by Carlo R. Lini and Julio A. Ramirez - pci.org Reports... · Daniel P. Jenny Research Fellowship...

Daniel P. Jenny Research Fellowship

On the Design for Torsion of Precast/Prestressed Concrete SpandrelGirders

by

Carlo R. Lini and Julio A. Ramirez

Purdue UniversitySchool of Civil Engineering

West Lafayette, IN 47907

Final Report Submitted To

Precast/Prestressed Concrete Institute209 West Jackson BoulevardChicago, Illinois 60606-6938

June 2004

Conclusions or recommendations in this report are the opinions of the authors. PCI assumes no responsibility for the interpretation or application of the information contained herein.

ii

TABLE OF CONTENTS

Page

LIST OF TABLES .............................................................................................................vi

LIST OF FIGURES ..........................................................................................................vii

NOTATION ........................................................................................................................x

INTRODUCTION ..........................................................................................................1

1.1 Problem Statement .............................................................................................2

1.2 Objective ............................................................................................................3

1.3 Summary ............................................................................................................3

DESIGN FOR TORSION ...............................................................................................9

2.1 Background ........................................................................................................9

2.1.1 Noncircular Sections .........................................................................10

2.2 Skew Bending Theory ......................................................................................11

2.2.1 Transverse Steel ................................................................................12

2.2.2 Longitudinal Steel .............................................................................14

2.2.3 Minimum Longitudinal Steel .............................................................14

2.3 Thin-Walled Tube and Space Truss Analogy ..................................................16

2.3.1 Thin-Walled Tube .............................................................................16

2.3.2 Space Truss .......................................................................................17


o

°oo111

Page

2.3.2.1 Transverse Steel Derivation ...............................................18

2.3.2.2 Longitudinal Steel ..............................................................19

2.3.2.3 Minimum Longitudinal Steel .............................................20

2.4 Equilibrium and Compatibility Torsion ...........................................................21

2.5 Warping Torsion ..............................................................................................22

2.6 Summary ..........................................................................................................23

EXAMINATION OF DESIGN PROCEDURES .........................................................33

3.1 Fourth Edition ..................................................................................................33

3.1.1 Critical Section ..................................................................................33

3.1.2 Threshold Torsion .............................................................................34

3.1.3 Maximum Combined Shear and Torque ...........................................35

3.1.4 Concrete Contribution .......................................................................35


3.1.6 Longitudinal Steel .............................................................................38

3.2 Fifth Edition .....................................................................................................39


3.2.2 Threshold Torsion .............................................................................39

3.2.3 Cross Section ....................................................................................40


3.2.5 Longitudinal Steel .............................................................................4l

3.3 Summary ..........................................................................................................42


iv

Page

4. DESIGN EXAMPLE ....................................................................................................53

4.1 Analysis ............................................................................................................53

4.2 Adjustments .....................................................................................................54

4.2.1 Level of Prestressing .........................................................................54

4.2.2 Concrete Contribution in Shear ........................................................55

4.2.3 Depth to Compression Steel ..............................................................57

4.2.4 Summary of Adjustments .................................................................57

4.3 Required Longitudinal Steel Area ...................................................................58

4.3.1 Impact of Simplification of Ao ..........................................................58


4.4 Minimum Longitudinal Steel Area ..................................................................60

4.5 Threshold Torsion ............................................................................................61

4.6 Summary ..........................................................................................................61

5. PARAMETRIC STUDY ..............................................................................................76

5.1 Precast Pretensioned Spandrel .........................................................................76

5.2 Post-Tensioned Rectangular Section ...............................................................77

5.3 Comparison With Test Results ........................................................................78

5.4 Summary ..........................................................................................................79

6. SUMMARY OF FINDINGS AND CONCLUSIONS .................................................93

6.1 Findings ............................................................................................................93

6.1.1 Adjustments ......................................................................................93

6.1.2 Concrete Cover Spalling ...................................................................94


v

Page

6.2 Recommendations ............................................................................................94

6.3 Future Work .....................................................................................................95

LIST OF REFERENCES ...................................................................................................96


vi

Table

4.1

4.2

4.3

5.1

5.2

5.3

5.4

5.5

LIST OF TABLES

Page

Adjusted At Values ................................................................................................63

Adjusted At-~n Values ...........................................................................................63

Comparison of At Values .......................................................................................63

L Spandrel Beam ....................................................................................................80

Rectangular Section ...............................................................................................80

Comparison of Tests to Prestressed Concrete Design Procedure ..........................81

Eligible Prestressed Concrete Specimens ..............................................................82

Ineligible Prestressed Concrete Specimens ...........................................................83


vii

Figure

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

2.1

2.2

LIST OF FIGURES

Page

Underside view of a precast/prestressed L spandrel beam. The inverted T-

beam that is resting on the spandrel beam (at midspan) creates torsion in the

restrained spandrel beam .........................................................................................5

A view from the outside shows multiple double-T beams that are resting on the

spandrel beams .........................................................................................................5

End sections of the spandrel beams appear left and right of the column. One web

is visible of the double-T beam at the end of each spandrel beam (similar to

Example 4.4.2 from the 5th Edition of the PCI Design Handbook) .........................6

Typical Spandrel Sections ........................................................................................6

Bottom Restraint of Dapped L Spandrel Beam .......................................................7

Top and Bottom Restraint of Dapped L Spandrel Beams ........................................7

Typical Detail of a Dapped L Spandrel Beam .........................................................8

Equilibrium of a Dapped L Spandrel Beam with End Restraints ............................8

Coulomb’s Device for Torsional Oscillation Tests of Thin Wires ........................25

Comparison of Coefficients ...................................................................................26


viii

Figure

2.3

2.4

2.5

2.6

2.7

2.8

3.1

3.2

3.3

3.4

3.5

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

Page

Skew-Bending Failure Diagram ............................................................................27

Coefficient as a Function of yx/x~ ..........................................................................28

Shear and Torsion Interaction ................................................................................29

Thin-Walled Tube and Space Truss .......................................................................30

Thin-Walled Tube Derivation ................................................................................31

Space Truss Cross Section .....................................................................................32

Example of a Divided Cross Section .....................................................................44

Threshold Torsion Comparison for a Prestressed L Shaped Section .....................44

Varying Angle 0 of the Inclined Compressive Strut of the Variable Angle

Space Truss and Hoop Spacing ..............................................................................45

4th Edition Design Procedure Flow Chart ..............................................................46

5th Edition Design Procedure Flow Chart ..............................................................50

Adjustment on Required Longitudinal Steel Reinforcement .................................64

Adjustment on Av/s ................................................................................................65

Adjustment on Available At/s ...............................................................................66

Adjustment on Compressive Strut Angle ..............................................................67

Equation (4.5) and the Required Amount of Longitudinal Steel Reinforcement ..68

Comparison of Ao and 0.85 Aoh .............................................................................69

Using Equation (4.5) and the Compressive Strut Angle ........................................70

Value of Design Torsion Using 4th and 5tu Edition ................................................ 71

Shear Contribution Values for Both Editions ........................................................72

Adjustment on Minimum Longitudinal Steel Reinforcement ...............................73


ix

4.8

4.9

4.10

4.11

4.12

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

Value of Design Torsion Using 4th and 5th Edition ................................................ 71

Shear Contribution Values for Both Editions ........................................................72

Adjustment on Minimum Longitudinal Steel Reinforcement ...............................73

Scaled Cross Sections ............................................................................................74

Threshold Torque Values Versus Scale Multiplier ................................................75

L Section Required At as a function of Strut Angle ..............................................84

L Section Required A~_min as a function of Strut Angle ......................................... 85

HSU Example Problem 5.1 ....................................................................................86

Example 5.1 Flexural Moment, Shear, and Torque Diagrams for Dead and Live

Loads ......................................................................................................................87

Rectangular Section Comparison of A~ Required for Strength ..............................88

Compressive Strut Angle .......................................................................................89

Section Example Required A~ for Strength as a Function of the Strut Angle ........90

Rectangular Section Required A~-rnin as a Function of Strut Angle ........................91

Rectangular Section (A1 5th)/(A1 4th) .................................................... 92


NOTATION

Acp ~

Ag =

A~ =

Ag-min =

Ao --

Aoh =

Astirrup =

At

Av --

Av-bar =

bw =

Ct =

area enclosed by outside perimeter of concrete cross section, in.2

gross area of section, in.2

total area of longitudinal reinforcement to resist torsion, in.2

minimum required area of longitudinal reinforcement to resist torsion, in.2

gross area enclosed by the center line of shear flow path, in.2

gross area enclosed by centerline of the outermost closed transverse

torsional reinforcement, in.2

area of one leg of chosen stirrup size, in2

area of one leg of a closed stirrup resisting torsion within a distance s, in.2

area of shear reinforcement within a distance s, or area of shear

reinforcement perpendicular to flexural tension reinforcement within a

distance s for deep flexural members, in.2

area of both legs of a closed stirrup, in.2

web width, or diameter of circular section, in.

torsional constant

bwd

Zx2y


xi

d

fpc

fy¥

distance from extreme compression fiber to centroid of longitudinal

tension reinforcement, but need not be less than 0.8h for circular sections

and prestressed members, in.

nominal diameter of bar, wire, or prestressing strand, in.

specified compressive strength of concrete, psi

square root of specified compressive strength of concrete, psi

stress due to unfactored dead load, at extreme fiber of section where

tensile stress is caused by externally applied loads, psi.

compressive stress in concrete (after allowance for all prestress losses) at

centroid of cross section resisting externally applied loads or at junction of

web and flange, psi. (in a composite member, fpc is resultant compressive

stress at centroid of composite section, or at junction of web and flange

when the centroid lies within the flange, due to both prestress and

moments resisted by precast member acting alone)

compressive stress in concrete due to effective prestress forces only (after

allowance for all prestress losses) at extreme fiber of section where tensile

stress is caused by externally applied loads, psi

stress in the hoop, psi

maximum principal tensile stress of concrete, psi

t’t -=15 ~c

yield strength of closed transverse torsional reinforcement, psi

yield strength of longitudinal torsional reinforcement, psi


xii

G

h

I

Kt

Mmax

Pcp

q

Smax

modulus of rigidity, psi

overall thickness of member, in.

moment of inertia of section resisting externally applied factored loads,

in.4

torsional stiffness

7t 02- lOfpc/f’c)

length of wire

moment causing flexural cracking at section due to externally applied

loads

(I/Yt) (6X/-~7+fp~-fa)

maximum factored moment at section due to externally applied loads, in.-

lb

outside perimeter of the concrete cross section, in.

perimeter of centerline of outermost closed transverse torsional

reinforcement, in.

shear flow, lb/in.

spacing of shear or torsion reinforcement measured in a direction parallel

to longitudinal reinforcement, in.

maximum allowable spacing, in.

torque, in-lb

nominal torsional moment strength provided by concrete, in.-lb

nominal pure torsional moment strength provided by concrete, in.-lb


xiii

Wcr

Tn(max)

Tu(min)

Tu

Tv

x

Xl

y

y~

Yt

Vc

Vcr

Vu

Vo

V’c

Vci

VCW

Vd

cracking torque of concrete, in.-lb

torsional resistance provided by the horizontal forces in the hoops, in.-lb

maximum allowable torque, in.-lb

threshold torsion value, in.-lb

factored torsional moment at section, in.-lb

torsional resistance provided by the vertical forces in the hoops, in.-lb

short side of a component rectangle, in.

short side of the stirrup, in.

long side of a component rectangle, in.

long side of the stirrup, in.

distance from centroidal axis of gross section, neglecting reinforcement, to

extreme fiber in tension, in.

