I. INTRODUCTION
Design of prestressed concrete beams is
based upon two distinct concepts which lead to
two design methods known as service load de-
sign or working stress design, and ultimate
design.
In service load design the stresses in
the beam are calculated on the basis of the
assumption that concrete is an elastic material,
These calculated stresses are to be less than,
or equal to, certain limiting stresses known as
allowable stresses. The allowable stresses are
chosen so that the structure will perform its
intended service satisfactorily under service
conditions while providing indirectly for the
safety of the beam.
In ultimate design the flexural strength
or ultimate moment of the section is calculated
based on the knowledge of behavior of the beam.
The calculated ultimate moment is to be equal
to, or greater than, the sum of moments of all
forces each multiplied by a factor. These are
known as load factors and are chosen so that
the structure will be sufficiently safe under
the service conditions. Moreover, ultimate
design also requires certain ductility in the
beam, so that prior to failure the beam will
deform sufficiently. Ductility is measured
by the deformation of the beam at failure.
In our present practice prestressed con-
crete beams are in most cases designed and
proportioned by working stress design. The
provisions of ultimate design are used to check
the flexural strength of a section that has
already been designed. Furthermore, the pro-
visions for ultimate design in our present
specifications are more suitable for calculat-
ing the flexural strength of a given section.
It should be pointed out that there is a
relationship between working stress design and
ultimate design. Although they have different
bases, in fulfilling the objective of one, the
objective of the other is satisfied to a
(1)*
certain extent.
It can be shown that the provisions of
ultimate design can be used to proportion a
section with a more rigorous control of duc-
tility. The provisions of working stress de-
sign can then be used to check working stresses
in the section so designed. Furthermore,
rational design of a section is considerably
simpler by ultimate design than by service
load design.
The purpose of this study is to develop a
method by which a prestressed concrete beam can
be proportioned by the provisions of ultimate
design. It is intended to show the importance
of ductility in its influence upon the di-
mensions of the beam. In addition it is intend-
ed to study the influence of compression steel
on ductility and the proportions of a section.
We will consider a simply supported bond-
ed beam and assume that the strength of the
beam is measured by flexure. We will assume
that the only loads acting, in addition to
the prestressing force, are the weight of the
beam, the superimposed dead load, and live load.
*The numbers in parentheses refer to the
entries in Chapter VII, References.
II. CALCULATION OF ULTIMATE MOMENT
A. ASSUMPTIONS
For the sake of simplicity the determi-
nation of ultimate moment is discussed here
for prestressed concrete beams which have an
idealized section as shown in Figure 1. The
section considered is flanged; the prestressed
steel is assumed to be bonded to concrete, and
in addition to prestressed steel the section
is assumed to have non-prestressed compression
steel.
Practical sections sometimes contain non-
prestressed tensile reinforcement which in-
crease the flexural strength of the section
and reduces its ductility. Non-prestressed
tensile reinforcement is not considered here
since it does not contribute to our primary
purpose which is to develop a method of design
which leads to the lightest section for a
given strength and ductility.
The calculation of the ultimate moment is
usually based upon the following assumptions:
1. The strain distribution in concrete
varies linearly with depth in the compression
zone of the beam.
2. The stress-strain diagrams for the
prestressed as well as non-prestressed rein-
forcement are known.
3. Failure occurs when the strain in con-
crete at the top fiber reaches a limiting value.
4. The strain in non-prestressed com-
pression steel is equal to the strain in con-
crete at the level of compression steel.
5. The average strain in steel is not
greatly different from the maximum strain.
In addition to the above assumptions, the
tension contributed by concrete is usually neg-
lected since it is small at ultimate.
The neutral axis at failure may be either
in the flange or below the flange depending
upon the dimensions of the beam, the amount of
steel and the properties of both steel and con-
crete. We will first consider the case in
which the neutral axis falls in the flange.
B. FLEXURAL STRENGTH OF SECTIONS IN
WHICH THE NEUTRAL AXIS AT ULTIMATE
FALLS IN THE FLANGE
In this case the width of the compression
flange is constant and is equal to b, as
shown in Figure 2.
We will take f = f(e) as the equation
for the stress-strain diagram for concrete and
assume that this equation is the same for all
the fibers in the compression zone of the beam.
Since the width of the compression flange is
constant and the strain distribution in the
compression zone is assumed to be linear with
depth, the ultimate moment can be written as
follows:
2 u
Mu= ab ef(e) de + Afsu(d - a)
u 0
+ A'f' (a - d')
s su
where a = distance from neutral axis to the
top fiber
e = the limiting strain
u
f = stress in prestressed steel
su
at failure
fl = stress in non-prestressed com-
su
pression steel at failure
b = width of compression zone or top
flange
d = distance from the center of gravity
of prestressed steel to the top
fiber
d' = distance from the center of gravity
of the non-prestressed compression
steel to the top fiber
A = area of prestressed steel
A' = area of non-prestressed compression
steel
The above equation shows the sum of moment
of all forces about the neutral axis which is
the same as the bending moment at the section
due to the external loads. The first term on
the right side of the above equation is the
compression force contributed by concrete and
does not take into account the area of concrete
replaced by compression steel. This effect is
usually small, but it can be included by sub-
tracting the term As f(-a (a - d')) (a - de)
from the right side of the equation.
It can be seen from Equation 1 that for a
given section Mu depends on three quantities
that are not known, namely a, the depth to
the neutral axis; f su, the stress in the pre-
stressed steel; and f' , the stress in the
non-prestressed compression steel -- if f =
f(e) the stress-strain diagram for concrete
were known. Therefore, it is necessary to
obtain other relations in order to be able to
compute these unknown quantities.
From the equilibrium of horizontal forces
in the section we have the following equation:
ab f f(e) de + A'f' = A f (2)
e j s Su S SU
u0
As before, in the above equation the effect
of the area of concrete replaced by compression
steel is neglected. It may be included by
adding the term (A' f(-6 (a - d')) to the
5 a
right side of the equation.
The strain in prestressed steel at fail-
ure can be expressed as the sum of the follow-
ing quantities:
1. The strain in prestressed steel due
to effective prestress designated as e
se
Effective prestress is the magnitude of pre-
stress after losses, or at the time of de-
termination of ultimate moment.
2. The additional strain in steel that
is induced by sufficient load to make the
strain in concrete at the level of prestressed
steel equal to zero. It can be shown that in
bonded beams this addition to the strain in
steel is equal to ece, the strain in concrete
at the level of the prestressed steel due to
effective prestress.
Hence the total strain in steel at the
load corresponding to zero strain in concrete
at the level of steel is
e + e.
se ce
3. Additional strain in steel from the
load corresponding to zero strain in concrete
at the level of steel to ultimate. This ad-
dition can be expressed as follows:
_u (d - a) F.
a
The quantity F in the above expression is
a compatibility factor. If concrete were not
cracked and were bonded with steel its magni-
tude would be one. Since at ultimate, con-
crete at the level of prestressed steel is
cracked, the value of F in a particular
section depends on the position of crack and
condition of bond. Its magnitude is usually
less than one at ultimate, however it is
difficult to predict its value for a par-
ticular section in a given beam. A detailed
discussion of its magnitude is available.(2)
Hence e su, the strain in prestressed
steel at failure, can be expressed as the sum
of the quantities listed above
e = e + e + - (d - a)F.
su se ce a
Since e
ce
is small in comparison with
e
e + - (d - a)F
se a
it is often neglected.
