bars, a structural steel ¯anged section (composite section) and one or more openings. Assume that the two concretes (in the section and the jacket) as well as the three kinds of reinforcement (structural steel, reinforcement bars of initial section and those of the jacket) have different compressive strengths. The concrete cross-section with the opening is described by the coordinates of its vertices in the centroidal coordinate system YGC ±ZGC, where GC is the geometrical centroid of the section. All longitudinal reinforcement bars are described in the same system by their center coordinates and diameter, B. Finally, the structural steel section is described in the same manner as the concrete cross-section. In the following, the notation C2, C1, S2, and S1 refers to the most outer concrete (C) and steel (S) vertices of the total

cross-section, normally to the neutral axis, which are in tension (1) and compression (2) state. If the section does not have a jacket, then points C2, C1, S2, and S1 are those of the initial cross-section. For the jacket case, they are replaced by the points C2J, C1J, S2J, and S1J (Fig. 3). Assuming stress-resultants space is independent of the loading path, the failure surface can be easily constructed for monotonic-proportional loading. Herein, the surface is constructed equator-by-equator. For given values of the axial load N (i.e. a speci®c equator), the angle u n and the position Zn of the neutral axis, increments Dw of curvature are applied (Fig. 3). This increment is controlled by the parameter 1 C2, which is the maximum compressive strain at the extreme and the most compressive vertex of the section. For this angle location, u n, and position, Zn, of the neutral axis, it moves normally to its direction at small steps of ^Dd until it ®nds a position for which the internal axial load is in equilibrium with the external one. For this ®nal position, Zn ^ Dd; primary moment My and secondary

Fig. 3. Strain and stress diagrams for imposed curvature.

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M.G. Sfakianakis / Advances in Engineering Software 33 (2002) 227±242

moment Mz (if any) are computed. Axes Y and Z are parallel and perpendicular to the neutral axis n±n; respectively. Finally, moments My-GC and Mz-GC are computed by a simple rotational transformation between systems Y±Z and YGC ± ZGC. The procedure is repeated for all possible neutral axis positions Zn, starting from a location of Zn # ZC2 (all sections in tension), until a position Zn $ ZC2 (all sections in compression). Intermediate values ZC2 , Zn , ZC1 cover the case where the neutral axis lies inside the cross-section. This range of Zn covers the complete range N of the axial load. For each value of Zn, u n is given in the range 0±3608 at prede®ned steps Du n (i.e. Du n ˆ 158). With this procedure, the full moment±curvature diagrams …M±w† are obtained. The top of these diagrams corresponds to points of the failure surface (Fig. 4). Thus, the failure surface is constructed point-by-point considering a large number of M±w diagrams and then taking the maximum (top) moment values of them. For the case of the conventional failure surface, according to the codes, the earlier procedure is accelerated by omitting the construction of the M±w diagrams and applying directly the prede®ned values of strains 1C2 and 1S1 (which correspond to prede®ned curvatures) as they are described by the codes. The range of Zn and u n values is taken into account as previously. If the section has one or more axes of symmetry, then only a part of the complete u n range is used. In the following sections, the assumptions and material properties, as well as the procedure for the computation of the internal moments and axial load are described in detail. 4. Assumptions and material properties The proposed method is based on the following ®ve assumptions: 1. According to the Bernouli±Euler assumption, plane

2.

3.

4.

5.

sections before deformation remain plane after deformation. Thus, in the Y±Z local system, the strain at any point of the cross-section, with coordinates …y; z†; is a linear function of its perpendicular to the neutral axis distance z (Fig. 3). The compression stress±strain relationship used for concrete is that given in Fig. 5(a). The ascending part of this curve, until the maximum strength, is represented by a parabola, while the descending part, until the limit strain 1 cu is represented by a straight line with slope Z p. Alternatively, for the conventional surface, the Eurocode 2 [3] uses simpli®ed stress±strain relationships. In both cases, the tensile strength of concrete is neglected. The structural steel (if any) and the steel reinforcement bars are assumed to be elastic until the yield strain 1 y, and perfectly plastic for strains between 1 y and the hardening strain or until the limit strain 1 su, according to the Eurocode 2 [3] (Fig. 5(b)). The failure surface corresponds to strains 1 cu of the most outer compressed vertex of the section, which in turn correspond to the top of the moment±curvature …M±w† diagrams. The conventional failure surface of the EC2 [3] corresponds to prede®ned strains 1 C2 and 1 S1, for the most outer compressed and tensioned vertices, respectively. Strains and stresses are taken into account with their signs (negative for compression and positive for tension).