contribution of concrete stress in shear, psi

shear cracking stress, psi

factored shear stress, psi

nominal shear strength provided by concrete, lb

nominal pure shear strength provided by concrete, lb

nominal shear strength provided by concrete when diagonal cracking

results from combined shear and moment, lb

nominal shear strength provided by concrete when diagonal cracking

results from excessive principal tensile stress in web, lb

shear force at section due to unfactored dead load, lb

factored shear force at section due to externally applied loads occurring


xiv

Vn(max) =

Up

Vu =

U~t

~tt =

0 =

)~ =

g =

"1~c

~cr

Tu ~---

q0 =

q)w =

simultaneously with Mmax, lb

maximum nominal shear strength, lb

vertical component of effective prestress force at section, lb

factored shear force at section, lb

torsion coefficient for Skew-Bending Theory

1/3

[0.66 + 0.33y1/x1 ] _< 1.5

a factor dependent on the level of prestress

angle of compression diagonals in truss analogy for torsion

correction factor related to unit weight of concrete

a material constant

contribution of concrete stress in torsion, psi

torsional cracking stress, psi

factored torsional stress, psi

strength reduction factor

twisting angle


1. INTRODUCTION

A spandrel beam is an edge beam that can be subjected to large torsional moments as a

result of carrying slab, joists, and/or beams from one side. A typical precast/prestressed

concrete spandrel beam may be found in parking deck structures (Figure 1.1-1.3). Loads

applied on the ledge of an L spandrel beam result in a torsional moment due to the

eccentricity of the load with respect to the centroid of the section when it is restrained at

the ends. An advantage of using a pocket spandrel beam versus and L beam is the

reduced eccentricity of the applied load, hence a lower torque. This is shown in Figure

1.4. Figure 1.5 and 1.6 shows a photo of a restrained dapped L spandrel beam. A typical

detail of the spandrel to column connection for restrained dapped L spandrel beams, as

that seen in Figure 1.5 and 1.6, is shown in Figure 1.7. Figure 1.8 shows the basic

equilibrium of the dapped L spandrel beam.

In this thesis, two different design procedures for precast/prestressed spandrel

beams will be discussed. One based on the skew-bending theory and the other on a thin-

walled tube analogy prior to cracking, and a variable angle space truss analogy after

cracking. Both of these approaches have been widely used at one time or another in the

United States. Considering only equilibrium torsion (torsional moment required for

equilibrium), transverse and longitudinal reinforcement placed in a spandrel beam help it

to resist torsional moments. Depending on what design procedure is used, a concrete

contribution to the torsional strength of a reinforced/prestressed beam may also be


2

included after cracking. A detailed discussion of these two design procedures is provided

in Chapters 2 and 3.

1.1 Problem Statement

Example 4.4.2 of the 4th and 5tu Edition of the PCI Design Handbook is of particular

interest. Example 4.4.2 is a shear and torsion design example of a prestressed spandrel L

beam. In the 4th Edition, the design procedure used in Example 4.4.2 is based on the

skew-bending theory. The 5th Edition uses the thin-walled tube analogy prior to cracking

and variable angle space truss analogy after cracking as the basis for design. The

examples are essentially meant to have the same design problem setup but solved with

different design procedures. However, as will be shown in Chapter 4, there are some

differences. For the same spandrel beam (the transverse steel size and spacing are also

the same) and design conditions, the 5th Edition approach resulted in a required

longitudinal area for torsional strength approximately three times greater than that from

the 4th Edition procedure. The 5th Edition approach also resulted in a minimum

longitudinal area for torsion is approximately twice as large as the result obtained with

the 4th Edition. It is important to establish the reason for the increase in the longitudinal

steel. It must be noted that the current provisions in the ACI 318-02 Code are based on a

thin-wall tube analogy prior to cracking and variable angle space truss analogy after

cracking, similar to the procedures used in the 5th Edition.


3

1.2 Objective

The objective of this report is to explain the stated differences in the longitudinal

reinforcement amounts required for torsion by the skew-bending approach in the 4th

Edition of the PCI Design Handbook and the thin-walled tube and space truss analogy

procedure in the 5th Edition of the same handbook.

1.3 Summary

Two design procedures are the focus of this report, one based on the skew-bending theory,

and the other, based on the thin-walled tube analogy prior to cracking and variable angle

space truss analogy after cracking. Comparisons of both design procedures indicate that

there are differences in the longitudinal reinforcement requirements for torsion. It is

important to establish the reason for the difference in the required amounts of

reinforcement to properly establish the adequacy of both procedures. This is particularly

important since the design for torsion in the ACI 318-02 Code is based on the thin-walled

tube and variable angle space truss analogy.

Chapter 2 provides a brief history of torsion design in the United States, and

describes in greater detail, the skew-bending theory, and the thin-walled tube and variable

angle space truss approaches. The development of the design procedures based on these

theories is presented. Chapter 3 provides a comprehensive review of the design

procedures used in both the 4th and 5th Edition of the PCI Design Handbook. In Chapter

4, a comparison of Example 4.4.2 from both editions is presented. Chapter 5 will provide

results from the parametric studies conducted to establish the sensitivity of the procedures


4

to various design parameters. A summary of the report and needs for further research is

given in Chapter 6.


Figure 1.1: Underside view of a precast/prestressed L spandrel beam. The inverted Tbeam that is resting on the spandrel beam (at midspan) creates torsion in the restrainedspandrel beam.

Figure 1.2: A view from the outside shows multiple double-T beams that are resting onthe L spandrel beams.


Figure 1.3" End sections of the spandrel beams appear left and right of the column. Oneweb is visible of the double-T beam at the end of each spandrel beam (similar to Example4.4.2 from the 5th Edition of the PCI Design Handbook).

L Beam Pocket Spandrel

Figure 1.4: Typical Spandrel Sections


7

Figure 1.5: Bottom Restraint of Dapped L Spandrel Beam

Figure 1.6: Top and Bottom Restraint of Dapped L Spandrel Beams


~B045 TYP.PSA SLOTTED INSERT(J-FINISh)

MAST]CORDLOAD BEARING SiDE

ERECTOR NOTE: CP35g CAST INTACK WELD NUT END OF SPANDRELTO "I]-IR’D. ROD OR LEDGE WHERE D.T.BURR THREADS STEM IS B" OR

LESS FROM COL.

COLUMN-TO-SPANDRELCDNNECTI:R SLEEVE W/ COVER(HIGH CONCRETE ACC,)

SECTION "C"

~’~’= (ASTU A193 GR.B7)) THREADED RODX 2’-1" Wi3,,/4"~ASTM A325] NUT, &

(1) t~, H~t_E

EP553 TYP.X 6" X 0’-6"

MI~sTICORD PAD ~W/ 1"� HOLE AND EP551 ~PLASTIC HORSESHOE SHIMS~AS REO’D

Figure 1.7: Typical Detail of a Dapped L Spandrel Beam

Figure 1.8: Equilibrium of a Dapped L Spandrel Beam with End Restraints


9

2. DESIGN FOR TORSION

Chapter 2 provides a summary of the background of the design for torsion of prestressed

concrete beam sections in the United States, and provides background information on the

skew-bending theory, and the thin-walled tube and space truss analogy. Emphasis will be

placed on the connection between the theories and key design procedures. Only selected

portions of the required derivations will be provided and discussed within the context of

the approaches presented in the 4t~ and 5th Editions of the PCI Design Handbook. It must

be noted that prior to the 1995 Edition of the ACI Building Code for structural concrete

(318-95), the code was silent with respect to the design for torsion of prestressed concrete

members.

2.1 Background

In 1784, Coulomb ran a series of tests while investigating torsion. A rendition of the test

setup used is pictured in Figure 2.1. Coulomb, based on these experiments, established

the following equation to describe torque:

dw4T = i.t--~-~ ~pw (2.1)

where:

dw = diameter of the metal wire

~w = length of the wire


10

g = a material constant

This equation, used to describe torsion for an elastic circular member, was not proven

theoretically until 40 years later by Navier. Based on equilibrium, compatibility and

stress-strain relationships, Navier derived the following formula (Hsu 1984):

= rp (2.2)

The material constant in Coulomb’s equation, ~t, is expressed in Navier’s equation as

(~/32)G. Therefore, Equations (2.1) and (2.2) are the same.

2.1.1 Noncircular Sections

Navier’s theoretical equation and Coulomb’s experimental equation matched. However,

problems arose when Navier derived a similar expression for torsion of noncircular

sections, or more specifically, rectangular sections. Based on tests done by Duleau six

years earlier, comparison between iron members with circular and square cross sections

expressed approximately a 20% difference between the moduli of rigidity, G (Hsu 1984).

This 20% difference, initially thought by Navier to have been caused by inconsistent

qualifies of iron, was not correctly accounted for until St. Venant’s Semi Inverse

approach. There are two general approaches to the solution of stresses in the theory of

elasticity, the direct approach and the inverse approach. St. Venant invented a "semi-

inverse" method to solve the torsion problem for noncircular cross sections (Hsu 1984).

Using this method, St. Venant derived the following expression for torsional stress of a

rectangular member:


11

Tu (2.3)"/’u = ~x2y

Since then, three different theories have been developed to expand upon St. Venant’s

theory to predict torsional strength of plain concrete. They are the elastic theory, the

plastic theory, and the skew-bending theory. The 4th Edition of the PCI Design

Handbook uses a design procedure based on the skew-bending theory. A skew-bending

design method is also in the ACI 318 Building Code from 1971 through 1989. Therefore,

only the skew-bending theory will be covered in detail next.

2.2 Skew-Bending Theory

Zia and McGee (Zia 1974) stated that "a prestressed concrete member without web

reinforcement fails under torsion, in the form of skew-bending." When subjected to pure

torsion, a reinforced concrete member cracks when the maximum principal tensile stress

reaches the diagonal tensile strength of the concrete. Using Equation (2.3) developed by

St. Venant, the cracking torque based on the maximum principal tensile stress, f’t, can be

written as:

Tcr = f~t (~x2y) (2.4)

Simplifying Equation (2.4), the cracking torque of a member subjected to pure torsion

can be expressed as:

(2.5)

The term ZxZy allows a cross section to be broken down into individual rectangles to

simplify the calculation. Further explanation is given in Chapter 3. It must be noted that

for the skew-bending theory, the torsion coefficient, a, is equal to 113; whereas for the


12

plastic and elastic theories, a varies as a function of the aspect ratio, y/x. Figure 2.2

shows a comparison of the torsion coefficient for each theory. It can be observed that

there is an almost 50% change between the elastic and fully plastic estimates depending

on increases of the y/x ratio. This observation will be revisited when discussing the role

of the concrete contribution to the torsional strength of a member in subsequent chapters.

That is for tall slender sections, cracking can significantly decrease the concrete

contribution and torsional rigidity of the section. On the other hand, spalling of the

concrete cover may not be as generalized for the same sections but concentrated at the

top and bottom of the section.

2.2.1 Transverse Steel

Figure 2.3 shows a skew-bending failure surface with inclined cracks on three sides and a

concrete compressive zone on the fourth side. For simplification, assume that the

member is subjected to pure torsion so that cracking angles 01 and 03 are equal. The

following expression can be written for the horizontal forces in the hoops:

Th ---- nh (Atfs) Ya (2.6)

where:

nh -- horizontal hoops crossed by the crack (Xl cot 01/s)

Equation (2.6) can be rewritten as:

Th = klAtfy (xl/yl (2.7)

where:

k~ = cot 0~ (fs/fy)


13

A similar expression can be created for the vertical forces in the hoops:

T, = nv (Atfs)(kzxl) (2.8)

where:

nv = vertical leg of the hoop crossed by the crack (Yl cot 02Is)

The lever arm between the stirrup forces and the compressive force, Pc, is represented as

Rewrite Equation (2.8) so that it has the same form as Equation (2.7):

where:

k3 -- k2cotO2 (fs/fy)

k2Xl.

(2.9)

Adding Equation (2.7) and (2.9) will give and expression relating torsional strength and

transverse steel. Replace kl + k3 with at.