In the following discussions the value of
F will be taken as 1.0 for bonded beams. Hence
the expression for the strain in prestressed
steel at failure can be written as follows:
e
s = e + e +-u (d - a).
su se ce a
Since we have assumed that the strain in
concrete at the level of non-prestressed com-
pression steel is equal to the strain in com-
pression steel, we have
e
e' = ~ (a - d') (4)
su a
where e' = the strain in the non-prestressed
su
compression steel at ultimate
We have also assumed that the stress-
strain diagrams for prestressed steel and non-
prestressed compression steel are known. The
equations for these diagrams are designated as
F(esu ) and G(e' ) respectively.
fsu = F(e su) (5)
su su
su= G(e' ) (6)
su su
In order to obtain M it is necessary to
solve Equations 1,2,3,4,5, and 6 simultaneously
for the six unknowns M , a, e f , e' and
u su su su'
fo .
su
In a special case in which the section has
no compression steel, e' and f' as well as
su su
Equations 4 and 6, vanish and the problem is
reduced to the solution of four equations for
four unknowns. In this case Equations 1, 2,
3, and 5 are the relations, and M , a, f ,
and e are the unknowns.
su
C. FLEXURAL STRENGTH OF SECTION IN
WHICH THE NEUTRAL AXIS AT ULTIMATE
FALLS BELOW THE FLANGE
When the neutral axis at ultimate falls
below the flange the expression for M is
the following:
b 2 u
M = u ef(e) de
0 e
+ (b - bi)a2 u ef(e) de
u e (a-t)/a
+ A f (d - a) + A'f' (a - dn)
s su s su
where b' = the web thickness
t = the flange thickness.
The first term on the right side of the
above equation is moment due to the compression
force developed by a rectangle of width b'
and depth a about the neutral axis. The
second term is the moment of the compression
force contributed by a rectangle of width
(b - b') and depth t about the neutral axis.
Figure 3 shows all the forces of a section in
which the neutral axis at ultimate is below
the flange. As before, the area of concrete
replaced by compression steel is neglected.
The expression f(e), the stress-strain
diagram in concrete, is assumed known and
applicable to each compressed fiber of the
beam. In Equation 7 there are quantities a,
f , f which are unknown and should be
su, su
determined by other relations.
From the equilibrium of horizontal forces
we have:
bla uf(,) de + (b -b f(e) de
u 0 u e (a-t)/a
+ A'f' = A f . (8)
s su s su
As before the area of concrete replaced by the
compression steel has been neglected.
Equations 3, 4, 5,and 6 are equally appli-
cable in this case. Hence in order to find
M it is necessary to solve Equations 7, 8,
3, 4, 5, and 6 simultaneously for the six
unknowns M , a, f , e , f' , and e .
u su su su su
D. SUMMARY OF GENERAL EQUATIONS
The ultimate moment in a section in which
the neutral axis at failure is in the flange,
or when a < t, is obtained by the simul-
taneous solutions of the following equations:
M / ef(e) de + A f (d - a)
u 2 j s su
u 0
+ A'f' (a - dl)
s su
ab f(e) de + Af' = A f
lul c
u 0
e = e + e + u (d - a) (3)
su se ce a
e, = - (a - d') (4)
su a
fu = F(e u )
f' = G(e' ). (6)
su su
The ultimate moment for a section in
which the neutral axis at failure is below the
flange, i.e., when t < a, is obtained by the
simultaneous solution of the following equations:
Mu b 2 u ef(e) de
u 0
+ (b - b')a2 u ef(e) de
eu e (a-t)/a
+ Af (d - a) + Asf' (a - d')
C C
a uf(e) de +(b -e b)a
eu f u f
0 e (a-t)/a
+ A'f' = A f
s su s su
e
e = e + e + -- (d - a)
su se ce a
e = -u (a - d")
su a
f = F(e )
f = G(e" ) .
su su
(7)
f(e) de
(8)
(3)
(4)
(5)
(6)
It should be remembered that in Equations
I and 2 as well as 7 and 8 the area of con-
crete replaced by compression steel is not
taken into account.
E. CALCULATION OF ULTIMATE MOMENT BY
THE PROPERTIES OF STRESS BLOCK
The procedure for the calculation of
ultimate moment discussed in the preceding
sections is the most general formulation for
the problem of computing ultimate moment and is
consistent with the assumptions made. These
assumptions are reasonable and are in agree-
ment with observations.
Practically, however, there are diffi-
culties in solving the problem by this pro-
cedure. Equations I and 7 for ultimate moment,
as well as Equations 2 and 8 for equilibrium
of horizontal forces, depend upon f(e) the
stress-strain diagram for concrete. The
stress-strain diagram for concrete is non-
linear and may change shape as concrete
strength changes. In addition the stress-
strain diagram obtained from a concrete cylin-
der may not truly represent this relationship
for all the fibers of the beam in the com-
pression zone, and obtaining a stress-strain
diagram for the concrete in the beam by test-
ing a beam is a lengthy procedure. Even if a
reasonable expression for f(e) were es-
tablished from tests, the non-linear nature of
the expressions for the compression force and
moment contributed by concrete, make the so-
lution of the non-linear simultaneous equations
tedious.
Since strain varies linearly with depth
in the compression zone, by adopting a stress-
strain diagram for concrete, the stress distri-
bution with depth will be defined. From the
shape of stress-distribution in the compression
zone both the force and the moment contributed
by concrete can be calculated for any section
even if the width of the section is variable.
The shape of the stress distribution in the
compressed zone of concrete is called the
stress block.
To simplify the problem for a practical
solution a somewhat different approach is in-
troduced. In the equations of equilibrium of
horizontal forces and moments, it is only
necessary to know the compression force and
moment contributed by concrete. If we can
somehow find a way of estimating the force and
moment contributed by concrete without knowing
the actual distribution of stress in the com-
pressed zone, there would be no need in having
the stress-strain diagram for concrete.
This can be achieved if the average stress
in the section and the point of action of the
compression force contributed by concrete were
known. That is, if the area and centroid of
the stress block were known, the ultimate
moment could be determined.
Let us first consider the case in which
the width of compression zone is constant,
i.e., the neutral axis at ultimate falls in
the flange. In this case let us express the
force and moment contributed by concrete as
follows:
e
u
ab f f(e) de = k k fc ab
u 0
2 b u
a2b f ef(e) de = k, k3 fc ab
u 0
(9)
(a-ak2)(10)
where kI = ratio of the average to maximum
stress in the compression zone
k3 = ratio of strength of concrete in
beam to that of cylinder
k2 = a ratio defining the position of
the center of gravity of the com-
pression force contributed by con-
crete; ak2 is the distance of
center of gravity from the top
fiber.