The s ± 1 law for concrete is that proposed by Tassios [21]. The parabolic and the linear part of this law are given by the relations ! 1c 1c s c ˆ bc fc p 2 2 p ; 1c $ 1p0 …1a† 10 10 
 s c ˆ bc fc 1 1 zp 1c 2 1p0 ;

Fig. 4. Typical moment-curvature diagrams, M± w .

1c , 1p0

…1b†

M.G. Sfakianakis / Advances in Engineering Software 33 (2002) 227±242

231

Fig. 5. Material s ± 1 laws for: (a) concrete, and (b) steel.

where

s s ˆ fy ;

1p0 ˆ b2c 1co ; zp ˆ

1co ù 20:002

bc 2 0:85
bc 0:10avw 1 0:0035 1 1p0

…1c† …1d†

The increase in the concrete strength due to con®nement by the transverse reinforcement is taken into account through the coef®cient b c. This coef®cient is a function of the mechanical volumetric ratio v w of stirrups (the total volume of con®ning stirrup legs times their yield strength fyw divided by the volume of the con®ned concrete core times its uniaxial compressive strength fc), and the effectiveness factor a, which takes into account the total number of bars that belong to stirrup corners. More speci®cally [21]

bc ˆ 1 1 avw

…1e†


lw Asw fyw sAc;core fc

P

vw ˆ

…1f†

where fyw is the yield stress of the transverse reinforcement, s is the clear space between the stirrups, Ac;core is the concrete core area inside the stirrups, Asw is the section area P of a stirrup, fc is the concrete nominal strength, and lw is the total length of hoops appearing in a section. In this way, the total amount of the transverse reinforcement as well as the way of the distribution of the longitudinal reinforcement bars are taken into account. The longitudinal reinforcement amount is expressed via the mechanical reinforcement percentage v tot

vtot ˆ rtot

fy A fy ˆ s fc Ac fc

…2†

where r tot is the geometrical reinforcement percentage, fy the yield stress of the longitudinal reinforcement, As the longitudinal reinforcement area, and Ac the cross-section area. The stress±strain law for the steel is given by the following relationships:

s s ˆ 1 s Es ;

u 1s u # u 1y u

…3a†

u1s u . u1y u

…3b†

Coming to the case of the conventional failure surface, Eurocode 2 [3] speci®es the design material strengths fcd and fyd, for concrete and steel, respectively, de®ned as follows: fcd ˆ

fc ; gc

fyd ˆ

fy gs

…4†

In Eq. (4), g c and g s are safety factors, usually having the values 1.50 and 1.15, respectively. Thus, in Eqs. (1a), (1b), (2) and (3b), fc and fy are replaced by their design values fcd and fyd, respectively. The coef®cient b c takes the role of a reduction multiplication factor of the concrete strength and takes the value of 0.85. For this value, the Z p slope of Eq. (1d) becomes zero and the descending part of the concrete s ± 1 law falls into the horizontal segment de®ned by Eurocode 2 [3]. In this case the limit strain 1 cu takes the value of 20.0035. It is clear, that Eurocode 2 neglects the increase in the concrete compressive strength due to con®nement. The same code speci®es the value of 0.020 for the steel strain limit 1 su. These strain limits are valid for the cases where all the sections are in tension or the neutral axis lies inside the section. For the case where the section is totally under compression, Eurocode 2 [3] reduces the values of these limits by dictating the strain pro®le to pass through a point which lies at a distance 3h=7 from the most compressed vertex of the concrete section and has the prede®ned strain 1 co < 20:002: The distance h, is the concrete section height measured normally to the neutral axis, and equals to ZC1 2 ZC2 (Fig. 6). 5. Graphic representation and computations The main idea of the proposed method comes from the ®ber method. Thus, the proposed algorithm for constructing the full moment±curvature diagrams starts with the de®nition of the ®ber mesh of the total cross-section, independent of the location of the reinforcement bars and/or structural steel inside it. According to the classic ®ber method, one has

