Bi Axial

ORIGINAL ARTICLE

Optimal reinforcement of RC columns for biaxial bending

Luisa Mara Gil-Martn Enrique Hernandez-Montes

Mark Aschheim

Received: 4 May 2009 / Accepted: 7 December 2009 / Published online: 15 December 2009

RILEM 2009

Abstract The Reinforcement Sizing Diagram

(RSD) approach to determining optimal reinforce-

ment for reinforced concrete beam and column

sections subjected to uniaxial bending is extended to

the case of biaxial bending. Conventional constraints

on the distribution of longitudinal reinforcement are

relaxed, leading to an infinite number of reinforce-

ment solutions, from which the optimal solution and a

corresponding quasi-optimal pragmatic is determined.

First, all possibilities of reinforcement arrangements

are considered for a biaxial loading, including sym-

metric and non-symmetric configurations, subject to

the constraint that the reinforcement is located in a

single layer near the circumference of the section.

This theoretical approach establishes the context for

obtaining pragmatic distributions of reinforcement

that are more suitable for construction, in which

distributions having double symmetry are considered.

This contrasts with conventional approaches for the

design of column reinforcement, in which a predeter-

mined distribution of longitudinal reinforcement is

assumed, even though such a distribution generally is

non-optimal in any given design. Column and wall

sections that are subjected to uniaxial or biaxial

loading may be designed using this method. The

solutions are displayed using a biaxial RSD and can be

obtained with relatively simple algorithms imple-

mented in widely accessible software programs such

as Mathematica and Excel. Several examples

illustrate the method and the savings in reinforcement

that can be obtained relative to conventional solutions.

Keywords Ultimate strength design

Optimal reinforcement Biaxial bending

List of symbols

Ac Cross sectional area of concrete section

As Area of bottom reinforcement

A0s Area of top reinforcement

Ap Area of prestressing tendon

N Nominal axial strength

Nd Design value of the applied axial force

M Bending moment applied at the center of

gravity of the gross section

Md Design value of the applied bending moment

Mx Flexural moment strength about x-axis

Mxd Design value of the bending moment applied

about the x-axis

My Flexural moment strength about y-axis

L. M. Gil-Martn (&) E. Hernandez-MontesDepartment of Structural Mechanics, University

of Granada, E.T.S. Ingenieros de Caminos,

Campus Universitario de Fuentenueva,

18072 Granada, Spain

e-mail: [email protected]

E. Hernandez-Montes


M. Aschheim

Department of Civil Engineering, Santa Clara University,

500 El Camino Real, Santa Clara, CA 95053, USA


Materials and Structures (2010) 43:12451256

DOI 10.1617/s11527-009-9576-x

Myd Design value of the bending moment applied

about the x-axis

fck Characteristic compressive strength of

concrete

fyk Characteristic yield strength of reinforcement

sh Distance between centroids of consecutive

bars of the top and bottom reinforcement

sv Distance between centroids of consecutive

bars of side reinforcement

x Depth to neutral axis from top fiber of cross

section

y Vertical coordinate measures from the center

of gravity of the gross section

rc Stress in concrete

rp Stress in prestressing tendon

rs Stress in bottom reinforcement

r0s Stress in top reinforcement

n Intersection of the neutral axis with the y-axis

u Angle of the neutral fiber

U Bar diameter

1 Introduction

The approaches, assumptions, and hypotheses com-

monly used for the design of reinforced concrete

sections subjected to combinations of axial load and

moment applied about one or both principal axes of the

cross section were established many years ago. Even

though conventional approaches for the analysis and

design of cross sections make use of pre-determined

patterns of longitudinal reinforcement distributed

symmetrically about the cross section, there are

significant differences in the approaches suggested

by different authors. For example, Bresler [1] and

Gouwens [2] have suggested different approximations

for design of rectangular sections subjected to biaxial

bending. Marn [3] addressed the biaxial bending of L-

shaped sections. Approximate methods for biaxial

bending were evaluated by Lepage-Rodrguez [4] and

Furlong et al. [5]. A rigorous application of the plane-

sections remain plane assumption for the sectional

analysis of rectangular sections is described by Leet

and Bernal [6]. Even more recently, Leps and Sejnoha

[7] used genetic algorithms to designRCcross sections.

The present work applies conventional assump-

tions and hypotheses at a fundamental level for the

design of longitudinal reinforcement, but relaxes the

conventional assumption that the reinforcement

should be distributed symmetrically. Unlike previous

approaches, the present work determines longitudinal

reinforcement solutions for the design problem as a

function of the neutral axis depth and inclination, and

thus identifies an infinite number of reinforcement

solutions from which the optimal reinforcement

solution may be selected for design. In the present

case, the fundamental assumptions and hypotheses

are given as represented in Eurocode 2 [8].