Ts = at xlYl Atfy (2.10)s

In the 4th Edition Ts is taken as (Tu/q0) - Tc when designing for At/s, where Tc is

the torsional concrete contribution term. The torsional concrete contribution term is

covered in Chapter 3. The coefficient at was developed by Thomas Hsu based on test

results of 53 beams (Hsu 1968). He stated that a rectangular cross-section can be defined

by two variables: the height-to-width ratio, and an absolute dimension, either the height,

width, or cross-sectional area. The latter variable is called size effect or scale effect. It is

this scale effect that was used to define at. However, it was not know whether the scale

effect was caused by the width, height, or area of cross-section. Hsu examined each one

separately. A total of five beam series were investigated to determine ~t. Hsu


14

determined for practical purposes that the coefficient is based on the area of cross-section

and the resulting expression is given below.

~xt = /0.66+ 0"33Yl/< 1.50\ x~

(2.11)

The limit of 1.5 is based on beam Series K which had an yl]Xl ratio of 4, the highest of

the five series investigated. Figure 2.4 shows ~t on the vertical axis as a function of yl/Xl

on the horizontal axis.

2.2.2 Longitudinal Steel

The horizontal component, Ph, of the compressive force shown in Figure 2.3 may be

equilibrated by the longitudinal steel. This will occur if the volume of longitudinal and

transverse steel is equal as was stated in the ACI 318-89 Code. Based on this assumption,

which only applies if the cracks are at 45° and the beam is subjected to pure torsion, the

following expression can be written for longitudinal steel:

(2.12)

2.2.3 Minimum Longitudinal Steel

Minimum torsional reinforcement is required to ensure the ductility of the beam when it

cracks. The interaction between torsion and shear must be included in the derivation of

minimum longitudinal steel and Figure 2.5 will be used to aid in this derivation. Beams

which lie in region (2) of Figure 2.5 must be designed for minimum longitudinal steel

such that the beam will develop an ultimate strength equal to the cracking torque, T, = T~r.


15

For a beam at point P, shear and torsion reinforcement must be provided for the stresses

(’lyr - "1¥) and (Vcr -- Vc). The torsional web reinforcement required to resist the torsional

stress is:

’lTcr -’1~c ) x2YAt-

3atXlYlfy(2.13)

Assuming at = 1.2 and x/xl = y/y~ = 1.2, Equation (2.13) can be simplified:

xsAt =-~-- E0.40 (’rcr -’to )~ (2.14)y

Shear web reinforcement required to resist the shear stress (Vcr - Vc) is ¯

bwS ,Av =--{Vcr-Vc) (2.15)

fy

Generally the height of a beam is greater than the width which allows the assumption that

bw = x for simplicity. Adding Equation (2.14) and (2.15) gives:

(2.16)

Hsu mentions that Equation (2.16) is to complicated for practical use since the stress

equations are complicated and derives the simplified equation (Hsu 1984):

200xs2At +Av-fy

(2.17)

Assume that the entire minimum web reinforcement is provided in the form of closed

stirrups allowing 2At to replace 2At+Av in Equation (2.17) which gives:

(2.18)


16

Since there is an equal amount of stirrups and longitudinal bars, Equation (2.18) can be

rewritten in terms of longitudinal steel:

s _ 200xs ( ~:z--~ (2.19)At’min (Xl +yl) fy

The total amount of minimum reinforcement includes both the stirrups and longitudinal

steel and can be obtained by adding Equations (2.18) and (2.19). Solving for !~-rnin and

replacing the stresses with forces yields the equation of minimum longitudinal steel:

3Ct ) J

The te~ 2At in equation (2.20) need not be t~en less than the ~nimum required

~ount of web reinforcement equal to 50bws/fy.

2.3 Thin-Walled Tube and Space Truss Analogy

The current design method, first adopted in the 1995 ACI Building Code, is based on a

thin-walled tube and space truss analogy. This method is generally viewed as being

easier to understand and use when compared with the skew-bending design method and

provides a comparable estimation of calculated capacities. This section will provide the

fundamentals of the thin-walled tube and the space truss analogy approach.

2.3.1 Thin-Walled Tube

Torsional strength provided by the interior of a prestressed concrete beam is viewed as

negligible prior to cracking due to the small moment arm from the centroid of the section.


17

If torsional strength provided by the interior concrete can be ignored, a solid beam could

be treated as an equivalent thin-walled tube as shown in Figure 2.6a. Torsion of a thin-

walled tube results in a shearing stress, x. This shearing stress can be expressed as shear

flow, q, if the thickness of the tube, h, is known. Shear flow is simply q = "c h. Figure 2.7

provides an illustration of a thin-walled tube subject to a torque, T, and its corresponding

shear flow. Based on Figure 2.7, torque can be expressed as:

T=q~ rdt (2.21)

Looking closely at Figure 2.7, rdt is twice the area of the shaded triangle. Therefore the

integral of rdt would be twice the gross area enclosed by the center line of shear flow, Ao.

~rdt=2Ao (2.22)

Substituting Equation (2.22) into (2.21), the following expression can be written for shear

Tq=-~o

(2.23)

2.3.2 Space Truss

Tests have shown that after cracking, solid and hollow beams end up with roughly the

same torsional strength (MacGregor and Ghoneim 1995). This suggested that once

cracking occurred, any contribution toward torsional resistance provided by the interior

concrete is minimal. Once the thin-walled tube has cracked, the beam can be modeled as

a space truss. A space truss is composed of three major components; compressive

concrete struts, transverse steel, and longitudinal steel. All three are put together to form

the space truss as pictured in Figure 2.6b. The dimensions Xo and Yo are measured to the


18

center of the corner bars (longitudinal reinforcement). By creating a relationship between

the space truss and the shear flow discussed in the previous section, direct equations

between the torque and the reinforcing steel can be established.

2.3.2.1 Transverse Steel Derivation

Observing one side of the space truss in Figure 2.8a, simple statics can be used to relate

stirrups with torque. The shear force, V2, acting on side two is just the shear flow, q,

times the length of the side, Yo. Equation (2.23) can be expressed as:

TV2 = ~--~oyo (2.24)

From statics, the force in the stirrups cut by the inclined crack must equal V2. The

number of stirrups that the crack crosses is equal to:

n = Y° cot0 (2.25)s

From Figure 2.8(a) and Equation (2.25), V2 can be expressed in the following manner.

Atf~YoV2 = cot0 (2.26)s

Substituting Equation (2.24) into Equation (2.26) gives the following expression for

torsional strength based on transverse steel:

2AoAIfT~ - ~ cot0 (2.27)

S

Typically Equation (2.27) is rearranged to solve for either cot 0, or for At/s. This depends

on the designer’s preference of selecting the desired compressive strut angle 0, or the


19

desired size and spacing of the hoops. This flexibility in the design approach does not

exist in the 4th Edition where the compressive strut angle is always taken as 45 degrees.

2.3.2.2 Longitudinal Steel

The longitudinal steel’s contribution to torsional resistance can be derived using the same

approach used for the transverse steel. Figure 2.8b shows two sides of the space truss.

As shown in Figure 2.8b, the diagonal compressive force D2, parallel to the concrete

struts can be resolved into a shear force, V2, and an axial tension force, N2, where V2 is

expressed in equations (2.24) and (2.26). The longitudinal steel must account for the

cumulative axial tensile force N, where N can be expressed as,

N= 2(N~+N2) (2.28)

The axial tensile force on side 2 can be expressed as a function of the shearing force, V2.

A similar expression can be used for the axial tensile force on side 1.

N2 = Vzcot0 (2.29)

N~ = V~cot0 (2.30)

force can be accounted for by the longitudinal steel by summingThe total axial

horizontal components and setting them equal to A~fyt, where At equals the area of

longitudinal steel placed inside the hoops around the perimeter of the section.

Substituting Equations (2.29) and (2.30) into Equation (2.28) results in:

A~fy~ = 2( V~ +V2 )cot0 (2.31)


2O

An expression similar to Equation (2.26) can be used for V1, only in this case, Xo is used

(2.31) results in ain place of Yo. Substituting these expressions into Equation

longitudinal steel equation as a function of the transverse steel, At/s.

(2.32)

If the yield strength of both reinforcements is the same, then the ratio of yield strengths,

fyv/fy~, can be taken as 1.

2.3.2.3 Minimum Longitudinal Steel

A minimum required amount of torsional reinforcement, both longitudinal and transverse

steel, is necessary in order to avoid a brittle torsional failure. To determine the minimum

required amount of torsional reinforcement, the post-cracking strength Tn will be set

equal to the cracking strength Tcr and the compressive concrete strut angle will be taking

as 45 degrees. The equation for the post-cracking torsional strength, Tn, can be obtained

from rearranging Equation (2.27):

2AoAtfy~T,- (2.33)s

The cracking torque of solid and hollow sections can be expressed by the same equation:

T==4~Ag Acp (2.34)Pep

Combining equation (2.34) and (2.33) gives:

Atf~ = 2~-~c AgAcpS (2.35)AoPcp


21

In order to simplify Equation (2.35), multiply both sides by ph/s. Hsu states that "for

sizes of cross sections normally used in buildings, the ratios Acp/Ao and ph/pcp can be

taken as 1.5 and 0.83 respectively" (Hsu 1997). Simplifying Equation (2.35) gives:

Atfyv Ph =2.5~Ag(2.36)

s

Since the compressive strut angle is assumed to be 45 degrees, there must be an equal

amount of longitudinal and transverse reinforcement.

A~fye = Atf~v P-~-~ (2.37)S

Substituting Equation (2.36) into Equation (2.37) gives:

Atfre =2.5~~ Ag (2.38)

Adding Equation (2.36) and Equation (2.38) and solving for At, which represents the

minimum amount of longitudinal steel, At-min, gives:

5@Ag I~) (fyv)At-n~n = fyt - Ph ~ (2.39)

The torsional transverse steel, At/s, shall not be taken less than 25bw/fyv.

2.4 Equilibrium and Compatibility Torsion

In structures, there are two types of torsion, equilibrium torsion and compatibility torsion.

Equilibrium torsion refers to the case where the torsional moment cannot be reduced by

the redistribution of internal forces. With compatibility torsion, the torsional moment can

be reduced due to the redistribution of internal forces after cracking as it arises from

restrained deformations. The redistribution of internal forces can result from torsion


22

created by the member twisting in order to maintain compatibility of deformations. For

equilibrium torsion, reinforcement must be provided to resist the total design torsional

moment as it is necessary for equilibrium of the structure. Equilibrium torsion can occur

in both statically determinate and indeterminate structures. Example 4.4.2 of the 4th and

5th Edition of the PCI Design Handbook represents a case of equilibrium torsion and

hence reinforcement is provided to resist the total design torsional moment.

Compatibility torsion only occurs in statically indeterminate structures and the current

code, ACI 318-02, allows for the reduction of the maximum factored torsional moment

for prestressed members to:

q~4~c, cp ÷ pc

~Pcp ~ 4~c’(2.40)

This allows for a reduction in the amount of torsional reinforcement required in the

spandrel beam. Equation (2.40) is the cracking torque value for a prestressed concrete

beam of a solid cross section. Equation (2.40) may also be used for hollow sections as

well. As specified in the ACI 318-02 Code, for hollow sections, Acp shall not be replaced

with Ag in Equation (2.40). Thus, the torque after redistribution is larger and hence more

conservative. This is unlike the calculation of the threshold torque for hollow sections

where Ag is used in place of Acp.

Chapter 3.

