426_theorem of optimal reinforcement for reinforced concrete

5/28/2018 426_Theorem of Optimal Reinforcement for Reinforced Concrete

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INDUSTRIAL APPLICATION

Theorem of optimal reinforcement for reinforced concrete

cross sections

E. Hernndez-Montes &L. M. Gil-Martn &

M. Pasadas-Fernndez &M. Aschheim

Received: 18 September 2006 /Revised: 16 July 2007 /Accepted: 30 August 2007# Springer-Verlag 2007

Abstract A theorem of optimal (minimum) sectional rein-

forcement for ultimate strength design is presented for

design assumptions common to many reinforced concrete

building codes. The theorem states that the minimum total

reinforcement area required for adequate resistance to axial

load and moment can be identified as the minimum

admissible solution among five discrete analysis cases.

Therefore, only five cases need be considered among the

infinite set of potential solutions. A proof of the theorem is

made by means of a comprehensive numerical demonstra-

tion. The numerical demonstration considers a large range

of parameter values, which encompass those most often

used in structural engineering practice. The design of a

reinforced concrete cross section is presented to illustrate

the practical application of the theorem.

Keywords Reinforced concrete . Beams . Columns .

Optimal reinforcement. Concrete structures

Notation

Ac cross-sectional area of concrete section

As area of bottom reinforcement

A0s area of top reinforcement

N axial force applied at the center of gravity of the

gross section

M bending moment applied at the center of gravity of

the gross section

b width of cross section

d depth to centroid of bottom reinforcement from top

fiber of cross section

d depth to top reinforcement from top fiber of cross

section

fc specified compressive strength of concrete

fy specified yield strength of reinforcement

h overall depth of cross-section

x depth to neutral axis from top fiber of cross section

x* depth to neutral axis corresponding to a compres-

sive strain of 0.003 at top fiber and a tensile strain

of 0.01 at bottom reinforcement

xb depth to neutral axis corresponding to a tensile

strain of y at bottom reinforcement

x0

b depth to neutral axis corresponding to a tensile

strain of y at top reinforcement

xbb depth to neutral axis given by (9)

Struct Multidisc Optim

DOI 10.1007/s00158-007-0186-3

E. Hernndez-Montes : L. M. Gil-Martn :M. Pasadas-FernndezUniversity of Granada,

Campus de Fuentenueva,

18072 Granada, Spain

E. Hernndez-Montes

e-mail: [email protected]

L. M. Gil-Martn


M. Pasadas-Fernndez


M. Aschheim (*)

Civil Engineering Department, Santa Clara University,

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


E. Hernndez-Montes : L. M. Gil-MartnDepartment of Structural Mechanics, University of Granada,



M. Pasadas-Fernndez

Department of Applied Mathematics, University of Granada,




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xc depth to neutral axis corresponding to a compres-

sive strain of y at bottom reinforcement

x0

c1 depth of neutral axis corresponding to a compres-

sive strain of y at top reinforcement and a tensile

strain of 0.01 at the bottom reinforcement

x0

c2 depth of neutral axis corresponding to a compres-

sive strain of y at top reinforcement and a

compressive strain of 0.003 at top fiber.x0 depth of neutral axis at the minimum of total

reinforcement

y vertical coordinate measures from the center of

gravity of the gross section

x discrete increments in the depth of the neutral fiber

strain

c strain of the concrete

c,

max

maximum compressive strain of the concrete

s strain at bottom reinforcement

"0

s strain at top reinforcement

u maximum allowable tension strain of the steel

y yield strain of the reinforcement

s stress of bottom reinforcement

s

0

s stress of top reinforcement

1 Introduction

The approaches commonly used for the design of rein-

forced concrete sections subjected to combinations of axial

load and moment applied about a principal axis of the cross

section were established many years ago. However, a new

design approach was presented recently by Hernndez-

Montes et al. (2004, 2005), which portrays the infinite

number of solutions for top and bottom reinforcement that

provide the required ultimate strength for sections subjected

to combined axial load and moment. Solutions obtained

with this new approach allow the characteristics of optimal

(or minimum) reinforcement solutions to be identified. The

characteristics of these optimal solutions have led to the

development of the theorem of optimal section reinforce-

ment (TOSR) presented herein.

