Development Rational Design Part 2
Transcript of Development Rational Design Part 2
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Editors quick points
This paper summarizes the results o an analytical research
program undertaken to develop a rational design procedure or
precast concrete slender spandrel beams.
The analytical and rational models use test results and re-
search fndings o an extensive experimental program pre-
sented in a companion paper.
The overall research eort demonstrated the validity o using
open web reinorcement in precast concrete slender spandrel
beams and proposed a simplifed procedure or design.
Development
of a rational
design
methodology
for precast
concrete
slender
spandrel
beams: Part
2, analysis
and design
guidelinesGregory Lucier,Catrina Walter, Sami Rizkalla,Paul Zia, and Gary Klein
Design and analysis o precast concrete slender spandrel
beams is not a simple task because o eccentrically applied
loads, lateral support conditions, and asymmetrical cross
sections. Signicant shear and torsion stresses develop in
the end regions and act in combination with in-plane and
out-o-plane bending stresses. When vertical eccentricloads are applied to the ledge or corbels o a typical slender
spandrel beam, it will defect downward, laterally inward
at the top edge o the web, and laterally outward at the
bottom edge o the web at midspan. The combination o
vertical and lateral defections results in a warped defected
shape and a tied arch cracking pattern. It is well docu-
mented that ace-shell spalling and spiral cracking do not
develop in slender spandrel beams at ailure, though these
limit-state behaviors are implicitly assumed in current
practice.15 The design method ound in the sixth edition o
the PCI Design Handbook: Precast and Prestressed Con-
crete6
oten results in conservative, heavy reinorcement
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results1 to study the various parameters that infuence the
behavior o slender L-shaped spandrel beams.
The nite element code ANACAP used in this study is
capable o analyzing plain, reinorced, or prestressed
concrete members in three dimensions. The program has
extensive nonlinear capabilities and includes an advanced
concrete material model. Additional details describing the
nite element code can be ound elsewhere.2
Nonlinear finite element model
Considering symmetry, the nonlinear nite element analy-
sis was based on modeling one-hal o a typical slenderspandrel beam using approximately 4,700 twenty-node
brick elements. The exact number o elements varied
depending on the specics o the case under study. The
large number o elements was necessary to maintain a su-
ciently ne mesh around all o the loading and boundary
conditions as well as in the end region, where ailure was
expected to occur. The modeled portion o a typical 45-t-
long (13.7 m) beam consisted o ve ledge loads, three
tieback connections, two lateral end restraints, one primary
vertical end reaction, and a symmetry condition at mid-
span. Figure 1 shows the nite element mesh and bound-
ary conditions used or a typical L-shaped spandrel.
when used or slender spandrel beams, which oten have
aspect ratios much greater than 3.0.3, 7
This paper discusses the design and analysis o precast
concrete slender spandrel beams. The experimental pro-
gram on which it is based was reported in the companion
paper.1
Objective
The objective o this research was to develop rational de-
sign guidelines or precast concrete slender spandrel beams
with an aspect ratio (height divided by width) o at least
4.6. The guidelines are expected to simpliy the reinorce-ment detailing required or slender spandrels, especially in
the end regions. Specically, the research ocused on inves-
tigating whether traditional closed stirrups are required or
the slender cross sections o typical precast concrete L- and
corbelled spandrels. The use o open web reinorcement in
lieu o closed stirrups would greatly simpliy abrication
and reduce the cost o production.
Nonlinear finite element analysis
A three-dimensional nonlinear nite element model (FEM)
was developed and then calibrated with experimental
Figure 1. Typical mesh conguration or nonlinear nite element analytical model (L-shaped spandrel).
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Figure 2 indicates that the predicted cracking patterns
matched well with those observed in experiments. Flexural
cracking was observed in the midspan region, primarily
on the outer web ace. A primary diagonal crack initiated
in the end region at the ace o the support. This crack ex-
tended upward at an angle o approximately 45 deg. Away
rom the end region, the crack angles began to fatten and
arch toward midspan.
The predicted cracking pattern or a reinorced concrete
L-spandrel shows extensive fexural cracking at midspan, as
expected or a nonprestressed concrete beam. In comparison,
analysis o a prestressed concrete L-spandrel showed little
fexural cracking at corresponding load levels. These predic-
tions were conrmed by observed cracking patterns. The pri-
mary diagonal crack developed rom the ace o the support
or all three selected beams. One interesting dierence is that
the predicted angle o this crack is almost 45 deg or the re-
inorced concrete beam, and as expected, it was fatter or the
two prestressed concrete beams closer to 35 deg. This nding
was not conrmed by the tests, as shown in the photographs
in the let column o Fig. 2. While cracks on the inner ace o
the prestressed concrete beams fattened out more rapidly than
did cracks on comparable nonprestressed concrete beams,
there was minimal dierence in the angle o the primary crack
extending rom the support o each specimen.
Failure mode
Figure 3 shows the ailure modes predicted by the nonlin-
ear FEM or the same three selected beams alongside the
observed ailure modes. The ailure mode determined rom
analysis is illustrated by plotting the predicted shearingstrain contours in the plane o the web ace. All beams in
Fig. 3 were reinorced with additional fexural and local re-
inorcement to avoid premature fexural ailure and induce
end-region behavior, as described in the companion paper.1
The comparison shown in the gure indicates that the
analytical model was capable o predicting the observed
skewed-diagonal end-region ailure mode or the L-shaped
and corbelled spandrels.
Deflections
Figure 4 compares a representative vertical load-defectionprediction with the experimental results. The initial sti-
ness o the predicted load-defection curves tended to be
slightly higher than the measured values; however, the pre-
dicted stiness tended to match closely to measured values
ater cracking. The analytical model closely predicted the
ailure load and ailure defection, with some deviation in
the nal stages o loading.
Figure 5 shows the predicted and measured lateral defec-
tions or a representative spandrel beam. The comparison
highlights the eect o short-term creep during the test,
as evidenced by horizontal portions o the load-defection
Several parameters were examined using the nonlinear FEM.
These parameters included closed versus open web rein-
orcement, web reinorcement type and ratio, cross section
dimensions, concrete strength, and the infuence o boundary
conditions such as bearing riction and deck connections.
Boundary conditions
The boundary conditions used in the model were chosen to
simulate the connections typically used to support spandrel
beams in the eld. The spandrel-to-column tiebacks were
simulated by restraining lateral movement at those two
locations in the model. Vertical movement was restrained
in the bearing area o the main vertical reaction. Sym-
metry boundary conditions were applied at midspan. Each
vertical load applied to the spandrel ledge was modeled as
uniorm pressure acting on an area equal to the size o the
bearing pads used in the experimental program. These ap-
plied loads were increased incrementally until ailure.