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M.G. Sfakianakis / Advances in Engineering Software 33 (2002) 227±242

Fig. 6. Eurocode 2 strain speci®cations for cross-section in compression.

to divide the whole section into a large number of squared (preferably) ®nite elements (the ®bers). The number depends on the desired accuracy. The computational disadvantage of this procedure is the large memory storage demands, especially in cases where the perimeter of the section is an arbitrary and sometimes nonconvex polygon. This disadvantage, in correlation with the high computational cost, makes the nonlinear models that incorporate the classic ®ber method not attractive for analyses of large structures. Herein, a new graphics method is proposed which fully eliminates the aforementioned disadvantages. According to this method, the grid mesh may consist of the picture-elements (pixels) of the computer monitor or the corresponding map of them on the graphics display card of the computer. Fig. 7(a) shows a scaled layout of the composite section of Example 2, which will be described later. Thus, the total section consists of a large number of pixels. Fig. 7(b) shows a magni®ed detail of the pixel grid. The whole graphic procedure consists of the following steps, which incorporate simple programming of computer graphics: (1) For a graphical user interface system (i.e. Microsoft Windows), a squared window is created with a selected resolution. For reasons that will be explained later and seen in Section 8.2, the resolution need not be high. A value of 300 £ 300 pixels is enough for usual sections. Larger resolutions are required for larger cross-sections, such as bridge box sections, etc. (2) For a given angle u n of the neutral axis, all of the section coordinates (including steel bars and/or structural steel) are simply rotationaly transformed from the YGC ±

ZGC system to the Y±Z system. The latter has the Y axis parallel to the neutral axis n±n (Fig. 3). The transformation is performed by a simple rotation of the system YGC ±ZGC by an angle 2u n, i.e. #" # " # " YGC Y cos un 2sin qn …5† ˆ Z sin un cos un ZGC (3) System Y±Z is attached to the created window and its origin is placed at the center of that window. The relation between the lengths of axes Y, Z and the selected resolution, de®nes the design scale. Each pixel is then considered of having equivalent dimensions dy ˆ dz: (4) The section is designed in the de®ned scale using different colors for each material. Because of the rotational transformation of step 3, the neutral axis n±n and the corresponding Y axis will always be parallel to the horizontal dimension of the computer's monitor. For the next step, the whole section is considered to consist of horizontal `ribbons' of pixels parallel to the Y axis (Fig. 7(c)). (5) The obtained color picture on the monitor is scanned (optical recognition) ribbon-by-ribbon retaining for each pixel its own color, for recognizing the kind of the material and the physical coordinates (on the monitor's physical coordinate system), which are directly transformed to coordinates of the Y±Z system. This transformation is performed by calling appropriate subroutines or functions of the programming language library that is being used. Since all pixels of a ribbon are under the same strain value, pixels of each material of this ribbon will have the same stress. Hence, for each ribbon, during the scanning process, the

M.G. Sfakianakis / Advances in Engineering Software 33 (2002) 227±242

233

Fig. 7. Sample of pixel grid map of a section: (a) composite section, (b) magni®ed detail, and (c) further magni®cation.

position of the resultant force of each material is being computed simply by taking area moments about the Z axis. Finally, for each ribbon the y-coordinate of the resultant force of each material, the common z-coordinate and the number of pixels per material are kept in different onedimensional arrays, i.e. T

‰Ym Š ˆ ‰ y1 ‰ZŠT ˆ ‰ z1 ‰Nm ŠT ˆ ‰ N1

y2 z2

¼ yr Š ; T

y3 z3

N2

¼ N3

zr ŠT ; ¼

m ˆ 1; 2; ¼; no: of materials;

Nr ŠT ;

