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Design Guide for Twisting Moments in SlabsReported by Join t ACI-ASCE Committee 447

ACI 447R-18

https://telegram.me/seismicisolation

First PrintingApril 2018

ISBN: 978-1-64195-010-7

Design Guide for Twisting Moments in Slabs

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This guide assists practitioners in understanding: 1) twisting

moments in two-way slabs, when twisting moments are an essential

consideration; 2) methods that can be used to account for twisting

moments in design; and 3) the options available for each method

of the various system geometries. Descriptions of twisting moments

are provided theoretically and visually in the guide, and six methods

of accounting for twisting moments in design are discussed. Appli-

cability of the various methods is evaluated through a comparison

of designs resulting from each method for a variety of two-way

slab types and geometries. The theories described in the guide also

apply to the design of two-way wall and two-way dome systems.

Keywords: fnite element analysis; shell design; slab design; torsion; twist; twisting moments; wall design.

CONTENTS

CHAPTER 1—INTRODUCTION AND SCOPE, p. 2

1 .1—Introduction, p. 21 .2—Scope, p. 2

CHAPTER 2—NOTATION AND DEFINITIONS, p. 2

2.1—Notation, p. 22.2—Defntions, p. 3

CHAPTER 3—BACKGROUND, p. 3

3.1—Qualitative introduction to twisting moments in slabs, p. 33 .2—Behavior of linear-elastic isotropic slabs, p. 43 .3—Equilibrium in slabs, p. 43 .4—Principal axes, p. 43 .5—Orthogonal reinforcement and equilibrium for

twisting moments, p. 53 .6—Efects of slab geometry on twisting moments, p. 53 .7—Traditional slab design methods, p. 63 .8—Finite element analysis (FEA)-based slab design

resultants, p. 6

Ganesh Thiagarajan, Chair Jian Zhao, Secretary

ACI 447R-1 8

Design Guide for Twisting Moments in Slabs

Reported by Joint ACI-ASCE Committee 447

Riadh S. Al-MahaidiGangolu Appa RaoAshraf S. AyoubZdenĕk P. BažantAllan P. BommerMi-Geum Chorzepa

Carlos Arturo Coronado

Gianluca CusatisMukti L. Das

James B. DeatonJason L. DraperSerhan Guner

Trevor D. HrynykJohn F. Jakovich

Song F. JanIoannis KoutromanosLaura N. Lowes

Yong LuYi-Lung Mo

Abbas Mokhtar ZadehWassim I. Naguib

Dan PalermoGuillermo Alberto RiverosMohammad Sharafbayani

Hazim SharhanSri Sritharan

Consulting Members

Ahmet Emin AktanSarah L. BillingtonJohan BlaauwendraadOral BuyukozturkIgnacio CarolLuigi Cedolin

Wai F. ChenChristopher H. ConleyRobert A. DameronFilip C. FilippouKurt H. GerstleWalter H. Gerstle

Robert IdingAnthony R. Ingrafea

Feng-Bao LinChristian MeyerHiroshi Noguchi

Gilles Pijaudier-Cabot

Syed Mizanur RahmanVictor E. SaoumaFrank J. VecchioKaspar J. Willam

ACI Committee Reports, Guides, and Commentaries are intended for guidance in planning, designing, executing, and inspecting construction. This document is intended for the use of individuals who are competent to evaluate the signifcance and limitations of its content and recommendations and who will accept responsibility for the application of the material it contains. The American Concrete Institute disclaims any and all responsibility for the stated principles. The Institute shall not be liable for any loss or damage arising therefrom.Reference to this document shall not be made in contract

documents. If items found in this document are desired by the Architect/Engineer to be a part of the contract documents, they shall be restated in mandatory language for incorporation by the Architect/Engineer.

ACI 447R-18 was adopted and published April 201 8.Copyright © 2018, American Concrete Institute.All rights reserved including rights of reproduction and use in any form or by

any means, including the making of copies by any photo process, or by electronic or mechanical device, printed, written, or oral, or recording for sound or visual reproduction or for use in any knowledge or retrieval system or device, unless permission in writing is obtained from the copyright proprietors.

1


CHAPTER 4—AVAILABLE DESIGN METHODS, p. 7

4.1—Finite element analysis (FEA)-based design ignoring twist, p. 74.2—Design using the Wood and Armer method, p. 74.3—Design using the sandwich model, p. 74.4—Design using element nodal forces, p. 84.5—Design using twist-free analysis, p. 9

CHAPTER 5—COMPARISON OF DESIGN

METHODS, p. 1 0

5.1—Sensitivity to angle of principal axes, p. 1 05.2—Typical design conditions, p. 11

CHAPTER 6—TWO-WAY WALLS, p. 21

6.1—General considerations, p. 216.2—Impact of twisting moment on walls exhibiting two

adjacent fxed edges, p. 21

CHAPTER 7—SHELL STRUCTURES, p. 24

7.1—General considerations, p. 247.2—Typical bulk material storage hemisphere, p. 247.3—Typical loading conditions, p. 257.4—Typical design regions, p. 25

CHAPTER 8—REFERENCES, p. 29

Authored documents, p. 29

CHAPTER 1—INTRODUCTION AND SCOPE

1.1—Introduction

Section 8.2.1 of ACI 31 8-1 4 allows slabs to be designed by any procedure that satisfes equilibrium and geometric compatibility, and requires that, at each section, the design strength exceeds the required strength and serviceability requirements are fulflled.Traditional strip design methods for slabs are based

on approximate analysis and provide neither a complete equilibrium load path or satisfy geometric compatibility. Nonetheless, these methods have been used successfully for many years to design slabs with supports arranged in a rectangular grid.From 1 995 to 201 5, design engineers transitioned from

predominantly using traditional slab analysis methods to using fnite element analysis (FEA). More recently, engi-neers use FEA to assist in the structural design of two-way concrete members. Twisting moments in two-way slabs can require additional reinforcement from those proportioned for bending moments, yet they are often misunderstood and sometimes ignored, neglected, or both, by practitio-ners in design. This is most likely due to their lack of being discussed comprehensively in design codes and frequent exclusion from college concrete design course curricula.Although FEA solutions provide a full equilibrium load

path and satisfy geometric compatibility, they determine load paths that require twisting moments for equilib-rium (Shin et al. 2009). Many designers using FEA have ignored these twisting moments—a possible unconserva-

tive assumption where twisting moments are high (Park and Gamble 2000). To provide designers with guidance related to this issue, methods for explicitly incorporating twisting moments determined from FEA in the design of slabs are discussed in this guide.The purpose of this design guide is to provide advice

to design engineers who analyze slab systems with fnite element methods and who need to ensure their designs are satisfactory for the twisting moments predicted by the anal-ysis. This guide provides background information regarding twisting moments and describes multiple approaches for consideration of twisting moments in design. It also provides advice for designers of walls and shells with twisting moment conditions similar to those in slabs.