Equation 9 gives the compression force
contributed by concrete, and Equation 10 gives
the moment of this force about the neutral
axis. Figure 4 shows the forces in the section.
If the ratios kl, k3, and k2 can be de-
termined, this procedure is very convenient.
Substituting Equation 9 for the com-
pression force contributed by concrete in
Equation 2 we have
k k3 f ab + Af'su = As su (2a)
A f - A'f'
or k k = 5s SU s su
S3 f ab ()
C
Equation 11 indicates that kI k3 can be
measured from tests, since all the quantities
at the right side are either known or can be
measured. The quantity kI k3 for the ma-
terials commonly used is in the neighborhood
of 0.7. In a similar fashion k2 can be
evaluated since the moment in the section can
be measured. The quantity k2 varies around
0.42 and has little influence on the ultimate
moment.(3)
This approach is convenient in cases where
the width of compression zone is constant.
Equation I can now be written as follows:
Mu = kl k3 f' ab (a - ak) + Asfsu (d - a)
+ A'f' (a - d').
S su
This equation represents the sum of
moment of all forces in the section about the
neutral axis. In this case it can also be
written in the following convenient form by
taking moments about the centroid of the com-
pression stress block:
Mu = As su(d - ak2) + Asf u(ak2 - d').(la)
Equation 2 for the equilibrium of hori-
zontal forces in the section becomes the same
as Equation 2a.
This method is inadequate when the width
of the compression flange becomes non-uniform.
This condition occurs when the neutral axis at
ultimate falls below the flange. For the
section shown in Figure 5 let us express the
compression force developed by the rectangle
b'a as
-a uf(e) de =k k f a b'
Cu 1 c
0
and the compression force developed by the
rectangle (b - b')t as
e
(b - b')a f(e) de = clfi (b-b')t,
u (a-t)/a
where cI is the ratio of average to maximum
stress for the flange. Since the stress
distribution is undefined, the quantity c1
is undefined. It can vary widely between
0.7 and 1.00.
The moments of these compression forces
about the neutral axis are
b a2 Cu
2 f(e)e de
u 0
= k1 k3 f' ab" (a - ak2)
and (b J b)a2 u ef() ) de
u C (a-t)/a
= cI f' (b - b') t (a - c2t)
where c2 is the ratio of the distance between
the point of action of flange force and the
top fiber, to the flange thickness. The
quantity c2 is undefined; however, it does
not influence the moment appreciably. It is
in the neighborhood of 0.5. Figure 5 shows
the forces in the section.
Equation 7 can now be written:
M = k k3 f abl (a - ak2
+ cl f (b - bi) t (a - c2t)
+ As su (d - a) + A'fs (a - d').
ssu S SU
The equation gives the sum of moments of
all forces in the section about the neutral
axis. Often it is more convenient to take
moments about the center of gravity of pre-
stressed steel. In this case we have
Mu = k k3 ab' (d - k2a)
+ c f' (b - b') t (d - c2t)
+ A'f' (d - d'). (7a)
It is also possible to take momentsabout
the compression force contributed by the web.
In this case the ultimate moment can be ex-
pressed:
Mu = Asfsu (d - k2a)
u + l f' (b - b) t (k2a - c2t)
I c (b bl) t (k2a - c2t)
+ A'f' (ka - d').
2 (k
su
Equation 8 for the equilibrium of hori-
zontal forces in the section can be written:
kl k f' a b' + c f' (b - b') t
1 c 1 c
+ A'f' = A f
S su s su
In summary, when the neutral axis at
failure is in the flange, or when t > a, the
ultimate moment in the section may be obtained
by a simultaneous solution of the following
equations:
Mu = Asfsu (d - ak2)
+ A'f' (ak2 - d') (la)
S SU
kI k3 f ab + A'f' = A f
1 3 c s su ssu
e
su se ce a (d a)
e' = _- (a - d')
su a
f = F(e )
su = F(su
fl = G(e ).
su su
When the neutral axis is below the flange,
or when t < a, the ultimate moment may be
calculated by a simultaneous solution of the
following six equations.
Mu = kI k3 f' ab' (d - k2a)
+ cl f' (b - b') t (d - c2t)
+ A'f' (d - d')
s su
k, k3 f' ab' + c, f' (b - b') t
+ A'f = A f
s su s su
e = e + e + _u (d - a)
su se ce a
Ce u- (a - d')
su a
f = F(e )
su su
f' = G(e' )
As before, the area of concrete replaced
by compression steel has been neglected in
Equations la, 2a, 7a,and 8a.
The present thinking and practice in pre-
stressed concrete is based upon the above
equations. Instead of defining the stress-
strain diagram the properties of the stress
block k,, k3, k2, c,, and c2 are specified.
It should be pointed out that while k1 k3 and
3
non-prestressed
compression steel- |
b
"II t
prestressed steel - I t
kb
FIGURE 1. IDEALIZED I-SECTION
FIGURE 2. FLANGED SECTION, NEUTRAL AXIS IN THE FLANGE
I
i E
*__^______^____
A,
du Eu(a-t)/a
FIGURE 3. FLANGED SECTION, NEUTRAL AXIS BELOW THE FLANGE
A's f;u
ki k3fc ob
Asfsu =pbdfsu
FIGURE 4. FLANGED SECTION, NEUTRAL AXIS IN THE FLANGE
FLANGED SECTION, NEUTRAL AXIS BELOW THE FLANGE
FIGURE 5.
t/h
FIGURE 6. RELATIONSHIP BETWEEN ,* AND GEOMETRIC PARAMETERS OF THE SECTION
)
0.02
228
0.03
0.04
0.05
Strain
FIGURE 7. STRESS-STRAIN DIAGRAM FOR STEEL
II
214 ,
/
0.01
150
10c
n
E
0.06
0.07
A
A
A
18s
FIGURE 8. SECTION OF A BEAM WITH
LOW DUCTILITY e = 0.01 - WEIGHT OF
BEAM = 300 LB/FT 36
4"
r 15"
17 1/2" Strands
^ ___.