Spring elements were used to simulate stem-to-ledge
bearing riction and deck-to-spandrel tieback orces. For
the deck connections, the stiness o the spring elements
was estimated based on the material and cross-sectional
properties o the weld plate. The spring constant simulating
the riction at bearing reactions was determined based on
the coecient o riction or the chosen bearing pads. The
coecient o riction or a given bearing pad was assumed
to remain constant at all levels o applied load. These as-
sumptions tended to slightly overestimate the riction orce
at high overload levels when the out-o-plane behavior
became signicantly nonlinear.
Selected results rom the nonlinear nite element analysis
are presented in the ollowing sections.
Deflected shapeand cracking pattern
Figure 2 shows the predicted and observed inner web ace
cracking patterns in the end region o a reinorced con-
crete L-spandrel, a prestressed concrete L-spandrel, and
a prestressed concrete corbelled spandrel. For clarity, the
observed cracking patterns were photographed ater ailure
with the deck sections removed. The analytical crackingpatterns are shown at the actored load level or clarity.
Only hal o each beam is shown in the gure because the
analysis was symmetric about the midspan.
In general, the predicted deormed shape was a combina-
tion o vertical and lateral defections, and matched well
to the observed behavior. The midspan o the beam moved
downward, and the top o the web rolled inward toward the
applied loads at midspan. The degree o lateral deormation
was infuenced by the deck connections, as discussed in
detail in the technical report.2
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Rational model
Based on the observed behavior and the nonlinear nite el-
ement analysis, a rational model was developed to describethe behavior o a slender spandrel beam. The rational
model orms the basis o the proposed simplied design
approach, presented later in this paper. The model is based
on equilibrium o orces at the diagonal skewed ailure
plane observed in the experimental program.
The design principles are only applicable or the modeling
o slender beams with an aspect ratio equal to or greater
than 4.6 as this was the smallest aspect ratio tested. The
beams considered had the ollowing controls:
behavior. Short-term creep was especially prominent over
the 24-hour sustained load tests. The FEMs do not account
or creep. I creep eects were removed, representing an
experimental test conducted without sustained loading, theexperimental results would match the predicted values more
closely. The predicted lateral defections matched well the
experimental values through most o the tested range. The
FEM captured the spandrel behavior with the top edge o a
slender spandrel beam moving inward and the bottom edge
moving outward at midspan, pivoting about the deck con-
nection. Further description o the nite element analysis
is presented along with the ull analytical results in the
research report.2
Figure 2. Comparison o experimental cracking and analytical cracking. Note: the detailed speciman names are or reerence to the companion paper.
Experimental cracking Analytical cracking
Reinforced concrete
L-spandrel
SP10.8L60.45.R.O.E.
Prestressed concrete
L-spandrel
SP12.8L60.45.P.O.E.
Prestressed concrete
corbelled spandrels
SP17.8CB60.45.R.O.E.
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Transition region
The transition region extends a distance 2h beyond the end
region. Test results indicate that the primary crack angle
is approximately 30 deg. Shear and torsion demands are
reduced in this region while bending moment demands are
increasing compared with the end region.
Flexure region
The fexure region is dened as the portion o the beam
beyond the transition region to the midspan o the beam.
The behavior in this region is marked by vertical cracking
on the inner and outer web aces due to in- and out-o-plane fexure and by horizontal cracking on the inner web
ace due to hanger loads. Shear and torsion demands are
relatively low, while moment demands are relatively high.
The loading demands considered in the rational model or
each region consisted o the actored bending momentMu,
the actored shear orce Vu, and the actored torque Tu. Ec-
centricity e or the calculation o torsion should be consid-
ered rom the point o load application to the center o the
web. Flexure design should ollow the recommendations in
the PCI Design Handbook.
Proposed rational model
for resisting the applied torsion Tu
The vertical eccentric loads acting on a slender spandrel
beam produce torque acting about the longitudinal axis o
the beam. The torsion demand Tu is calculated by multiply-
ing the applied actored ledge loads by the distance rom
the applied load to the center o the web. The Tuvector at
any diagonal crack can be resolved into two equivalent
orthogonal vectors (Fig. 7). One component o the torque
vector, dened in the analysis as Tub, will act to bend the
spandrel web out o plane about the diagonal axis de-
ned by the diagonal crack angle (Eq. [1]). The second
component o the torque vector, dened in the analysis asTut, acts to twist the web about an axis perpendicular to that
diagonal crack (Eq. [2]).
Tub = Tucos (1)
Tut= Tusin (2)
Figure 4. Measured and predicted vertical defections at midspan. Note: 1 in. = 25.4 mm; 1 kip = 4.448 kN.
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Designing for the plate-bending
component of torsion Tub
Equation (1) provides the actored out-o-plane bending
moment demand about the diagonal axis dened by angle
. The resistance to Tub is developed by plate bending in
the spandrel web. The nominal resisting moment can be
closely determined by Eq. (3).
Tub = A f ds y wm (3)
where
Tub = plate-bending component o torsion
Asm = total area o steel required on the inner web ace
crossing the critical diagonal crack in a direction
perpendicular to the crack
fy = yield stress o the inner-ace reinorcing steel
dw = eective depth rom the outer surace o the web to
the centroid o the combined horizontal and vertical
steel reinorcement o the web; usually taken as web
thickness less concrete cover less the diameter o the
inner-ace vertical steel bars
The lateral bending resistance must satisy the lateral
bending demand (Eq. [4]).
TubfTnb (4)
where
f = strength reduction actor or fexure = 0.9
Tnb = nominal plate-bending resistance o the web
Combining Eq. (1), (3), and (4) into Eq. (5) results in thetotal required area o steel A
s sm .
Asm
cos
f d
T
f y w
u
z
i(5)
Although Tub is a component o torsion, the strength reduc-
tion actor or fexure is considered appropriate because
Tub is resisted by out-o-plane fexure o the web, which is
plate bending.
Because the web steel resisting plate bending is usually
Figure 5. Measured and predicted lateral defections at midspan. Note: 1 in. = 25.4 mm; 1 kip = 4.448 kN.
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= 45 deg = 30 deg
h
h
2h
a
End region Transition region Flexure region
Variable to midspan
Figure 6. Inspection o the cracking pattern o all tested slender spandrel beams loaded to ailure allowed or identication o three distinct zones: end region, transition
region, and fexure region. Note: a= vertical distance rom bottom o spandrel to center o lower tieback connection; h= height o spandrel; = angle o critical diagonalcrack or region under consideration with respect to horizontal.
End region
Typical inner-face cracking pattern
Typical outer-face cracking pattern
Generic regions of behavior identified in rational model
Transition region
Flexure region
Midspan
End region Transition region Flexure region
Midspan
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lch tan
h
i(9)
The required area o plate-bending reinorcement dis-
tributed over the horizontal crack projectionAsv ./s can be
determined by Eq. (10).
Asv/s cos sin
f d h
T
y w
u
z
i i(10)
Similarly, the distributed longitudinal steel required on theinner web ace or plate bendingAsl/s can be determined
by Eq. (11).