1.2—Scope

This design guide applies to slabs of both uniform and nonuniform thicknesses, including drop caps and drop panels, except where noted in the text. This guide does not apply to wafe slabs, or the beams of beam-and-slab foor systems. Chapters 3 through 6 address slabs and walls in which the response is determined purely by bending. Chapter 7 addresses shells for which the response is determined by bending and membrane action. Chapter 6 and the theory sections of this guide are applicable to walls. Chapter 7 and the theory sections of this guide are applicable to shells, with the caveat that equations presented in Chapter 3 are not valid for curved shells.

CHAPTER 2—NOTATION AND DEFINITIONS

2.1—Notation

ci, j = fraction for consideration of sections partially crossing element to apply to forces in local node j

in element iD = fexural rigidity of plate, in.-lb (N·mm)E = Young’s modulus, psi (MPa)F = force vector, lb (N)fi, j = nodal force vector for local node j in element ih = thickness of slab or plate, in. (mm)L = width of design section, in. (mm)M = bending moment, or moment vector, in.-lb (N·mm)Md = design bending moment, in.-lb (N·mm)Mi = bending moment from isotropic analysis, in.-lb

(N·mm)Mtf = bending moment from twist-free analysis, in.-lb

(N·mm)Mu = design moment for slab cross section, in.-lb (N·mm)m i, j = nodal moment vector for local node j in element i,

in.-lb/in. (N·mm/mm)mr = bending moment causing stresses parallel to r-axis,

per unit length of slab or plate, in.-lb/in. (N·mm/mm)

mrs = twisting moment relative to r-s-axes per unit length of slab or plate, in.-lb/in. (N·mm/mm)

ms = bending moment causing stresses parallel to s-axis, per unit length of slab or plate, in.-lb/in. (N·mm/mm)

2 DESIGN GUIDE FOR TWISTING MOMENTS IN SLABS (ACI 447R-1 8)


mux = design moment causing stresses parallel to x-axis, per unit length of slab or plate, in.-lb/in. (N·mm/mm)

mux+ = positive design moment causing stresses parallel

to x-axis, per unit length of slab or plate, in.-lb/in. (N·mm/mm)

mux– = negative design moment causing stresses parallel to

x-axis, per unit length of slab or plate, in.-lb/in. (N·mm/mm)

muy = design moment causing stresses parallel to y-axis, per unit length of slab or plate, in.-lb/in. (N·mm/mm)

muy+ = positive design moment causing stresses parallel

to y-axis, per unit length of slab or plate, in.-lb/in. (N·mm/mm)

muy– = negative design moment causing stresses parallel

to y-axis, per unit length of slab or plate, in.-lb/in. (N·mm/mm)

mx = bending moment causing stresses parallel to x-axis, per unit length of slab or plate, in.-lb/in. (N·mm/mm)

mxy = twisting moment relative to x-y-axes per unit length of slab or plate, in.-lb/in. (N·mm/mm)

my = bending moment causing stresses parallel to y-axis, per unit length of slab or plate, in.-lb/in. (N·mm/mm)

nx = membrane tension in x-axis direction per unit length, lb/in. (N/mm)

nxy = membrane in plane-shear in x-y-axes direction per unit length, lb/in. (N/mm)

ny = membrane tension in y-axis direction per unit length, lb/in. (N/mm)

q = transverse load per unit area, lb/in. 2 (N/mm2)T = torsional moment, in.-lb (N·mm)V = shear force, lb (N)Vd = design shear force, lb (N)

vx = transverse shear on x-face per unit length, lb/in. (N/mm)

vy = transverse shear on y-face per unit length, lb/in. (N/mm)

w = transverse defection, in. (mm)xi, j = distance vector from section centroid to local node

j in element i, in. (mm)ν = Poisson’s ratio

2.2—Defntions

ACI provides a comprehensive list of defnitions through an online resource, ACI Concrete Terminology. Defnitions provided herein complement that source.anticlastic bending—curvature caused by the Poisson

efect and curvature about a perpendicular axis.strip design method—a method of designing slabs by

dividing them into two sets of approximately perpendicular strips, with each strip analyzed and designed independently from each other.

CHAPTER 3—BACKGROUND

3.1—Qualitative introduction to twisting moments

in slabs

Twist exists in most every slab, except those theoretical-case-only slabs whose moments at any point are identical about any axis. Figure 3 .1 illustrates an extreme case of twist; a square slab with supports at three corners and a load at the fourth corner.

From equilibrium, it can be shown that the bending moment about the A-A and B-B axes in the fgure is zero, although this slab is clearly supporting a load and needs to be reinforced. Looking at the C-C axis, whose bending moment is nonzero, gives us insight to the load-carrying

Fig. 3. 1—Twisting moment example free body diagram.

DESIGN GUIDE FOR TWISTING MOMENTS IN SLABS (ACI 447R-1 8) 3


mechanism of the slab. What appears as twist about A-A and B-B is bending about C-C and D-D, as shown in the defected shapes along C-C and D-D.

3.2—Behavior of l inear-elastic isotropic slabs

Like other components in concrete structures, the design forces for slabs are typically determined using some form of a linear-elastic analysis. For slabs that are not thick (span-to-thickness ratio of more than 1 0), linear-elastic slab behavior can be predicted by well-known plate theory.The defection of a thin, linear-elastic, isotropic plate

subjected to loads perpendicular to its plane can be expressed by a fourth-order partial diferential equation (Timoshenko and Woinowski-Krieger 1 959)

∂

∂+

∂

∂ ∂+

∂

∂=

4

4

4

2 2

4

42

w

x

w

x y

w

y

q

D (3 .2a)

where w = w(x, y) is the transverse (out-of-plane) displacement feld in the direction of loading; q = q(x, y) per unit area; and D is the fexural rigidity of the plate, which is expressed as

DEh

=−

3

212 1( )ν

(3 .2b)

in which E is the modulus of elasticity, h is the thickness of the plate, and ν is Poisson’s ratio.The deformations of the plate can be interpreted as two

curvatures ∂

∂

2

2

w

x and

∂

∂

2

2

w

y, and one twist

∂

∂ ∂

2w

x y. Corre-

sponding to these three deformations, there are two bending moments and one twisting moment as shown in Eq. (3 .2c).

m

m

m

D D

D D

D

w

x

w

y

x

y

xy

= −

−

∂

∂

∂

∂

ν

ν

ν

0

0

0 0 1 2

2

2

2

( ) /

22

2

2∂

∂ ∂

w

x y

(3 .2c)

From Eq. (3 .2c), note that unless ∂

∂ ∂

2w

x y is zero, there will

be twisting moments in the plate. In general, ∂

∂ ∂

2w

x y is zero in

only a small subset of locations in a slab, so twisting moments will exist almost everywhere in a linear-elastic isotropic slab.