…6†

r ˆ no: of ribbons

In this way the overall cross-section problem is reduced to a problem of concentrated points at speci®c locations …y; z† and speci®ed stress, strain and area. Fig. 8 shows the positions of the resultant forces for each ribbon and material, for the cross-section of Example 2 for un ˆ 08: (6) Because of the squared nature of pixels, it is obvious that further criteria need to be speci®ed in order to eliminate any possible loss of accuracy of the area values of each material. This loss is mainly focused on the value of the area of the circular reinforcement bars. A simple but very effective idea to overcome this problem is the recomputation of the equivalent pixel dimensions dy; dz since, the exact value of the area Am of each material and the total

number of the pixels constituting this area are known v u A m dy ˆ dz ˆ u u iˆr uX t N iˆ1

…7†

i

In summary, the described graphic procedure is performed only once for each angle location u n of the neutral axis. Although this graphic-computational procedure is extremely fast, it can be more accelerated if the graphical result is not displayed in the monitor. In this case, the required computer time is only that of the graphic processor. The resulting and very small arrays of Eq. (6) are then used for the computation of moments about the Y and Z axes as it is described in Section 6. 6. Computation of normal actions and curvature As stated in Section 3, for a given value of the external axial load N (i.e. a speci®c equator), the neutral axis is rotated in the plane of the section in prede®ned angle steps Du n (Fig. 3) in order to cover all possible directions of the moment. For every current angle location of the neutral axis, the strain 1 C2 of the most outer compressed ®ber is controlled by giving successive strain increments in the range …0; 1cu †: Then, for every strain value 1 C2, the neutral axis moves step-by-step normally to its direction in

Fig. 8. Position of ribbon resultant forces for: (a) concrete, (b) steel reinforcement bars, and (c) structural steel.

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M.G. Sfakianakis / Advances in Engineering Software 33 (2002) 227±242

the cross-section plane until it ®nds a unique position for which the sum of the internal normal stresses becomes equal to the external given axial load N. A tolerance is speci®ed for reaching this axial load equilibrium. When the equilibrium is obtained, the moment components are computed. In this way, for a speci®c value of the axial load N, one can obtain the full record of the moment values, where everything corresponds to a value of the strain 1 C2. After the determination of the depth Zc of the compressive zone and the corresponding value of 1 C2, one can ®nd the curvature value:

wˆ

1C2 Zc

…8†

Thus, the full moment±curvature diagram for a speci®c value of the axial load N can be constructed. Because of the linear nature of the strain pro®le, the value of the strain at any ®ber-point …y; z† is a linear function of its z-coordinate, i.e.

1…z† ˆ

8 z 2 zn > > < 1C2 zC2 2 zn

z 2 zC2 > > : 1S1 zS1 2 zC2

…9†

1

Z ACJ

s CJ …y; z†dACJ 1

ns X iˆ1

s C …y; z†dAC

asi s si 1

…10†

Z Ast

s st …y; z†dAst

Eq. (10) refers to the general case of Fig. 3, where AC is the clear (without any reinforcements) concrete area of the initial cross-section, ACJ is the corresponding clear concrete area of the jacket, asi is the area of each reinforcement bar and Ast is the structural steel area in the initial cross-section. Stresses s C, s CJ, s si and s st are the corresponding material stresses. Similarly, the moments My and Mz about axes Y and Z are given by: M y;int ˆ My;C 1 My;CJ 1 My;s 1 My;st ˆ

Z AC

1

zs C …y; z†dAC 1

ns X iˆ1

zsi asi s si 1

Z

Z Ast

ACJ

AC

1

ys C …y; z†dAC 1

ns X

ysi asi s si 1

iˆ1

zs CJ …y; z†dACJ

zs st …y; z†dAst

…11†

Z

Z Ast

ACJ

ys CJ …y; z†dACJ

ys st …y; z†dAst

…12†

Because of the geometrical reduction of the problem, as it was stated in step 5 of Section 5, the computation of the integrals of Eqs. (10)±(12) is simpli®ed by the use of Eq. (9) as follows: Nint ˆ NC 1 NCJ 1 Ns 1 Nst ! iˆr iˆr iˆr iˆr X X X X 2 ˆ dy NCi s Ci 1 NCJi s CJi 1 Nsi s si 1 Nsti s sti iˆ1

iˆ1

iˆ1

iˆ1

…13† My;int ˆ My;C 1 My;CJ 1 My;s 1 My;st iˆr X

1

iˆr X iˆ1

Thus, from Eq. (9) one can compute the strains of all ribbons in the section for given values of the control parameters 1 C2 or 1 S1. These strains are then replaced in Eqs. (1a), (1b), (3a) and (3b) in order to compute the pixel stresses of each material and for each ribbon. Computationally, for a given value of 1 C2 and a speci®c location (w , Zc) of the neutral axis, the equilibrium of the internal stresses with the external load N is given by the integrals of the normal stresses of each material over the total cross-section,