* U U U U U U if
5
* U U U U U U
16"
I_ _32"
4 I
5
16 1/2" Strands
-- --- t_.m
S. . . . jj
16"
FIGURE 9. SECTION OF A BEAM WITH VERY HIGH DUCTILITY
(eu = 0.03) - WEIGHT OF BEAM = 377 LB/FT
56"
- I
16= L
No. 9
36"
o.5
Z 6A
16
FIGURE 11. END SECTION OF THE BEAM WITH COMPRESSION REINFORCEMENT
2F
78
7A-
FIGURE 10. SECTION OF A BEAM IN WHICH THE
HIGH REQUIRED DUCTILITY (E = 0.03)
IS OBTAINED BY THE USE OF COMPRESSION
REINFORCEMENT - WEIGHT OF BEAM = 300 LB/FT
0
z
to -
0
z
~0
LL
0
z
0
I-
ULJ
-i
LU
0
LU
U-
LU
-j
LU
L-
LU
C,
I-
V
TABLE I SUMMARY OF SECTION PROPERTIES AND STRESSES
FOR SECTIONS OBTAINED BY ULTIMATE DESIGN
(All section properties are based upon the transformed section assuming n = 7)
(Negative stresses are tensile)
Stress Before Stress
A y y I A A' Weight Losses (transfer) After Losses
SA t b A s s ksi ksi
Section 2 4 2 .-2
in in in in in in lb/ft top bottom top bottom
(tens) (comp) (comp) (tens)
Illustrative
Example la 302 18.14 17.86 52,190 2.44 - 300 -0.14 2.56 2.38 -0.29
e = 0.01
su
Illustrative
Example lb 376 15.90 20.10 65,360 2.29 - 377 -0.09 2.20 1.68 -0.36
E = 0.03
su
Illustrative
Example 2 330 16.62 19.38 59,150 2.15 5.00 300 -0.14 2.29 1.90 -0.41
S = 0.03
su
Allowable Stresses
American Concrete Code (318-63)
(f' = 5 ksi; f'. = 4 ksi) -0.19 2.40 2.25 -0.42
A = transformed area, yt = distance from centroidal axis to top fiber; Yb = distance from
centroidal axis to bottom fiber; I = moment of inertia; As = area of prestressed steel;
A' = area of non-prestressed compressive steel; n = modular ratio for both types of steel;
fc = strength of concrete; f'. = strength of concrete at transfer; prestress at transfer
150 ksi; effective prestress after losses 128 ksi.
k2 can be estimated from tests with a suf-
ficient degree of precision, there is no way
to estimate c1 and c2 generally, so that they
will be applicable to all sections. In spite
of this weakness our specifications at present
assign specific values to these coefficients.
F. PROVISIONS OF THE AMERICAN CONCRETE
INSTITUTE BUILDING CODE
The provisions of the American Concrete
Institute Code (318-63) do not consider the
effect of compression steel on ultimate moment
or strain in steel, and instead of giving the
stress-strain diagram for concrete, the coef-
ficients of the stress-block are specified. 4)
In the following paragraphs the expressions
given in the Code will be derived and discussed.
When the neutral axis is in the flange and
there is no non-prestressed compression steel
from the preceding section, the ultimate moment
may be obtained by a simultaneous solution of
the following equations.
Mu = Af su (d - ak2)
k k f' ab = A f
1 3 c s su
e
(2b
e = e + e +-u (d- a) (3)
su se ce a
fsu = F(e su) (5)
Elimination of a between Equationslb
and 2b and between Equations 2b and 3 permits
the reduction of the above four equations to
three equations in which M , f , and e
u su su
are the unknowns. We will introduce p =
s
--, as percentage of steel. From Equations
lb and 2b we have:
k2 fsu
m = A d ( p ) (13
1u s su k k3 c
from Equations 2b and 3:
f kI k3 e
su 1 3 u
f e -e -e + e
c su se ce u
fsu = F(e su).
su 'su'
The American Concrete Institute Code
k2
(318-63) gives k k = 0.59 and requires
that the expression Tor the ultimate moment
be multiplied by a capacity reduction factor
as follows:
f
Mu = [Afsu d (1 - 0.59 p-Sf)
c
where 1 is a capacity reduction factor,
taken as 0.9.
The Code implies that when the stress
strain diagram for steel is available, f
su
and e can be obtained by a simultaneous
su
solution of Equations14 and 5; if not, f
su
in Equation 15 may be calculated by the follow-
ing expression:
f = f' (1 - 0.5 p f'/f),
su s s c
where f' is the ultimate strength of pre-
) stressing steel.
The Code controls the ductility of the
beam by requiring that the ratio p f su/f'
be less than or equal to 0.3. In cases in
which p fsu /f is greater than 0.3, the
ultimate moment may be calculated by Equation
15 provided that p f /f' is taken as 0.3.
This resuts in the foowing equation:
This results in the following equation:
M = t (0.25 f' bd2).
u c
(17)
By limiting the value of p f su/f the
Code in effect requires that the strain in
steel at failure be not less than a limiting
) value, since from Equation 14 we have
su 3 < 0.3.
Pf - - e + --
c su se ce u
The above requirement sets an acceptable
range for e su, roughly in the neighborhood
of 0.01, below which it cannot go as far as
calculation of M is concerned.
u
For sections in which the neutral axis at
ultimate falls below the flange and there is
no non-prestressed compression steel, from the
preceding section the ultimate moment may be
obtained by a simultaneous solution of the
following equations:
M = k k f' ab' (d - k a)
+ cl f' (b - b') t (d - c2t)
k k f ab' + c, f' (b-b') t = Asf (8b)
1 3 c c ssu
e
s = e + e + - (d - a)
su se ce a
f = F(e ).
su su
Elimination of a between Equations 7b
and 8b and between Equations 8b and 3 will
result in three equations in which Mu fsu,
and e are the unknowns. We have
su
M = d[A fsu - C f' (b- b') t]
u s su 1 c
Af - c f' (b - b')t
[l- s su c k21
k k fP bid
13 c
+ c, fj (b - b') t (d - c,t)
f kI k3 e b
(su 1 3 u
P fr e - e - e + e (1
c su se ce u
+ c (l -_ ) (19)
fsu = F(e su) (5)
The American Concrete Institute Code (318-
63) designates the quantity
A f = k k f' ab'
sr su 1 3 c
as that part of the force contributed by the
prestressed steel that develops the rectangle
a b' in the compression zone of a flanged
section. The quantity
A f = c f' (b - b') t
sf su 1 c
is designated as that part of steel that
develops the rectangle (b - b') t in the
compression zone.
Since A = A + A we have
s sr sf
Af - c f' (b - b') t = A f .
s su 1 c sr su
By substituting the above quantity in
Equation 18 we have
k A f
M = A f d (1 2 sr su
u sr su kI k3 b'd f
1 3 C
Scf (b - b') t (d - c2t).
The Code gives k2/kI k3 = 0.59, cl =
0.85, and c2 = 0.50 and requires that the
expression for ultimate moment be multiplied
by a capacity reduction factor as follows:
A f
M = [Asr f d (1 - 0.59 -- u)
u sr su b'd f
+ 0.85 f' (b - b') t (d - 0.5t)].
The arbitrary assumption of cI = 0.85 and
c2 = 0.5 by the Code may result in some in-
accuracies when the neutral axis is very
slightly below the bottom fiber of the flange.
For sections of small b'/b the assumption of
cI = 0.85 may even lead to a negative quantity
for A
sr
The Code implies that when the stress-
strain diagram for the prestressed steel is
available f and e can be obtained by a
su su
simultaneous solution of Equations 19 and 5.
If the stress-strain diagram is not available
the Code permits calculation of f by
su
Equation 16.
