RC BIAX Software

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RC-BIAX Software

Summary

This program performs analysis and design of RC sections at ultimate state under biaxial bending and axial force according to Saudi SBC and American ACI codes.

Developed by Professor Abdelhamid Charif King Saud University, Civil Engineering Department.

Last edited in October 2009.

Analysis option includes drawing of moment - axial force interaction surface at ultimate state, as well as moment - curvature - stiffness relationships under an increasing load.

Various standard section types as well any user-defined section can be designed or analyzed. Positive or negative moments and axial forces as well as slenderness effects are all considered. Optimal (minimum steel) design is based on an original and powerful fast-converging re-analysis algorithm.

Steel reinforcement is described by bar coordinates. Several generation schemes are available. RC analysis requires bar diameters whereas RC design requires bar ratios relative to the total steel area. The user must first choose "design" or "analysis" option, fix the section type and dimensions, before entering the relevant reinforcement data. Concrete displaced by steel in compression may be considered (default, recommended) or not. Nominal and design P-Mx-My

interaction surfaces are produced with the code compression limit.

Various 2D scans can be obtained (P = Constant, Mx = Constant, My = Constant). A scan for a constant neutral axis angle may also be viewed.

Design option delivers the required optimum steel, the diagrams of strain and stress distributions showing the neutral axis angle and depth. Detailed contributions of concrete and steel bars to nominal and design forces and moments are also delivered.

The interaction surface is not limited to compressive forces but also includes axial tensile forces. It is useful to remind that lateral loading (earthquake, wind) may cause some RC columns to be subjected to axial tension forces.

Combination of loops over material strain and neutral axis depth is used to track all the interaction surface points. This technique is more accurate than the method seeking the interaction points for various levels of axial forces. The latter is particularly inappropriate in the transition zone where variation of the strength reduction factor generates in some cases non-convex design interaction surfaces, thus leading to many interaction points for the same axial force. The strain compatibility method used in RC-BIAX with the adequate evaluation of the strength reduction factor (expressed in terms of steel strain), captures all the nominal and design interaction points, whatever the section anti-symmetry. Such a construction of the interaction surface including all tension controlled cases requires the definition for steel of a fracture (ultimate) strain. This parameter depends on steel grade and varies from some 20 millistrains for high yield steel to over 150 millistrains for mild steel and is not defined in ACI and SBC codes. This omission assumes in fact an infinite value for steel strain, and has no major effect on the tension capacity of the section. It discards however completely all failures by steel fracture which may yet happen even in beam bending with no axial tension. It also overestimates ductility in moment-curvature relations. European and international codes set conservative values for the

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steel ultimate strain ranging from 0.010 to 0.020. An unlimited value (ACI / SBC) is assumed by default but the user can change this and enter any positive value greater than or equal to 0.010. This allows interested users to investigate the effect of this parameter.

With the drawn interaction surface, the user may check any other combination of ultimate axial force & moments. Many load combinations may be checked at the same time. Biaxial checks using approximate methods such as Bresler equation, or the equivalent eccentricity method, may also be performed. This offers research options in the limitations of these approximate methods. Detailed material contribution for particular points or any user-defined point on the interaction surface may be obtained. This option combined with possibilities of analyzing unreinforced sections / using zero-strength concrete offers investigation opportunities in RC material contributions. Other investigation options include effects of section shape, displaced concrete, value of steel ultimate strain and moment-curvature.

Moment-curvature figures can be obtained for any value of the axial force (compression or tension), and any neutral axis angle, using a simplified equivalent rectangular concrete block. Design results are sent to the output file: "design-biax.out"

Interaction surface results are in output file: "inter-biax.out" Combination check results are in file: "check-biax.out" Slenderness check results are sent to file: "slender-biax.out"

The same filenames are used for all runs. Long term saving requires renaming of appropriate output files after the run.

Results of 3D scans and moment-curvature can be sent to any user-defined file.

The software offers original and unmatched features and is presented in compact form with one single executable file.

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THEORETICAL BACKGROUND

In the analysis of RC sections subjected to biaxial bending combined with an axial force, much attention and emphasis have been given to the stress integration techniques. Layer and fiber models [1-7] have been successfully used for over two decades, while boundary integration techniques [8-10] have recently been proven to be better alternatives. Little interest was accorded to iterative strategies or design problems, which remain the most important engineering concern. Non-convexity of design interaction curves and surfaces, in ACI and similar codes, has also not received any focus.