AC

Z

iˆ1

if zn ˆ zC2

Z

ˆ

ˆ dy2

if zn ± zC2

Nint ˆ NC 1 NCJ 1 Ns 1 Nst ˆ

Mz;int ˆ Mz;C 1 Mz;CJ 1 Mz;s 1 Mz;st

zCi NCi s Ci 1 !

iˆr X iˆ1

zCJi NCJi s CJi 1

iˆr X iˆ1

zsti Nsti s sti

zsi Nsi s si (14)

Mz;int ˆ Mz;C 1 Mz;CJ 1 Mz;s 1 Mz;st ˆ dy2

iˆr X iˆ1

1

iˆr X iˆ1

yCi NCi s Ci 1 !

iˆr X iˆ1

yCJi NCJi s CJi 1

ysti Nsti s sti

iˆr X iˆ1

ysi Nsi s si (15)

Index m of Eq. (9) takes the values C, CJ, s and st for the four different materials, respectively. The ®nal step is the computation of the moments of the desired Cartesian coordinate system. The expression of the moments in the YGC ±ZGC system is given by the following simple rotational transformation: " # " #" # My My cos un sin qn ˆ …16† Mz YGC ±ZGC 2sin un cos un Mz Y±Z The maximum permissible axial load limits are those which correspond to pure tension or compression of the section. In these cases, the neutral axis tends to in®nity. The following relations give these axial load limits for compression and tension, respectively: N 2 ˆ AC fC 1 ACJ fCJ 1 As fs 1 Ast fst

…17†

N 1 ˆ As fs 1 Ast fst

…18†

In Eqs. (17) and (18) various strengths are taken into account by their signs, while the areas are the clear ones for each material. Because of the section unsymmetries that have been mentioned, the application points of the forces of Eqs. (17) and (18) may not coincide with the origin of the Cartesian system in which bending moments are referred. In

M.G. Sfakianakis / Advances in Engineering Software 33 (2002) 227±242

this case, secondary bending moments are produced. This means that the points of the axial load limits always have no zero moment components. Some exceptions to this general rule are the orthogonal cross-section with equally distributed reinforcement along the four sides, or the circular and circular-ring sections with equally distributed reinforcement along their perimeter. It is worth mentioning that some numerical algorithms in the literature exhibit the problem of nonconvergence. Most of those algorithms compute the moments about a Cartesian system whose origin coincides with the plastic center PC (Fig. 3) of the section, which is the point of application of the axial forces of Eqs. (17) and (18). Because of its nature, the present method can compute moments at any desired Cartesian system. The coordinates of the plastic center of a section are computed by the following relations: YPC ˆ

YC AC fC 1 YCJ ACJ fCJ 1 Ys A s fs 1 Y st Ast fst AC fC 1 ACJ fCJ 1 As fs 1 Ast fst

…19†

ZPC ˆ

ZC AC fC 1 ZCJ ACJ fCJ 1 Zs As fs 1 Zst Ast fst AC fC 1 ACJ fCJ 1 As fs 1 Ast fst

…20†

Usually, for reasons of generality, axial load N, and bending moments My and Mz are normalized according to the following relations:

nˆ

N ; Ac;tot fc

my ˆ

My ; Ac;tot hz fc

mz ˆ

Mz Ac;tot hy fc …21†

235

In Eq. (19), Ac;tot is the total section area including any reinforcement and/or openings, while hy and hz are the total section heights parallel to axes Y and Z, respectively. It is reminded that for the case of the conventional failure surface, the strength fc is replaced by its design value fcd according to Eq. (4). For this case, and for a section without concrete jacket and structural steel reinforcement, the normalized limit axial loads of Eqs. (17) and (18), with the aid of Eqs. (2), (4) and (19), become ! fyd fcd 2 ; 1co # 21y ˆ 2 n ˆ 20:85 2 vtot 1 2 0:85 fyd Es …22†

n1 ˆ vtot

…23†

7. Computer program development A computer program, named biax, which incorporates the procedure described in this paper, has been developed. The program is written in Visual Fortran-90/95 [4] and can be used for any arbitrary cross-section shape. Its speci®cations, besides the formation of the failure surfaces and the design charts, include capabilities for parametric studies based on the section moment±curvature diagrams (i.e. curvature ductility factors). Moreover, it can be used in the design practice for the analytical study of a given cross-section with its reinforcements. In particular, for the