The longstanding conventional approaches treat the

design of sectional reinforcement in one of two ways.

One approach utilizes the distinction between large and

small eccentricities based on an approach taken by Whitney

and Cohen (1956), as described in Nawys (2003) textbook.

The second approach uses NM interaction diagrams,

which have been widely used since their initial presentation

by Whitney and Cohen (1956). These diagrams provide

solutions for the reinforcement required to resist a specified

combination of axial load, N, and moment, M, under the

constraint that the reinforcement is arranged in a predeter-

mined pattern. Typically, a symmetric pattern of reinforce-

ment is used. However, experience with beam design

suggests that an asymmetric pattern of reinforcement would

be more economical for small axial loads for cases in which

the applied moment (or eccentricity) acts in one direction

only.

The family of solutions for combinations of top and

bottom reinforcement required to confer adequate strength

to a cross section constitutes an infinite set of solutions thatincludes the symmetric reinforcement solution obtained

using conventional NM interaction diagrams. The family

of solutions is displayed graphically on a Reinforcement

Sizing Diagram (RSD) as described by Hernndez-Montes

et al. (2005). Using an RSD, an engineer can rapidly select

the reinforcement to be used in reinforced and prestressed

concrete sections subjected to a combination of bending

moment and axial load. Reinforcement may be selected to

achieve whatever may be dictated by the design objectives,

such as minimizing cost, facilitating construction, or

providing a structure that has a very simple and uniform

pattern of reinforcement.

RSDs were used in a recent investigation by Aschheim

et al. (2007) to characterize the optimal (minimum)

reinforcement solutions for a cross section over the two-

dimensional space of design axial load and moment. This

study identified domains in NM space for which the

optimal solution for nominal strength is characterized by

either constraints on reinforcement area (As=0, A0

s 0, orAs A

0

s 0) or constraints on the strains at the reinforce-ment locations (s=y, s equal to or slightly greater than

y, or"0

s "y) for stresses and strains taken as positivein compression), where As=the cross-sectional area of

bottom reinforcement, A0s =the cross-sectional area of top

reinforcement, s=the tensile strain in the bottom rein-

forcement, "0

s =the compressive strain in the top reinforce-

ment, and y=the yield strain of the reinforcement. The

optimal domains approach uses the characteristics of the

optimal solution to solve directly for the minimum rein-

forcement required for a given combination of axial load

and moment.

The present paper puts forth a TOSR and demonstrates

its validity by argument and computationally using numer-

ical results obtained for a large range of parameter values

representative of those commonly encountered in practice

(Table1).

2 Assumptions used in flexural analysis

and strength design

The design problem for combined flexure and axial load

involves the simultaneous consideration of equilibrium,

compatibility, and the constitutive relations of the steel and

concrete materials at the section level.

Hernndez-Montes et al.


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Equilibrium equations must be satisfied at the section

level (e.g., Fig. 1) for the governing combination of

bending moment, M, and axial force, N:

N N AccdAc A0s0

sdA0

s AssdAs

M M AccydAc A0s0

sydA0

s AssydAs1

where M is computed relative to the location that N is

applied, with y being the distance of each differential area

(dAc,dAs, dA0

s, ordAp) from the location of the point about

which the stress resultants act, as illustrated in Fig. 1.

Stresses and axial forces in (1) are positive in compression

and negative in tension. Without loss of generality, the axial

load, N, and moment, M, that equilibrate the internal stress

resultants are presumed to act about the center of gravity of

the gross section (see Fig. 1). The moment, M, is con-

sidered positive if it produces tensile strain on the bottom

fiber. For consistency, in the case that the applied loads

cause compression over the depth of the section, the

moment is considered positive if the compressive strain at

the bottom fiber is smaller than the compressive strain at

the top fiber.