In design problems, the number of unknown parameters is in general greater than the number of equations and solution can only be found by performing successive cycles of trial design and analysis checks using P-M or P-M-M interaction curves or surfaces. Variation of the strength reduction factor in all ACI inspired codes, to account for ductile and brittle behaviors, generate design curves and surfaces which are sometimes non-convex and their tracking requires appropriate iterative procedures. Methods based on axial force level may miss the non-convex parts and deliver inappropriate results.

The present work deals with a systematic analysis method of arbitrary RC sections under biaxial bending combined with an axial force. Compression forces in concrete are determined using the equivalent rectangular block, as allowed by all codes of practice. No layer or fiber modeling or any section subdivision is therefore required. Areas of arbitrary compression blocks and sections as well as centroids and moments of inertia are computed using exact border integrals. Possible holes in sections are easily taken into account. RC design is then performed using a simple and powerful fast-converging re-analysis algorithm.

Analysis of RC sections subjected to biaxial bending and axial force

Biaxial bending moment is a vector with two rectangular components about perpendicular axes. The resulting moment M_x2 M_y2 may have any direction in the plane and the bending direction is defined by the following loading angle:

_        ux uy M M 1 tan  (1) The inclination of the neutral axis is an extra unknown and with the engineering sign convention shown in Figure 1a, the neutral axis angle  and the bending angle  vary in the same direction from zero to 360 degrees. In general the two angles are different. They are however equal if the bending moment is applied about a direction perpendicular to an axis of symmetry. The two angles are fully coincident for an axi-symmetric section such as the circular shape.

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Figure 1: Biaxial bending (a): Inclined neutral axis in global axes

(b): Rotated section and use of variables in local axes

Figure 2: Strains and stresses in local axes

With respect to the sign convention shown in Figure 1a, the nominal force and moments in biaxial bending are given by:







B is the area of the concrete compression block. Xb and Yb are coordinates of the centroid of the compression block with respect to X and Y axes having the origin as the centroid of the gross section. Steel bars are described by their coordinates Xsi and Ysi.

a cu c    su s    c dmin dmax h 0.85 f _c' v1 v2 a c x y xsi ysi h dmin dmax  X Y Mx My Ysi Xsi c a c x y xsi ysi h dmin dmax

5 Interaction surface P-Mx-My

The interaction between the three components P-Mx-My is a three dimensional surface (Figure 3), which can be generated in several ways:

1. From equators obtained as Mx-My interaction curves for various axial force values 2. From meridians corresponding to various bending angles  given by (1)

3. From meridians corresponding to various neutral axis angles 

The first two techniques generate equators or meridians forming a very regular pattern but require multiple axis transformations. Moreover, generation of the interaction surface using equators with various levels of the axial force is not appropriate in the transition zone, which is frequently non-convex. Meridians generated using the neutral axis angle (method 3) are generally irregular but are more efficient as all calculations are performed in one set of local axes. Using small enough angle increments (five degrees), the interaction surface is more accurately and more easily generated by scanning the neutral axis angle from 0 to 360 degrees.

Figure 3: Drawing of the 3D interaction surface Interaction curve meridian

For each value of the neutral axis angle, an interaction meridian curve P-Mx-My is determined using a method similar to unixaial bending.

Calculations are performed in local axes (Figures 1b and 3) before transformation to global axes. Coordinate transformation (for vertices of concrete section and compression block, and for steel bars) are given by:

xXcosYsin and yXsinYcos (5) The section thickness h and the extreme bar positions, for each neutral axis angle, are updated as follows:

6 min

max y

y

h  d_min y_max y_smax d_max  y_max y_smin (6) For each neutral axis angle and each depth, the form of the compression block is tracked and its area computed by exact border line integration. The compression block may have more than one part (Figure 1b) and possible voids in the section are easily accounted for.