3.3—Equilibrium in slabs

By considering the rotational equilibrium of the slab element shown in Fig. 3 .3 , Eq. (3 .3), describing the inter-relationship of mx, my, and mxy, can be derived.

∂

∂+

∂

∂ ∂+

∂

∂= −

2

2

2 2

22

m

x

m

x y

m

yqx xy y (3 .3)

Equation (3 .3), which holds for all slabs regardless of material behavior, reinforcement, or cracking, shows that any change in mxy must be ofset by some change in mx, my, or both. Hence, mxy can only be reduced to zero by changing the load path of the slab.

3.4—Principal axes

The values of mx, my, and mxy are related via Mohr’s circle (Timoshenko and Woinowski-Krieger 1 959); the moments mr, ms, and mrs for any set of perpendicular r-s axes can be determined from mx, my, and mxy, as shown in Eq. (3 .4a) through (3 .4c).

m m m m m mr x y x y xy= + + − +1

2

1

22 2( cos sin) ( ) θ θ (3 .4a)

m m m m m ms x y x y xy= + − − −1

2

1

22 2( ( cos sin) ) θ θ (3 .4b)

m m m mrs x y xy= − − +1

22 2( sin cos) θ θ (3 .4c)

For every point in a plate, there is one set of principal axes, where mrs is zero and about which the slab is in pure bending. The orientation of these axes changes from point to point in the slab, and the principal axes are typically more valuable for general understanding than for determining

Fig. 3. 3—An infnitesimal plate element shown with resulting

shear forces and moments due to transverse loading.



design forces. Figure 3 .4 illustrates the construction of Mohr’s circle for Eq. (3 .4a) through (3 .4c).

3.5—Orthogonal reinforcement and equilibrium for

twisting moments

How can reinforcement be used to resist twisting moments? This question is most conveniently addressed by considering the reinforcement required to resist the moments about the principal axes. Figure 3 .5 illustrates the conceptual approach.

Figures 3 .5(a) and 3 .5(b) illustrate the slab diferential element to be reinforced, and its principal moments m1 and

m2. The values of m1 and m2 cause tension in the bottom of the slab. Figure 3 .5(c) shows bottom reinforcement forces aligned with the principal axes, selected using stan-dard bending design approaches, and the x- and y-axis force vectors the reinforcement provides. Figure 3 .5(d) shows the reinforcement forces along the 1 - and 2-axes replaced by reinforcement forces along the x- and y-axes that, when combined, provide vector forces equivalent to the 1 - and 2-axes reinforcement forces.

3.6—Effects of slab geometry on twisting

moments

The slab in Fig. 3 .6a illustrates how diferent slab regions and geometries afect the importance of the consideration of twisting moments when designing a slab. The fgure high-lights three slab regions that are discussed as follows. The slab is assumed to be designed with reinforcing bars parallel to the x- and y-axes.

Region 1 of the slab is regular with all columns arranged in a rectangular grid. Regions similar to this are not twist-sensitive. Ignoring twisting moments in these regions can be proved safe through yield line theory. Although yield line theory is an upper-bound approach, for a slab with regular supports such as those in Region 1 , the number of yielding patterns to consider is small. If top reinforcement is clus-tered at the columns, only the folded-plate pattern in each

Fig. 3. 4—Mohr’s circle for situation where mx and my are

both positive and mx > my.

Fig. 3. 5—Orthogonal reinforcement from twisting moments.



direction shown in Fig 3 .6b need be considered (Kennedy and Goodchild 2003 ). The folded plate yield line patterns show that any reinforcement design that supports the total span static moment (wL2/8 for a uniform load) in each direc-tion is safe.Region 2 of the slab is where there is a major shift in

the column layout. Regions similar to this are moderately twist-sensitive. The slab tends to span along a 45-degree line between the closest columns. This diagonal spanning reduces the bending moments along the x- and y-axes while increasing twisting moments. Ignoring twisting moments in design of similar regions could lead to a slab load capacity that is approximately 20 percent less than calculated. This percentage is calculated by comparing integrated slab strip bending moments using isotropic plates and using ortho-tropic plates with zero twist stifness. This calculation assumes that the slab has no reliable twist capacity if twist has not been considered in the design of reinforcement, and is performed by dividing the bending moment determined by a twist-free analysis, Mtf, by the bending moment determined by an isotropic analysis, Mi. For Region 2, Mtf/Mi = 1 .2.Region 3 of the slab is a cantilevered corner. Regions

similar to this are highly twist-sensitive. The slab tends to span between, and cantilever of, the line between the two columns. Although this diagonal cantilevering reduces the bending moments along the x- and y-axes, it adds large twisting moments. Ignoring twisting moments in design of similar regions could lead to a slab load capacity that

is approximately 40 percent less than calculated. This percentage is calculated by comparing integrated slab strip bending moments using isotropic plates and using ortho-tropic plates with zero-twist stifness. This calculation assumes that the slab has no reliable twist capacity if twist has not been considered in the design of reinforcement (Mtf/Mi = 1 .4). The strip design methods referred to in this docu-ment are those commonly used by practicing engineers, not the more sophisticated strip approaches discussed in academic literature.

3.7—Traditional slab design methods

Traditionally, slab design is performed using design forces determined by simplifed analysis methods that idealize the slab as a set of intersecting strips (efectively supports and wide beams) in two perpendicular directions. The required reinforcement is calculated using the strip bending moments and standard beam bending design approaches.These methods have two primary faws. The frst is that

equilibrium is not fully satisfed because there is no consis-tency between the load paths of the two sets of strips. The second faw is that deformation compatibility between both parallel and perpendicular strips is ignored.Despite these faws, slabs designed using these methods

have generally performed well. For slabs with supports arranged in a rectangular grid it is often shown, using yield-line theory or lower-bound methods, that the traditional strip methods provide adequate capacity when reinforcement is distributed appropriately. (Burgoyne 2004; Kennedy and Goodchild 2003). For those slabs with supports not arranged in a rectangular grid, engineering judgment is necessary to determine if the traditional methods will produce a safe design.

3.8—Finite element analysis (FEA)-based slab

design resultants

To complete slab design, an engineer should determine the quantity of reinforcement required for each design cross section. Although design cross section locations and lengths are often guided by code rules, in general, sections are needed at peak stress locations and their lengths based on the extent of the slab that can be assumed to act as a unit in resisting internal forces. The width of each section should be chosen so that the moment distribution along the section is reasonably uniform, does not change sign, and can be resisted by uniformly distributed reinforcements.When using FEA to support slab design, the engineer

should convert the slab analysis element results to resultant forces and moments acting on these sections. At this stage, the engineer should frst transform the results of FEA into a coordinate system orthogonal to the section, and then inte-grate/sum all forces and moments acting on the section to determine the design moment. Consideration of twisting moments is the most difcult aspect of this conversion. ACI 31 8 does not explicitly address twisting moments, nor does the commentary provide guidance on their consid-eration. However, Section 6.1 .1 of ACI 31 8.2-1 4, which addresses shell reinforcement, states that reinforcement

Fig. 3. 6a—Representative regions of a slab foor plan.