Asl/s cos sin
f d h
T
y w
u
z
i i(11)
Equations (10) and (11) are identical irrespective o crack
angle or beam region. The required vertical plate-bending
steel distributed over the horizontal crack projection equals
the required horizontal plate-bending steel distributed over
the vertical crack projection in a given beam region.Asl
orthogonal to the beam (not perpendicular to a crack), thetotal required area must be expressed in terms o vertical
steel and longitudinal steel. The total required area o verti-
cal steel crossing the diagonal crackAsv can be determined
by Eq. (6).
Asv = Asmcos (6)
Using Eq. (5) and (6), the requiredAsv can be determined
by Eq. (7).
Asv cos
f d
T
f y w
u
2
z i(7)
Similarly, the total required area o longitudinal steel on
the inner ace to resist plate bendingAsl can be determined
by Eq. (8).
Asl cos sin
f d
T
f y w
u
z
i i(8)
The area o vertical steel required to resist plate bending
Asv over the horizontal crack projection lch can be deter-
mined by Eq. (9).
Figure 7. Components o applied torque along diagonal crack within end region. Note: The double-headed torque vectors indicate moments according to the right-hand
rule. Tu= actored torque; Tub= plate-bending component o torsion; Tut= twisting component o torsion; = angle o critical diagonal crack or region under consideration
with respect to horizontal.
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Commentary (ACI 318R-08)8 recommendation or tor-
sion design. The twist demand must be resisted by a cross
section through a diagonal crack, normal to the Tutvector,
having dimensions oh/sin and thickness b (Fig. 8).
Evaluation o the twisting resistance in slender spandrel
beams is based on the well-accepted mechanism used to
transer unbalanced moments rom reinorced concrete
slabs to columns. In the case o a column-slab interace,
ACI-318-08 section 11.11.7.2 recommends that the shear
stress shall vary linearly about the centroid o the critical
section. The same concept o linear shear distribution is
applied to the inclined cross section o a slender spandrel
beam (Fig. 8). The linear model assumes a shear-stress dis-
tribution with a maximum value at the extreme end o the
section. More complex models or twisting resistance wereevaluated using rational models and FEMs. The complex
models and nite element analysis closely match the linear
approximation, as discussed in the technical report.2 In all
cases, the reduced nominal twisting resistance must exceed
the actored twist demand, as shown in Eq. (12).
TutTnt (12)
Based on the linear stress distribution in Fig. 8, the out-o-
plane resultant orcesHntare separated by a distanceLtwist
andAsv should be calculated or each region at the point
o maximum torsion Tu in that region. In the end region,
not all possible diagonal cracks intersect the bottom o the
beam. Thereore, it is appropriate to consider a diagonalcrack extending upward rom the lower lateral tieback,
where crack projections would be calculated using the
distance h a in place oh. Further details are presented in
the technical report.2
The vertical reinorcement resisting plate bendingAsv
is based on the assumption that the inner-ace web steel
yields under the eect o the applied actored plate-bend-
ing moment Tub. Calculation indicates that, in practical
designs, the reinorcement requirements or plate bending
are substantially less than the tension-controlled limit.
Thus, plate-bending web steel will yield with signicantductility required or tension-controlled ailure, and a value
o 0.9 or f is justied.
Designing for the twisting
component of torsion Tut
The proposed rational model requires that the twisting
resistance o the cracked sectionsTnt(where s is the
strength reduction actor or shear and Tnt is the nominal
twist resistance o the section) exceed the twisting demand,
with s equal to 0.75, according to theBuilding Code
Requirements for Structural Concrete (ACI 318-08) and
Figure 8. Linear distribution o shear stresses on inclined cross section. Note: b= thickness o web; h= height o spandrel; Hnt = out-o-plane orce resultants; Ltwist = lever
arm; Tut = twisting component o torsion; = critical diagonal crack angle or region under consideration with respect to horizontal.
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equal to 2/3 the height o the diagonal crack. The magni-
tude o these resultant orces is determined in Eq. (13).
Hnt=sin
hd X f
2
1
2
1w ci
lf p (13)
where
Hnt = out-o-plane orce resultants rom a linear shear-
stress distribution
fcl = specied concrete compressive strength
X = out-o-plane shear-stress coecient (calibrated
later in this paper rom experimental data)
X fcl = maximum shear stress o the linear distribution
(calibrated rom the experimental data later in this
paper)
Accordingly, the nominal twist resistance o the section Tnt
can be evaluated as the product oHntandLtwist(Eq. [14]).
Tnt=sin
X fh
d6
1c w2
2
il (14)
To ensure that the twisting resistance according to the
linear model exceeds or equals the applied actored torque
demand, Eq. (15) should be satised. Equation (15) is
derived by substituting Eq. (14) and (2) into Eq. (12).
Tusin
X fh
d6
1s c w
2
3z
il (15)
where
s = 0.75
Calibration of the twist-resistancemodel
O the 16 tested beams, seven ailed in out-o-plane modesin their end regions. For each o these seven beams, the
measured lateral reactions at ailure were used to calculate
the out-o-plane shear stresses resisted by a given beam
just beore ailure. It is recognized that there are interac-
tions among the fexural, shear, and torsional stresses on
the ailure plane. Because the proposed twist resistance
is calibrated to the measured test data, it accounts or the
eects o the fexural stress and the vertical shear and
torsional stresses. However, the proposed method neglects
the eect o the torsional stress on the vertical shear stress
based on the experimental evidence or beams tested.
The twist-resistance model was calibrated by rst deter-
mining the torsion acting on the end region o a given
beam at ailure. For each tested beam, all lateral orces
were included to determine the twisting moment at ailure
Ttl . Lateral orces not directly measured during the tests
were evaluated by considering the equilibrium o the beam
as a whole about a selected origin, point O (Fig. 9).
The sum o lateral orces at the deck connections H2 and
the riction coecient were determined by considering
moment equilibrium about point O and orce equilibrium
in the horizontal direction. The analysis was perormed
or each o the seven beams that ailed in its end region.
The range o calculated riction coecients corresponds
well to the value o 0.05 determined by the nonlinear nite
element analysis described previously. In addition, the
calculated values correspond well to the range published in
the PCI Design Handbookor Tefon-coated bearing pads
similar to those used in the tests.
With all lateral orces determined, the twisting component o
torsion resisted by each beam at ailure was evaluated. Figure
10 shows a ree body diagram taken along a 45 deg crack
plane extending upward rom the ace o the support (line 1-1
in the gure). All dimensions are known rom the geometry
o a given beam, and all orces were measured or determined
rom equilibrium. There are no deck connections to the let o
crack plane 1-1; thereore, the deck connection orcesH2 do
not appear in the ree body diagram to the let o the crack.
