development rational design part 2

7/28/2019 Development Rational Design Part 2

1/28Fal l 2011 |PCI Journal6

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


2/28 10PCI Journal|Fal l 2011

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).



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



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.


5/28



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.



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.



= 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



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.



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.



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



(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.



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.



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.



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.



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.



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.


18/28



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-



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



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



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.



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.



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.



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).



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


28/28

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

development rational design part 2

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