Threshold torque is discussed in greater detail in

2.5 Warping Torsion

Warping of a thin-walled, non-circular, open section generally occurs since plane

surfaces perpendicular to the longitudinal axis will not remain plane. In other words, not


23

only does a particular cross section rotate, but it also deforms out of its plane. An

unrestrained L-shaped spandrel beam under an applied torque will experience some

warping. If this warping deformation is restrained by supports, diaphragms, or the

requirement of symmetry, a phenomenon called warping torsion will occur. Both

editions neglect warping torsion. Hsu has stated that "warping torsion will drastically

modify the torsional behavior and stresses of a thin-walled open section" (Hsu 1984). St.

Venant’s solution for elastic noncircular sections does account for warping of

unrestrained beams. However, if a beam were restrained creating warping torsion, an

additional normal stress in the longitudinal direction would exsist that is not accounted

for in St. Venant’s derivation that resulted in Equation (2.3) shown earlier.

For a thin-walled tube, the total differential warping displacement around the

whole perimeter of a closed section must be equal to zero. There can be no warping

torsion if there is no warping displacement. The 5th Edition of the PCI Design Handbook

is based on the thin-walled tube analogy. But the thin-walled tube analogy does not

represent the actual behavior of an L shaped spandrel beam where warping torsion in a

spandrel beam is restrained at the ends. However, this type of torsion is neglected

because the added complexity of including warping torsion is not justified on the basis of

the variability of concrete tensile strength and the actual influence of the end restraints.

2.6 Summary

Two main methods have been developed for the design for torsion of structural concrete,

one based on the skew-bending theory and the other based on the thin-walled tube and

variable angle space truss analogy. The design procedure used in the 4tr’ Edition of the


24

PCI Design Handbook is based on the skew-bending theory and uses Equations (2.10),

(2.12) and (2.20) derived in this chapter. The 5th Edition design procedure is based on the

thin-walled tube prior to cracking and a variable angle space truss analogy after cracking,

and uses equations (2.27), (2.32) and (2.39). Therefore, the basis of the respective design

procedures has been established. It must be remembered that there is a significant

reduction in the torsional rigidity of the section after cracking and the contribution of the

concrete is neglected for ultimate strength considerations.

Chapter 3 provides a comprehensive review of the design procedures themselves.

In Chapter 4, the results of Example 4.4.2 from the 4th and 5t~ Edition are compared. It

will be seen that there are some differences between both procedures. Chapter 5 provides

results of the parametric studies conducted to study the influence of critical parameters in

both design procedures. Chapter 6 summarizes the report, and indicates areas of needed

research.


25

(a) /

90

Figure 2.1: Coulomb’s device for torsional oscillation tests of thin wires (Hsu 1984)


0.6

0.5

0.4

0.3

0.2

0.1

Nadai’s Plastic Coefficient

St. Venant’s Elastic Coefficient

5 6 7

y/xFigure 2.2 Comparison of coefficients

0.48

5O%

0.31

10


Xl

Pe= Concrete Compressive ForcePv= Vertical Component of Compressive Force

Ph= Horizontal Component of Compressive Force

T

Figure 2.3: Skew-Bending Failure Diagram


1.8

1.6

1.4

,q.~ 1.2

0.8

Beam Series N and G

Beam Series B ~~

’Beam Series C

Beam Series K

1 1.5 2.5 3 3.5

yl/xl

4

Figure 2.4: Coefficient as a Function of yl/xl


29

INTERACTION CURVEFOR CRACKING

P

INTERACTION CURVE FORCONTRIBUTION OF CONCRETE

Figure 2.5: Shear and Torsion Interaction


30

T

Shear Flow Path

T

Stirrups

Cracks

Shear Flow, q

LongitudinalBar

ConcreteCompressiveStruts

Note: The height and width of the truss are Yo and xo, measured between thecenters of the comer bars.

(b)

Figure 2.6: (a) Thin-Walled Tube; (b) Space Tress


h

Figure 2.7: Thin Walled Tube Derivation


32

V2 Y0

Y0

Yo cot e =I

Side 1

Side 2

N1/2

N1/2

N2/2

N~

N2/2

Note: The height and width of the truss are Yo and xo, measured between thecenters of the comer bars.

Figure 2.8: Space Tress Cross Section


33

3. EXAMINATION OF DESIGN PROCEDURES

The following review of the 4th and 5th Edition procedures is provided in that order. Flow

charts for each procedure have also been provided on the basis of the example provided

in both editions of the PCI Design Handbook. In Chapter 4, a detailed comparison of the

design approaches is given.

3.1 Fourth Edition

Based on the design constraint of Example 4.4.2 in the 4th Edition of the PCI Design

Handbook, such as loading conditions, L beam sectional dimensions, compressive

strength of the concrete, yield strength of the reinforcing steel and information on the

prestressing strands, the following steps can be followed to design for torsion and shear.

3.1.1 Critical Section

From the loading conditions provided, the factored shear, Vu, and torsional moment, Tu,

at the critical section are calculated. For a prestressed member designed for shear and

torsion, the critical section can either be a distance h/2 from the face of the support or at

the face if a concentrated load or torsional moment occurs within the distance h/2. A

designer must use judgment when deciding if a load or torsional moment occurring

within 13/2 is concentrated or not.


34

3.1.2 Threshold Torsion

Threshold torsion is the level of torsion above which torsional effects must begin to be

accounted for in the design. For reinforced concrete beams, this is just one-fourth of the

cracking torque of a cross-section. Equation (2.5) gives the cracking torque. For

prestressed beams, torsional cracking loads increase due to the prestress force acting on

the beam. This prestress factor is expressed

I lOf~yt= 1+~ (3.1)

f’c

As stated earlier, the threshold torsion is simply one-fourth of the cracking torque, Tcr.

Therefore, the following equation can be written for threshold torsion.

1WThreshold = q)’~ ]/t ~c Z X

2y (3.2)

The expression Xx2y can be calculated based on the dimensions of the cross section being

designed. If an L beam is being designed, then the cross section would be split up into

two rectangular sections. The x term in the expression represents the shorter dimension

of the rectangle and y, the longer. The L shaped cross section would be split such that

XxZy gives the highest attainable value. Figure 3.1 gives a simple example of two

different ways an L beam could be divided. Whichever of the two setups shown in

Figure 3.1 provide a higher y~x2y should be used. If a threshold value greater than Tu is

obtained, then torsion can be neglected. Otherwise, it must be considered in design.


35

3.1.3 Maximum Combined Shear and Torque

Maximum torsional moments and shear force limits must be checked. This calculation

helps prevent compressive failure of the concrete due to over-reinforcing of the member.

That is, compressive failure before yielding of the steel reinforcement.The following

equations are for this exact purpose.

Kt)~c ~ x2y/3 (3.3)Tn(max) =Ii+(~.30CtTumtVu ./2

IOL~-~ bwd (3.4)Vn(max) ~ ]l+~30CtTu.)2

The design shear, Vu, and design torque, Tu, are not allowed to exceed maximum values

shown in Equation (3.3) and (3.4).

3.1.4 Concrete Contribution

Next is the assessment of the concrete contribution in shear Vc, and torsion Tc. By taking

into account Vc and Tc, a reduced value of the torsional moment and shear force can be

designed for. In turn, less longitudinal steel, At, will be required. If one looked strictly

at a pure torsion case, that is to say, a particular member is subjected only to a torsional

moment, the resistance of that moment provided by the concrete would act entirely

against Tu. Therefore, the entire torsional moment strength of the prestressed concrete,

T’c, can be used. A similar analogy can be used for the concrete contribution in shear,

V’c. The equations for T’c and V’~ are expressed below.


T’c = 0.8)~@-c Y’,x2y(2.5Yt -1.5)

miniVowV’o =[V~i

Vcw = (3.5@-~ + 0.3fr~)bwd + Vo

Vci = 0.6@cbwd+ Vd +-ViM~rMmax

36

(3.5)

(3.6)

(3.7)

(3.8)

Since the critical section is near the support or at the face of the support, one should

expect web shear cracking to control, Vcw. Section 11.4.3 of ACI 318-02 states, "In a

pretensioned member in which the section at a distance h/2 from face of support is closer

to the end of a member than the transfer length of the prestressing steel, the reduced

pregtress shall be considered when computing Vcw." Development length of the

prestressing strand must be taking into consideration when using Equation (3.7)

When dealing with combined shear and torsional loading, T’c and V’c must be

proportioned accordingly. This can be done by using the following circular interaction:

+ -c = 1 (3.9)W’c

Assume that the ratio of shear stress to torsional stress remains constant for various loads.

This allows for the following expression:

Vc _ Vo (3.10)

Substituting Equation (3.10) into Equation (3.9) and solving for Tc and V~ gives:


37

T’c (3.11)

V’c (3.12)

1+ V’c/VuT’c

From Equations (3.11) and (3.12), the shear force and torsional moment from concrete

contribution can be determined based on different levels of Tu and V,.


Now that the values for Tc and Vc are know, the T, and Vu values can be reduced, and the

remaining torsional moment and shear force can be used to design the transv~rse steel.

From this, the transverse area of reinforcement required by shear and torsion are

Av (Vo/ )-Vos df~

(3.13)

At _ (Tu/q0)-Wc

s atxlylfyv(3.14)

There is a minimum value for the hoop size and spacing that needs to be checked to make

sure that the ductility of the beam is adequate when it cracks. This check is made with

the following equation:

(~ 2At)> 50bwd+T)-- fy----~

(3.15)


38

For the calculation of the hoop size and spacing, the designer chooses a bar size and can

then calculate the correspond spacing of the stirrups for that particular bar size. If a

different spacing is desired, a different bar size can be selected. After a bar size is

selected, the area of the bar, Av-bar, Can be used to calculate the spacing.

Av-b~ (3.16)

However, spacing of hoop for torsion must not exceed

Sin,× -- Xl + Yl < 12in. (3.17)4

The spacing of the hoops is limited to ensure the development of the ultimate torsional

strength of the beam, to prevent excessive loss of torsional stiffness after cracking, and to

control crack widths.


With the design of the transverse steel in place, the required amount of longitudinal steel,

At, can be calculated. As discussed in Chapter 2, the 4t~ Edition is based on a concrete

compressive strut angle of 45 degrees, to achieve this angle, an equal volume of

longitudinal and transverse steel must be provided. This is expressed in Equation (2.12).

After calculating At, the minimum amount of longitudinal steel, At-rain, must be checked

using Equation (2.20). Once the required amount of longitudinal steel is known, the

designer can select the number and size of the longitudinal bars to be used, so long as all

code requirements are met. The flow chart in Figure 3.4 summarizes the 4th Edition

design procedure.


39

3.2 Fifth Edition

The major differences between the approaches in the 4th Edition and 5th Edition are the

varying angle as opposed to the fixed 45 degree compressive concrete strut angle and the

exclusion of the concrete contribution in torsion, Tc. Since both procedures follow a

similar path with similar equations, a much more condensed explanation will be given.

However, important parts of the design procedure will be emphasized. The design

constraints are the same in both editions of the handbook.


The calculation of the contribution of concrete is simplified in this edition. Torsional

moment contributed by concrete under torsion and shear loads is neglected in the thin-

walled tube and variable angle truss analogy. As discussed in Chapter 2, resistance

provided by the interior concrete is considered minimal and neglected.

Since there is no Tc to account for, there is no shear and torsion interaction. So

essentially the shear force contributed by concrete under torsion and shear, Vc, is simply

equal to V’c, as shown in Equation (3.6). The shear reinforcement required, Av/s, can

now be calculated using Equation (3.13).

3.2.2 Threshold Torsion

The threshold torsion for the 5th Edition is slightly different from the 4th Edition.

equation to calculate the threshold torsion is:

~/I fPc

q) ~c Acp~ ~ 1 ÷ 4~-~-~TThreshold = ~c pcp

The

(3.18)


40

Comparing the threshold torque values for the 4th and 5t~ Edition, the 4th Edition

threshold torque value is higher than the 5th Edition. This relationship is shown in Figure

3.2 for a prestressed L shape section. The difference between the two editions would

increase for a rectangular section having the same gross area of concrete. Threshold

torque will be explained in further detail in Chapter 5.