The present work represents a simplification of the

more general approach for ultimate strength design

presented by Aschheim et al. [9], wherein a single

model applicable to a rectangular section and solution

approach was used for the design of reinforced

concrete beams, columns and walls for uniaxial or

biaxial loading. The model uses a conjugate gradient

search method to solve the nonlinear optimization

problem of minimizing the total reinforcement area.

The solution approach allows additional constraints

to be imposed such as single or double symmetry of

the reinforcement distribution.

The theory for strength design of biaxial columns

can be found in several references or text books [6,

10]. Beal and Pannell [11] shows that multibar

rectangular concrete columns can be designed using

the published curves for four-bar columns if the

effective depth of the multibar column is transformed

into an equivalent four-bar effective depth. Rafiq and

Southcombe [12] presented an interesting approach to

optimal design of RC biaxial columns using genetic

algorithms. Extensions to RC moment frames have

been made. For example, Liu [13] considered a

quadratic expression for the relationship between the

area of longitudinal reinforcement and the fully

plastic moments of cross-sections in order to mini-

mize the total volume of reinforcing steel in the frame.

Once the reinforcement has been determined,

existing software such as PCAColumn [14] or

BIAX-2 [15] may be used to analyse the strength of

the section, producing interaction diagrams that are

useful for validating trial designs. Ehsani [16] shows

a valid algorithm that can be used to obtain the

interaction diagrams. These programs, which analyse

sections with known reinforcement, do not provide a

direct solution for the reinforcement required to

provide a section with adequate strength according to

a governing building code.

Recent work by the authors has emphasized a

unique solution strategy in which reinforcement

1246 Materials and Structures (2010) 43:12451256

solutions are obtained as a function of the neutral axis

depth, allowing optimal reinforcement solutions to be

characterized and used for design. The resulting

reinforcement distributions generally are not sym-

metric, but conform to building code requirements

and may result in significant savings of reinforce-

ment, and thus advance the aims of sustainability in

construction. In one instance, Reinforcement Sizing

Diagrams are applied to the design of sections

subjected to uniaxial bending in conjunction with

axial load [17]. Characteristics of the optimal solu-

tions obtained with Reinforcement Sizing Diagrams

are the basis of the Optimal Domains identified in

Aschheim et al. [18]. The treatment of multiple load

combinations is addressed by Lee et al. [19]. A

general solution to the biaxial bending problem was

described by Aschheim et al. [9]. This reference

includes a comparison of an optimal solution with a

conventional textbook solution for the biaxial bend-

ing of a rectangular section column.

The present paper extends the uniaxial Reinforce-

ment Sizing Diagram concept to sections subjected to

biaxial bending. Rather than considering the general

solution that is described in [9], this solution reflects a

view that rectangular cross sections should have

reinforcement patterns in which double symmetry is

present (i.e. parallel sides have equal reinforcement).

Reinforcement configured in this way, with double

symmetry, is often used for rectangular sections

subjected to biaxial bending [20]. The consideration

of double symmetry forces to a reduction in the

number of variables, with this restriction, admissible

solutions for strength design depend only on one

variable, so the minimum reinforcement area can be

identified on a 2D (two-axis) diagram, this point will

be develop later on the paper. In this paper,

commercial software is used to show that reinforce-

ment solutions, plotted on biaxial RSD diagrams

provide the section with sufficient resistance to the

specified combination of axial load and moment. The

minimum total reinforcement area solution, under the

constraint of double symmetry, is readily identified,

as evident in several examples.

2 Ultimate strength evaluation of cross sections

The design problem for combined biaxial flexure and

axial load involves the simultaneous consideration of

equilibrium, compatibility, and the constitutive rela-

tions of the steel and concrete materials at the section

level. In the present biaxial formulation, the orthog-

onal axes x and y are aligned with the principal axes

of the cross section, and the applied moment, M, is

represented in terms of its components about the x

and y axes, Mx and My, respectively.

The compatibility conditions make use of Ber-

noullis hypothesis that plane sections remain plane

after deformation (see Fig. 1). The plane sections

hypothesis allows the distribution of strain over the

cross section to be defined by two variables in the

case of uniaxial bending (usually strain at the center

of gravity of the gross section and curvature, or

alternatively, depth to the neutral axis and curvature).