The out-o-plane shear stress induced by the twisting
moment at ailure Ttl was assumed to be distributed
linearly across the selected crack plane 1-1 (Fig. 10). Themaximum values at the extremes o the distribution are
identied asX* fcl and X f
cl, whereX* is the height
o the centroid and X is the extreme value o the stress
distribution. The location o the centroid o the twist-
ing moment was determined by an iteration process that
satised moment equilibrium about the selected centroid
and equilibrium o the horizontal orces. The tendency o
the vertical reaction to shit inward (toward the ledge) as a
spandrel deorms under load was conservatively neglected.
Based on the two equilibrium equations, the height o the
centroidX* and the extreme values o the stress distribution
X were determined or each experimental case (Table 1).X* and X are related by the linear distribution.
The calculated height o the twist axisZt is consistently
just above the midheight o all specimens. Thereore, as-
suming a balanced linear stress distribution or design is an
acceptable approximation.
Given the results in Table 1, a conservative value o 2.4 is
recommended orXin Eq. (15). Because the centroid o
the linear distribution assumed or design will be consid-
ered at the midheight o the cross section, it is appropriate
to compare the selected value oXto the values oXavg
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(averages oX* and X [Table 1]). Selecting a value below
Xavg is a sae choice or design, especially considering the
conservative nature o the linear distribution and the as-
sumptions made in the calibration.
Figure 11 plots the ratio o twisting moment at ailure to
nominal twisting capacity or all seven spandrels that ailedin their end regions. These seven spandrel beams were
specially congured to induce end-region ailure modes
by over-reinorcing other controlling ailure modes.1 The
plot shows that Eq. (15), with a value o 2.4 orX, conser-
vatively predicts the twisting capacity o each tested beam.
Thus, Eq. (15) is recommended or design with 2.4 or the
value oX. This recommendation provides a conservative
prediction o the twisting moment capacity o a slender
spandrel and is consistent with the historical association
(ACI 318-719) o 2.4 fcl with torsion. Thus, Eq. (16) gives
the proposed twist resistance using the linear shear-stress
distribution model.
Tu .sin
fd h
2 46
s c
w
3
2
zi
lf p (16)Shear and torsion stresses
Due to the loading and support conditions, shear and tor-sion stresses act in the same direction on the inner web
ace (the ace on the ledge or corbel side) but oppose each
other on the outer web ace10 (Fig. 12). The concrete on the
inner ace is more vulnerable to diagonal cracking. This
behavior is clearly identied by experimental cracking
patterns1 and is veried by linear nite element analysis
presented in the technical research report.2
Designing for the applied shear
As discussed previously, the rational model considers the
design o shear and torsion independently. Thus, design
Figure 9. Equilibrium o generic slender spandrel beam. Note: O = origin; XV= distance rom outer web ace to center o main vertical reaction, estimated to remain at
center o the web; XVa= distance rom outer web ace to center o ledge reactions, estimated to remain at center o ledge bearing pads; XVs= distance rom outer web ace
to center o sel-weight reaction, estimated to remain at center o gravity; ZH1 = height o lower lateral reaction; ZH2 = height o te deck connection; ZH3 = height o upperlateral reaction; ZV= height to Tefon surace o main bearing pad; ZVa= height to Tefon surace o ledge (or corbel) bearing pad; H1 = sum o measured lower lateral
reactions at ailure; H2 = sum o lateral orces at deck connections; H3 = sum o measured upper lateral reactions at ailure; V= sum o measured vertical reactions at
ailure; Va = sum o loads applied to spandrel by double-tee decks and jacks; Vs= sel-weight o spandrel beam; V= sum o lateral orces at main vertical bearing due
to riction; Va= sum o lateral orces at ledge (or corbel) due to riction.
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The minimum requirements or area o shear reinorcement
Av specied by ACI 318-08 may control over Eq. (17).
Proportioning the webreinforcement
The previously calculated quantities o steel reinorcement
required to resist torsion and shear must be provided in the
spandrel web according to the nature o shear and torsion
stresses illustrated in Fig. 12.
The total quantity o distributed vertical steel required onthe inner web aceAsi/s is the quantity required or the
plate-bending component o torsion plus hal o the total
quantity required or shear (Eq. [18]).
Asi/s /cos s in
f d h
T
f d
V V
2f y w
u
y
u s c
z
i i z+
-(18)
where
d= distance rom the extreme compression ber to the
centroid o the longitudinal reinorcement
or shear should ollow traditional practice. Equation (17)
determines the uniormly distributed vertical shear rein-
orcementAv/s required at a given cross section to resist the
actored shear orce Vu.
Av/s /
f d
V V
y
u s cz -
(17)
where
s = 0.75
Vc = nominal concrete shear strength equal to the lesser o
Vcior Vcw, as given by Eq. (11-11) and (11-12) o ACI
318-08
Vci = nominal shear strength provided by concrete when
diagonal cracking results rom combined shear and
moment
Vcw = nominal shear strength provided by concrete when
diagonal cracking results rom high principal tensile
stress in the web
Figure 10. Forces contributing to torque on a slender spandrel end region. Note: Forces acting into the page are identied by a circle with a cross, while orces acting out o
the page are identied by a circle with a dot. f c = specied concrete compressive strength; H1 = measured lower lateral tieback reaction; H2 = deck connection orces; H3
= measured upper lateral tieback reaction; T t = torque at ailure; V= measured main vertical end reaction; Va = experimentally applied stem loads; X* = height o centroid;
X
= extreme values o stress distribution; Zt = height o twist axis; = riction coecient.
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The remaining hal o the required shear reinorcement is
allocated to the outer spandrel ace where shear and torsion
stresses oppose each other. On the outer ace, the vertical
component o the plate-bending compressive orce reduces
the amount o required vertical reinorcement or shear
(Fig. 13).
Thus, the total distributed vertical reinorcement required
on the outer web aceAso can be theoretically reduced
by the component o vertical plate bending (Eq. [19]),
provided that the total amount o transverse reinorcement
in the beam is sucient or shear. However, it is recom-
mended thatAso be provided as hal oAv.
Table 1. Results rom twist model calibration
Test specimen Zt, in. X*, psi/ fcl X
, psi/ f
cl Xavg, psi/ fcl
SP3.8L60.45.P.O.E 34.5 2.36 3.19 2.77
SP4.8L60.45.P.O.E 33.9 2.02 2.62 2.32
SP10.8L60.45.R.O.E 36.7 2.48 3.91 3.20
SP12.8L60.45.P.O.E 35.9 1.90 2.83 2.36
SP17.8CB60.45.P.O.E 37.0 1.82 2.94 2.38
SP18.8CB60.45.P.S.E 36.8 2.17 3.45 2.81
SP20.10L46.45.P.O.E 26.4 2.11 2.83 2.47
Mean 2.62
Minimum 2.32
Standard deviation 0.32
Note: X* = height o the centroid; X = extreme values o stress distribution; Xavg = average o X
* and X ; Zt = height o twist axis.
1 in. = 25.4 mm; 1 psi = 6.895 kPa.
Figure 11. Ratio o ailure twisting moment to proposed nominal twisting moment.
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steel required or shear and torsion. Only the greater o the
two should be used.