3.2.3 Cross Section

To ensure that the cross section is adequate, the following check is made. The

commentary of ACI 318-02, section R11.6.3.1 states that, "The size of a cross section is

limited for two reasons, first to reduce unsightly cracking and second to prevent crushing

of the surface concrete due to inclined compressive stresses due to shear and torsion."

This is achieved using the following equation for solid sections:

Vu T.p~, V~+ < q0 + 8 (3.19)1.7A o~, 2 -

For hollow sections:

(3.20)


The 5th Edition approach to the design of transverse steel differs from the 4th Edition. In

the 4th Edition, based on a selected bar size, the spacing of the hoop could be calculated

since the concrete compressive strut angle is held constant at 45 degrees. In the 5th


41

Edition, this angle can vary between some prescribed limits. A trial bar size and spacing

is selected, and from this, the transverse steel can be calculated.

2(Ahoop leg/S)-- A v/SAt/s = (3.21)

2

After calculating At/s, the compressive concrete strut angle can be calculated.

cot0 = Tu/~P1.7Aoh (At/s)fyv (3.22)

As stated in section 11.6.3.6 of ACI 318-02, "0 shall not be taken smaller than 30 deg nor

larger than 60 deg." From Equation (3.22), At/s, is transverse steel available to resist

torsion. The ductility check expressed in Equation (3.15) is also applied in the 5th Edition.

Figure 3.3 illustrates the 5th Edition approach in further detail. It also illustrates

the important implication of allowing the concrete compressive strut angle to vary. In the

4th Edition, the angle of the concrete compressive strut is 45 degrees, therefore an equal

amount of longitudinal and transverse steel is needed. The 5th Edition allows more

flexibility in the design. For instance, a designer could decrease the number of stirrups

required to create a 45 degree compressive strut angle as is done in Figure 3.3 (b). This

in turn lowers the cracking angle and the concrete compressive strut now requires a larger

tension force for equilibrium for the same level of torsion. Conversely, steeper angles

require more hoops and reduce the tension force requirements on the longitudinal steel

(Figure 3.3 (c)).


Once the design for the transverse steel is set, the longitudinal steel requirements can be

calculated. The longitudinal steel is


42

AAt = --~ phcotZ0 (3.23)

S

Equation (3.23) is essentially the same as Equation (2.32) in Chapter 2, with the ratio

fyv/fyt = 1.0. Equation (2.39) gives the minimum amount of required longitudinal steel.

The transverse steel, At/s in Equation (2.39) shall not be less than (25bw)/fy~. Where Tu

exceeds threshold torque, the minimum area of transverse closed stirrups shall be

computed by:

(Av+2At) : 0.75,f~bws < (50bwS)fyv - fr,

The flow chart in Figure 3.5 summarizes the 5th Edition design procedure.

(3.24)

3.3 Summary

The 4th Edition provides a design procedure that is based on the skew-bending theory. It

is important to note that this design procedure allows concrete contribution, To, to be

included in the torsional resistance. The 5th Edition does not include this concrete

contribution. Remember that the 5th Edition design procedure for torsion is based on the

thin-walled tube and variable angle truss analogy. It is essentially a method based on a

variable angle as opposed to the 45 degree fixed angle used in the 4th Edition. In the 4th

Edition, the amount on transverse steel needed to resist torsional moments can directly be

calculated since the angle of the concrete compressive struts (45 degrees) is known.

From this, hoop size and spacing can be determined. The 5th Edition has a different

approach. A trial size and spacing is selected for the hoops. Based on a certain level of

torsion and the hoop size and spacing selected, the concrete compressive strut angle


43

would be a value that would allow for the optimization of the hoop. A lower and upper

limit of the concrete compressive strut angle is given as 30 and 60 degrees respectively.

Allowing this angle, 0, to vary between 30 and 60 degrees permits designers to optimize

the assigned amounts of hoop reinforcement. In other words, 0 will be equal to a value

that would result in the complete utilization of the hoops in both shear and torsion.

A comprehensive review of the design procedures used in the 4th and 5th Editions of the

PCI Design Handbook has been covered. Having a clear understanding of both design

procedures will help in understanding the comparison of Example 4.4.2 made Chapter 4

and the parametric studies presented in Chapter 5. A summary of the report and needs

for further research will be given in Chapter 6.


44

X

Y2

X2

////////

Y2

Figure 3.1: Example of a divided cross section

400

350

300

250

200

150

100

4th Edition

10 20 30 40 50

hledg~ (in.)

60

Figure 3.2: Threshold Torsion Comparison for a Prestressed L Shaped Section


45

A~1 TOTAL

S1

r.’/ A

0 = 45°

A~ TOTAL

82

,/ ../

(a)

’~ 30° < 0 < 45°

I!1>A [3 TOTAL

// / /,i /

/ / /< 0 < 60°

A~2 TOTAL > At~ TOTAL > At~ TOTAL

S2 > S1 > S3

Figure 3.3: Varying Angle 0 of the Inclined Compressive Strut of the Variable AngleSpace Truss and Hoop Spacing


46

4th Edition Design Example Flow Chart

Given: f’c, fy, d and Cross Section

Calculate Tu and Vu at the critical section(At face of column or h/2)

~/~alculate developed prestress force at )

~.~ritical section, fpc.

= 41+ 10fpc/f’c I~/t

Ta~reshola = q~/t (1.5~t~,~-c)~ X 2y

Where: ~t = 1/3

Tu > TThreshold

T’c

ViMcr

V’ min Wci = 0"6~-~-~ bwd + Va -t[Vcw (3.5~f~-~ + 0.3fr.:)b w d + Vp

Neglect Torsion

Figure 3.4:4th Edition Design Procedure Flow Chart


47

Kt = yt (12-10fp~/f’c)

bwdCt -- y,~x2y

Kt X~-f~ ~-’~ x 2 y/3

1 + ~30--CtT~

10~L~b ,~ d

~ Desi~ may continue if~Tu < Tn(~x) and Vu < Vn(max)

Figure 3.4:4th Edition Design Procedure Flow Chart (Continued)


48



49

A, =2At (x~ +y,)S

If At > At-min, Then use AtOtherwise, use At-rain



50

5th Edition Design Example Flow Chart

Given: f’c, fy, d and Cross Section

Calculate Tu and Vu at the critical section(At face of column or hi2)

tViMcr

Vc = Min.Wci = 0"6~-~-~bwd + Vd -t Mm,x

[Vcw (3.5~-~ + 0.3fp~)bw d + Vp

S fyd

~//~alculate developed prestress force at ~

~.~ritical section, fpc.

T.l~reshold = q) Acp~ 1 +

Use Ag in place of Acp for hollow sections

Neglect Torsion

Figure 3.5:5th Edition Design Procedure Flow Chart


51

Solid Section:+ . T~pn:. <_q) Vc 8~c

~.l’7Aoh ) b--f-d+

Hollow Section: ~ ~’l’7A°hj_ ~bwd J

Select Trial Bar Size and Spacing, Abar/S ~-~

(Av + 2At ) _ Abar

S S

At/s = s2

cot0 = TuhP1.7Aoh (At/s)fyv

YES

A~/s>50 bw ~fyv

Change Bar Sizeand/or Spacing



52

A~ n~n = 5~-~ Acpfy~

At

If At > At-rain, Then use At

Otherwise, use At-rain



53

4. DESIGN EXAMPLE

Design Example 4.4.2 of the 4th and 5th Editions of the PCI Design Handbook covers the

design for shear and torsion of the same prestressed concrete spandrel girder using two

different approaches. The difference in the area of longitudinal steel required is about

three times greater in the 5th Edition, indicating a need for a close examination of both

approaches. Some adjustments were needed in order to permit a fair comparison of both

procedueres.

4.1 Analysis

Varying the unfactored dead load provided a range in which key outcomes of the design

procedure were compared on the basis of the design problem constraints. A list of these

outcomes is summarized below:

1. Total area of longitudinal reinforcement to resist torsion, A~.

2. Minimum area of longitudinal reinforcement to resist torsion, Ag-min.

3. Transverse steel required to resist torsion, At/s.

4. Transverse steel required to resist shear, Av/s.

5. Angle of compression diagonal in truss analogy for torsion, 0, in the 5t~’

Edition approach.

A key constraint used in the PCI example is that the total amount of transverse steel for

shear and torsion, E(Av +2At )/s_~, is to be the same for both editions for the same level of


54

unfactored dead load. It must be noted that this assumption directly affects the amount of

required longitudinal steel.

4.2 Adjustments

To provide a proper comparison of the design procedures used in the 4th and 5tr’ Edition,

it is important that the shear and torsion design example used in each edition be the same.

This is to ensure that any differences found in the calculated longitudinal steel values be

the result of the different theories. In the following sections, the adjustments made to

Design Example 4.4.2 of the 4th and 5th Edition of the PCI Design Handbook are

discussed in more detail.

4.2.1 Level of Prestressing

The compressive stress in the concrete, fpc, is 204 psi in the 4th Edition whereas in the 5th

Edition is 88 psi. The large difference between the two values is due to the fact that the

5th Edition has accounted for the strands as partially developed at the critical section for

torsion and shear located at h/2, while the 4tt~ Edition has not. A concentrated torque

occurs within a distance of h/2, hence the critical section in both editions is taken at the

support face, 11 inches from the beam end. It must be noted that the equation for

calculating shear strength in the 4t~ Edition adjusts the fpc value to account for transfer

length as shown in Equation (4.1). Changes in fp~ affect the shear strength, V~, which in

the 4th Edition, is equal to the controlling web shear cracking, V~w. The following

equation is used in the 4th Edition design example to calculate Vow.


55

The fraction 6/50db, estimates the compressive stress at the support face which results in

a value equal to 48.9 psi, about half of the 88 psi used in the 5th Edition. The level of

prestressing in the 5th Edition is based on Section 12.9.1 of the ACI 318-02 Code.

ga=/~) db +(fps-fse) db(4.2)

The first term in Equation (4.2), (fse/3)db represents the transfer length in the strand that

would develop the prestress fse. The developed prestress in the 5th Edition is calculated

by multiplying f~e by the of the length to the critical section divided by (fs~/3)db. For this

analysis, fpc was changed to 48.9 psi in the 5t~’ Edition. This change had no impact on the

required longitudinal steel for the 5~ Edition and only affected the threshold torque value.

The threshold torque value, originally equal to 183 kip-in., after this modification is 173

kip-in, for the 5tr’ Edition compared to 226 kip-in, in the 4tr~ Edition. The difference in the

threshold torque values is discussed in Section 4.5.

4.2.2 Concrete Contribution in Shear

There is a discrepancy between the two editions related to the shear strength, Vc.

Equation (4.1) shows that V~ is equal to Vow in the 4th Edition, since web shear cracking

In the 5th Edition, Ve was taken as:

Vc = 2,f~-~-~ bw d

controlled at the end region.

(4.3)

This results in a much lower value for V~, 81.5 kips versus 157.8 kips used in the 4th

Edition. In the direct comparison of the 4th and 5th Edition results, the same concrete


56

contribution in shear is used for both editions. Equation (4.3) represents the shear

strength provided by concrete for nonprestressed members. A more appropriate

representation of the web shear cracking mechanism observed at end regions of

prestressed concrete beams such as that being designed in the 5th Edition example.

Therefore, the controlling Vow will be used in both examples.

Using Vow significantly reduces the required amount of longitudinal steel in the

5th Edition as shown Figure 4.1. Figure 4.1 shows the required longitudinal steel, A~, on

the vertical axis for different levels of unfactored dead load shown on the horizontal axis.