As in uniaxial bending, when subjected to a positive or negative moment, and to compression or tension axial force, the nominal capacity of the section will be reached when either of the two materials (concrete or steel) reaches an ultimate state (Figure 2). Either compressive concrete strain is equal to the ultimate value _cu = 0.003 or tension steel strain reaches its ultimate value_su. There is no limitation on tension strain of concrete. Compression failure is controlled by concrete crushing while tension failure is controlled by ultimate steel strain. The design strength is related to the nominal strength through the strength reduction factor. This variable factor is defined by ACI code to account for the fundamental difference between ductile and brittle behaviors. It is related to the tensile steel strain as shown in Figure 4 and cannot be known in advance even in beam bending.

The possible mechanisms of failure when combining a positive or negative bending moment with a compression or tension axial force are represented in Figure 5, with a neutral axis depth c varying from infinity to minus infinity and considering that the top zone of the section is either the most compressed or least tensioned. Four different parts are defined (Figures 5a to 5e) according either to the value of the strength reduction factor or the type of failure. These parts are described by variation in the neutral axis depth or material strain.

 Part 1 (Fig.5a-5b): Neutral axis depth c varying from infinity to balanced conditions. Concrete strain is equal to the ultimate value _cu while steel strain varies until it reaches the yield point. This part corresponds to the compression controlled zone (brittle concrete crushing failure) with a strength reduction factor ₀ (0.65 for tied columns and 0.70 for spiral columns).

 Part 2 (Fig.5b-5c): Tension steel strain varies from yield value y to 0.005. This

corresponds to the transition zone with a strength reduction factor  varying linearly from ₀ to 0.90. Failure is again controlled by concrete crushing.

 Part 3 (Fig.5c-5d): Tension steel strain varies from 0.005 to the ultimate value _su. This corresponds to the tension controlled zone ( = 0.90) with ductile failure controlled by concrete crushing.

 Part 4 (Fig.5d-5e): Concrete strain varies from the compression ultimate value until pure tension where the neutral axis depth is minus infinity. This also corresponds to the tension controlled zone ( = 0.90) but with failure controlled by steel breaking.

7 Tied column: ₀= 0.65

Spiral column: ₀= 0.7

Figure 4: Variation of the strength reduction factor  with tension steel strain_s

Figure 5: Various parts and possibilities of strain distribution (a): Pure compression – (b): Balanced point – (c): 0.005 steel strain

(d): Double failure point – (e): Pure tension

In the first three parts controlled by concrete crushing, variation of the strain and neutral axis depth is performed by rotation about concrete pivot C whereas in the fourth, controlled by steel breaking, rotation is performed about steel pivot S (Figure 5).

In parts 1 to 3, the rectangular concrete block is used with a stress 0.85 f and a depth given by: c'

Compression block depth a₁c with 0ah (7) Coefficient 1 is given by:





Compression control Transition zone Tension controlled zone

Concrete crushing control Steel breaking control

Part 1 Part 2 Part 3 Part 4

90 . 0   0    Strain _s s  = 0.005 y s    Factor Tension control

Compression control Transition (Brittle) Transiti on (Ductile) Transiti on

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In part 4 however, concrete strain _c is less than the ultimate value _cu and the stress block is not rectangular with a value of 0.85f . In this case, we either integrate the parabolic equation or _c'

use a rectangular block with a stress 0.85f_c where fc is the concrete stress related to concrete strain _c by a second degree parabolic equation.

Thus for strain _c

cu c c c f f   '  (9) Exact tracking of the full nominal and design interaction curves requires using an iterative procedure to scan these four parts as well as inverting the section to consider negative moments or more generally cases where the top zone of the section is either the least compressed or the most tensioned.

Parts 1 and 4 may each be subdivided into two subparts by separating the partial tensioned and partial compression case from the case with a section either fully in compression (part 1) or fully in tension (part 4).

Iterations can be performed either using the neutral axis depth c or the material strain. In order to avoid useless points in part 1 and part 4 (with infinite values of c), alternative maximum and minimum values of c are used. They are defined as follows:

In part 1, instead of plus infinity, the neutral axis depth c will vary from a maximum value ensuring that the section is fully in the compression block and that the top steel layer has yielded (if steel yield strain yis less than concrete ultimate strain cu, which in is general always the

case).           y cu cu d h Max c     max 1 max , (10) In part 4, instead of minus infinity, the neutral axis depth c will vary to a minimum value ensuring that the section is fully in tension and that the top steel has yielded.

            y su su d d d Min c    ) ( ,

0 max max min

min (11)

It must be noted that steel ultimate strain _su is not defined in ACI and some other codes, and this omission is in fact equivalent to considering an infinite value, which thus eliminates steel breaking failure. The software developed in this work allows varying _su from 0.01 to infinity (any large number).