First cracking load
The anticipated load causing initial cracking in the endregion o a slender spandrel beam is oten o interest to the
designer. While minor hairline cracking is usually not a
concern rom a structural or durability standpoint, it may
be desirable to minimize or eliminate such cracking in
cases where aesthetics are relevant. The rst cracking load
will be the same whether open or closed transverse web
reinorcement is used.
Equation (20) gives the diagonal tensile stress due to shear
o a slender spandrelfcr1.
fcr1 =bh
V
2
3(20)
where
fcr1 = diagonal tension stress due to shear
V = the applied shear orce at the level o interest
b = width o web
h = height o the spandrel
Aso/s / cos sin
f d
V V
f d h
T
2y
u s c
y w
uz i i-
- (in.2/in.) (19)
On the outer web ace, plate bending about a crack extend-
ing downward rom the top lateral reaction can, in theory,control a design; however, this orthogonal plate bending is
counteracted by vertical shear. Failure about this potential
plane could only develop i torsional stresses, acting up-
ward on the outer ace, greatly exceeded the shear stresses,
acting downward. Further details and analysis related to
orthogonal plate bending are presented in the technical
report.2
The longitudinal steelAsl calculated by Eq. (8) is to be
provided on both the inner and the outer web aces. Provid-
ing longitudinal steel on the outer ace protects against
possible plate bending on the outer ace. In addition,longitudinal steel provides dowel orces benecial to twist
resistance, particularly in the end region.
Potential additional failure modes
Other reinorcement requirements may control the design
o web reinorcement and must be checked according to
procedures specied in the PCI Design Handbook. By
experience, it was ound that hanger steel requirements will
oten control the design o vertical reinorcement on the in-
ner web ace, especially outside o the end region. Hanger
steel requirements need not be combined with the vertical
Figure 12. Directions o shear and torsional stresses acting on slender spandrel cross section.
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In actual structures, cracking can be infuenced by han-
dling, curing conditions, thermal exposure, and other ac-
tors. Also, concentration o stresses near the bearing area
increases the likelihood o cracking just inside the sup-
port. As such, cracking may occur sooner than expected.
However, i designed as recommended herein, end-region
diagonal cracks will be narrow and will not adversely a-
ect strength or durability. The aesthetic impact should alsobe minimal.
Proposed simplified designguidelines
To assist the designer, a step-by-step simplied design
procedure was developed based on the rational model. The
procedure is intended or slender precast concrete spandrel
beams subject to the ollowing restrictions:
A simply supported precast concrete spandrel is
loaded along the bottom edge o the web.
Normalweight concrete is used.
The web is laterally restrained at two points at each
end.
Applied loads are evenly spaced along the bottom
edge o the web.
The aspect ratio (height divided by web thickness) is
equal to or more than 4.6.
Equation (21) calculates the diagonal tensile stress due to
plate bending o a slender spandrel.
fcr2 =b h b h
VeT3 3
2 2= (21)
where
fcr2 = diagonal tension stress due to plate bending
T = applied torque
e = eccentricity contributing to torsion
Combining Eq. (20) and (21) and assuming a limiting ten-
sile stress o concrete o 6 fcl, Eq. (22) can determine the
shear orce at cracking Vcr.
Vcr=1 2 /e b
fbh
4c
+
l
a k
R
T
SSSS
V
X
WWWW
(22)
Comparison with experimental data indicates that Eq.
(22) provides a conservative estimate o the load at which
cracking is rst likely to appear. First cracking loads can be
increased by increasing the web thickness, increasing the
concrete strength, or distributing prestressing orce through
the height o the web. Prestressing orce concentrated only
near the bottom o the section is not eective in controlling
diagonal cracks near the support.
Figure 13. Compression on outer web ace due to plate bending.
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Step 1: Determine the loading
demands
Construct the actored bending moment, shear, and torsion
diagrams. Eccentricity or calculating torsion should be
taken rom the point o applied load to the center o the
web.
Step 2: Divide the slender spandrel
into the three regions shownin Fig. 14
Step 3: Verify that the cross section
can sustain the twisting componentof torsion
Veriy that the maximum torque Tu in the end region does
not exceed the twist limit or this region by considering Eq.
(23).
With taken as 45 deg or the end region, Eq. (16) be-
comes Eq. (23).
Tu . f d h1 13s c w2z lc m (23)
where
s = 0.75
Conservatively, Eq. (23) can also be used to check twist
capacity in the transition region. However, twist in the tran-
sition region will not control the design or a simply sup-
ported beam with a uniorm cross section. Similarly, there
is no need to check the twist limit in the fexure region.
Because the proposed twist resistance is calibrated to the
measured test data, it accounts or the eects o the fexural
stress and the vertical shear and torsional stresses.
I the lateral end reactions are spaced vertically at least
0.6h, a check or twist resistance on a secondary ailure
plane in the end region (line 2-2 in Fig. 14) is not required.
Otherwise a check is necessary by using the height o the
section above the lower connection (distance h a, where
a isthe vertical distance rom the bottom o a spandrel
to the center o the lower tieback connection) in Eq. (23)
in lieu oh. Calculations involving the transition region
would still use the ull section height h. I this check or
the twisting component o torsion is not satised, the con-
crete strength or cross-section dimensions would have to
be increased; otherwise, closed reinorcement would have
to be used.
Step 4: Design the beam for flexure
Design and proportion the longitudinal steel reinorcement
(mild or prestressed) in accordance with section 4.2 o the
PCI Design Handbook.
Step 5: Calculate the required shear
steel at all sections along the beam
Design the beam or shear according to all provisions in
section 4.3 o the PCI Design Handbook. For the vertical
shear design in this step, ignore eccentricity.
Determine the amount o vertical shear reinorcementrequired per unit length (Av/s) at all locations along the
length o the beam.Av will likely change at each applied
point load along the ledge and will also be infuenced by
the development o prestressing strands in the end region.
All requirements or minimumAv still apply.
The proposed method neglects the eect o the torsional
stress on the vertical shear stress based on successul
perormance o the test specimens, which were designed
without consideration o the infuence o torsion on shear
strength. However, Zia and Hsu7 indicated that torsion
reduces shear strength, especially in compact sections. As
Figure 14. Regions o slender spandrel beam. Note: end region = region rom end o beam to distance hbeyond inner ace o support; transition region = 2hbeyond end
region; fexure region = remainder o beam; a= height rom bottom o spandrel to center o lower lateral tieback; h= height o web; = critical diagonal crack angle or
region under consideration with respect to horizontal = 45 deg or end region = 30 deg or transition region.
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and extend or the ull length o that region. Generally,
horizontal U-shaped bars are eective in the end regions
because they allow or development at the end o the beam.
In the end region, longitudinal steel below the level o the
lower lateral reaction should not be counted toward the
plate-bending requirement.
In the transition region, ully developed excess longitudinal
reinorcement not required or fexure may be used as part
o the plate-bending requirement.