The vertical dashed line at an unfactored dead load of 89.5 psf represented the value used

in both editions of the handbook. Arrows depicting the change in the required amount of

longitudinal steel due to Vcw have been placed in the figure. By increasing the concrete

contribution resisting shear, Vc, in the 5th Edition, less transverse steel is needed for shear

resistance, Avis, as seen in Equation (3.13). This decrease in Avis is also displayed in

Figure 4.2. Figure 4.2 also displays the unfactored dead load on the horizontal axis. The

required amount of transverse steel reinforcement for shear, A,,/s is plotted on the vertical

axis. Since the same total amount of transverse reinforcement, /(Av +2At )/s~, is used in

both editions, if Avis decreases as shown in Figure 4.2, then the amount of transverse

reinforcement available to resist torsion, At/s, should increase for the given amount of

total transverse steel. This increase in the available amount is shown in Figure 4.3. In

this figure, the available amount of transverse steel reinforcement for torsion, At/s, is

plotted on the vertical axis versus unfactored dead load on the horizontal axis. An

increase in the available At/s for the same unfactored dead load will result in an increase

in the concrete compressive strut angle required to resist torsion, 0, seen in Equation


57

(3.21) and in Figure 4.4. Figure 4.4 plots the concrete compressive strut angle, 0, on the

vertical axis versus unfactored dead load on the horizontal axis. The concrete

compressive strut angle is equal to 32.7° prior to the adjustment to Vc, and 41° after the

adjustment was made for a value of unfactored dead load of 89.5 psf. The increase in 0

results in a decrease in At. Equation (3.23) shows the 5th Edition calculation for At.

Therefore, by using the appropriate equation to represent concrete contribution in shear,

Vow, in the 5th Edition, less longitudinal steel reinforcement is required to resist the same

level of torsion as more transverse steel reinforcement is available to resist from the total

used in both examples.

4.2.3 Depth to Compression Steel

The distance from the extreme compression fiber to the centroid of longitudinal tension

reinforcement, d, was taken as 72 inches in the 5th Edition and 69 inches in the 4th Edition.

For the purpose of comparing both editions, 69 inches was used in place of 72 inches in

the 5th Edition design example. This adjustment has very little impact on the required

longitudinal steel in the 5th Edition example.

4.2.4 Summary of Adjustments

The use of V~w instead of Vc in the 5th Edition had the greatest impact of the three

adjustments, as it decreased the required longitudinal steel values in the 5th Edition.

Table 4.1 summarizes the results of all three adjustments.


58

4.3 Required Longitudinal Steel Area

After all necessary adjustments to Example 4.4.2 of the 4th and 5th Edition of the PCI

Design Handbook have been made; the comparison between the two approaches indicates

that there are two key items leading to the stated differences in A~:

1. The simplification of the lever arm area, Ao = 0.85Aoh, in Equation (3.22)

2. Concrete Contribution Term, Tc

4.3.1 Impact of Simplification of Ao

In Section 11.6.3.6 of the ACI 318-02 Code, it is stated that "Ao shall be determined by

analysis except that it shall be permitted to take Ao equal to 0.85Aoh."

A° = 0.85Aoh

(4.4)

Equation (4.4) over-estimates the amount of concrete spalling of tall slender sections

since spalling will occur near the top and bottom of the section. This over-estimation in

spalling will result in a greater amount of reinforcing steel for torsion. The ACI 318-02

Code also states in Commentary Section R11.6.3.6 that, "The expression for Ao given by

HSU11.32 may be used if greater accuracy is desired." Hsu (Hsu 1997) noted that Equation

(4.4) may "under-estimate the torsional strength of lightly reinforced small members by

up to 40 percent and over-estimate the torsional strength of heavily reinforced large

The following expression was proposed by Hsu:

2TupcpAo = Aop- q)f~A~p

members by up to 20 percent."

(4.5)

Using Equation (4.5), the term 1.7Aoh in Equation (3.22) is replaced with 2Ao which

gives the following expression for calculating the compressive concrete strut angle:


59

cot0 = T,/q~ (4.6)2Ao (A,/s)f~

Using Equation (4.6) with the procedure in the 5th Edition results in a required

longitudinal steel amount of 1.33 in.z, similar to the value obtained using the 4th Edition.

Table 4.3 shows the comparison between the 4t~ and 5t~ Edition values for At using

Equation (4.6). The difference in At is plotted in Figure 4.5 for various values of

unfactored dead load. Figure 4.6 shows the large difference between Equation (4.4) and

(4.5). Using Equation (4.5) also affects the compressive strut angle as shown in Figure

4.7, and it is this change in the compressive strut angle that causes the decrease in

Using Equation (4.5) in the 5t~ Edition results in comparable At values to those obtained

using the 4t~ Edition approach.


Another important factor in the difference between A~ values in the 4t~ and 5th Edition

examples is the concrete contribution, To. Since the 4t~ Edition includes concrete

contribution in torsion, a shear and torsion interaction must be used. As can be seen in

Figure 4.8, the amount of torsion to be designed for increases at about the same rate in

both editions suggesting that At/s for both editions should increase at the same rate.

However, the 5t~ Edition design procedure will require less transverse reinforcement for

shear, Av/s, than the 4t~ Edition. This is the result of using a larger concrete contribution

in shear in the 5t~ Edition than in the 4t~ Edition, since there is no shear and torsion

interaction in the 5tr’ Edition. Figure 4.9 shows the concrete contribution in shear for both

the 4th and 5th Edition along with the design value for shear, Vu. From Equation (3.13)


60

and the use of Figure 4.9, as the unfactored dead load increases, the Av/s term for the 4th

Edition increases while none is required for the 5th Edition. Since the total amount of

transverse reinforcement is kept the same between both editions, the At/s value from the

5th Edition is increasing at a greater rate than the At/s value from the 4th Edition. This

results in an increase in the cracking angle and a decrease in the rate of required amount

of longitudinal steel, ,44.

4.4 Minimum Longitudinal Steel Area

Initially, the values for the minimum amount of longitudinal steel reinforcement, At-min,

also differed significantly. The 5th Edition approximately required double the amount of

minimtun longitudinal steel reinforcement when compared to the 4th Edition. However,

unlike the required longitudinal steel reinforcement for strength, the initial difference in

the minimum values, A~-min, between both editions appears to mainly be a result of the

discrepancy in the shear concrete contribution reviewed in Section 4.4.2. Figure 4.10

shows the result of changing Vc in the 5th Edition to Vow. As the unfactored dead load

increases, the minimum amount of longitudinal steel is decreasing due to the increase in

transverse steel to resist torsion, At/s. From Figure 4.8, the 4th and 5t~ Edition approach

provide similar minimum longitudinal

both initial and adjusted A~-min values.

steel reinforcement values. Table 4.2 provides

Since more or less At/s can be used so long as the

cracking angle remains between 30 and 60 degrees, the amount of A~_min can vary in the

5th Edition depending on what cracking angle or hoop size and spacing is desired.


61

4.5 Threshold Torsion

The adjusted threshold values from the 4th and 5th Edition of the PCI Design Handbook

are 199 kip-in, and 173.3 kip-in, respectively. The adjustments of fpc described earlier

only affected the 5th Edition threshold torque value which originally was 183.3 kip-in.

This change was a result of using 48.9 psi as the compressive stress, fpc, instead of 88 psi.

The 4th Edition uses an fpc value based on fully developed prestressing and the 5th Edition

uses an fpc that is accounted for development length. An fpc value of 48.9 psi was used in

the 4th Edition and accounts for the decrease in the threshold torque value to 199 kip-in.

Equation (3.2) and (3.18) are used to calculate the threshold torque values in the 4th

Edition. The threshold torque obtained from the 4th Edition is higher than that obtained

from the 5th Edition, but the difference is not significant. A model was set up in which

the L-shaped cross section from Example 4.4.2 of the PCI Design Handbook was scaled

to provide different cross sections while still maintaining the original shape. Figure 4.11

provides an example of how the cross sections were scaled. Threshold torsion values

were calculated for each scaled cross section and the results were used to generate Figure

4.12. It can be seen from Figure 4.12 that both editions provide similar threshold torsion

levels. The 4th Edition results in slightly higher, but comparable threshold torsion values

in comparison to the 5th Edition results for L shaped spandrel sections.

4.6 Summary

The initial difference between the 4th and 5th Edition values for the amount of

longitudinal steel can be considerably less if the same estimate of V~ is used in both

editions. Use Vcw and V~i where permitted in place of the minimum value of 2@b,~d


62

for the nominal shear strength provided by the concrete. Even after Example 4.4.2 of the

PCI Design Handbook was adjusted in both editions, so the 5th Edition still required

larger amounts of longitudinal steel compared to the 4th Edition. This can be attributed to

the assumption behind Equation (4.4) regarding the amount of concrete cover spalling

compared to Equation (4.5). Using Equation (4.5) in the 5th Edition resulted in similar

results for longitudinal steel for strength. Furthermore, threshold torque values were also

similar in both approaches. Differences found in the required A~ value for strength, after

all adjustments including the use of Equation (4.5), were attributed to the concrete

contribution term for torsion and the resulting shear and torsion interaction in the 4th

Edition which reduced the concrete contribution in shear. Chapter 5 will present the

results of parametric studies conducted to further examine differences between both

procedures. Chapter 6 summarizes the findings of the study and indicates areas of

needed research.


63

Table 4.1: Adjusted A~ Values

Initial Values Adjusted Values

4th Edition

5th Edition

% Difference

1.58 in2

4.96 in2

214%

1.59 in2

3.65 inz

130%

Table 4.2: Adjusted A~

Initial Values

4th Edition 1.10 in2

5th Edition 2.03 in2

% Difference 85%

Table 4.3:Comparison of A~ Values

Eq. (4.4) Eq. (4.5)

4th Edition 1.59 in2 1.59 in2

5th Edition 3.65 in2 1.33 inz

% Difference 130% 19%

-rain ValuesAdjusted Values

1.09 in2

1.29 in2

18%


Calculated (A~/s+2A~/s)from 4th Edition was usedto calculate/~ and Al.min in

the 5"’ Edition

PCI Example, Equation (11-18) of ACI 318-02Limits Maximum Unfactored Dead LoadTu = 707 in.-kips

" " " 5th (Original)~,,, , , ,

, , ,

.~l~ll~~~1~1 -",I.V Ad, u ,

., , , stment

30 Degree Minimuim ’, t ’,for Original Values

, ,~,~,.~,: ....... 4 (Original and Adjusted) ,

+ Ongtnal 4th Edition ~1’, :’~" ,-!--Original 5th Edition............... Adjusted 4th Edition~ Adjusted 5th Edition ’

20 40 60 80 100 120 140 160

Unfactored Dead Load (psf)

180

Figure 4.1: Adjustment on Required Longitudinal Steel Reinforcement


0.035

0.030

0.025

0.020

0.015

0.010

0.005

0.000

Calculated (A~/s+2A~/s) from the 4th Edition was usedto calculate AI and Ai.min in the 5th Edition

0

+ 4th Adjusted+ 5th Adjusted......... 4th Original---x-- 5th Original

V~ Adjustment

40

~~ .~,~ 5t~ (Original)

.... ~’ .~~

5th (Adjusted)

60 80 100 120 140 160 180


Figure 4.2: Adjustment on Av/s


0.035

0.030

0.025

0.020

0.015

0.010

0.005

0.000

Calculated (,~,/s+2At/s) from 4th Edition was usedto calculate A~ and Ai.min in the 5th Edition 5th (Adjusted)

i--e--Adjusted 4th Edition~Adjusted 5th Edition....... :,,,,, Original 4th Edition

---X--Original 5th Edition

Vow Adjustment

(Original)

20 40 60 80 100 120 140


160 18O

Figure 4.3: Adjustment on Available At/s


60

55

35

30

Calculated (Av/s+2A~/s) from 4th Edition was used

to calculate/~ and ~-r~in in the 5th Edition

-’~"Adjusted 5th Edition

Original 5th Edition

0

Angle in the 4~h Edition

20 40

Adjustment

60 80 100 120 140 160 180


Figure 4.4: Adjustment on Compressive Strut Angle


68

0 -~hi-

(~’u!) lee.l.S leu!pn~.!6uo"l

0

o

o

oo


69Conclusions or recommendations in this report are the opinions of the authors. PCI assumes no responsibility for the interpretation or application of the information contained herein.