As before, the Code limits the quantity
Asr su
bd- to 0.3 or less. If this ratio is
b-d fr
c
more than 0.3, the ultimate moment may be
calculated by Equation 20 provided that
A f /f'd f' is taken as 0.3. This results
in the following equation:
M = < [0.25 f' b' d2
u c
+ 0.85 f (b - b') t (d - 0.5t)] (21)
c
since we have
A f su/b d f
sr su c
1 3 u < 0.3.
e -e -e +e -
su se ce u
Limitation of this quantity is equivalent to
requiring a minimum value for e su, which as
before, is in the neighborhood of 0.01.
It should be pointed out that the limi-
tations on the ductility of the beam as given
by the American Concrete Institute Code do not
include the effect of any compression steel
that may be present at the top of the beam.
It will be shown in Chapter V, Section C that
this effect is not negligible.
The neutral axis may either fall in the
flange or below the flange. The distinction
between the two cases is made according to the
following inequalities:
If t > a = p f d/kl k f' the neutral
su I 3 c
axis falls in the flange. Since the American
Concrete Institute Code gives kl k3 = 0.7
this condition can be restated as follows:
fsud
When t > 1.4 p - , the neutral axis
falls in the flange. f d
su
When t < 1.4 p , the neutral axis
falls below the flange.c
III. PROVISIONS FOR SAFETY AND DUCTILITY IN ULTIMATE DESIGN
In ultimate design a section is
proportioned in such a way that the ultimate
moment is greater than the moment developed
under service loads by a prescribed quantity,
and that it deforms a certain amount before it
fails.
These concepts may be stated in the form
of the following requirements:
M > N (M + M ) + N M (22)
and
su > e (23)
where M = the required flexural strength
of the beam
Nd = the load factor for the dead load
M = moment due to the weight of the
g
beam
M = moment due to the superimposed
dead load
N = the load factor for the live load
M. = moment due to the live load
e = strain in steel at ultimate
su
es9 = limiting strain in steel
Expression 22 states that the required
flexural strength of the beam should be at
least equal to Nd(M + Ms) + NMA, which is
a requirement for the strength of the beam.
For the type of loads considered here, the
American Concrete Institute Code (318-63) gives
Nd = 1.5 and N = 1.8.
Expression 23 states that the ductility
of the beam should be large enough so that the
strain in steel at ultimate will be at least
equal to a given limiting value designated
as e sA
From the discussions in the preceding
section we know that the prediction of failure
on the basis of moment depends upon the limit-
ing strain e , i.e., a value for e is
U U
required for an unambiguous definition of
failure.
There are many ways that ductility of the
section can be measured. In Chapter II,
Section F, it was shown that the American
Concrete Institute Code defines ductility by
the following quantity
kl k3 eu
e - e - e + e
su se ce u
Ductility may also be measured by the
curvature at ultimate which may be defined as
follows:
e e - e - e + e
u su se ce u
a d
where cp is the curvature of the section.
Both of these measures of ductility are
based upon the magnitude of e which is the
strain in prestressed steel at failure. Hence
esu may be used as a measure of ductility.
This method is based upon the assumptions
that we have a value for 6 which defines
u
flexural failure, and we have a minimum limit-
quantity for e su, designated as es5. In
addition we have the stress-strain diagram for
prestressed as well as non-prestressed com-
pression steel.
IV. ULTIMATE DESIGN OF SECTIONS WITHOUT COMPRESSION REINFORCEMENT
A. METHOD OF ANALYSIS USED
The design procedure developed here does
not depend on the method of analysis. The
following are adopted.
When the neutral axis is in the flange,
Equations 15, 14, and 5 will be adopted for the
calculation of ultimate moment. Equation 15 is
the expression for the ultimate moment given by
the American Concrete Institute Code (318-63).
We will further assume kI k3 = 0.70.
These equations are rewritten here for
fsu d
convenient reference when t > 1.4 p f su
c
f
M = [Afsu d (1 - 0.59 p -)] (15)
us su fo
f 0.7 e
su u
P f e - e - e +e
c su se ce u
(14a)
f 0.7 e
P f- = e - e - e + e (b)
c su se ce u
+ 0.85 (1 - -) -
b~- d
fsu = F(e su).
In addition to the above equations we
also know that e > e6 This condition
eliminates the need of considering Equations
17 and 21.
Introducing q = pf /f', the above
expressions may be written in dimensionless
form as follows:
When q < 0.7 t/d:
M
2----- = q (1 - 0.59q)
cbd f
c
(5) where
fsu = F(e su).
When the neutral axis is below the flange,
Equations 20, 19, and 5 will be adopted for
calculation of the ultimate moment. Equation
20 is the expression of the ultimate moment
given by the American Concrete Institute Code.
We will assume kI k3 = 0.70, c1 = 0.85, and
c2 = 0.5. f d
When t < 1.4 p --- we have:
c
A f
M = $ [A fs d (I - 0.59 b fsu)
u sr *su b'd f
c
+ 0.85 f' (b - b') t (d - 0.5 t)] (20)
c
(15a)
(14a)
V.,
Se - e - e +
su se ce u
When q > 0.7 t/d:
M
u I + (t - ) - 0.59 - q]
*bd2f d k
.085 t )
0.7 e (b'/b)
where q = u- - +
su se ce u
+ 0.85 (1 - bl/b) t/d.
(20a)
(19a)
(19a)
(5)
B. DETERMINATION OF AREA OF THE BEAM
Let us write M /$bd2 f' = Q. Expression
U C
22 can then be written as an equation in the
following form:
Qebd2ft = Nd (M + Ms) + NM M.
Substituting A/hi for b where A is the
gross cross-sectional area of the beam, h is
the overall depth, and * is a dimensionless
shape factor, we have:
M yAL =, A
m = 8 = *
Ne
or Ms +- M
A =d N d
d2 f'Q 2
h*N d 8
2 N1 N
d fc - - M - M
c N N d e
hence they should be made as small as possible
without causing the dimensions of the beam to
become unreasonably thin.
From the expression for Q' and Equation
25, it can be seen that Q' decreases with
k. However, since the bottom flange of the
beam should be large enough to permit the
placing of steel, k cannot be reduced in-
definitely. It should be made as small as
possible.
The quantity d/h should be made as
large as possible; however, it is doubtful
that in most practical cases it can be made
greater than 0.9.
Since Q' increases with q it is de-
sirable to make q as large as possible;
however, Expression 23 sets the upper limit
for q.
where y is the unit weight of concrete.
For the idealized I-section shown in
Figure 1 the following general expression may
be used for 4:
S= A- = -t (1 + k) + - (l - 2 ). (25)
Sbh h b h
The quantity k in the above equation is
the ratio of the width of bottom flange to that
of the top flange. Equation 25 is plotted in
Figure 6 for a few typical cases. A study of
Equation 24 shows that for a given depth and
type of concrete A depends upon * and Q
only. It can be seen that A decreases with
Q and increases with t. In order to decrease
the area of the beam it is necessary to in-
crease Q and decrease *.