The compatibility conditions make use of Bernoullis

hypothesis that plane sections remain plane after deforma-

tion and assume no slip of reinforcement at the critical

section. The Bernoulli hypothesis allows the distribution of

strain over the cross section to be defined by just two

variables. The strain at the center of gravity (cg) of the

gross section and the curvature () of the cross section may

be used to define the strain diagram, as illustrated in Fig.1.For strength design, the reinforcement usually is mod-

eled to have elasto-plastic behavior, and the concrete

compression block usually is represented using a rectangu-

lar, trapezoidal, or parabolic stress distribution. ACI-318

(2005) allows the use of a rectangular stress block having

depth equal to the product of a coefficient, 1, and the

depth of the neutral axis, where 1varies between 0.85 and

0.65 as a function of the specified compressive strength of

the concrete. Eurocode 2 (2002) specifies that the stress

block has a constant compressive stress of fcd having a

depth equal to the x, wherex =the depth of the neutral axis

and fcd=the design strength of the concrete, for concrete

whose resistance is between 25 and 55 MPa. The factor

defines the effective height of the compression zone, and

the factor defines the effective strength. The design

strength of the concrete is given as a function of the

specified characteristic strength, fck, where fcd=ccfck/c.

The termccaccounts for long term effects on strength and

the rate at which the load is applied. The term c is the

partial safety factor for concrete, taken as 1.5. For the range

contemplated in Table1, we have chosen =0.8 and=1.0

and cc=0.85, as these represent fairly typical values.

Perhaps the greatest difference in code provisions for

ultimate strength determination is the treatment of cross

section strains. The maximum usable strain at the extreme

compression fiber is 0.003 in ACI 318 (2005), and there is

no limit on the strain in the tensile reinforcement. Con-

sequently, the neutral axis depth is a positive number. In

Eurocode 2 (2002), the maximum usable strain in the ex-

treme compression fiber ranges between 0.002 and 0.0035,

and the tensile reinforcement strain cannot exceed 0.01.

The Eurocode 2 (2002) approach invokes the concept of

strain domains, wherein the strain diagram pivots about

certain points located on the boundaries between adjacent

Stresses due to external loads

Ap

s

xCenter of gravityof the gross section

As

p

cM

N

cg

Strains due to external loads

As

s

Neutral fiber

y

Fig. 1 Strains and stresses

diagrams at cross section level

Table 1 Range of variables studied for rectangular cross sections

Variable Range

Lower limit Upper limit

Concrete strength resistance (fc) 25 (MPa) 55 (MPa)

Height (h) 200 (mm) 2,000 (mm)

Width (b) 200 (mm) 2,000 (mm)

Yield strength (fy) 275 (MPa) 500 (MPa)

Yield strain (y) 275/200,000 500/200,000

Distance from extreme

compression fiber to centroid

of compression reinforcement (d)

hd

Mechanical cover condition dh/4 and d3/4 h

Modulus of elasticity of

reinforcement (Es)

200,000 (MPa)

Axial load 0.2bhfc bhfcFlexural moment 0 0.25bh2fc

TOSR for reinforced concrete cross sections


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domains; consequently, the neutral axis depth may assume

positive or negative values. The Swiss Concrete Code SIA-

162 (1989) establishes the maximum compressive strain

c,max=0.003 and a tensile steel strain limit ofs,max=0.005.

For the demonstration of the theorem, assumptions

intermediate between ACI 318 (2005) and Eurocode 2

(2002) were adopted. It is assumed that plane sections

remain plane, the maximum usable strain for concretein compression is given by c,max=0.003, and the max-

imum usable strain for steel in tension is given by s,max=

0.01. These assumptions are similar to those used in

the Swiss Concrete Code; the only difference is that the

tensile strain limit used for the demonstration is 0.01

and the Swiss Concrete Code uses 0.005. Walther and

Miehlbradt (1990) indicate that the choice of a tensile

strain limit of 0.005 or 0.01 has little effect on strength

design.

These strain limits are illustrated in Fig. 2, where three

domains are identified. In domains I and II, the extreme

compression fiber is at a strain of 0.003; for domain I, the

bottom reinforcement is either in compression or is at a

tensile strain less than the yield strain, while in domain II,

the bottom reinforcement is yielding in tension. In domain

III, the bottom reinforcement is at a strain of 0.01, and the

top fiber is either in tension or at a compressive strain less

than 0.003. Thus, for domains I and II, the neutral axis

depth,x, assumes a positive value, and can approach + asthe strains approach 0.003 over the entire section. The

neutral axis depth may be positive or negative in domain

III, and approaches as the strains approach 0.01 (intension) over the entire section.