6O

55

5O

45

4O

35

3O

Calculated (A~/s+2A~/s) from 4t" Edition was usedto calculate/~ and Ai.min in the 5th Edition

--,i’--Cracking Angle with Eq. (4.5)--I--Cracking Angle with Eq. (4.4)

20 40 60 80 100 120 140 160


180

Figure 4.7: Using Equation (4.5) and the Compressive Strut Angle


1200

1100

1000

900

8OO

700

600

5OO

400

300

200

100

0

--*--Tn-Tc (4th Edition)

~Tn (5th Edition)

Tc (4th Edition)

_.....--

_.....--

mn 5th Edition

0 20 40 60 80 100 120 140


160 180

Figure 4.8: Value of Design Torsion Using 4t~ and 5t~ Editions


200

150

100

50

0

~ Vu (4th & 5th Edition)

~ Vow (4th & 5th Edition)............. , Vc (4th Edition)

Vow

Me

20 40 60 80 100 120 140


160 180

Figure 4.9: Shear Contribution Values for Both Editions


7

6

5

4

3

2

Calculated (Av/s+2A~/s)

from 4th Edition was usedto calculate/~ and Ai.min in

the 5m Edition

PCI Example,Tu = 707 in.-kips

30 Deg. Minimum foAdjusted Values

30 Degree Minimuimfor Original Values

+Original 4th Edition

--I--Original 5th Edition....... Adjusted 4th Edition

--X--Adjusted 5th Edition

,,

Equation (11-18) of ACl 318-02Limits Maximum UnfactoredDead Load

Vc,, Adiustment !,

0 20 40 60 80 100 120 140 160


180

Figure 4.10: Adjustment on Minimum Longitudinal Steel Reinforcement


30O

250

5O

00.6

--X---5th Edition L-Spandrel Design Example

0.7 0.8 0.9 1 1.1

Scale Multiplier

Figure: 4.12: Threshold Torque Values Versus Scale Multiplier

1.2


76

5. PARAMETRIC STUDY

Chapter 5 provides results of further studies using two design examples featuring

rectangular and L beam sections to help verify the findings in Chapter 4. It should be

noted that both examples represents cases of equilibrium torsion. The previous chapter

established that with some modifications, the 5th Edition could result in similar amounts

of required longitudinal steel reinforcement as those obtained using the approach in the

4th Edition of the handbook. Finally, a comparison of test data versus calculated values

from the report entitled "Evaluation of Design Procedures for Torsion in Reinforced and

Prestressed Concrete," (Ghoneim 1993) will be provided at the end of this chapter.

5.1 Precast Pretensioned Spandrel

Additional analysis was preformed on the design example in the 4t~ and 5th Edition of the

PCI Design Handbook studying the effect of using different strut angles in the 5th Edition

design procedure. First the spandrel beam is designed using the 4t~ Edition design

procedure. It is important to note that adjustments made in the previous chapter are also

included here. There are a total of four different designs carried out using the 5tr~ Edition

design approach. One in which the transverse steel is kept the same, and three additional

designs based on compressive concrete strut angle values of 30, 45, and 60 degrees. The

results of these designs are shown in Table 5.1 and represent values for a section located

at the face of the .support. It can be seen from Table 5.1 that the 5tr~ Edition consistently


77

provides similar amounts of reinforcement as the 4th Edition approach. The required

longitudinal steel values shown in Table 5.1 can be reduced using 6-1/2 in. diameter,

270K strands, since there is no moment at the supports. The strand is only assigned the

capacity to develop 60 ksi at the support face as per Section 11.6.3.10 in the ACI 318-02

Code. Figure 5.1 and 5.2 show the strength and minimum requirements for longitudinal

steel for various values of unfactored dead load. From Table 5.1, Figure 5.1 and 5.2, it

can be summarized that for PCI Design Example 4.4.2, the 5th Edition procedure results

in similar amounts of reinforcement.

5.2 Post-Tensioned Rectangular Section

The next design example studied was taken from the book entitled "Torsion of

Reinforced Concrete" (Hsu, 1984). It consists of a post-tensioned, rectangular beam

section under shear and torsion. This example was chosen to illustrate the results of the

two approaches in the case of a rectangular cross and compare the findings with those of

the L shaped spandrel beam. Figure 5.3 provides an illustration of the design constraints

and Figure 5.4 gives the moment, shear, and torque diagrams. For continuous beams,

negative moments at the supports must be accounted for if prestressing steel is to be used

to resist torsion at the critical section. The negative moment reduces the prestressing

steel stress at the critical section.

example, Vci controls and not Vow.

It should also be noted that unlike the PCI design

The web-shear cracking region is smaller for a beam

near a continuous support and larger near a simple support. Results of this analysis have

been placed in Table 5.2. Once again, the 5th Edition design procedure provides similar

design values as those obtained with the 4t~ Edition approach.


78

Figure 5.5 shows the positive affect of using Equation (4.5) on the required longitudinal

steel values for strength. The results shown in Figure 5.5 are based on the enforcement of

the same amount of transverse reinforcement in the 4th and 5th Edition approaches. The

calculated strut angle in the 5th Edition, where the total amount of transverse

reinforcement is the same between both editions, is shown in Figure 5.6. The minimum

required longitudinal steel values for various dead load levels as a function of the strut

angle are shown in Figure 5.7 and 5.8. As was the case for the L-shaped spandrel beam,

when using Equation (4.5), the 5t~ Edition design procedure provided similar results as

those of the 4t~ Edition approach.

5.3 Comparison With Test Results

In Ghoneim and MacGregor (1993), test data was gathered and used to compare test

versus calculated capacities obtained with what is now the current design procedure for

torsion of prestressed concrete in the ACI 318-02 Code. In this report, no comparison

was made between the 4th and 5th Edition design procedures. In this section, the 5th

Edition comparison was done twice, once using Equation (4.4) and a second time using

Equation (4.5). The ratio obtained using the approach in the 4th Edition is also included

in Table 5.3. Table 5.3 summarizes the results, and Tables 5.4 and 5.5 provide the test

versus calculated torsion values for each individual beam. Some beams were not

included from the original report because the authors were not able to reproduce the

calculated values provided in the report. Furthermore, the beams listed in Table 5.4 are

considered eligible beams whereas the beams in Table 5.5 are ineligible for because they

do not meet the ACI Code requirement for stirrups spacing; hence, these beams were not


79

included in the analysis. Based on the results of this comparison, it was found that the 5th

Edition using Equation (4.5) provided the best agreement with the test values than the 4t~

Edition as well as the 5t~ Edition with Equation (4.4) calculated values. A parametric

study conducted to evaluate the role of the cross section dimensions and the applicability

of the findings stated in this chapter are valid through a wide range of section sizes is

summarized in Figure 5.9.

5.4 Summary

The comparisons conducted in this chapter show that the 5th Edition design values for

reinforcement are more comparable with 4th Edition design values when using Equation

(4.5). This follows in line with results from Chapter 4 where it was determined that the

5th Edition design procedure resulted in a similar amount of longitudinal steel

reinforcement compared to the 4th Edition design procedure. Chapter 6 summarizes the

report and indicates areas of needed research.


80

Table 5.1" L Spandrel Beam¯ 2Required At (in2) Required At-m~ (in)

4th Edition 1.59 1.09

5t~ Edition 1.33 1.29

Condition

45 Deg. CompressiveConcrete Strut Angle(Av+2At)/s equal for

4t~ and 5t~ Edition

30 Deg. CompressiveConcrete Strut Angle

45 Deg. CompressiveConcrete Strut Angle60 Deg. CompressiveConcrete Strut Angle

3.35 2.98

1.93 2.17

1.12

Hoop Spacing (in.)

12.1 (12)

12.1 (12)

30.4 (12)

17.6 (12)

10.2 0.75

Note: Use No. 4 Closed Stirrups and maximum spacings in ( )

Table 5.2: Rectangular Section

4t~ Edition

5th Edition

Condition45 Deg. CompressiveConcrete Strut Angle

(Av+2At)/s equal for4~ and 5~ Edition

30 Deg. CompressiveConcrete Strut An~le

45 Deg. CompressiveConcrete Strut An~le60 Deg. CompressiveConcrete Strut Angle

Required At (in2) Hoop Spacing (in.)

2.77 6.1

RequiredAt-rain (in2)

0

2.77 6.1 0

5.42 11 0.85

3.13 6.8 0

4.11.81

Note: Used No. 5 Closed Stirrups


81

Table 5.3: Comparison of Tests to Prestressed Concrete Design Procedures

Loading

Combined Torsion,Shear, and Bending

No. ofTests

31

5t~ EditionEq. (4.3)

Mean

1.89

Ttest!Tcal

5th EditionEq. (4.4)

Mean

1.3

4t~ Edition

Mean

1.56

4t~ EditionFrom Report

Mean

1.773

Note: 4t~ Edition mean taken from the report is based on 63 tests


82

Table 5.4Eligible Prestressed

and Bending) (Ghoneim 1993)

Source TestSpecimen

Concrete Specimens (Combined Torsion, Shear

Ttest Ttest

T4~Ed.Mode of Failure

IV-1-3

IV-2-3

IV-3-3

IV-4-3

IV-l-4

IV-2-4

IV-3 -4

IV-4-4

IV-l-5

1V-2-5

IV-3-5

IV-l-6

IV-2-6

IV-3-6

IV-4-6

V-1

V-5

V-7

VA102

VA104

VA105

VA107

Yielding of stirrups

(in-kips)

120

102

100

75

122

100

100

90

140

120

115.0

105.0

140.0

125.0

115.0

100.0

89.4

89.4

87.9

75.9

128.3

128.7

Under-reinforced

Crushing of concrete


Under-reinforced

Ttest

TsthEd.Eq. (4.3)

2.19

1.86

1.82

1.51

2.22

1.82

1.82

1.64

2.55

2.19

2.10

1.91

2.55

2.28

2.10

1.82

1.72

1.65

1.604

1.37

1.69

1.69

1.23

Ttest

T5th gd.

Eq. (4,4)

1.33

1.07

1.05

0.96

1.49

1.12

1.12

0.97

1.82

1.44

1.35

1.19

1.94

1.61

1.41

1.15

0.82

0.83

0.804

1.32

1.79

2.09

1.17

Henry and

Zia, (1971) Yielding of stirrups

2.08

1.57

1.70

1.35

1.997

1.56

1.64

1.52

2.27

1.84

1.86

1.54

2.28

1.97

1.81

1.49

1.15

0.797

0.77

1.09

1.33

1.34

1.02

McMullen andWoodhead,

(1973)

35.3

Crushing of concrete





Yielding of stirrupsYielding of bottomlonllit. And stirrup

Mukherjee andWarwaruk, (1970)


Yielding of bottomlonsit. And stirrup

Yielding of stirrupsYielding of bottomlon~it. And stirrup


Note: Ttestfr4th Ed. matches values from report.


83

Table 5.5

Source

McMullen andWoodhead,

(1973)

Ineligible Prestressed Concrete Specimens (Combined Torsion, Shear

and Bending) (Ghoneim 1993)

TestSpecimen

TtestTtest

W4th Ed.(in-kips)

69.9

28.3

90.0

105.0

35.3

81.9

105.0

135.0

78.9

81.9

78.9

75.9

Wrest

Eq. (4.3)

3.41

1.38

4.39

5.13

1.36

3.15

4.04

5.19

4.55

6.30

7.59

1.59

Ttest

Tsth Ed.