Step 12: Detail the beam
Recommendations in the PCI Design Handbookregarding
ledge design, connection detailing, hanger steel, vehicular-
impact steel, and reinorcement or other possible local-
ized ailure modes are to be considered. In addition, it is
recommended that continuous bars or strands be placed in
the longitudinal direction at all our corners o the web to
control the possible cracking rom lateral bending away
rom the ends.
Step 13: Check service-level
cracking
I desired, the load at which diagonal cracking in the end
region is rst likely to appear may be conservatively esti-
mated according to Eq. (31). In many cases, beams will be
cracked at service load, regardless o end-region reinorce-
ment. First cracking load is independent o web reinorce-
ment conguration (that is, open or closed stirrups). The
eects o prestressing are ignored in Eq. (31) becausesome o the prestressing strands are not ully developed
across the potential crack.
Vcr=/e b
fbh
1 2
4c
+
l
a k
R
T
SSSS
V
X
WWWW
(31)
where
Vcr= shear orce at cracking
Abstract
A rational design method is recommended or the design o
precast concrete slender spandrel beams with aspect ratios
greater than or equal to 4.6. The procedure was developed
using data rom an extensive research program includ-
ing 16 ull-scale tests, extensive nite element modeling,
and a rational analysis. It is recommended that the design
or shear and torsion o slender precast concrete spandrel
beams use the concept o resolving the applied torsion into
two orthogonal vectors: a plate-bending component and a
twisting component. Equations were presented or evalu-
ating the capacity o a slender precast concrete spandrel
beam to resist both components o torsion. The research
demonstrates that slender precast concrete spandrel beams
can be saely designed to resist the combined eects o
fexure, shear, and torsion without the use o traditional
closed web reinorcement. Use o open web reinorcement
could greatly reduce reinorcement congestion in the end
regions o spandrel beams and provide signicant savings
in abrication cost.
Acknowledgments
This research was sponsored by the PCI Research and
Development Committee. The work was overseen by an
L-spandrel advisory group chaired by Donald Logan. The
authors are grateul or the support and guidance provided
by this group throughout all phases o the research. In ad-
dition, the authors would like to thank the numerous PCI
Producer Members who donated test specimens, materials,
and expertise in support o the experimental program.
References
1. Lucier, G., C. Walter, S. Rizkalla, P. Zia, and G. Klein.
2011. Development o a Rational Design Methodol-
ogy or Precast Concrete Slender Spandrel Beams:
Part 1, Experimental Results. PCI Journal, V. 56, No.
2 (Spring): pp. 88112.
2. Lucier, G., C. Walter, S. Rizkalla, P. Zia, and G. Klein.
2010. Development o a Rational Design Methodol-
ogy or Precast Slender Spandrel Beams. Technical
report no. IS-09-10. Constructed Facilities Laboratory,
North Carolina State University, Raleigh, NC.
3. Logan, D. 2007. L-Spandrels: Can Torsional Distress
Be Induced by Eccentric Vertical Loading? PCI Jour-
nal, V. 52, No. 2 (MarchApril): pp. 4661.
4. Klein, Gary J. 1986.Design of Spandrel Beams. PCI
Specially Funded Research and Development program
research project no. 5 (PCISFRAD #5). Chicago, IL:
PCI.
5. Raths, Charles H. 1984. Spandrel Beam Behavior and
Design. PCI Journal, V. 29, No. 2 (MarchApril): pp.62131.
6. PCI Industry Handbook Committee. 2004. PCI Design
Handbook: Precast and Prestressed Concrete. MNL-
120. 6th ed. Chicago, IL: PCI.
7. Zia, P., and T. Hsu. 2004. Design or Torsion and
Shear in Prestressed Concrete Flexural Members. PCI
Journal, V. 49, No. 3 (MayJune): pp. 3442.
8. ACI Committee 318. 2008.Building Requirements
for Structural Concrete (ACI 318-08) and Commen-
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e = eccentricity
fcl = specied concrete compressive strength
fcr = diagonal tensile stress due to plate bending o a
slender spandrel
fy = yield strength o mild steel
h = height o spandrel section
h/b = aspect ratio
H1 = measured lower lateral tieback reaction
H2 = measured deck connection orces
H3 = measured upper lateral tieback reaction
Hnt = out-o-plane orce resultants
lch = length o horizontal projection o diagonal crack
Ltwist = lever arm
Mu = actored bending moment
Pu = actored stem load
s = spacing o vertical reinorcement
T = applied torque
Tnb = nominal plate-bending resistance o web
Tnt = nominal twist resistance o section
Ttl = twisting moment at ailure
Tu = actored torque
Tub = plate-bending component o torsion
Tut = twisting component o torsion
V = experimentally measured main spandrel end reaction
Va = experimentally applied stem reaction
Vc = nominal concrete shear strength = lesser oVci or Vcw
as given by Eq. (11-11) and (11-12) o ACI 318-087
Vci = nominal shear strength provided by concrete
when diagonal cracking results rom combined
shear and moment
Vcr = shear orce at cracking
tary (ACI 318R-08). Farmington Hills, MI: American
Concrete Institute (ACI).
9. ACI Committee 318. 1971.Building Code Require-
ments for Structural Concrete (ACI 318-71) and
Commentary (ACI 318R-71). Farmington Hills, MI:
American Concrete Institute (ACI).
10. Hsu, T. C. 1984. Torsion of Reinforced Concrete. New
York, NY: Van Nostrand Reinhold Publishers.