Assuming that compression is positive, for each strain distribution, nominal values of axial force and bending moments given by equations (2) to (4) are rewritten here:







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Bar inside compression block (d_i a): F_si  A_si





fsi is the stress in steel bar i given by:

si s

si E

f   with  fy  fsi  fy (13)

The expression of the strain _si in steel bar i, depends on the failure type: For concrete crushing control (_c _cu, parts 1 to 3):

c d c _i cu si    (14a)

For steel breaking control (_s _su, part 4):

c d d c i su si    max   (14b)

It must be noted that in part 4, concrete strength fc given in expression (9) is used in equations (2), (3) and (12b) instead of ultimate strength f . _c'

Generation of the interaction surface

The various meridian curves are then assembled to generate the 3D interaction surface. A hidden face algorithm is used to draw the 3D interaction nominal and design surfaces. Its implementation required subdivision of the interaction surface in many “shell finite elements” and typical results are shown in Figures 6 to 10. Different colors are used to distinguish the various parts. It can be seen in Figure 3 how the nominal and design surfaces for a square section are regular. The meridians obtained by increments of five degrees, are regularly spaced. Various 2D Scans can then be extracted from the 3D surface (for constant value of axial force P or moment Mx or My or neutral axis angle). Figure 7 shows a typical scan at a given value of the axial force generating Mx-My interaction curve which can be viewed in an isometric or plane view. The isometric scans can also be viewed as isolated slices as shown in Figure 8. Analysis results of a dissymmetric section (about horizontal and vertical axes) show irregular meridians as well as non-convexity of the design surface (Figure 9). The design surface viewed from the 45 degrees axis of symmetry shows a more regular pattern. A scan for a given neutral axis depth can also be obtained (Figure 10).

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Design of RC sections subjected to biaxial bending and axial force Design equations in biaxial bending are given by:

u

n P

P 

 M_nx M_ux M_ny M_uy (15) The reinforcement pattern is fixed and only total steel is unknown. The various steel bars are given by their ratios as follows:

A_si A_stR_i with



i i

R 1 (16) The re-analysis strategy used does not scan all the previous four parts. The procedure is confined to a small relevant zone according to the values of ultimate moments and axial force.

The iterative procedure used is presented graphically. The minimum steel required will be reached when equations (15) are satisfied otherwise the loading point is on the border of the safe design region in an optimal way.

As for the total steel area, the distance to the origin is used as an interpolation factor to shoot the solution (Figure 11).

The next total steel section (step k+1) is thus given by:

k u k st k st D D A A 1  (17) This simple acceleration procedure is efficient whether the initial solution is greater or less than the exact solution. The distances Du and Dk (Figure 11) are determined using reduced force and moment:

For the given ultimate axial force, the neutral axis angle is scanned to determine the design point

k having the same bending angle as the loading angle (1). The moments at point k are such that

(Figure 11): ux uy xk yk M M M M  (18) For this point, the distances are again used to obtain the next solution as in (17) until equilibrium equations (15) are satisfied.

The 3D distances used for convergence acceleration are expressed as follows:

2 2 2 y x m m p D   (19) with g A P p g xg x x A I M m  g yg y y A I M m  (20)

Ag and Ig are the area and moment of inertia of the concrete gross section.

Convergence can also be accelerated by confining the neutral axis scan around an appropriate value instead of sweeping from zero to 360 degrees. The loading angle (1) can be used as the reference value. The modified loading angle defined below is however a better alternative as it includes effects of moments of inertia in x and y directions.