Eq. (4.4)

1.46

0.72

1.98

2.50

0.91

1.36

1.84

2.63

1.98

2.65

3.16

0.70

I1-2 2.11

11-3 0.75

II-4 3.02

11-5 3.55

III-2 0.89

UI-3 2.01

1II-5 2.48

III-6 3.79

V-2 2.54

V-3 3.63

V-4 3.84

V-6 0.83

Mode of Failure

Excess stirrupsspacing

Note: Ttest]T4th Ed. matches values from report.


6 4 2 0

At

for

5th

Ed

itio

n w

as

calc

ulat

ed fr

om s

elec

ting

ast

rut a

ngle

of 3

0, 4

5 an

d 60

degr

ees.

PC

I Exa

mpl

e,T

u =

707

in.-

kips

Equ

atio

n (1

1-18

) of

AC

I 318

-02

Lim

its M

axim

um

Un

facto

red

~D

ead

Load

4th

--@

-- A

I (4t

h)

AIAI (5

th)(5t

h) 453

0 DegD

eg

| ~ A

I (5

th)

60

De

g

45 D

eg.

60 D

eg.

06

08

010

012

014

0

Unf

acto

red

Dea

d Lo

ad (

psf)

Fig

ure

5.1

" L

Se

ctio

n R

eq

uire

d A

t a

s a

fu

nct

ion

of S

tru

t A

ng

le

160

180


00

At.,

~i,

for

5th

Edi

tion

was

calc

ulat

ed fr

om s

elec

ting

ast

rut a

ngle

of 3

0, 4

5 an

d 60

degr

ees.

PC

I Exa

mpl

e,T

u =

707

in.-

kips

Equ

atio

n (1

1-18

) of

AC

I 318

-02

Lim

its M

axim

um

Un

facto

red

~D

ead

Load

-"~-

- A

I (4t

h)...

.. "A

I (5t

h) 3

0 D

eg~

AI

(5th

) 4

5 D

eg

,--x

--A

I (5

th) 6

0 D

eg

20

40

60

80

100

120

140

Unf

acto

red

Dea

d Lo

ad (

psf)

Fig

ure

5.2:

L S

ectio

n R

equi

red

At-

min

as

a fu

nctio

n of

Str

ut A

ngle

30 D

eg.

45 D

eg.

160

180


JJ

II


M T

(in-k

)

V (

k)

( in-

k )

2363

28.9

DE

AD

LO

AD

1 1

PD=

42 k

tW

d= 5

25 p

lf

I 1

ITd

= 10

08 in

-k

30’

2126

504

2363

1620 18

28.9

504

LIV

E L

OA

D

PL=

36k

TL=

864

in-k

30’

1620

432

1620

18

432

Fig

ure

5.4:

Exa

mpl

e 5.

1 fle

xura

l mom

ent,

shea

r, a

nd to

rque

dia

gram

s fo

r de

ad a

nd li

ve lo

ads


2 0

Cal

cula

ted

(Av/

s+2A

t/s) f

rom

4th

Edi

tion

was

use

dto

cal

cula

te A

I and

AI-m

in in

the

5th

Edi

tion

~--e

-- 4t

h Ed

ition

~ 5t

h Ed

ition

with

Eq.

(4.4

)~

5th

Editi

on w

ith E

q. (4

.5)

Eq.

(4.4

)

~ ~

Eq.

(4.

5)

10

20

30

40

50

60

Unf

acto

red

Dea

d Lo

ad (

kip)

Fig

ure

5.5:

Rec

tang

ular

Sec

tion

Com

paris

on o

f At R

equi

red

for

Str

engt

h

7O


6O 55 35

3O

Cal

cula

ted

(A~/

s+2A

t/s) f

rom

4th

Edi

tion

was

use

dto

cal

cula

te A

~ an

d A

i.mi n

in th

e 5t

h Ed

ition

Ang

le in

the

4th

Ed

itio

n

--~-

-5th

Edi

tion

Cra

ckin

g A

ngle

with

Eq.

(4.

5)t

10

20

30

40

50

Unf

acto

red

Dea

d Lo

ad (

kip)

Figu

re 5

.6: C

ompr

essi

ve S

trut A

ngle

8O


+ 4

th E

ditio

n...

......

.. 5t

h E

ditio

n (3

0 D

egre

es)

--I-

5th

Ed

itio

n (

45

De

gre

es)

--X

--5t

h E

ditio

n (6

0 D

egre

es)

Fig

ure

5.7:

20

30

40

50

60

Unf

acto

red

Dea

d Lo

ad (k

ip)

Sec

tion

Exa

mpl

e R

equi

red

At f

or S

tren

gth

as a

Fun

ctio

n of

the

Str

ut A

ngle

7O


_~

~--

4th

Ed

itio

n

~ 5

th E

diti

on

(3

0 D

eg

ree

s)...

.... 5

th E

ditio

n (4

5 D

egre

es)

~ 5

th E

diti

on

(60

Deg

rees

)

Min

mum

Val

ues

for t

he 4

~ Ed

ition

and

for t

he 5

th E

ditio

n at

stru

tan

gles

of 4

5 an

d 60

Deg

rees

had

neg

ativ

e va

lues

for A

~.m

~n d

ue to

the

incr

ease

in tr

ansv

erse

ste

el a

s le

vel o

f unf

aoto

red

dead

load

incr

ease

d.

10

20

, ~ ~

x

, .

30

40

50

Unf

acto

red

Dea

d Lo

ad (

kip)

7O

Fig

ure

5.8

: R

ect

an

gu

lar

Se

ctio

n R

eq

uire

d A

t-ra

in a

s a

Fu

nct

ion

of

Str

ut

An

gle


2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

Are

a C

oncr

ete

= 50

4 in

.2

+ R

ecta

ngul

ar S

ectio

n (E

q. 4

.3)

,, R

ecta

ngul

ar S

ectio

n (E

q. 4

.4)

0.2

0.4

0.6

0.8

Rat

io (

x/(A

c^.5

)

1.2

Figu

re 5

.9: R

ecta

ngul

ar S

ectio

n R

atio

(A1

5th)

/(A1

4th)


93

6. SUMMARY OF FINDINGS AND CONCLUSIONS

6.1 Findings

The 5t~ Edition example of the PCI Design Handbook required an additional amount of

longitudinal steel equal to about three times the amount required for the same example in

the 4th Edition for strength and almost double for minimum required longitudinal steel.

The design approach for torsion in the 4tla Edition of the PCI Design Handbook is based

on the skew-bending theory and on the thin-walled tube and space truss analogy

procedure in the 5t~ Edition of the same handbook. This observation triggered the

examination of both examples. A key design constraint was that in both approaches the

same total amount of transverse steel reinforcement is used for shear and torsion.

6.1.1 Adjustments

Some modifications to Design Example 4.4.2 of the PCI Design Handbook had to be

made in order to make proper comparisons of the two design procedures. As seen in

Chapter 4, the initial difference in required longitudinal steel for torsional strength

decreased to 1.3 times the amount in the 4th Edition after all necessary adjustments had

been made so that the two design examples were conducted under the same basic

conditions. A key adjustment was the use of the web-shear cracking concrete

contribution, Vow, in the 5th Edition instead of the minimum value, Vc--2~-~bwd, as it is

more appropriate for the spandrel beam from the PCI design example.


94

6.1.2 Concrete Cover Spalling

The assumption of concrete cover spalling around the entire cross section was examined.

It was found that using the expression given in Equation (4.5) to calculate the concrete

area enclosed by the shear flow path, Ao, basically eliminated the remaining difference in

the required amount of longitudinal steel for strength. When Equation (4.5) was used

instead of Equation (4.4), the difference between A~ for both design procedures was

further reduced to 0.2 times the amount in the 4th Edition of the corrected value with Vow.

This finding was confirmed in Chapter 5. It would appear based on the results of the

parametric studies that the current assumption regarding spalling of the concrete cover is

too conservative with respect to the 4t~ Edition approach in the case of sections which are

tall and slender such as the spandrel beam studied in both editions of the PCI Design

Handbook.

6.2 Recommendations

The following recommendations are made regarding shear and torsion design of

precast/prestressed concrete spandrel beams.

1 .) Use Vow and Vci instead of the minimum value of 2xf~-~bwd for the nominal shear

strength provided by the concrete. Using the expression is appropriate since they

account for the effect of prestressing and are more relevant to the shear behavior

of prestressed members. Further, use of V~w and Vci will result in a decrease in

the required amount of torsional steel reinforcement for a given total amount of

transverse steel.


95

Use Equation (6.1) in estimating the gross area enclosed by the shear flow path,

Ao, for spandrel beams:

2TL’PcP (6.1)A° = Acp q)~cncp

It is stated in Commentary Section R11.6.3.6 of the ACI Code 318-02 that, "The

expression for Ao given by Hsu1132 may be used if greater accuracy is desired"

when compared to the simplified equation of Ao = 0.85Aoh. The Hsu expression

referred to in Commentary Section R11.6.3.6 is Equation (6.1) shown above.

6.3 Future Work

1.) Additional work is needed in the form of tests of precast, prestressed concrete

spandrel beams to properly evaluate the end effects of warping torsion on restrained L

spandrel beams at end regions and more appropriately represent these effects in the

design of such members. It is possible that the current design approaches because of the

particular end restraint conditions and the geometry of the member do not properly

represent the ultimate mode of failure of precast prestressed spandrels such as those used

in parking garages and examined in this report.

2.) Examine if the 60 ksi limit, stated in Commentary Section Rll.6.3.10,

prestressing longitudinal steel used for torsion is applicable.

for


96

ACI318,2002,

LIST OF REFERENCES

American Concrete Institute, Farmington Hills, MI

Henry, R.L. and Zia, P., "Prestressed Beams in Torsion, Bending, and Shear," Journal

of the Structural Division, Vol. 100, No. 5, May 1974, pp. 933-952

Hsu, T.T.C., "Torsion of Reinforced Concrete," Van Nostrand Reinhold Company,

Inc., 1984

Hsu, T.T.C., "ACI Shear and Torsion Provisions for Prestressed Hollow Girders,"

ACI Structural Journal, November-December 1997

Johnston, D.W. and Zia, P, "Prestressed Box Beams under Combined Loading,"

Journal of the Structural Division, Vol. 101, No.7, July 1975, pp. 1313-1331

Klein, G.J. "Design of Spandrel Beams," Final Report to the PCI, Research Project

No. 5, WJE Associates, Inc., 1986

MacGregor, J.G. and Ghoneim, M.G., "Design For Torsion," ACI Structural Journal,

March-April 1995, pp. 211-217


97

MacGregor, J.G. and Ghoneim, M.G., "Evaluation of Design Procedures for Torsion

in Reinforced and Prestressed Concrete," Structural Engineering Report, No. 184,

University of Alberta, February 1993, pp. 52-54

McMullen, A.E. and Woodhead, H.R., "Experimental Study of Prestressed Concrete

Under Combined Torsion, Bending, And Shear," PCI Journal, September-October

1973, pp. 85-100

Mukherjee, P.R. and Warwaruk, J., "Torsion, Bending, and Shear in Prestressed

Concrete," Journal of the Structural Division, Vol. 97, No. 4, April 1971, pp. 1063-

1079

PCI, 4th Edition, "PCI Design Handbook," Raths, Raths & Johnson, Inc., 1992 pp. 4-

34 to 4-36

PCI, 5th Edition, "PCI Design Handbook," Raths, Raths & Johnson, Inc., 1999, pp. 4-

43 to 4-45

Zia, P. and McGee, W.D., "Torsion Design of Prestressed Concrete," PCI Journal,

March-April 1974, pp. 47-58


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Transcript of by Carlo R. Lini and Julio A. Ramirez - pci.org Reports... · Daniel P. Jenny Research Fellowship...