Region 1 is twist-sensitive, Region 2 moderately twist-sensi-

tive, and Region 3 highly twist-sensitive.

Fig. 3. 6b—Folded plate yield lines.



must be designed to resist tension from bending and twisting moments.

Design can also be performed considering the FEA results at each point in the slab, independently and foregoing the integration along a section. A zero-length section should still be considered to provide the correct axes for the design resultants and reinforcement. Design code requirements, such as those in ACI 31 8, can be difcult to apply to these point results, as the codes are written for section-based results. Another potential difculty with using FEA results at a point for design is that the results at certain locations, such as under a point load, will approach infnity as the mesh is refned. In general, the approaches described in this guide apply to either point-based or section-based resultants.

CHAPTER 4—AVAILABLE DESIGN METHODS

This chapter discusses various options to consider for twisting moments. Chapters 5 through 7 evaluate some of these options in sample structures. Design methods discussed in this chapter are typically used with the results from linear-elastic analyses.

4.1—Finite element analysis (FEA)-based design

ignoring twist

A simple, commonly-used approach to determine the design moment for a slab cross section is to integrate the FEA-predicted moment about the axis of the section along the length of the section.

Mu = ∫mrds (4.1 )

In Eq. (4.1 ), mr is the moment per unit length along the section. This approach ignores twisting moments in the equi-librium load path and may signifcantly underestimate the design demand.

The efect of twisting moments can be safely ignored in this manner when sections are approximately aligned orthogonally to the principal bending directions at all loca-tions, as the twist on the section vanishes in this case. An instructive rule of thumb is that twist can be ignored if it is smaller than 1 0 percent of the primary bending moment (Deaton 2005). If this criterion is not satisfed, neglecting the efect of twist could lead to unconservative results.

4.2—Design using the Wood and Armer method

Wood (1 968) and Armer (1 968) (referred to in this docu-ment as Wood and Armer) proposed one of the most popular methods for explicitly incorporating twisting moments in slab design. The method seeks to prevent yielding in all direc-tions and was developed by considering the normal moment and Johansen’s yield criteria (Park and Gamble 2000). At any point in the slab, for any arbitrary direction, the design moment determined from FEA results must not exceed the ultimate normal resisting moment in that direction. The ulti-mate normal resisting moments calculated from the rein-forcement in the x- and y-directions are typically compared to adjusted design moments mux and muy. Design moments based on the Wood and Armer method are computed

mux = mx ± |mxy| (4.2a)

muy = my ± |mxy| (4.2b)

where all plus signs are used to compute required bottom reinforcement, all minus signs are used to compute required top reinforcement, and mux and muy will be negative when the top reinforcement is in tension. This assumes a sign conven-tion where positive moment causes tension on the bottom surface of the slab and negative moment causes tension on the top surface of the slab. For non-slab systems such as walls and domes, top and bottom should be defned such that they are consistent throughout and consistent with loading assumptions.

Where mux or muy is found to have the opposite sign from what is expected (negative for bottom reinforcement or positive for top reinforcement), the design moment can be conservatively set to zero. A less conservative calculation option for this case is shown in Table 4.2.

The Wood and Armer approach requires that a fne regular mesh be used to produce accurate slab moment and twist predictions. Also, because moments, twists, or both, can be theoretically infnite at concentrated loads or reactions (such as slab column connections), integration of the moments or twists at these locations is difcult to perform accurately. Previous studies have indicated that the Wood and Armer method could produce unconservative results for slabs with high reinforcement ratios (approximately more than 0.75 percent) at regions of signifcant twisting moments, espe-cially near restrained slab corners (May and Lodi 2005). Note also that this method cannot be directly applied to slabs with beams or drop panels. The Wood and Armer method has been modifed and implemented in more conservative forms in both CSA-A23.3-04 and EN 1 992-1 -1 :2004 .

4.3—Design using the sandwich model

Designs based on the normal moment yield criterion, such as the Wood and Armer method, do not account for trans-verse shear or membrane forces within the slab. The sand-wich model approach allows for the computation of rein-forcement to resist not only bending and twisting moments, but also the efects of shear and membrane stresses. In this approach, membrane efects (nx, ny, and nxy) and bending and twisting moments (mx, my, and mxy) are resisted by the sand-wich exterior layers, whereas the shear efects (vx and vy) are resisted via the sandwich core. Figure 4.3 (Marti 1 990) shows this concept.

To design by using the sandwich model, a slab section is divided into three layers; the depth of each layer can be determined such that the middle planes of the outer layers coincide with the center of the top and bottom reinforce-ment layers. Then, bending moments are decomposed into a couple of tensile and compressive normal forces, and twisting moments are decomposed into a couple of in-plane shear forces acting at top and bottom layers of the slab. The sandwiched inner layer is used to resist transverse shear. These normal and shear forces due to moments are combined with membrane (in-plane) forces. Finally, slab reinforce-



ment is proportioned to satisfy equilibrium conditions at each top and bottom layer in the two orthogonal directions. Further descriptions of this method have been presented by fb Bulletin 45 (Comité Euro-International du Béton 2008), Brøndum-Nielsen (1 985), Lourenço and Figueiras (1 995) , and the CEB-FIP Model Code (Comité Euro-International du Béton 1 993 ).The sandwich model is not typically used for slab design

for commercial structures or generally available in slab design software. The assumed lever arm between the outer layers is very conservative for thin slabs, as the ratio of true ultimate lever arm to assumed lever arm is signifcantly greater than 1 .0 (although this conservatism is reduced as the slab thickness is increased).

4.4—Design using element nodal forces

4.4.1 Nodal forces approach—The methods discussed in 4.1 , 4.2, and 4.3 use slab stresses as the analysis quanti-

ties from which slab design forces are calculated. Another approach for slab design using FEA results is to compute results for the design cross section from element nodal forces and moments. Using this approach, the section design forces are guaranteed to be in equilibrium with the externally applied nodal loads (Deaton 2005). Slab design methods based on element nodal forces have been imple-mented in various FEA software. These methods are attrac-tive because results are relatively accurate (even for very coarse or irregular meshes), as they can be used for slabs containing beams or drop panels and are easily extended to design post-tensioned foors.