Since both Q and * are functions of
t/h, b'/b, and d/h, in order to study their
variation with A it will be more convenient
to study the variation of Q' = - Q with A.
h *
In order to obtain the least area it is
necessary to make Q' as large as possible.
The quantities t/d and b'/b decrease with Q'
Since Expression 23 sets the required
minimum ductility of the beam as a strain in
steel equal to es£, the required maximum q
consistent with the required ductility can be
computed from Equations 14a and 19a as follows:
0.7 e
u
qmax e - e - e + e
si se ce u
0.7 e
or max e - e - e e+ e
s- se ce u
+ 0.85 (1 - ) (-)
bs- (ý)
(14b)
(19b)
whichever applies.
It should be pointed out that Equations
14b and 19b contain the additional parameter
Cse, the strain in steel due to effective
prestress. It can be seen that since e
se
increases with qmax it should be taken as
large as practicable. The practical upper
limit for e for the materials used in pre-
tensioned construction is about 0.005.
C. ILLUSTRATIVE EXAMPLE I.
The following example is presented to
illustrate the procedure for the ultimate
design of a prestressed concrete beam and to
show the influence of the required ductility
on the dimensions of the beam so designed.
It is necessary to design a simply sup-
ported beam of 54 foot span subjected to a
superimposed dead load of 1.0 kips per linear
foot (klf) and a live load of 0.6 klf. The
load factors are given as Nd = 1.5 and N
= 1.8, and the capacity reduction factor is
4 = 0.9. Design the section for: 1. a minimum
ductility corresponding to e = e = 0.01,
and 2. a minimum ductility corresponding to
es = e = 0.03.
su s2
The effective prestress may be taken as
the prestress after losses which in this
problem is given as 128 kips per square inch
(ksi). This value corresponds to a prestress
of 150 ksi at transfer if the effectiveness
is taken as 0.85. The strain due to effective
prestress is e = 0.0044, and e may be
se ce
approximated as 0.0006. This approximation
may be verified after the section is designed.
Also for the purposes of this problem assume
fl = 5 ksi, e = 0.004, y = 0.15 kips per cubic
c u
foot (kcf), and h = 36 in. The stress strain
diagram for steel is shown in Figure 7.
1. Section with Minimum Required Ductility
Corresponding to e = 0.01
In Chapter IV, Section B it was shown
that the quantities t/h, b'/b, and k in-
crease with A, hence they should be taken
as small as possible. Here they will be taken
as t/h = 1/6, b'/b = 1/4, and k = 0.8.
The shape factor of the section * is
obtained using Equation 25 as follows:
- =-1 (1 + 0.8) +1 (1 - 2 ) = 0.467.
Assuming d/h = 0.9, for h = 36 inches, we
obtain d = 32.4 inches and t/d = 0.185. The
values of q and Q can be computed from
max
Equations 19b and 20a since in this case q
> 0.7 t/d and the neutral axis at ultimate is
below the flange. From Equation 19b we have:
q 0.7 x 0.004 x 1/4
max 0.010 - 0.0044 - 0.0006 + 0.004
+ 0.85 (1 - I) 0.185 = 0.196
and from Equation 20a:
M
Q ---u- 0.196 [1 + 0.185 (4-1)
<bd fl
c
- 0.59 x 4 x 0.196] - 085 (0.185)2(4-1)
= 0.170.
The area A of the section can be
obtained using Equation 24 with the following
values:
M =- x 54 x 1.0 x 12 = 4370 k-in
s 8o
M = I x 54 x 0.6 x 12 = 2620 k-in
1 -9 4 .
-- = 1.2
Nd
d 2f' Q (32
_ _c (32
h*Nd
.4)2 x 5 x 0.170 x 0.90
36 x 0.467 x 1.5
= 31.8 k-in
2 2
15 x 54 = 4.6 k-in.
Therefore
4370 + 1.2 x 2620 276 in
A =- 31.8 - 4.6 =276 in
b = A = 276 = 16.4 in
b-h 0.467 x 36
kb = 0.8 x 16.4 = 13.1 in
b' =- x 16.4 = 4.1 in.
4
The stress in the steel at ultimate can be
found from the stress-strain diagram for steel
shown in Figure 7.
f = 214 ksi.
su
The amount of prestressing can be found from
the definition of q to be
0.196 x 5
p- 2142 - 0.00458
A = 0.00458 x 16.4 x 32.4 = 2.44 sq. in.
(Use seventeen 1/2-inch strands)
A total of seventeen 1/2-inch strands are
needed. Each 1/2-inch strand has an area of
0.1438 square inch. The final dimensions of
the section in this solution are shown in
Figure 8. The bottom flange has been widened
to accommodate the reinforcement. Both top
and bottom flanges are tapered to facilitate
construction. The properties of the trans-
formed section as well as the stresses at the
top and bottom fibers before and after losses
are listed in Table 1.
2. Section with Minimum Required Ductility
Corresponding to e = 0.03
The ultimate strain in the steel required
for this example is very large and is not used
frequently in actual practice. It has been
selected to show that direct design for the
largest levels of ductility is possible and to
study how it affects the shape of the section.
All the quantities are the same as in
part 1 of this example except that in this
case e = 0.03, and k, the ratio of the
width of bottom flange to width of top flange,
is different. The bottom flange needs only be
large enough to accommodate the reinforcement.
Due to the fact that the higher the ductility
the wider the top flange has to be to provide
the required area under compression, it is
necessary to select k small enough so that
the bottom flange is not overdesigned. A
value of k = 0.5 is selected. From Equations
24 and 25 it can be seen that the area of the
section decreases with the web thickness.
However, the web thickness cannot be reduced
indefinitely, since the cover requirement for
the draped reinforcement and the shearing
strength determine the minimum thickness. In
this case let b'/b = 1/6.
In view of the large ductility required
the neutral axis is bound to be closer to the
top fiber than in the preceding example. De-
termination of this position affects the se-
lection of the thickness of the top flange.
The value of a can be obtained from Equation
3 as follows:
0.030 = 0.0044 + 0.0006 + 0a004(32.4-a)
a
from which a = 4.47 inches. Use of t >
4.47 inches would result in an overdesigned
top flange, the bottom fibers of which would
not be subjected to compressive stresses at
ultimate. The value of t = 4.5 inches is
selected as a practical dimension. Then t/h
= 0.125 and Equation 25 yields the value of
as follows
= (0.125)(1 + 0.5) + (1/6) [1-(2)(0.125)]
= 0.312.
The values of q and Q can be obtained
from Equations 14a and 15a as follows:
0 = (0.7)(0.004) =0 0965
q = 0.03 - 0.0044 - 0.0006 + 0.004
Q = 0.0965 [1 - (0.59)(0.0965)] = 0.091
The above equations are applicable because
q < 0.7 t/d = 0.0973, and the neutral axis is
in the flange of the resulting section.
The area of the section can be computed
from Equation 24 using the known values of
case 1 and 25.5 k/in as the modified value
of d2f'QO/htNd.
