426_theorem of optimal reinforcement for reinforced concrete
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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: emontes@ugr.es
L. M. Gil-Martn
e-mail: mlgil@ugr.es
M. Pasadas-Fernndez
e-mail: mpasadas@ugr.es
M. Aschheim (*)
Civil Engineering Department, Santa Clara University,
500 El Camino Real,Santa Clara, CA 95053, USA
e-mail: maschheim@scu.edu
E. Hernndez-Montes : L. M. Gil-MartnDepartment of Structural Mechanics, University of Granada,
Campus de Fuentenueva,
18072 Granada, Spain
M. Pasadas-Fernndez
Department of Applied Mathematics, University of Granada,
Campus de Fuentenueva,
18072 Granada, Spain
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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
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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
TOSR for reinforced concrete cross sections
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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
TOSR for reinforced concrete cross sections
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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
top related