4.4.2 Calculation of nodal forces—The nodal forces approach transforms all element nodal forces and moments acting at nodes on the design cross section into resultant forces and moments acting at the centroid of the section, as shown in Fig. 4.4.2. Because transformation of nodal forces to the centroid considers the eccentricity of the nodes from the centroid, equilibrium of all acting forces and moments is provided.The equilibrium equations in vector form are:

Table 4.2—Less conservative design moments (Wood 1 968; Armer 1 968)

mx > |mxy| mx ≤ |mxy| mx < –|mxy|

my > |mxy|

muxbot = mx + |mxy|

muxtop = 0

muybot = my + |mxy|

muytop = 0

muxbot = mx + |mxy|

m mm

mux

top

x

xy

y

= −2

muybot = my + |mxy|

muytop = 0

muxbot = 0

m mm

mux

top

x

xy

y

= −2

m mm

muy

bot

y

xy

x

= +2

muytop = 0

my ≤ |mxy|

muxbot = mx + |mxy|

muxtop = 0

muybot = my + |mxy|

m mm

muy

top

y

xy

x

= −2

muxbot = mx + |mxy|

muxtop = mx – |mxy|

muybot = my + |mxy|

muytop = my – |mxy|

muxbot = 0


m mm

muy

bot

y

xy

x

= +2


my < –|mxy|

m mm

mux

bot

x

xy

y

= +2

muxtop = 0

muybot = 0

m mm

muy

top

y

xy

x

= −2

m mm

mux

bot

x

xy

y

= +2


muybot = 0


muxbot = 0


muybot = 0


Fig. 4. 3—Sandwich model.

Fig. 4. 4. 2—Design section forces derived from element

nodal forces.



? ?

F c fi j

i j i j= ∑∑ ⋅, , (4.4.2a)

?

? ?

?

M c m x fi j

i j i j i j i j= ∑∑ ⋅ + ×

, , , ,( ) (4.4.2b)

where i is the iterator over the elements crossed; j is the iterator over the local nodes in element i (on one side of the

section); ?

F and ?

M are the force and moment vectors at the

section centroid, respectively; ?

fi j, and ?

mi j, are nodal force and moment vectors, respectively;

?

xi j, is a vector distance from the section centroid to node i, j; and ci, j is a multiplier that equals 1 .0 unless the section does not fully cross element i, in which case it varies between 0.0 and 1 .0, typically based on the fraction of the element crossed.The forces and moments to be used in the slab design

are found by transforming the ?

F and ?

M vectors into the coordinate system of the design cross section. While six resultants are determined, typically only transverse shear V, bending moment M, and torsion T are of interest. There is no resultant quantity that directly corresponds to the twisting moments on the section. Methods available for consider-ation of the calculated torsion are as follows.

4.4.3 Torsion as bending—One approach to incorporating T is to directly combine M with ±T, as shown in Eq. (4.4.3). Efectively, this method uses T as a proxy for integrated twist and then applies the Wood and Armer approach.

Md = M ± T (4.4.3)

This approach can lead to unconservative design moments in two conditions:(1 ) Condition 1—Twisting moments along the cross

section change sign and partially cancel in their integration(2) Condition 2—Torsion due to shear forces is of the

opposite sign as torsion due to twistCondition 1 can be detected visually and avoided by

breaking the design section into two design sections; however, this can be difcult when considering numerous loading conditions, as the location of the sign change could vary. Condition 2 rarely occurs without Condition 1 also occurring. When neither Condition 1 or 2 occur, the design moments derived from considering torsion as bending can be conservative. For a simple torsional cantilever slab, the design moments are twice those derived from the Wood and Armer approach (Den Hartog 2014).

4.4.4 Torsion as shear—The total torsion T acting on the design cross section can be decomposed into an assumed linearly varying transverse shear per unit length, as shown in

Fig. 4.4.4. This torsional shear stress has a maximum value equal to 6T/L2, where L is the width of the section—that is, the width of section cut in Fig. 4.4.2. This is analogous to the bending stress in a rectangular beam being calculated as 6M/bh2.By multiplying this maximum shear stress assumed due to

torsion by the width of the section, an equivalent, but very conservative, resultant design shear force due to torsion, which is equal to 6T/L, can be considered during the shear design of the section, as is shown in Eq. (4.4.4).

V V

T

LL V

T

Ld= ± = ±

6 62

(4.4.4)

Considering torsion as shear is generally not applicable when signifcant twisting moments exist. However, applying the torsion-as-shear design approach to the results of a twist-free analysis (3 .5) has the potential for providing improved vertical shear designs while simultaneously fully consid-ering twisting moments.

4.4.5 Nodal forces with twisting moment integrations—While the nodal forces approach does not provide twisting moments, it can be supplemented with element twisting moment integrations to provide a full set of design forces suitable for the Wood and Armer method or other design methods. The beneft of using such a hybrid approach is to gain accuracy of the nodal forces approach for all quantities to which it applies.

4.5—Design using twist-free analysis

Slab analysis and design software capable of analyzing orthotropic slabs (with diferent properties in two orthogonal directions) typically allows engineers to separately modify the stifness corresponding to each of the bending moments about the two orthogonal axes and the twisting moment (Shin et al. 2009). Setting the twist stifness to zero leads to an analytical solution and load path in which all the twisting moments are zero. The behavior and load path of a twist-free slab is somewhat similar to that of a grillage of beams. The design philosophy behind twist-free analysis is similar to that for the compatibility torsion approach that has long been accepted in ACI 31 8.Bending and twisting moments are interrelated through

a Mohr’s circle equilibrium constraint. Because of this constraint, the twist stifness can only be set to zero about a single set of perpendicular axes at each location. The most practical axes for this purpose are the axes of the design cross sections, which are perpendicular to the reinforcement. In a typical slab that is reinforced in only two perpendicular direc-

Fig. 4. 4. 4—Assumed shear stress distribution causing torsion.



tions, the entire slab can be given the same modifed stifness. For slabs where the reinforcement directions vary from region to region, multiple stifness modifcations are required.

Twist-free slab analyses result in larger defections than those predicted by analyses with nonzero twist stifness. This is to be expected, as the load path with nonzero twist stif-ness is more efcient in terms of elastic energy than one with zero twist stifness. Comparing slab defections computed by analyses with and without twist stifness is a quick means of gauging how much the forces will be redistributed to mobilize the twist-free load path. If the defections from the twist-free analysis are signifcantly greater than those from the analysis with twist stifness—for example, by a factor of 2—the potential for twist-related cracking should be inves-tigated. Twist-free analysis can be used in combination with nodal force or moment resultant methods.

CHAPTER 5—COMPARISON OF DESIGN

METHODS

This chapter compares the available twisting moment design approaches in various scenarios. Although numer-ical comparisons are provided, the intent is to qualitatively illustrate how well or poorly each design method considers support confgurations and design section orientations. The slabs investigated in this chapter are thin (8 in. [200 mm]), with small bars (0.5 in. [1 2 mm]) and moderate cover (1 in. [25 mm]); other slab parameters have little infuence over the comparative results.

5.1—Sensitivity to angle of principal axes

This section investigates the sensitivity of the considered design methods to the angle between the design section and the principal zero-twist axes for an isotropic linear-elastic slab. Figure 5.1 a shows a slab with uniform moments (and zero twist) about the x- and y-axes (one moment causing defection upward and the other causing defection down-ward), causing uniform twist and zero bending moments along the 45-degree axis.