In the case of biaxial bending, three variables are

sufficient to define the distribution of strain over the

cross section (usually strain at the center of gravity of

the gross section, angle of inclination of the neutral

fiber relative to a principal axis of the section, and

curvature). For example, the maximum compressive

strain of the gross section, the height of the neutral

axis (along the y-axis) from the center of gravity of

Neutral axis

Tension

Compression

b

h

Neutral axis

Tension zone

M

N

y

xMx

My

C ompression zone

Fig. 1 Strains at cross section level in biaxial bending

Materials and Structures (2010) 43:12451256 1247

the gross section (n), and the angle between the x-axis

and the neutral fiber (u) may be used to define the

strain diagram, as illustrated in Fig. 1. Under the

constraint of ultimate strength design, one of these

variables is given, because the maximum compres-

sive strain is established by code.

Lets consider the following example to clarify the

previous paragraph: h = 700 mm and b = 400 mm

are given for the cross section in Fig. 1. Steel grade is

500 (fyk = 500 MPa) and concrete strength, fck, is

30 MPa (C-30). Reinforcement consists of five bars

along each side of the cross-section and mechanical

cover is 70 mm. In order to resist Nd = 200 kN,

Mxd = 400 kN m and Myd = 100 kN m, the bars

must have a diameter of 16.1 mm (so 20 mm is

used) and the sectional strains are defined by three

variables: maximum compression strain of 0.035,

u = 438 and n = 0.07 m.

The strain diagram defines the strains at every fiber

of the cross section. Based on the constitutive

relationships for the component materials, or useful

design approximations thereof, the nominal axial

strength, N, and flexural strengths Mx and My may be

determined as

N X

internal

N

Z

Ac

rcdAc

Z

A0s

r0sdA0s

Z

As

rsdAs

Z

Ap

rpdAp

Mx X

internal

Mx

Z

Ac

rcydAc

Z

A0s

r0sydA0s

Z

As

rsydAs

Z

Ap

rpydAp

My X

internal

My

Z

Ac

rcxdAc

Z

A0s

r0sxdA0s

Z

As

rsxdAs

Z

Ap

rpxdAp

1

where N acts at the point where x and y are measured

from andP

internal

Mx is the summation of internal

bending moments about the x-axis (idem forP

internal

My

about the y-axis). The convention adopted in this paper

is to measure x and y from the center of gravity of the

gross section (see Fig. 2). In Eq. 1, extended from

Collins and Mitchell [21] to biaxial bending, rc = the

normal stress on a differential element of concrete,

Ac = the area of concrete in the cross section,

r0s = the stress and A0s = the area of the compression

reinforcement, rs = the stress andAs = the area of the

tension reinforcement, and rp = the stress and

Ap = the area of the prestressing steel, if present, with

x and y being the respective distances of each

differential area (Ac, As, A0s or Ap) from the center of

gravity of the gross section, as illustrated in Fig. 2.

Axial forces and stresses are positive in compression

and negative in tension. The moments,Mx andMy, are

considered positive if they produce tensile strain on the

upper and left portions of the section, respectively.

3 Reinforcement Sizing Diagrams for uniaxial

bending

In the event thatMx orMy is zero, the biaxial problem

reduces to the simpler case of uniaxial bending in

conjunction with axial load. In this case bending

moments are applied about a principal axis of the

cross section and only two variables are required to

define the strain distribution, because the neutral axis

is aligned with one of the principal axes of the cross

section. Thus, design for uniaxial bending requires

that only two equilibrium equations be satisfied

(N and either Mx or My).

The reinforcement is used most efficiently when it

is located as far from the neutral axis as possible. Thus,

for uniaxial bending problems it is desirable to have

reinforcement at the bottom of the section (As) and/or

at the top of the section (A0s). In this case, the number of

unknowns in the two governing equilibrium equations

is three: As, A0s and the neutral axis depth, x. Since the

number of unknowns exceeds the number of equilib-

rium equations, an infinite number of solutions are

possible, and constraints of some kind must be

imposed in order to identify a particular solution. For

beam design, A0s = 0 is often imposed, while for

column design A0s = As is often imposed. More

generally, however, one may observe that solutions

for As and A0s may be determined for any neutral axis

depth, x. Alternatively, since top reinforcement may be

used to satisfy strength requirements or for other

reasons, such as to increase the curvature ductility of


the member, to control long term deformations due to

creep and shrinkage, or to aid in constructability, one

may solve for the required value of As given a

predetermined value of A0s.