Notation
a = vertical distance rom the bottom o a spandrel to
the center o the lower tieback connection
Asm = total area o steel required on the inner web ace
crossing the critical diagonal crack in a direction
perpendicular to the crack
Ash = required hanger steel
Asi = total required area o vertical steel on inner web
ace
Asi/s = total quantity o distributed vertical steel required
on inner web ace
Asl = total required area o horizontal steel on inner ace
to resist plate bending
Asl/s = distributed longitudinal steel required on inner
web ace or plate bending
Aso = total distributed vertical reinorcement required on
outer web ace
Asv = total required area o steel crossing diagonal crack
in vertical direction
Asv2 = total quantity o vertical steel crossing line 2-2 on
inner ace
Av = area o shear reinorcement
Av/s = uniormly distributed vertical shear reinorcement
b = thickness o web
d = distance rom extreme compression ber to cen-
troid o longitudinal reinorcement
dw = eective depth rom outer surace o web to
centroid o combined horizontal and vertical steel
reinorcement o web; usually taken as web thick-
ness less concrete cover less diameter o inner-
ace vertical steel bars
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V = sum o measured vertical reactions at ailure
Va = sum o loads applied to the spandrel by the
double-tee decks and jacks
Vs = sel-weight o the spandrel beam
V = sum o lateral orces at the main vertical bearingdue to riction
Va = sum o lateral orces at the ledge (or corbel) due
to riction
f = strength reduction actor or fexure = 0.9
s = strength reduction actor or shear = 0.75
Vcw = nominal shear strength provided by concrete when
diagonal cracking results rom high principal
tensile stress in web
Vu = actored shear orce
X = out-o-plane shear-stress coecient
X* = height o centroid
X = extreme values o stress distribution
Xavg = average oX* and X
X fcl = eective out-o-plane shear stress
XV = distance rom outer web ace to center o main
vertical reaction, estimated to remain at center o
the web
XVa = distance rom outer web ace to center o ledge
reactions, estimated to remain at center o ledge
bearing pads
XVs = distance rom outer web ace to center o sel-
weight reaction, estimated to remain at center o
gravity
ZH1 = height o lower lateral reaction
ZH2 = height o deck connection
ZH3 = height o upper lateral reaction
ZV = height to Tefon surace o main bearing pad
ZVa = height to Tefon surace o ledge (or corbel) bear-
ing pad
Zt = height o twist axis
= critical diagonal crack angle or region under
consideration with respect to horizontal
= 45 deg or end regions
= 30 deg or transition regions
= 0 deg or fexure regions
= riction coecient
H1 = sum o measured lower lateral reactions at ailure
H2 = sum o lateral orces at the deck connections
H3 = sum o measured upper lateral reactions at ailure
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Given information
Web thickness b = 8 in. (200 mm)
Web depth h = 60 in. (1500 mm)
Eccentricity e = 10 in. (250 mm)
Span = 44 t 6 in. (13.6 m)
Depth o web steel dw = 6.5 in. (165 mm)
Concrete design strength fcl = 6000 psi (40 MPa)
Appendix: Design example
Problem statement
Determine the reinorcement required in the web o the
simply supported, L-shaped spandrel beam (Fig. A1). The
spandrel supports nine double-tee stems, each spaced 5 t
(1.5 m) apart. One stem is centered at midspan. Each stemreaction comprises a dead-load reaction o 10.74 kip (47.8
kN), a live-load reaction o 6 kip (27 kN), and a snow-load
reaction o 4.5 kip (20 kN). Stem loads are centered 6 in.
(150 mm) rom the inside web ace. 1.2D + 1.6L + 0.5S
is the controlling load case. The spandrel sel-weight is
567 lb/t (8.27 kN/m).
Figure A1. L-shaped spandrel considered in design example. Note: a= vertical distance rom bottom o spandrel to center o lower tie-back connection; b= thickness
o web; dw = eective depth rom outer surace o web to centroid o combined horizontal and vertical steel reinorcement o web; usually taken as web thickness less
concrete cover less diameter o inner-ace vertical steel bars; e= eccentricity; h= height o spandrel; Pu = actored stem load; Vu = actored shear orce. 1 in. = 25.4 mm;1 t = 0.305 m; 1 kip = 4.448 kN.
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The maximum actored torque at the support Tu = 1113
kip-in. (125.8 kN-m)
Tu = . . . . .1 2 10 74 1 6 6 0 5 4 52
910+ +a a a f ak k k p k; E
Tu = 1113 kip-in. (125.8 kN-m)
The eccentric sel-weight o the ledge is neglected.
Step 2: Divide the slender spandrel
into three regions
Figure A3 shows the divisions o the slender spandrel into
three regions.
Step 3: Verify that the cross section can
sustain the twisting component of torsion
Veriy that the maximum torque Tu in the end region does
not exceed the twist limit or this region by consideringEq. (23).
Typically, a section is checked or the twisting component
o torsion about the 1-1 crack.
Tu < 1.13 f d hs c w2z lc m (23)
Tu
f d h
T
2 f y w
u
z
(24)
Asv/s >. , .
, ,
2 0 9 60 000 6 5 60
1 113 000
a a a ak k k k
Asv/s > 0.0264 in.2/in. (670 mm2/m)
Asv/s > 0.317 in.2/t (670 mm2/m)
For the transition region (maximum Tu in this region is 866
kip-in. [97.8 kN-m]):
Asv/s >. f d h
T
2 3f y w
u
z(25)
Tu < 1536 kip-in. (173.5 kN-m)
Tu = 1113 < 1536 kip-in. (125.8 < 173.5 kN-m) The sec-
tion is OK for twist
Check that lateral connection spacing exceeds 0.6h.
Spacing = 60 12 4 = 44 in. (1100 mm)
44 in./60 in. = 0.73
0.73h > 0.6h
The lateral connection spacing is sucient; thus, a check
or twist on the 2-2 crack is not required.
Step 4: Design the beam for flexure
The beam is designed or fexure using the techniques
in section 4.2 o the PCI Design Handbook(not shown).
Thirteen longitudinal prestressing strands are selected to
provide fexural resistance.
Step 5: Calculate the required shear steel
at all sections along the beam
The beam is designed or vertical shear using the tech-niques in section 4.3 o the PCI Design Handbook(not
shown). For the vertical shear design in this step, eccen-
tricity is ignored. The vertical shear steel required (Av/s)
is determined to be 0.08 in.2/t at all locations (Table A1).
Vc is relatively high in this example; thus, minimum steel
requirements govern.
Figure A3. Elevation view o vertical bars provided on inner web ace (step 7). Note: no. 4 = 13M; 1 in. = 25.4 mm.
Table A1. Vertical shear steel requirement
Distance rom let end o beam, t 1 2.75 6.0 7.75 12.75 16.0 17.75 22.75
Shear steel required, in.2/t 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
Note: 1 in. = 25.4 mm; 1 t = 0.305 m.
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Step 8: Check the plate-bending capacity
about line 2-2 in the end region
Plate bending about line 2-2 may be more critical than
plate bending about line 1-1 in the end region (Fig. A1 and
A2). Whether line 2-2 is critical or plate bending depends
on the location o the lateral tieback reactions and on the
quantity o vertical shear steelAv. Ater proportioning the
vertical steel in step 7, veriy that the total quantity o verti-
cal steel crossing line 2-2 on the inner ace exceeds Eq. (27).
Asv2
f d
T
2f y w
u
z
(27)
Asv2 . , .
, ,
0 9 60 000 6 5
1 113 000
2a a a ak k k k
Asv2 1.59 in.2 (1030 mm2)
Eight vertical no. 4 (13M) reinorcing bars cross the 2-2
Asv/s >. . , .
,
2 3 0 9 60 000 6 5 60
886 000
a a a ak k k k
Asv/s > 0.0179 in.2/in. (460 mm2/m)
Asv/s > 0.215 in.2/t (460 mm2/m)
Step 7: Proportion the vertical steel on the
inner web face
Vertical steel is provided or shear and torsion on the inner
web ace (ledge or corbel side) to satisy Eq. (26) at alllocations:
Asi/s > /A A s2
1sv v+f p (26)
Table A2 lists the values o the shear and torsion steel
requirements on the inner web ace.
Vertical L-shaped bars are spaced on the inner web ace
(Fig. A4) to satisy the shear/torsion and hanger steel
requirements.