Fig. 9. s ± 1 diagrams by the biax computer program.

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Fig. 10. Eurocode 2 design charts.

Fig. 11. Eurocode 2Ðdesign failure surfaces.

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8. Examples In order to illustrate the proposed method and its accuracy and ef®ciency, four representative examples have been worked out and the results compared with existing ones in the literature. 8.1. Example 1

Fig. 12. Composite concrete cross-section.

cases of jackets for repairing damaged beam or column members, the existence of such a computational tool is extremely useful. By the use of this program design charts for L, T and U-shaped sections, as well as other complicated sections in addition to the classic orthogonal, circular, etc. sections, are being constructed (Sfakianakis, in preparation). The design charts include the cases were these section shapes are reinforced columns or shear walls. These design charts are then introduced in a database for fast design of a section. This database has been designed in an explicit way that it can be linked to the most popular computer packages for analysis and design of concrete structures. As shown in Fig. 9, the program optionally displays the complete strain and stress diagrams during the computations, if the user so desires. Many options are included in order to be a user friendly program.

The developed program is used for the construction of the Eurocode 2 [3] design charts for a rectangular cross-section with the total reinforcement amount placed equally at the two opposite sides of the section. Fig. 10(a) shows the complete design charts for uniaxial bending, and Fig. 10(b) shows one quarter of the equator at n ˆ 20:20; for biaxial bending. Fig. 11 (which has been drawn using the mathematical software [16,22]) shows two different views of one quarter of the conventional failure surfaces for biaxial bending with un ˆ 0±908 at steps of 158, and for vtot ˆ 0:00= 0:50=1:00=1:50=2:00: Due to the unsymmetry in the range un ˆ 15±758; the corresponding meridians to these angles are not plane. The accuracy of the results is identical to that of Eurocode 2 [3] and for this reason the curves shown in Figs. 10 and 11 correspond to both the present results and those of the Eurocode 2 [3]. 8.2. Example 2 This is a numerical example studied by Chen et al. [2]. The composite steel±concrete cross-section of Fig. 12 consists of the concrete matrix, 15 reinforcement bars of diameter 18 mm, a structural steel element and a circular opening. The material properties are as follows: Characteristic strengths for concrete, structural steel and reinforcement bars are fc ˆ 30 MPa, fst ˆ 355 MPa and fs ˆ 460 MPa, respectively. These characteristic strengths are reduced by dividing them with the corresponding safety

Fig. 13. Eurocode 2ÐIsoload contour My ±Mz and conventional failure surface.

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Fig. 14. Eurocode 2ÐMeridians of conventional failure surface at Mz ˆ 0:

factors gc ˆ 1:50; gst ˆ 1:10 and gs ˆ 1:15: The section is analyzed by the use of Eurocode 2 [3] s ± 1 laws as described in Section 4. Fig. 13 shows the My ±Mz interaction curve for a value of the axial load N ˆ 24120 kN and the complete conventional failure surface. The computations have been done using b c ˆ 0:85: For this value of b c the present method gives the inner interaction curve with the dashed line. The values of Ref. [2] are recovered by the present method for bc ˆ 1:00: The same results are obtained by using an older classic ®ber model for biaxial bending. The bending moments in this example are

computed about axes y±z; which pass through the plastic center of the cross-section. As it was stated in Section 5, some different resolution values have been selected for this section, in order to clarify the minimum resolution demands of the model. Fig. 14 shows meridians of the conventional failure surface at Mz ˆ 0; un ˆ 0; 1808. The window graph areas were of resolution 50 £ 50 (extremely small), 100 £ 100 and $ 200 £ 200 pixels (the shaded area). It is clear that the two small resolutions converge to the correct solution, which corresponds to

Fig. 15. Reinforced concrete cross-section with jacket.