The design sections shown in Fig. 5.1 a(c) are rotated about their crossing point and the design quantities and results investigated. At angle zero, there is pure bending and no twist. As shown in Fig. 5.1 b, all methods, except for the sandwich method, lead to the same results. The sandwich method requires additional reinforcement due to its conser-vative lever arm assumption. At 45 degrees, there is pure twist and no bending. Between zero and 45 degrees, there is a combination of twist and bending.

Figure 5.1 b shows the relative total quantities of rein-forcement required as the design sections are rotated from zero to 45 degrees. The y-axis is normalized by the required reinforcement for the no-twist zero angle. Although this plot shows reinforcement quantities for a particular design scenario, general behaviors can be observed.

The Wood and Armer method and the torsion-as-bending method have similar curves that approach a ratio of 2 as the slab approaches pure twist. As explained in 3 .2, the Wood and Armer approach has a strong theoretical basis that ensures a safe design.

Fig. 5. 1a—(a) Uniform moment feld; (b) defected shape; and (c) design sections to rotate

in evaluation of twist-sensitivity.



It is expected that the torsion-as-bending approach will give similar results to the Wood and Armer approach, provided that the total torsion is approximately the same as the total twist. Figure 5.1b shows that the torsion-as-bending gives the same design as Wood and Armer for pure twist and pure bending, and similar designs for cases between those two extremes. The diferences between Wood and Armer and torsion-as-bending in the plot are caused by the of the ci, j factor of Eq. (4.4.2b) not exactly matching the mxy stress integration results.Ignoring torsion and treating torsion-as-shear have iden-

tical curves that approach zero as the slab approaches pure twist. In this case, the torsional moments were not large enough for the torsion-as-shear method to require transverse reinforcement. Therefore, it is expected that its design will be the same as ignoring torsion. Both methods are unsafe for the pure twist condition.Although the sandwich model follows a similar pattern

to Wood and Armer, it requires more reinforcement due to its conservative lever-arm assumption. The test slab is thin (8 in. [200 mm]), which accentuates this conservatism. As explained in 4.3 , the sandwich method has a strong theo-retical basis that ensures a safe design.The twist-free method provides a clear, safe equilibrium

load path and is efcient for angles less than approximately 25 degrees. However, at angles greater than 25 degrees, the reinforcement layout creates an inefcient load path, where it becomes questionable if this load path can be attained without some failure or serviceability distress. At 45 degrees, the requirements theoretically approach infnity and no equi-librium is found.Figure 5.1 c shows the predicted relative maximum defec-

tions as sections (and the slab behavior in the twist-free case) are rotated from zero to 45 degrees. The y-axis is normalized by the isotropic defection value. For all methods, except twist-free analysis, the predicted defections are constant, as the design method does not afect the isotropic analysis. For twist-free analysis, the slab orthotropic properties are oriented parallel to the design sections, and the slab becomes increasingly more fexible as the axes approach the pure-twist axes of 45 degrees. Above approximately 25 degrees,

the defections become very large, and signifcant cracking would be expected for the slab to deform from the uncracked isotropic load path to the cracked orthotropic load path. This plot shows that the ratio of twist-free defection to isotropic defection can be used as a proxy to determine qualitatively the amount of redistribution and cracking necessary to reach a twist-free load path.

5.2—Typical design conditions

This section investigates regions of slabs with plan geom-etries that occur frequently, comparing how the considered design methods difer for each of the geometries. Isotropic and orthotropic (twist-free) linear-elastic behaviors are investigated. While the slab confgurations discussed do not cover every scenario, they provide a range of conditions that expose the strengths and weaknesses of each design method. Figure 5.2 locates the typical conditions on a slab plan and lists the section numbers where the conditions are discussed.Note that while many design codes, including ACI 31 8,

would suggest using column strips and middle strips for slab design, the design sections investigated in this section have been chosen for illustrative purposes and might not follow any particular code requirements.The slab regions analyzed for in 5.2 are simplifed and are

intended to represent typical behavior patterns. Only gravity loading efects are evaluated. Lines of symmetry have been used where the slab continues into an adjacent region.

5.2.1 Notes on fgures and tables used in 5. 2—Sections 5.2.2 through 5.2.7 each contain one fgure with four images. The legend for each set of four images is the same and presented in Fig. 5.2.1 .The tables presented in 5.2.2 through 5.2.7 contain the

same rows and columns presented for each of the design conditions. Each row reports for a design method, and the meaning of the columns is:a) M represents the bending moment at the cross section.

This will be the same for all methods except for twist-free. The value is normalized by the isotropic result.b) T represents the torsion (due to both twist and eccen-

tric shear) at the cross section. This will be the same for all methods except for twist-free. The value is normalized by the isotropic M value.

Fig. 5. 1b—Relative calculated total reinforcement versus

section angle.

Fig. 5. 1c—Relative maximum defection versus section

angle.



Fig. 5. 2—Key map for typical design conditions discussions.

Fig. 5. 2. 1—Legends for fgures in 5. 2. 2 to 5. 2. 7.

1 2 DESIGN GUIDE FOR TWISTING MOMENTS IN SLABS (ACI 447R-1 8)


c) |Twist| represents the integral of the absolute value of twist (∫|mxy| ) along the cross section. This will be the same for all methods except for twist-free. The value is normal-ized by the isotropic M value.d) Def represents maximum defection in the slab region.

This will be the same for all methods except for twist-free. The value is normalized by the isotropic defection value.e) Md represents design moment. The value is normalized

by the Wood and Armer design moment. Design moments are not provided for the sandwich model, as it reduces the design demand to a design force instead of a design moment.

f) Ab represents the bottom reinforcement area. The value is normalized by the maximum of the Wood-Armer top and bottom reinforcement areas.g) At represents the top reinforcement area. The value is

normalized by the maximum of the Wood-Armer top and bottom reinforcement areas.h) Av represents the requirement for shear reinforcement.

The only method that can require shear reinforcement is torsion-as-shear.i) Appl summarizes the applicability of the method to the

confguration. The values used are:i. Yes — The method is applicable, and within 5 percent of

the most efcient applicable method.ii. Cons — The method is applicable, and within 20

percent of the most efcient applicable method.iii. Ex. Cons — The method is applicable, although it is 20

percent or more conservative than the most efcient appli-cable method.iv. Serv? — The method is applicable for strength, but

there are serviceability concerns (this value only applies to the twist-free analysis approach).v. ? — The reinforcement calculated is not less than one of

the applicable methods, but the assumed mechanisms of the method do not match the slab behavior, so the applicability of the method to similar confgurations is questionable.vi. No — The method is not applicable.The discussion in 5.2.2 through 5.2.7 references the correct

quantity of reinforcement. In the context of those discussions, the correct quantity of reinforcement is the minimum quantity of reinforcement required by a method that is known to be appropriate for the conditions being discussed.