Therefore,
A 4370 + (1.2) (2620) 360 in2
25.5 - 4.6 360 in
b = A _ 360 -32 in
b = = (0.32)(36) = 32 in
h (0.312) (36)-
kb = (0.5)(32) = 16 in
bi =-- = 5.34 in.
From Figure 7, the stress strain diagram for
steel, f may be obtained
f = 228 ksi
su
p = .96 5 = 0.00212
2
A = 2.20 in .
(Use sixteen 1/2-inch strands)
Figure 9 shows the final section of the
beam. The dimension of the bottom flange is
the minimum required to accommodate the pre-
stressing steel at the required depth. It
coincides with the calculated value of kb
thereby requiring no adjustments. If kb
turns out to be larger than necessary, only the
minimum required should be used, as the bottom
flange contributes nothing to ductility and
strength. If the adjustment of the dimensions
is large, recalculation may be necessary to
improve the shape of the section. The proper-
ties of the transformed section and the
stresses before and after losses for this part
are also given in Table 1.
A comparison of Figures 8 and 9 indicates
that a large ductility results in a heavy
section. In this particular example increas-
ing the required e from 0.01 to 0.03
causes the weight of the beam to increase by
26 per cent. There is a 6 per cent saving in
the amount of prestressing steel as the more
ductile section requires one 1/2-inch strand
less. This is because the larger stress in
the steel at ultimate not only compensates
the additional weight of the heavier section
but also results in less required area of
steel.
V. ULTIMATE DESIGN OF SECTIONS WITH NON-PRESTRESSED COMPRESSION STEEL
A. METHOD OF ANALYSIS USED
Determination of flexural strength of
prestressed concrete beams with non-prestressed
compression steel was discussed in Chapter II,
Sections D and E. It was shown that the
ultimate moment of a given section in which
the neutral axis falls below the flange can
be calculated by a simultaneous solution of
Equations 7a, 8a, 3, 4, 5, and 6.
For design purposes the ultimate moment
will be computed by Equation 7b assuming that
the stress in compression steel has reached the
yield point, and taking cI = 0.85, c2 = 0.5,
and k2 = 0.42. In addition the expression for
the ultimate moment will be multiplied by the
capacity reduction factor t.
Mu = [A f (d - 0.42a)
+ 0.85 f' (b - b') t (0.42a - 0.5t)
c
+ A' f* (0.42a - d')]
s y
where f* = f - 0.85 f', and f is the
y y c y
yield point of non-prestressed compression
steel.
Here it is assumed that non-prestressed
compression steel is American Society for Test-
ing and Materials Billet Steel A-15 with a flat
stress-strain diagram beyond the yield point.
The stress in the area of concrete replaced by
compression steel is taken into account by the
term 0.85 f' which is an approximation.
c
In Equation 26 the quantities a and
f are unknowns, and for their determination
we need Equations 8a, 3, and 5. In Equation
8a we will take kI k3 = 0.7 and cI = 0.85.
0.7 f' a b' + 0.85 f' (b - b')t
c c
+ A' f* = A f
s y s su
e = e + e + - (d - a)
su se ce a
fsu = F(e su)
Elimination of a between Equations 8b
and 3 will result in the following:
0.70 f' e (b'/b)
f = c u
su p [ - e - + e
su se ce u
fl A'
+ 0.85 (1 - b-) - f s. (27)
p b d yA
The condition that the compression steel
has yielded is satisfied by the following
inequality:
e
e = u (a - d') > e .
su a - y
We will substitute for a from Equation
8b in the above inequality and rearrange it to
arrive at:
e F. Asf A' f*
d' < (1 - -1) s su - s Y
e 0.7 fS b' 0.7 f' b1
u c c
- 1.21 ( - 1)t .
Hence for the solution of unknowns Mu,
f su, and esu we have available Equations 26,
5, and 27.
Equation 26 may conveniently be expressed
in the dimensionless form:
S=q 1 + (1 - -) (--- 1)
c
C
- 0.59 q (1 - -) -
-q d' 1 0.85 ()2 2b
q d 2d
f
where q = p -l
c
A' f*
, = bs f
q bd fl
c
and
0.7 e (b'/b)
q e - e - e + e
su se ce u
+ 0.85 (1 - -L) .
1) (26a)
The preceding equations were developed
for the case of a T flanged section in which
the neutral axis falls in the web. This
condition can be stated
t > 1.4 (q - qi) d.
When the above inequality is not satisfied,
the neutral axis falls in the flange and the
flanged section becomes a rectangular section.
For this case, b' = b and Equations 26a,
27a, and 28a yield
M , 2
-- = Q = q [I - 0.59q (1 - -)
*bd 2f q
c
q d
0.7 e
q = q + C u
e - e - e + e
su se ce u
0.7 e
q max = q' + - e -u
s£ se ce u
(27a)
Expression 28 can similarly be presented
in a dimensionless form as follows:
e b
d' < d (1 - )[(q- q') 0.7 b'
u
- 1.21 - ( - 1)]. (28a
The expression for the required maximum
value of q consistent with the required
ductility corresponding to e = es is
given by the following:
0.7 e
max e -6 -6e +- (-
si se ce u
+ o.85 (1 - .) (27b
when q' = 0, the beam has no non-prestressed
compressive reinforcement, and Equations 26a,
27a, and 27b become identical with Equations
20a, 19a, and 19b respectively.
d' < 1.4d (1 - -Y) (q - q').
u
(26b)
(27c)
(27d)
(28b)
Equation 27a implies that for a large
required ductility corresponding to e >
su ss£
it is possible to increase q, hence to de-
crease the area of the beam, by increasing q'.
This relationship is very useful when the re-
) quired ductility is high.
B. ILLUSTRATIVE EXAMPLE 2
In order to show that the non-prestressed
compressive reinforcement increases the
ductility without increasing the area of the
section, the following example is presented.
It is required to design the section in
"Illustrative Example 1" in such a way that
for a ductility corresponding to e = 0.03,
) the area of the section will be the same as
that for a ductility corresponding to e
su
= 0.01.
The yield point stress of the compressive
reinforcement may be assumed as f = 50 ksi.
The section designed in part 1 of
The section designed in part 1 of
"Illustrative Example 1" has a ductility corre-
sponding to e = 0.01. The problem is to
determine how much compressive steel of the
type given should be placed, so that the
ductility of the section will reach that corre-
sponding to e = 0.03.
The neutral axis was determined in Example
1, part 2 for the same required e as being
at a distance from the top fiber given by a =
4.47 in. In this case a < t = 6 in. and the
T section behaves as a rectangular beam. Using
Equations 27c and 26b with Q = 0.170 as in
Example 1, part 1, and d' = 2 in. the follow-
ing independent relations between q and q'
are obtained:
q = q1 + (0.7)(0.004)
q 0.03 - 0.0044 - 0.0006 + 0.004
0.170 = q - 0.59 (q - q') - q' .2
Solution of the above equations yields
q = 0.181 and q1 = 0.084.