The general solution of As and A0s as a function of y

for a cross section subjected to a single combination

of axial load, N, and moment, Mx, can be portrayed

on a Reinforcement Sizing Diagram. This diagram,

proposed by Hernandez-Montes et al. [17], provides a

convenient way to represent the infinite solutions

given by the equilibrium equations (N P

internal

N and

Mx P

internal

Mx) for the case of uniaxial bending. In

addition to plotting As and A0s as a function of the

neutral axis depth, y, the sum (As ? A0s) is also

plotted, in order to allow the minimum total longi-

tudinal reinforcement solution to be identified. The

RSD also portrays solutions corresponding to sym-

metric reinforcement and the solution for one steel

area given a known value for the other steel area.

Fig. 3 illustrates the RSD for an example according

to design provisions of ACI-318 [17]. Cross section

analyses indicate that ductility capacities are not

significantly reduced, and may be enhanced in some

cases, when optimal (minimum) reinforcement is

selected relative to conventional (symmetric) rein-

forcement [22].

The RSD is intended for use in design, that is, for

determining necessary and sufficient reinforcement

quantities for a given cross section for the governing

center of gravity of the gross

N

y

x

As1 s1

fcu

As2 s2

As3 s3

1

2

s1

c,max

s2

s3

3

Neutral axis

MMy

Mx

c

Fig. 2 Stresses and

resultants in biaxial bending

for an arbitrary reinforced

concrete section

Neutral axis depth, x (mm)

Steel area (mm2)

Total area (As + As)

As

As

225 250 275 300 325 350 375 400

2000

4000

6000

8000

10000

h=406 mm (20 in)

b= 508 mm (16 in)

0.75h

Pn=3559 kN (800 kips) e=178 mm (7in)

AsAs

Fig. 3 Example of RSD for uniaxial bending


load case. NM interaction diagrams [23] may then

be developed for the section having a particular

combination of As and A0s in order to establish that the

given design has adequate resistance for the various

combinations of N and M associated with the relevant

load combinations. In this way, an interaction

diagram for an asymmetrically reinforced section is

developed, in contrast to the conventional use of NM

interaction diagrams, which are prepared for a

predetermined reinforcement pattern. Conventional

interaction diagrams enforce ad hoc assumptions

about the distribution of reinforcement in the cross

section and provide no information about the range of

acceptable solutions, and thus do not aid the designer

in identifying a truly optimal reinforcement solution.

4 Optimization problem for rectangular RC

concrete under biaxial bending

In the case of rectangular RC cross sections, the

general problem of reinforcement distribution can be

considered in terms of independent quantities of

reinforcement along each side: As,t, As,b, As,l and As,r.

For strength design of rectangular RC cross section,

equilibrium provides three equations while there are

six unknowns (As,t, As,b, As,l, As,r, n, and u), therefore

an infinite number of satisfactory combinations of

reinforcement steel areas may be identified.

Aschheim et al. [9] presented a procedure to easily

obtain the optimal solutions using nonlinear conju-

gate gradient search methods; the optimum could be a

global minimum area or minimum area subjected to

additional constraints, such as the use of equal

reinforcement on opposite faces or equal reinforce-

ment on all faces. Smeared reinforcement was

considered by using plates along each side. To

illustrate the application, an example previously

evaluated by McGregor and Wight [24] was used.

Aschheim et al. [9] presented minimum reinforce-

ment required to provide the section with the same

nominal strength (Nn = 1700 kN, Mnx = 225 kN m

and Mny = 111 kN m), determined for different val-

ues of n and u (see Fig. 4). Minimum reinforcement

solutions are shown for each value of n and u

(uniformly spaced) where such solutions could be

obtained, along with contours. The global minimum

is given by qg = 2.22%, and occurs for n/h = 0.750

and u = 38.8. Small sketches superimposed on

15 20 25 30 35

0.6

50.7

00.7

50.8

00.8

50.9

0

2.25

2.25

2.37

2.22

2.64

2.47

2.33

2.25

2.23

2.27

2.39

2.62

2.99

4.75

5.95

2.59

2.43

2.33

2.26

2.25

2.3

2.43

2.66

3.04

3.63

5.14

2.59

2.44

2.36

2.29

2.28

2.34

2.48

2.72

3.11

3.71

4.67

2.66

2.51

2.42

2.36

2.36

2.43

2.58

2.85

3.26

3.9

4.91

2.88

2.72

2.59

2.54

2.55

2.65

2.84

3.16

3.66

4.43

5.67

3.2

2.94

2.81

2.77

2.82

2.97

3.23

3.66

4.33

5.39

7.1710.51

2.41%

0.20% 0.33%

0

0.89%

0.12%4.38%

0

0.04%

2.11%0

0.28%

Symmetric2.72%

Optimal

0.09%

0.78%2.76%

0

/h

Fig. 4 Contours of total

reinforcement (expressed as

a percent of gross section

area) required to provide

Nn = 1700 kN,

Mnx = 225 kN m, and

Mny = 111 kN m as a

function of n/h and u


Fig. 4 indicate the distribution of reinforcement over

the section for four combinations of n/h and u.