3 Design solutions

The algebraic form of the integrals of (1) allows the internal

stress resultants to be determined as the product of the

stresses and the corresponding areas. Using the above

Fig. 2 Possible strain distribu-

tions in the ultimate limit state.

Stress distributions according to

rectangular block assumption

and equilibrium of applied N

and M with internal stress

resultants forx >0

Hernndez-Montes et al.


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simplifying assumptions, the compressive force carried by

the concrete,Nc, can be expressed as

Nc x 0 if 0:8x 0

0:85fc0:8x b if 0 0:8x< h0:85fchb if 0:8x> h

8d), the stress assigned to the

rectangular concrete block should not be counted in

determining the force carried by the top reinforcement,

N0

s. Similarly, if the compressive stress block extends below

the bottom reinforcement, the stress carried by the

rectangular stress block should not be counted in determin-

ing the force carried by the bottom reinforcement, Ns.

Therefore, N0

s and Ns can be determined as:

N

0

s x

0

s x A

0

s x Ns x s x As x 3

where A0

s =the cross-sectional area of steel located at a

distance d from the top of the section and As=the cross-

sectional area of steel located at a distance dfrom the top of

the section, and

0

s x

0

s x 0:85fc if 0:8x> d0

0

s x otherwise

s x s x 0:85fc if 0:8x> d

s x otherwise

4

The stress s0

s and s are positive in compression.

The internal stress resultants Nc, N0

s, and Ns equilibrate

the applied load, N, and moment, M. For a given neutral

axis depth, material properties, and reinforcement areas Asand A

0

s, the internal stress resultants can be determined and

equations of equilibrium can be applied to the free body

diagram of Fig. 1 to determine the axial load and moment

resisted by the section. The equations of equilibrium forN

and M applied at the center of gravity of the gross section

are:

N Nc x N0

s x Ns x

M Nc x

h2

0:4x

N0

s x h

2 d0

Ns x d

h2

if 0:8x h

N0

s x h

2 d0

Ns x d

h2

otherwise

5

Alternatively, the equations of equilibrium can be solved

to determine the steel areas As and A0

s required to provide

the section with sufficient capacity to resist the applied

loadsNand M. In particular, the sum of moments about the

location of the top reinforcement results in (6), while the

sum of moments about the location of the bottom

reinforcement results in (7).

As MN h

2d0 Nc x d00:4x s*

x dd0 if 0:8x< h

M N0:85fcbh

h2d0

s* x dd0 otherwise

6

A0

s MN dh

2 Nc x d0:4x

0s*

x dd0 if 0:8x< h

M N0:85fcbh d

h2

0s*

x dd0 otherwise

7

The solutions for reinforcement areas given by (6) and

(7) are functions of the neutral axis depth, x. Some values

of x result in negative values of As and A0

s, and therefore

must be considered inadmissible. The admissible solutions

for As and A0

s, obtained with (6) and (7), are plotted on a

RSD. Such a plot clearly indicates that the minimum total

reinforcement solution generally differs from the symmetric

reinforcement solution that typically is represented using

conventional NMinteraction diagrams. More information

on RSDs can be found in Hernndez-Montes et al. (2005)

where an example of the design of a column from ACI

Publication SP-17 (1997) is used (see Fig. 3). ACI

Publication SP-17 presents only the symmetric reinforce-

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)

Fig. 3 Example RSD



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ment solution, but the RSD indicates that a significant

savings in cost can be obtained when optimal (minimum

total) reinforcement solutions are used in place of tradi-

tional symmetric reinforcement solutions.

Although it is possible to obtain zero reinforcement

solutions from (6) and (7), code-required minimum rein-

forcement requirements also must be considered in design.The theorem addresses only equilibrium considerations.