From Figure 7, the stress-strain diagram
for steel, e = 0.03 corresponds approxi-
mately to f = 228 ksi. We have
su
S= q su (0.18)(5) = 0.00397
A = (0.00397)(32.4)(16.3) = 2.10 sq. in.
(Use fifteen 1/2-inch strands)
Also
P' = q' f/f = 50-(0.85)(5) = 0.0093
or
A' = (0.0093)(32.4)(16.3) = 4.9 in.2
(Use five #9 bars of hard grade steel)
The distance of these bars from the top
must be such that e' > e , if yielding of
s y
the compressive reinforcement is to occur at
ultimate. This condition can be checked using
Expression 28b:
d' = 2 in. < (1.4) (32.4) (1 - 0.00167
(0.181 - 0.084) = 2.6 in.
If d' greater than 2.6 inches had been
selected, compressive steel at ultimate would
not yield, requiring use of the actual value
of stress in compression steel which is less
than the yield stress.
The length of these non-prestressed bars
need not be the total span of the beam. Theo-
retically they are not needed at a section
where the required q is that of the section
without the compression reinforcement. Assum-
ing the distribution of the required q to be
the same as the distribution of bending
moments, the theoretical section at which the
bars are no longer needed can be determined by
the distance XI from the center line as
follows:
54 0.084 , .
X =-2- 18 = 18.4 ft.
Further economy can be achieved if the non-
prestressed compression bars are separated in
two groups. A group of three short bars re-
presenting a q of .050 could then be cut at
a section theoretically at a distance X2 =
14.2 feet from the center line. Taking into
consideration the additional length required
to develop bond the non-prestressed bars may
be specified as 2 # 9 x 40' and 3 # 9 x 32'.
Figures 10 and 11 show the section of the
beam at midspan and at the end respectively.
Figure 12 shows the profile of the prestressed
and non-prestressed steel. Three web strands
have been draped to prevent overstressing of
the end sections of the beam. In addition to
the 5 # 9 non-prestressed top reinforcement
an end # 5 has been added for practical con-
struction purposes. Stirrups have been de-
signed according to American Concrete Institute
Code (318-63). As before the properties of
the transformed section as well as the stresses
before and after losses are given in the table.
A reduction in the amount of non-pre-
stressed compression reinforcement is possible
with a section having a wider top flange. The
parameter q' is related to q by Equation
27b. Selection of a smaller value of q' or
q'/q would fix q and permit the determi-
nation of the required Q by Equation 26b.
The area of the section and its final shape can
be determined as usual from Equation 24. If
the proper values of t/h, b'/b, and k were
selected the new section will present a flange
wider than that of Example 1, part 1, but not
as large as that of Example 1, part 2. Also
the compressive reinforcement required will be
smaller than that of Example 2. This solution
would show that to obtain high ductility a
compromise section can be obtained if some
increment of weight is tolerated with a smaller
amount of non-prestressed compression steel.
C. COMPARISON OF THE THREE SOLUTIONS
It has been shown that ultimate strength
design provides a convenient procedure which
leads to well proportioned sections. The de-
sired ductility and strength were used as the
fundamental constraints for proportioning the
sections, while the stresses at transfer and
under service loads were checked.
An examination of Table I shows interest-
ing details. The beam of Example 1, part I
with a required ductility corresponding to
e = 0.01 required more prestressing steel
(17 strands) than the beams of Example 1,
part 2 and Example 2 with a required ductility
corresponding to e = 0.03.
For the stress-strain diagram of pre-
stressed steel adopted in these examples, any
increase in ductility is accompanied with an
increase in stress in steel at ultimate. For
the larger ductility considered here the stress
in steel increases at ultimate from 214 ksi
to 228 ksi. This increase in steel stress
causes a decrease in the required area of
prestressing steel.
The beam of Example 1, part 2 shows that
by increasing the width of the top flange and
thereby adding concrete area to the compression
zone, high ductility can be obtained. This,
however, increases the weight of the section
by 26 per cent, but decreases the amount of
prestressing steel to 16 strands. The in-
crease in stress in steel at ultimate not
only supports the additional weight of the
beam, but also permits a reduction in the re-
quired area of steel. Under the service loads
this beam shows, however, a tendency for a
large tensile stress at the bottom fiber due
to the smaller amount of prestressing force.
The beam of Example 2 shows a different
way of obtaining high ductility. Five #9 bars
are added to the top flange of the low
ductility section of Example 1, part 1. This
increment in compression area raises the
neutral axis and increases the lever arm of
the resisting couple by approximately five
per cent. In addition the stress in the
steel at ultimate is increased from 214 ksi
to 228 ksi, approximately seven per cent.
These two factors combined explain the 12 per
cent reduction in the number of prestressing
strands (from 17 to 15), since the required
tensile force at ultimate can be obtained with
less area of steel at a higher stress and a
larger lever arm. The non-prestressed bars
also provide additional tensile strength for
the top part of the beam at transfer and during
handling operations. Furthermore, they have a
tendency to reduce the inelastic deflections
due to creep.
VI. SUMMARY AND CONCLUSIONS
The work reported here presents a study
of ultimate design of prestressed concrete
beams. It consists of a detailed discussion of
various methods for calculating ultimate moment
of practical sections including sections with
non-prestressed compression reinforcement. A
method is presented by which a prestressed
concrete beam can be proportioned by ultimate
design. Particular emphasis has been placed
upon the requirement of ductility and its
influence upon the dimensions of the section.
The design examples presented show the actual
method of proportioning as well as the in-
fluence of ductility on the dimensions of the
beam.
The following conclusions may be drawn
from the study presented in this work.
1. Methods with varying degrees of
accuracy can be developed for the determination
of ultimate moment in terms of the properties
of the beam section. For design purposes the
ultimate moment may be expressed conveniently
in a dimensionless form.
2. The expressions for the calculation
of the ultimate moment and ductility given in
the American Concrete Institute Code (318-63)
do not include the effect of non-prestressed
compression steel. The influence of com-
pression steel on the flexural strength is
small and may be ignored. However, neglect-
ing the effect of compression steel upon the
ductility of the section is unreasonable.
Compression steel contributes appreciably to
the ductility of the section and should be
taken into account. "Illustrative Example 2"
shows that the most expeditious way for in-
creasing the ductility of a section is by
placing non-prestressed compression reinforce-
ment as near the top fiber as possible.
3. A prestressed concrete beam can be
proportioned for a given required minimum
flexural strength and ductility. The stresses
at transfer and at service conditions may be
checked in a section thus obtained.
4. The dimensions of a section are in-
fluenced greatly by the required ductility.
An increase in the required ductility results
in an increase in the required area of the
section, unless compression steel is provided.
5. For a large required ductility con-
siderable saving in the area of the beam may
be effected by use of non-prestressed com-
pression steel. Compression steel has ad-
ditional advantages such as its contribution
to the crack stability of top fiber, its use
as spacer for the web reinforcement and its
function in providing more safety for the beam
during transportation and erection.