Clearly, very different patterns of reinforcement may

be used to provide a section with a desired nominal

strength. For comparison, the symmetric reinforce-

ment solution (equal reinforcement on all faces)

requires qg = 2.72%, and is obtained for n/h = 0.784

and u = 30.0.

Constraints may be imposed on the reinforcement

areas to obtain solutions mimicking conventional

column reinforcement distributions. Where equal

reinforcement areas are desired on opposite faces

(As,t = As,b and As,l = As,r), the variables As,b and As,rmay be set equal to the values used on the opposite

faces, and values of As,t and As,l are sought that

minimize the objective function (As,t ? As,b ?

As,l ? As,r, for example). Similarly, if equal rein-

forcement areas on all faces is desired, three

reinforcement areas may be set equal to a fourth,

and values of the fourth are sought that minimize the

objective function.

5 Reinforcement Sizing Diagrams for biaxial

bending

Unsymmetrical bar arrangements are not suitable for

common practice in columns. Reinforcement arrange-

ments having double symmetry may be acceptable in

some circumstances.

Authors of this paper [25] have investigated

unsymmetrical reinforcement arrangements for sev-

eral years. Some configurations have been found to

be very attractive, such as circular piers for retaining

walls (protected by international patents property of

the University of Granada), see Fig. 5.

These piers can be installed prior to excavation

and in some circumstances are very cost effective.

Solutions like this may save more than 50% of the

reinforcement and 1015% of total (materials and

labor) cost for the reinforced concrete portions of the

work.

Of interest in this paper is the case that a rectangular

RC section is used with equal reinforcement along

opposite sides, an arrangement that aids in construc-

tability of reinforced concrete columns. For this case,

common computing software can be used for the

optimization process described above. IECA 3.0 [26]

is used in this paper for the design verification of

reinforcement, which assures the fulfilment of Euro-

code 2 [8] requirements.

Relative to the case of uniaxial bending, strength

design for biaxial bending requires an additional

variable to define the strain distribution. Chosen here

are the intersection of the neutral fiber with the y-axis

(n) and the angle between the x-axis and the neutral

axis (u). The steel areas can be parameterized in

several ways including some that require a large

number of variables. Constructability considerations

require that the reinforcement pattern not be exces-

sively complex. Thus, a single longitudinal bar size is

used, and the spacing of the bars along the top and

bottom faces (sh) is allowed to differ from the spacing

along the side faces (sv). Consequently, bottom and

top reinforcement areas are equal, and may differ

from the reinforcement on each side, as represented

in Fig. 6a. Consequently, for a given set of assump-

tions regarding ultimate strength analysis, the strain

distribution is defined by two unknowns (n and u,

which define the position of the neutral axis) in

5 32 @ 64mm + 9 10 @ 270mm

5 32 @ 64mm

Fig. 5 Cross section used for tangent wall piles


conjunction with the strain limit specified in the

applicable design code for ultimate strength analysis,

while the reinforcement distribution is defined by two

unknowns (the diameter of the reinforcement, U, and

the bar spacing ratio, sh/sv). Thus, the equilibrium

equations can be stated as:

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Top and Bottom Reinforcement (mm2)

To

tal

Sid

e R

ein

forc

em

en

t (m

m2)

(b)

(c) Biaxial Reinforcement Sizing Diagram

Biaxial Reinforcement Sizing Diagram

4200

4300

4400

4500

4600

4700

4800

4900

5000

0 1000 2000 3000 4000 5000 6000

Top and Bottom Reinforcement Area (mm2)

To

tal

Re

info

rce

me

nt

Are

a (

mm

2)