5.2.2 Regular slab – interior panel at column—Figure 5.2.2 shows an interior panel of a regular slab and the loca-tion of a critical design section at the support in a region with large bending moments and signifcant twist.Table 5.2.2 notes the normalized key design quantities and

results for the design section shown in Fig. 5.2.2.While there is a signifcant twist peak near the support,

overall the integrated twist for this section is only 9 percent of the integrated moment. Due to symmetry of the slab confguration, torsion on the section is zero. Note the following regarding design methods:a) Ignore twist: Ignoring twist results in the correct quan-

tity of reinforcement, although it is questionable, as it does not guarantee a mechanism exists to resist the twist.b) Wood and Armer: This approach is applicable, although

it results in a 9 percent excess reinforcement penalty.

c) Torsion as bending: Condition 1 discussed in 4.4.3 applies here; the twisting moments change sign along the section, so torsion-as-bending is not expected to apply. Treating torsion as bending results in the correct quantity of reinforcement. Due to symmetry, however, the total torsion is zero; therefore, torsion is a poor proxy for integrated twist and torsion-as-bending questionable.d) Torsion-as-shear: Treating torsion as shear results in

the correct quantity of reinforcement. However, it does not guarantee a mechanism for resisting the twist that exists; therefore, torsion-as-shear is questionable.e) Sandwich model: Although this approach is appli-

cable, it results in a 32 percent excess reinforcement penalty. This excess would be smaller in thicker slabs.

f) Twist-free analysis: This approach results in defec-tion predictions that are 4 percent larger than a regular (isotropic) analysis. This increase is very small, so there are no concerns of signifcant cracking required to achieve the twist-free load path. Twist-free analysis is the most efcient applicable method for this case.

5.2.3 Regular slab – edge panel at column—Figure 5.2.3

shows an edge panel of a regular slab and the location of a design section at the support in a region with large bending moments and signifcant twist.Table 5.2.3 notes the normalized key design quantities and

results for the design section shown in Fig. 5.2.3 .While there is a signifcant twist peak near the support,

overall the integrated twist for this section is only 1 4 percent of the integrated moment. Due to support at one end of the design section, torsion on this section is large—149 percent of the integrated moment. This torsion is almost entirely caused by eccentric shear and not twisting. Note the following regarding design methods:a) Ignore twist: Ignoring twist results in the correct quan-

tity of reinforcement, although it is questionable as it does not guarantee a mechanism exists to resist the twist.b) Wood and Armer: This approach is applicable, although

it results in a 1 5 percent excess reinforcement penalty.c) Torsion-as-bending: Neither Condition 1 or 2 discussed

in 4.4.3 applies here, so torsion-as-bending is expected to apply. Treating torsion-as-bending results in a very conser-vative quantity of reinforcement, as torsion in this case is a very conservative proxy for integrated twist.d) Torsion-as-shear: Because the torsion is primarily

caused by eccentric shear, torsion-as-shear largely matches the actual slab behavior and provides a much more appro-priate shear design than the other approaches. However, although this approach results in the correct quantity of rein-forcement, it does not provide a mechanism for resisting the 1 0 percent of torsion caused by twist; therefore, torsion-as-shear is considered questionable.e) Sandwich model: This approach is applicable but

results in a 40 percent excess reinforcement penalty. This excess would be smaller in thicker slabs.

f) Twist-free analysis: This approach results in defection predictions that are 6 percent larger than a regular (isotropic) analysis. The increase is small, so there are no concerns of signifcant cracking required to achieve the twist-free load

DESIGN GUIDE FOR TWISTING MOMENTS IN SLABS (ACI 447R-1 8) 1 3


path. Twist-free analysis is the most efcient applicable method for this case.5.2.4 Regular slab – corner panel at column—Figure

5.2.4 shows a corner panel of a regular slab and the location of a critical design section at the support in a region with small bending moments and signifcant twist.

Table 5.2.4 notes the normalized key design quantities and results for the design section shown in Fig. 5.2.4.There is a signifcant twist peak near the support and

the bending moment is very small; overall, the integrated twist for this section is 1 57 percent of the integrated (small) moment. Due to the support at one end of the design section,

Fig. 5. 2. 2—Regular slab, interior panel.

Table 5.2.2—Analysis and design quantities for regular slab interior panel

Design

approach M T |twist| Def. Md Ab At Av Appl.

Ignore twist 1 .00 0.00 0.09 1 .00 0.92 0.00 0.92 No ?

Wood and Armer

1 .00 0.00 0.09 1 .00 1 .00 0.00 1 .00 No Cons.

Torsion as bending

1 .00 0.00 0.09 1 .00 0.92 0.00 0.92 No ?

Torsion as shear

1 .00 0.00 0.09 1 .00 0.92 0.00 0.92 No ?

Sandwich model

1 .00 0.00 0.09 1 .00 N/A 0.00 1 .21 No Ex. Cons

Twist-free analysis

1 .00 0.00 0.00 1 .04 0.92 0.00 0.92 No Yes



the torsion on the section is very large at 525 percent of the integrated moment. Although torsion is mostly caused by eccentric shear, the twisting component is signifcant. Due to the corner column being much less stif than the slab, the overall reinforcement demands at the corner column are

very small; minimum reinforcement provisions can govern in this region. Note the following regarding design methods:a) Ignore twist: Ignoring twist results in the correct quan-

tity of reinforcement, although it is questionable as it does not guarantee a mechanism exists to resist the twist.

Fig. 5. 2. 3—Regular slab, edge panel.

Table 5.2.3—Analysis and design quantities for regular slab, edge panel

Design

approach M T |twist| Def. Md Ab At Av Appl

Ignore twist 1 .00 1 .49 0.1 4 1 .00 0.87 0.00 0.87 No ?

Wood and Armer

1 .00 1 .49 0.1 4 1 .00 1 .00 0.00 1 .00 No Cons

Torsion as bending

1 .00 1 .49 0.1 4 1 .00 2.1 8 0.42 2.23 No Ex. Cons

Torsion as shear

1 .00 1 .49 0.1 4 1 .00 0.87 0.00 0.87 No ?

Sandwich model

1 .00 1 .49 0.1 4 1 .00 N/a 0.00 1 .22 No Ex. Cons

Twist-free analysis

1 .00 1 .43 0.00 1 .06 0.87 0.00 0.87 No Yes



b) Wood and Armer: This approach is applicable, although it results in a 220 percent excess reinforcement penalty.c) Torsion-as-bending: Neither Condition 1 or 2 discussed

in 4.4.3 applies here, so torsion-as-bending is expected to apply. Treating torsion as bending results in a very conser-

vative quantity of reinforcement, as torsion in this case is a very conservative proxy for integrated twist.d) Torsion-as-shear: Because the torsion is primarily

caused by eccentric shear, the torsion-as-shear approach largely matches the actual slab behavior and provides a much more appropriate shear design than the other approaches.