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Modified loading angle: _

        yg ux xg uy m I M I M 1 tan  (21)

It was found that the final neutral axis angle was not much different from this modified loading angle (less than five degrees difference) provided both moments are different from zero. It can thus be used as an acceleration scheme in biaxial bending design. The neutral axis angle tolerance used is 0.05 degree. Typical design results are shown in Figures 12 and 13. The L section in Figure 12 is designed for a moment about x-axis only (My = 0) combined with an axial force. It can be seen how section dissymmetry has shifted the neutral axis from the horizontal. Figure 13 illustrates the design of a complex section with holes. The neutral axis angle of 57.1o is close to the modified loading angle of 52.9o.

Figure 11: RC biaxial design by re-analysis

Mx

My

Mux

Muy

Interaction curve k for axial force = Pu

Mxk

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Figure 12: Design of an L section

Figure 13: Design of a complex section with holes in biaxial bending

Other options of RC-BIAX

Moment magnification method for slender columns (braced or un-braced) according to SBC / ACI specifications is fully integrated in the software.

The details are described in RC-TOOL document.

Bresler reciprocal method for checking biaxial method is fully integrated in the software for comparison purposes. Bresler method is a simple and approximate method used by many codes and is described in several codes of practice and textbooks. It has limitations however. RC-BIAX users can investigate these limitations.

18 Conclusions

A systematic method of analysis and design of arbitrary shaped RC sections subjected to axial or biaxial bending combined with an axial force is presented. An efficient design technique based on a simple and powerful re-analysis approach is described. It was successfully used for various types of sections including complex geometries with holes.

The method used allows tracking complex failure surfaces including non-convex parts.

The modified loading angle was found to be a convenient acceleration scheme by searching the neutral axis angle around this value only in biaxial design.

RC-BIAX offers unmatched features for design of RC columns under biaxial bending and integrates both SBC and ACI codes.

References

[1] Virdi KS, Dowling PJ. “Ultimate strength of composite columns in biaxial bending.” Proc Inst Civ En (London) 1973;55(Part 2):251–72. [2] Basu AK, Suryanarayana P. “Reinforced concrete columns under biaxial bending.” Int J Struct 1982;2(3):99–114.

[3] Brondum T. “Ultimate limit states of cracked arbitrary cross sections under axial loads and biaxial bending.” ACI Concr Int 1982;4(11):51–5. [4] Yen JR. “Quasi-Newton method for reinforced concrete column analysis and design.” J Struct Eng, ASCE 1991;117(3):657–66.

[5] JA Rodriguez-Gutierrez. Dario Aristizabal-Ochoa J. “Biaxial Interaction Diagrams for Short RC Columns of Any Cross Section. ” J Struct

Eng, ASCE 1999;125(6):672–83. June.

[6] C.T.T Hsu “Failure surface of structural concrete members”, Proceedings, 8th ASCE Conf. on Electronic Computation, Houston, Feb.1983, pp.671-682

[7] C.T.T Hsu: “RC members subject to combined biaxial bending and tension” ACI Journal, January-February 1986, pp.137-144

[8] Fafitis A. Interaction surfaces of reinforced-concrete sections in biaxial bending. J Struct Eng, ASCE 2001;127(7):840–6.

[9] Romero ML, Miguel PF, Cano JJ. “A parallel procedure for nonlinear analysis of reinforced concrete three dimensional frames.”

Computers and Structures 2002;80(16-17):1337–50.

[10] J.L. Bonet , M.H.F.M. Barros, M.L. Romero. “Comparative study of analytical and

numerical algorithms for designing reinforced concrete sections under biaxial bending”, Computers and Structures, Vol.84, 2006, pp.2184-2193

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USER’S MANUAL

The main RC-BIAX screen is shown in Figure 14.

Figure 14: RC-BIAX main screen

The screen is self-explanatory. The user has to choose either the Analysis or Design option. The appropriate section button must be pushed in order to enter the dimensions. The Reinforcement Data button must then be pushed in order to enter the steel information. Figure 15 shows bar generation in case of a Rectangular Section Analysis with eight bars. The graphical echo is shown in Figure 16. The same bar layout may be generated using corner bar coordinates as shown in Figure 17. Figure 18 shows how a user-defined section with holes may be generated and the graphical echo is shown in Figure 19.

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Figure 15: Bar generation in analysis of a rectangular section with eight bars

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Figure 18: User defined section with two holes

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Figure 20 shows the slenderness screen using ACI/SBC moment magnification factor. The method is used in both planes of bending.