(a) 400 mm

700 mm

Neutral Axis y

x

sv

sh

fck = 30 MPa; fyk = 500 MPa Mechanical cover = 7 mm

Case 1 Case 2Nd = 200 kN 200 kN Mxd = 300 kNm 400 kNm Myd = 250 kNm 100 kNm

Fig. 6 Reinforcement

Sizing Diagrams for biaxial

bending. a Cross section

and load cases. b Total side

reinforcement area plotted

against top and bottom

reinforcement area. c Total

reinforcement area plotted

against top and bottom

reinforcement area


N X

Nn;u;/; sh=sv

Mx X

Mxn;u;/; sh=sv

My X

Myn;u;/; sh=sv

2

where summations are used in place of integrals to

reflect the discrete locations of the resultant forces,

determined in conformance with the applicable

design provisions. The right side of the first equation

in (2) represents the internal contribution to the axial

force, and includes concrete and steel. The same

types of terms contribute to the flexural moments Mxand My, in the second and third equations of (2),

respectively. The number of variables considered is

four (n, u, / and sh/sv), which cover all the possible

cases for double symmetry reinforcement arrange-

ments. Solutions may be obtained by calculating the

required bar diameter (together with n and u) for

different values of sh/sv. In practice, the required bar

diameter may be determined assuming an integer

number of bars on the top and bottom faces and a

different integer number of bars along the side faces.

This represents a form of discrete optimization, and

differs from the smeared reinforcement distribution

(as plates) considered by Aschheim et al. [9]. One

useful graphical representation, named Biaxial Rein-

forcement Sizing Diagram (BRSD), plots the total top

and bottom reinforcement area on the abscissa and

the total reinforcement area on the ordinate, as

illustrated subsequently.

For a set of axial load and biaxial load bending

moments a check is required to ensure that the cross-

section resists this given set. This can be done with

PM interaction diagrams or with the Bresler load

contour method [1], although the later is a conserva-

tive approximation.

6 Biaxial bending example

The 700 9 400 mm rectangular cross section of

Fig. 6a was designed with doubly symmetric rein-

forcement according to Eurocode 2 [8], for a concrete

strength (fck) equal to 30 MPa and steel strength (fyk)

of 500 MPa. To illustrate the potential under different

loading conditions, two different design loadings are

considered: (1) Nd = 200 kN, Mxd = 300 kN m and

Myd = 250 kN m and (2) Nd = 200 kN, Mxd =

400 kN m and Myd = 100 kN m. Column cross

section width and height are given so these values

are not included as optimization parameters.

Eurocode 2 [8] assumptions for ultimate strength

analysis are used, wherein the maximum usable

compressive strain is 0.0035, the maximum usable

tensile strain in the reinforcement is 0.01, and

admissible strain diagrams are limited to strain

domains as described by Eurocode 2 [8]. The

reinforcement is assumed to be elastoplastic, with

design yield strength fyd equal to the nominal yield

strength fyk divided by a material safety factor, cs,

which is equal to 1.15. The parabola-rectangle

diagram for concrete compressive stress defined in

Eurocode 2 [8] was used, with a maximum stress of

0.85 fcd. The design strength fcd is given by the

nominal strength fck divided by a material safety

factor cc, which is equal to 1.5. The depth of the

concrete stress block is given by k times the depth of

the neutral axis, even where the neutral axis is

inclined at an angle u. In this example k is taken

equal to 0.8.

Table 1 shows that the minimum total reinforce-

ment consists of 16 /11.6 bars distributed equally to

the top and bottom layers with 13 bars added to each

side, resulting in a total reinforcement area of

4439 mm2 (Case 9). The reinforcement areas shown

in Table 1 are plotted in Fig. 6b and c. Figure 6b

shows the variation in side reinforcement area as a

function of top and bottom steel reinforcement area;

total reinforcement area is plotted in Fig. 6c. One can

observe that the optimum (or minimum) reinforce-

ment area is obtained for a specific amount of bottom

and top reinforcement area. Because significant

amounts of reinforcement are required on each face,

the minimum reinforcement solution is similar to the

solution obtained for equal reinforcement on each

face, given by Case 5 of Table 1. The optimum

reinforcement solution requires 97.3% of the rein-

forcement area obtained with equal reinforcement on

each face (4560 mm2). In this procedure the bar

diameter has been obtained from Eq. 2 and does not

match the available range of bars in the market, so a

slightly larger, commercially available, bar would be

used in construction.