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Fig. 16. Strain pro®les used in analysis.

values greater than 200 £ 200 pixels, with very small discrepancies about it. Finally, for usual sections, a minimum resolution of 300 £ 300 pixels is recommended and may be thought of as the golden rule for combining maximum accuracy with minimum computing time. 8.3. Example 3 Consider the rectangular section in Fig. 15 in which a jacket has been constructed for repairing purposes. Detailed dimensioning is shown in Fig. 15. The minimum reinforcement requirements as mentioned in Ref. [17] have been considered. The concrete compressive strength of the initial section is approximately fc ˆ 16 MPa and its four reinforce-

ment bars are of S220 steel grade. The corresponding strengths for the concrete jacket are fc ˆ 25 MPa and the reinforcement is of S400 steel grade. For this composite section, the conventional failure surface has been determined. Moreover, a detailed study has been done, which includes the cases of relative slip between old and new concrete interface and the case where the assumption of plane sections is slightly violated. Both cases are considered on an empirical basis and the scope herein is to see how they affect the results obtained by omitting them. Fig. 16 shows the assumed strain pro®les for the considered cases. For the cases where a relative slip exists (cases 3 and 4), it is assumed that the jacket box section and

Fig. 17. Meridians for uniaxial bending.

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Fig. 18. Meridians for biaxial bending.

the initial (core) section have a common neutral axis. Moreover, the maximum strain deviations from the ®rst case (plane section) are empirically assumed to be about 7%, while in the second and fourth case, the deformed shape of the strain pro®le is obtained by the use of an empirical sinusoidal function. The strain pro®les of Fig. 16 have been applied for the two uniaxial bendings and a biaxial bending with un ˆ 458: Note, that the diagonal bending of the initial section is at 49.48, and that of the composite section at 48.48. Figs. 17 and 18 show the meridians of the conventional failure surfaces of the two sections, for the considered bending conditions. From these graphs, it is clear that the relative slip, which sometimes may occur at the interface of the two concretes, can be thought of as an effect of minor importance in the ¯exural strength of reinforced concrete sections. The opposite phenomenon occurs for the effect of nonplanar sections. In this case, the ¯exural strength is slightly increased in a region of the meridians which corresponds to neutral axis inside the section. As it is expected, for the section in this example, this small increase in the ¯exural strength is observed mainly in the case of biaxial bending where the moment arms of the internal forces are increased in comparison with those of uniaxial bending.

20:0038; g ˆ 0:15; and fy ˆ 357:15 MPa. For this example, the descending part of the concrete s ± 1 law of Eq. (1b) is replaced by the following relation:    1 2 1co s c ˆ fc 1 2 g c 1cu 2 1co Fig. 20 shows the required interaction curve which is in excellent agreement with that of Ref. [19] and in quite good agreement with the experimental data of Ref. [11]. The complete failure surface of this section (Fig. 20) is computed for neutral axis angle increments Dun ˆ 158: Fig. 21 shows the normalized values of isoload contours of this surface viewed from top to bottom with reference to the equator at N ˆ n ˆ 0 (shaded area).

8.4. Exampe 4 This example concerns the L-shaped column section of Fig. 19. For this section Hsu [11] presented experimental and analytical studies, and later, Rodriguez and AristizabalOchoa [19] numerical solutions. The problem here is the determination of the biaxial interaction curves, and for un ˆ 458 using the centroidal axes. It is assumed that fc ˆ 24:13 MPa, bc ˆ 1:00; 1co ˆ 20:002; 1cu ˆ

Fig. 19. Reinforced concrete L-shaped section.

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Fig. 20. Meridian of biaxial bending and failure surface.