The no. 4 (13M) reinorcing bars spaced at 6.5 in. (165
mm) satisy the 0.357 in.2/t (750 mm2/m) required in the
end region. The no. 4 reinorcing bars spaced at the hanger
steel limit o 8 in. (200 mm) satisy the 0.255 in.2/t (540
mm2/m) required in the transition zone and also the 0.222
in.2/t (470 mm2/m) required at all locations or hanger
steel. Two C-shaped bars are provided at each end o the
beam to conne the connection zone and control longitu-
dinal splitting at the ends o the prestressing strands. The
remaining bars are L-shaped bars.
Table A2. Shear and torsion steel requirements, inner web ace
Beam region End Transition Flexure
Distance rom let end o beam, t 1 2.75 6.0 7.75 12.75 16.0 17.75 22.75
Shear steelAv/s, in.2/t 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
1/2(Av/s), in.2/t 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04
Asv/s, in.2/t 0.317 0.317 0.317 0.215 0.215 0.215 0 0
Steel required on inner web ace or shear and
torsion, in.2/t*0.357 0.357 0.357 0.255 0.255 0.255 0.04 0.04
*Other inner-ace steel requirements, such as hanger steel, may still control design. Using section 4.5 o PCI Design Handbook, required hanger steelAsh
is 0.222 in.2/t (470 mm2/m) (not shown). In addition, hanger steel design dictates that hanger bar spacing not exceed ledge depth o 8 in. (200 mm).
Hanger steel requirement is not additive. Thus, designer should choose larger o hanger steel requirement or shear/torsion steel requirement at given
location.
Note:Asv = area o vertical plate-bending reinorcement;Av = area o shear reinorcement; s= spacing o vertical reinorcement. 1 in. = 25.4 mm;
1 t = 0.305 m.
Figure A4. Prole view o web steel (step 9). Note: no. 4 = 13M; 1 in. = 25.4 mm.
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For the transition region (maximum Tu in this region is 866
kip-in. [97.8 kN-m]):
Asl >. f d
T
2 3f y w
u
z(30)
Asl >2.3 . , .
,
0 9 60 000 6 5
886 000
a a a ak k k k
Asl > 1.07 in.2 (690 mm2)
Step 11: Proportion the longitudinal web steel forplate bending
Six no. 5 (16M), horizontal U-bars provide the required
longitudinal steel on each ace in the end region. All bars
are located above the lower lateral reaction.
Sucient ully developed excess fexural reinorcement
may be used in the transition zone to satisy the longitudi-
nal steel requirement. In this example, prestressing strands
not required or fexural reinorcement are used to meet the
longitudinal plate-bending steel requirement in the transi-
tion zone. The longitudinal steel required in this zoneAsl(as previously calculated assuming 60 ksi [420 MPa]) can
be reduced proportionally according to the increase in steel
strength (rom 60 ksi to 270 ksi [420 MPa to 1860 MPa]).
Thus, 0.24 in.2 (150 mm2) o 270 ksi (1860 MPa) prestress-
ing steel would satisy theAsl requirement in the transition
zone or this example.
Step 12: Detail the beam
Recommendations in the PCI Design Handbookregarding
ledge design, connection detailing, hanger steel, impact
steel, and reinorcement or other possible localized ailure
crack plane, providing 1.60 in.2 (1030 mm2) o steel and
satisying theAsv2 requirement.
Step 9: Proportion the vertical steel
on the outer web face
Vertical steel is provided on the outer web ace to satisy
Eq. (28) at all locations.
Aso A2
1v
(28)
Av was calculated in step 5. Thus, the vertical steel requiredon the outer aceAso or shear and torsion is 0.04 in.
2/t (85
mm2/m) at all locations. A continuous sheet o 6 in. 6 in.
(150 mm 150 mm), W2.1 W2.1 welded-wire reinorce-
ment will satisy this requirement (Fig. A5).
Step 10: Calculate the longitudinal
reinforcement required to resist
the plate-bending component of torsion
The total quantity o longitudinal steel required to resist
plate bending is determined.
For the end region (maximum Tu in this region is 1113 kip-
in. [125.8 kN-m]):
Asl >f d
T
2f y w
u
z(29)
Asl >2 . , .
, ,
0 9 60 000 6 5
1 113 000
a a a ak k k k
Asl > 1.59 in.2
(1030 mm2
)
Figure A5. Horizontal U-shaped bars provided in end region (step 11).
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modes are to be considered. In addition, it is recommended
that continuous bars or strands be placed in the longitudi-
nal direction at all our corners o the web to control the
possible cracking rom lateral bending away rom the ends.
Step 13: Check service-level cracking
I desired, the load at which diagonal cracking in the endregion is rst likely to appear may be conservatively esti-
mated according to Eq. (31). In many cases, beams will be
cracked at service load, regardless o end-region reinorce-
ment. First cracking load is independent o web reinorce-
ment conguration (that is, open or closed stirrups). The
eects o prestressing are ignored in Eq. (31) because
some o the prestressing strands are not ully developed
across the potential crack.
Vcr=/e b
fbh
1 2
4c
+
l
a k
R
T
SSS
S
V
X
WWW
W
(31)
where
Vcr= shear orce at cracking
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About the authors
Gregory Lucier is a laboratorymanager or North Carolina State
University in Raleigh, N.C.
Catrina Walter is an engineer or
BergerABAM in Federal Way,
Wash.
Sami Rizkalla is a distinguished
proessor or North Carolina State
University in Raleigh.
Paul Zia is a distinguished
university proessor emeritus or
North Carolina State University in
Raleigh.
Gary Klein is executive vice
president and senior principal or
Wiss, Janney, Elstner Associates
Inc. in Northbrook, Ill.
Synopsis
This paper summarizes the results o an analyticalresearch program undertaken to develop a rational de-
sign procedure or normalweight precast concrete slen-
der spandrel beams. The analytical and rational models
use test results and research ndings o an extensive
experimental program presented in the companion pa-
per Development o a Rational Design Methodology
or Precast Concrete Slender Spandrel Beams: Part 1,
Experimental Results, which appeared in the Spring
2011 issue oPCI Journal. The overall research eort
demonstrated the validity o using open web reinorce-
ment in precast concrete slender spandrel beams and
proposed a simplied procedure or design. The webs
o such slender spandrels, particularly in their end
regions, are oten heavily congested with reinorcing
cages when designed with current procedures. The
experimental and analytical results demonstrate that
open web reinorcement designed according to the
proposed procedure is sae and eective and provides
an ecient alternative to traditional closed stirrups or
precast concrete slender spandrel beams..
Keywords
Open reinorcement, slender spandrel beam, torsion,
twist.
Review policy
This paper was reviewed in accordance with the
Precast/Prestressed Concrete Institutes peer-review
process.
Reader comments
Please address any reader comments to journal@pci.
org or Precast/Prestressed Concrete Institute, c/o PCI
Journal, 200 W. Adams St., Suite 2100, Chicago, IL60606. J