The second loading case better illustrates the

potential of biaxial RSDs to save reinforcement. For

this case, Nd = 200 kN,Mxd = 400 kN m andMyd =

100 kN m. The optimum reinforcement is given by

Case 1 of Table 2, for which 16 longitudinal bars are


Table 1 Analyses used to computer the biaxial Reinforcement Sizing Diagram for the example cross section, for Nd = 200 kN,

Mxd = 300 kN m and Myd = 250 kN m

Analysis

case

Bar diameter (mm) Number

of top and bottom

barsa

Number

of side barsbsh/sv Total area

of top and

bottom bars (mm2)

Total area

of side bars

(mm2)

Total area

of bars

(mm2)

1 19.9 8 0 0.08 4976 0 4976

2 18.5 8 1 0.16 4301 538 4839

3 17.3 8 2 0.24 3761 940 4701

4 15.6 8 4 0.40 3058 1529 4587

5 14.4 8 6 0.56 2606 1954 4560

6 13.4 8 8 0.72 2256 2256 4512

7 12.6 8 10 0.88 1995 2494 4489

8 11.9 8 12 1.04 1780 2669 4449

9 11.6 8 13 1.13 1691 2748 4439

10 11.9 6 14 1.69 1335 3114 4449

11 12.6 4 14 2.81 998 3491 4489

12 13.4 2 14 8.44 564 3949 4513

13 10.8 2 23 13.50 366 4214 4580

14 9.6 2 30 17.44 290 4343 4633

The minimum total reinforcement solution is shown in bold text; that obtained using equal amounts of steel on each face is shown by underlining

a Number of top and bottom bars is the number of bars at the top layer, which is the same as the number of bars in the bottom layer in this

implementation of biaxial RSDs

b The number of side bars, excluding the corner bars (which are counted as top or bottom bars)

Table 2 Analyses used to computer the biaxial Reinforcement Sizing Diagram for the example cross section, for Nd = 200 kN,

Mxd = 400 kN m and Myd = 100 kN m

Analysis

case

Bar

diameter

(mm)

Number

of top and bottom

barsa

Number

of side barsbsh/sv Total area

of top and

bottom bars (mm2)

Total area

of side bars

(mm2)

Total area

of bars

(mm2)

1 15.3 8 0 0.08 2942 0 2942

2 14.6 8 1 0.16 2679 335 3013

3 14.0 8 2 0.24 2463 616 3079

4 12.9 8 4 0.40 2091 1046 3137

5 12.1 8 6 0.56 1840 1380 3220

6 11.4 8 8 0.72 1633 1633 3266

7 10.8 8 10 0.88 1466 1832 3298

8 10.3 8 12 1.04 1333 2000 3333

9 10.1 8 13 1.13 1282 2083 3365

10 10.4 6 14 1.69 1019 2379 3398

11 11.2 4 14 2.81 788 2759 3547

12 12.1 2 14 8.44 460 3220 3680

13 9.8 2 23 13.50 302 3470 3771

14 8.7 2 30 17.44 238 3567 3805

The minimum total reinforcement solution is shown in bold text; that obtained using equal amounts of steel on each face is shown by underlining

a Number of top and bottom bars is the number of bars at the top layer, which is the same as the number of bars in the bottom layer in this

implementation of biaxial RSDs

b The number of side bars, excluding the corner bars (which are counted as top or bottom bars)


distributed equally to the top and bottom faces and

without side bars. The total required reinforcement is

2942 mm2. This is 91.4% of the total reinforcement

required if equal reinforcement is provided to each

face (Case 5). Figure 7 illustrates the reduction in

total reinforcement area as the top and bottom

reinforcement area increases.

7 Conclusion

The infinite number of reinforcement area solutions

for the classic flexural design problem has led in

current and past practice to the imposition of

constraints on the reinforcement pattern in order to

make column design practicable. With the advent of

modern computing tools, however, the constraints on

the reinforcement pattern can be relaxed, allowing a

larger number of solutions to be identified. By

portraying these solutions on a biaxial Reinforcement

Sizing Diagram, the engineer is given the freedom to

select the solution that is optimal in any particular

context. An example illustrated the use of biaxial

RSDs under the constraint that the same bars are

used, at one spacing along the top and bottom faces,

and at a different spacing along the side faces. The

examples illustrated the potential to reduce the

reinforcement required by 2.7 or 8.6%, depending

on the relative values of the moments Mx and My.

Greater savings may potentially be achieved depend-

ing on the relative values of the moments, the axial

load, and the willingness of the engineer or contractor

to construct reinforcement cages having an uncon-

ventional layout of longitudinal reinforcing bars.

Savings in longitudinal reinforcement not only reduce

the cost of construction, but also reduce the carbon

footprint, helping to improve the sustainability of

reinforced concrete construction.