9. Conclusions A new method based on the ®ber model concept and computer graphics is proposed for the study of arbitrary reinforced concrete cross-sectional shapes under biaxial bending with axial load. The proposed method is in general, fast, stable, accurate and does not include any numerical procedures, which sometimes fail in accuracy or become unstable, under certain circumstances. The method can

analyze any cross-sectional shape including the case of composite sections, sections with reinforced concrete jackets and sections with openings. Because of these advantages, it can be successfully used for both the nonlinear analysis and the design of concrete beams and structures. Due to the fact that the integration procedure is fully independent of the shape of the stress distribution, and the shape of the section of each material, one can use any other stress± strain laws for steel and concrete. Moreover, the results of

Fig. 21. Isoload contours (equators) of the complete failure surface: (a) bottom view, (b) top view.

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the method as applied to various examples were found to be in excellent agreement with those in the literature. Acknowledgements The author is grateful to Professors D.E. Beskos and D.D. Theodorakopoulos for helpful discussions and suggestions during the course of this work. References [1] Al-Noury SI, Chen WF. Behavior and design of reinforced and composite concrete sections. J Struct Div, ASCE 1982;108(6):1266±84. [2] Chen SF, Teng JG, Chan SL. Design of biaxially loaded short composite columns of arbitrary section. J Struct Engng, ASCE 2001;127(6):678±85. [3] Concrete structures. Euro-design handbook. Ernst and Son, ed., 1995. [4] COMPAQ Visual Fortran, ver. 6.1A. [5] Dafalias YF. A model of soil behavior under monotonic and cyclic loading conditions. Transactions of Fifth International Conference on Structural Mechanics and Reactor Technology. K1/8, Berlin, 1979. [6] Dafalias YF, Popov EP. A model of nonlinearly hardening materials for complex loading. Acta Mech 1975;21:173±92. [7] De Vivo L, Rosati L. Ultimate strength analysis of reinforced concrete sections subject to axial force and biaxial bending. Comput Meth Appl Mech Engng 1998;166:261±87. [8] El-Tawil S, Deierlein GG. Nonlinear analysis of mixed steel± concrete frames. II. Implementation and veri®cation. J Struct Engng, ASCE 2001;127(6):656±65.

[9] Fardis MN, Alibe B, Tassoulas JL. Monotonic and cyclic constitutive law for concrete. J Engng Mech, ASCE 1983;109:516±36. [10] Fardis MN, Chen ES. A cyclic multiaxial model for concrete. J Comput Mech 1986;1(4):301±15. [11] Hsu CTT. Biaxially loaded L-shaped reinforced concrete columns. J Struct Engng, ASCE 1985;111:2576±629. [12] Hsu CTT. T-shaped reinforced concrete members under biaxial bending and axial compression. ACI Struct J 1989;86:460±8. [13] Marin J. Design aids for L-shaped reinforced concrete columns. ACI J 1979;November:1197±216. [14] Munoz PR, Hsu CTT. Behavior of biaxially loaded concrete-encased composite columns. J Struct Engng, ASCE 1997;123(9):1163±71. [15] Munoz PR, Hsu CTT. Biaxially loaded concrete-encased composite columns: design equation. J Struct Engng, ASCE 1997;123(12):1576±85. [16] Papadakis KE. Tziola A, editors. A guide to mathematica, 1st ed. Greece: Thessaloniki, 2000 (in Greek). [17] Penelis GG, Kappos AJ. Earthquake-resistant concrete structures. E and FN Spon ed., 1997. [18] Ramamurthy LN, Hafeez Khan TA. L-shaped column design for biaxial eccentricity. J Struct Engng, ASCE 1982;109(8):1903±17. [19] Rodriguez JA, Aristizabal-Ochoa JD. Biaxial interaction diagrams for short RC columns of any cross-section. J Struct Engng, ASCE 1999;125(6):672±83. [20] Sfakianakis MG, Fardis MN. Bounding surface model for cyclic biaxial bending of R/C sections. J Engng Mech, ASCE 1991;117(12):2748±68. [21] Tassios TP. Con®ned concrete constitutive law. Justi®cation Note No. 13, Background Docs for Eurocode No 8, Part 1, vol. 2. Committee of the EC, DGIII/8076/89 EN, 1988. [22] Wolfram S. The mathematica book. 4th ed. Wolfram Media Inc, Cambridge University Press, USA, 1999. [23] Yau CY, Chan SL, So AKW. Biaxial bending design of arbitrary shaped reinforced concrete column. ACI Struct J 1993;90(3):269±78.