4 Theorem of optimal section reinforcement

Theorem The top (As) and bottom A0

s

reinforcement

required to provide a rectangular concrete section with

adequate ultimate strength1

for a combination of axial load

and moment applied about a principal axis of the cross

section has the following characteristics:

(1) An infinite number of solutions forAs and A

0

s exists.(2) The minimum total reinforcement area As A

0

s

occurs for one of the following cases: As=0, A

0

s 0,As A

0

s 0, sequal to or slightly greater than y,=s="

0

s =c,max=0.003, and =s="0

s 0:01.

Corollary The minimum reinforcement area for a specific

combination of axial load and moment may be determined

by:

(1) Evaluating the cases:As=0,A0

s 0, s=y, =s="0

s =

c,max=0.003, and =s="0

s 0:01.

(2) Selecting the minimum of the admissible solutions,where an admissible solution is one in which the value

of x0 is real and the areas As and A0

s are each non-

negative, subject to the following:

a. If As=0 produces a negative value forA0

s and A0

s

0 produces a negative value for As, then the

minimum reinforcement solution is given by

As A0

s 0.

b. If s=y produces an admissible solution andAs(xb)


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of 2x0

b; 2xc

for fixed values of N and M. Discrete

values ofx are considered in increments of 1 mm, and the

minimum value of the total area As A0

s

is retained and

plotted as a function ofx0 [As(x0), and A0

s(x0)]. Thus, each

point in Fig. 4a,b is the retained (optimal) solution for a

fixed value of N and M. Inspection of these plots reveals

that there are regions where either As or A0

s are equal to

zero, regions where both As and A0

s appear to be nonzero,

Fig. 4 a Values ofAs(xo/d)

determined for optimal

solutions.b Values of deter-

mined for optimal solutions



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and values of x0/d for which either many or no optimal

solutions occur. The values ofxb,x*,xc, x0

b,x0

c1 andx0

c2 are

identified on Fig. 4a,b and in Table 2.

Inspection of Fig. 4a,b reveals several singularities and

zones as follows. These are discussed sequentially in order

of increasingx0/d, with respect to Fig. 4a,b:

a) x0/d=0.11, which corresponds to x0 x0

b, (i.e., "0s

"y for Grade 400 reinforcement). For this case, bothAs and A

0

s assume positive values. It is interesting to

observe that for given values ofNand M, anyx smaller

than x0

b results in As(x)=As x0

b

and A

0

s x A0

s x0

b

.

Thus, if an optimal solution is found for x0/d


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x0/d=xb/d=0.6. Figure 6a shows this region for the subset

of data represented in Fig.4a for whichb =200 andb =1,000.

One may further observe that the minimum values ofAsare

aligned based on the value of the parameterb. This result is

not an artifact of the step size x.

Wherexb


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of the section using the TOSR. The selected values of N

and M are those for which the optimal solutions occur foreach of the possible cases. These values are presented in

normalized form in Table 3.

One may observe that in general, the minimum of the

admissible solutions corresponds to the optimal solution.

Two special cases are described as follows:

1) For the first case, no admissible solutions are found,

and in particular, the case As=0 produces a negative

value for A0

s, and second, the case A0

s 0 produces a

negative value for As. In this case, no reinforcement

is required for the section to resist the combined

axial load and moment, and the optimal solution is

As A0

s 0.2) For the last case of Table3, three admissible solutions

were identified. Among these, x=xb appears to be the

optimal solution, but there is a possibility that the true

optimal solution is in the range xb


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To illustrate the comparison ofAs (xb) andAs(xb) for a case

in which x0=xb, we consider the fourth case (N=0.2Agfc,

M=0.20Aghfc) of Table 3. To determine if the minimum

might be in the range xb


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Therefore, the stress in the bottom reinforcements(x) is

s x fy if x< xb

0:003Esdx

x if xb x xc

fy if x> xc

80.0715d.

Case (b) is defined by yielding of the top reinforcement in

compression in concert with the bottom reinforcement having

a strain of 0.01. More generally, the top reinforcementmaybe responding elastically or maybe yielding in compres-

sion or in tension while the bottom reinforcement at strain of0.01 (in tension). Any of these conditions may occur for

d0

426_theorem of optimal reinforcement for reinforced concrete

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sectionx depth

xc depth

reinforcementxb depth

reinforcementx0b depth

reinforcementxbb depth

university of granada

keywords reinforced

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