References

1. Bresler B (1960) Design criteria for reinforced columns

under axial load and biaxial bending. ACI Proc 57:481490

2. Gouwens AJ (1975) Biaxial bending simplified, reinforced

concrete columns, ACI Special Publication SP-50. Amer-

ican Concrete Institute, Detroit

3. Marn J (1974) Resistencia de las secciones de concreto

armado sometidas a flexo compresion: un metodo algo-

rtmico general y sus aplicaciones en el diseno de colum-

nas. tesis de Doctor de Ingeniera, Univeristy of Central

Venezuela, Caracas

4. Lepage-Rodrguez A (1991) Evaluacion de metodos para

el diseno de columnas rectangulares de concreto armado en

flexion biaxial. tesis de Magister en Ingeniera Civil,

Universidad Simon Bolvar, Caracas

5. Furlong RW, Hsu C-TT, Mirza A (2004) Analysis and

design of concrete columns for biaxial bendingoverview.

ACI Struct J 101(3):413423

6. Leet KM, Bernal D (1996) Reinforced concrete design.

McGraw-Hill, Blacklick

7. Leps M, Sejnoha M (2003) New approach to optimization

of reinforced concrete beams. Comput Struct 81(1819):

19571966

8. Eurocode 2 (2002) Design of concrete structuresPart 1:

General rules and rules for buildings prEN 1992-1-1 (July).

European Committee for Standardization, Brussels

9. Aschheim M, Hernandez-Montes E, Gil-Martn LM (2008)

Design of optimally reinforced RC beam, column and wall

sections. J Struct Eng 134(2):169188

10. Pannell FN (1966) Design charts for members subjected to

biaxial bending and trust. Concrete Publishing Ltd., Lon-

don, 52 pp

11. Beal AN, Pannell FN (1992) Simple design of multibar

concrete columns. Struct Eng 70(23/24):434435

12. Rafiq MY, Sothcombe C (1998) Genetic algorithms in

optimal design and detailing of reinforced concrete biaxial

columns supported by a declarative approach for capacity

checking. Int J Comput Struct 69:443457

13. Liu I-W (1998) An approximate resizing system for the

preliminary design of reinforced concrete frames. Struct

Optim 15:242250

14. PCAColumn, Version 3.6.4 (2005) Portland Cement

Association, Skokie

15. Wallace (1992) JBIAX-2: analysis of reinforced concrete

and reinforced masonry sections. http://nisee.berkeley.edu/

elibrary/getdoc?id=BIAX2ZIP. Accessed 16 Feb 2008

16. Ehsani MR (1986) CAD for columns. Concr Int 8(9):4347

17. Hernandez-Montes E, Gil-Martn LM, Aschheim M (2005)

The design of concrete members subjected to uniaxial

bending and compression using reinforcement sizing dia-

grams. ACI Struct J 102(1):150158

2500

2700

2900

3100

3300

3500

3700

3900

0 500 1000 1500 2000 2500 3000 3500

Top and Bottom Reinforcement Area (mm2)

To

tal

Re

info

rce

me

nt

Are

a (

mm

2)

Fig. 7 Reinforcement Sizing Diagram for biaxial bending, for

Nd = 200 kN, Mxd = 400 kN m and Myd = 100 kN m


18. Aschheim M, Hernandez-Montes E, Gil-Martn LM (2007)

Optimal domains for strength design of rectangular sec-

tions for axial load and moment according to Eurocode 2.

Eng Struct 29(8):17521760

19. Lee HJ, Aschheim M, Hernandez-Montes E, Gil-Martn

LM (2009) Optimum RC column reinforcement consider-

ing multiple load combinations. Struct Multidiscip Optim

39(2):153170

20. Bonet JL, Romero ML, Fernandez MA, Miguel PF (2007)

Design method for slender columns subjected to biaxial

bending based on second-order eccentricity. Mag Concr

Res 59(1):319

21. Collins MP, Mitchell D (1991) Prestressed concrete

structures. Prentice Hall, New Jersey

22. Hernandez-Montes E, Aschheim M, Gil-Martn LM (2004)

The impact of optimal longitudinal reinforcement on the

curvature ductility capacity of reinforced concrete column

sections. Mag Concr Res 56(9):499512

23. Whitney CS, Cohen E (1956) Guide for ultimate strength

design of reinforced concrete. ACI J 28(5):445490 (pro-

ceedings vol 53)

24. MacGregor JG, Wight JK (2005) Reinforced concrete:

mechanics and design, 4th edn. Pearson Prentice Hall,

Upper Saddle River, p 525

25. Hernandez-Montes E, Gil-Martn LM, Aschheim M (2010)

Pilotes Asimetricos para contencion de tierras. Revista de

Obras Publicas (in press)

26. Corres Peiretti H, Martnez Martnez JL, Perez Caldentey

A, Lopez Agu JC (2001) Prontuario Informatico del

Hormigon Estructural v. 30. IECA. www.ieca.es. Accessed

16 Feb 2008


Documents

Bi Axial