Fig. 5. 2. 4—Regular slab, corner panel.

Table 5.2.4—Analysis and design quantities for regular slab, corner panel

Design


Ignore twist 1 .00 5.25 1 .57 1 .00 0.39 0.00 0.39 No ?

Wood and Armer

1 .00 5.25 1 .57 1 .00 1 .0 0.22 1 .00 No Ex. Cons

Torsion as bending

1 .00 5.25 1 .57 1 .00 2.44 1 .67 2.47 No Ex. Cons

Torsion as shear

1 .00 5.25 1 .57 1 .00 0.39 0.00 0.39 No ?

Sandwich model

1 .00 5.25 1 .57 1 .00 N/a 0.34 1 .29 No Ex. Cons

Twist-free analysis

0.99 5.1 0 0.00 1 .1 0 0.38 0.00 0.38 No Yes



However, it does not guarantee a mechanism for resisting twist, which causes 30 percent of the torsion; therefore, torsion-as-shear is considered questionable.e) Sandwich model: This approach is applicable but

results in a 340 percent excess reinforcement penalty.f) Twist-free analysis: This approach results in defec-

tion predictions that are 1 0 percent larger than a regular (isotropic) analysis. This increase is small, so there are no concerns of signifcant cracking to achieve the twist-free load path. Twist-free analysis is the most efcient applicable method for this case.5.2.5 Slab with unaligned supports – interior panel at

column—Figure 5.2.5 shows an interior panel of an irregular slab with supports staggered by half the span length and the location of a critical design section at the support in a region with large bending moments and signifcant twist.Table 5.2.5 notes the normalized key design quantities and


overall the integrated twist for this section is only 11 percent of the integrated moment. Due to the symmetry of the slab confguration, torsion on the section is zero. Note the following regarding design methods:a) Ignore twist: Ignoring twist neither provides the correct

quantity of reinforcement or a mechanism to resist the twisting moments, so it is not applicable for this confguration.b) Wood and Armer: The Wood and Armer method is the

most efcient applicable method for this case.c) Torsion-as-bending: Condition 1 discussed in 4.4.3

applies here; the twisting moments change sign along the section, so torsion-as-bending is not expected to apply. Treating torsion as bending does not provide the correct amount of reinforcement, as torsion in this case is a poor proxy for integrated twist. Torsion-as-bending is not appli-cable for this confguration.d) Torsion-as-shear: Treating torsion as shear neither

provides the correct quantity of reinforcement nor a mecha-nism to resist the twisting moments, so it is not applicable for this confguration.e) Sandwich model: This approach is applicable but


f) Twist-free analysis: The twist-free analysis approach for this slab results in defection predictions that are 1 8 percent larger than a regular (isotropic) analysis. This increase is small, so there are no concerns of signifcant cracking to achieve the twist-free load path. However, twist-free analysis results in a 9 percent reinforcement penalty.5.2.6 Two-way cantilever slab—Figure 5.2.6 shows

a two-way cantilever and the location of a critical design section at the support in a region with very large bending moments and signifcant twist.Table 5.2.6 notes the normalized key design quantities and


overall, the integrated twist for this section is only 4 percent of the integrated moment. Due to the length of the canti-lever (creating a large moment), the torsion on the section is also small: 3 percent of the integrated moment. Note the following regarding design methods:a) Ignore twist: Ignoring twist results in the correct quan-

tity of reinforcement, although it is questionable as it does not guarantee a mechanism exists to resist the twist.b) Wood and Armer: This approach is also applicable,

although it results in a 4 percent excess reinforcement penalty.c) Torsion-as-bending: Condition 1 discussed in 4.4.3

applies here; the twisting moments change sign along the section, so torsion-as-bending is not expected to apply. Treating torsion as bending results in a safe quantity of rein-forcement, but the total torsion is a little less than integrated twist. Therefore, torsion-as-bending is questionable.d) Torsion-as-shear: Treating torsion as shear results in

the correct quantity of reinforcement, but eccentric shear only causes 25 percent of total torsion. Therefore, torsion-as-shear is questionable.e) Sandwich model: This approach is applicable but


f) Twist-free analysis: Unexpectedly, the twist-free anal-ysis for this slab results in a peak defection prediction that is 1 5 percent less than a regular (isotropic) analysis. One asks the question of how can this be; how can removing a slab stifness reduce the defection? The answer is that the average defection for the twist-free analysis is 1 2 percent

Table 5.2.5—Analysis and design quantities for slab with misaligned supports, interior panel

Design


Ignore twist 1 .00 0.00 0.11 1 .00 0.90 0.00 0.89 No No

Wood and Armer

1 .00 0.00 0.11 1 .00 1 .00 0.00 1 .00 No Yes

Torsion as bending

1 .00 0.00 0.11 1 .00 0.90 0.00 0.89 No No

Torsion as shear

1 .00 0.00 0.11 1 .00 0.90 0.00 0.89 No No

Sandwich model

1 .00 0.00 0.11 1 .00 N/a 0.00 1 .29 No Ex. Cons

Twist-free analysis

1 .21 0.00 0.00 1 .1 8 1 .09 0.00 1 .09 No Cons



Fig. 5. 2. 5—Slab with misaligned supports, interior panel.



more than for the regular (isotropic) analysis; however, the load path is diferent and there is less of a peak defection at the cantilever tip. Overall, there are no concerns of signif-cant cracking to achieve the twist-free load path. Twist-free analysis is the most efcient applicable method for this case.

5.2.7 Reentrant corner—Figure 5.2.7 shows a reentrant unsupported slab corner with the design section at the corner crossing a region of moderate bending moments and signif-cant twists.

Fig. 5. 2. 6—Two-way cantilever.

Table 5.2.6—Analysis and design quantities for two-way cantilever

Design


Ignore twist 1 .00 0.03 0.04 1 .00 0.96 0.00 0.96 No ?

Wood and Armer

1 .00 0.03 0.04 1 .00 1 .00 0.00 1 .00 No Yes

Torsion as bending

1 .00 0.03 0.04 1 .00 0.98 0.00 0.98 No ?

Torsion as shear

1 .00 0.03 0.04 1 .00 0.96 0.00 0.96 No ?

Sandwich model

1 .00 0.03 0.04 1 .00 N/a 0.00 1 .1 9 No Ex. Cons

Twist-free analysis

1 .00 0.03 0.00 0.85 0.96 0.00 0.96 No Yes



Table 5.2.7 notes the normalized key design quantities and results for the design section shown in Fig. 5.2.7.There is a signifcant twist peak near the notch and the

bending moment is moderate; overall, the integrated twist for

this section is 30 percent of the integrated moment. Torsion, 47 percent of the integrated moment, is caused primarily by twist, but also has a signifcant component due to eccentric shear. Note the following regarding design methods:

Fig. 5. 2. 7—Reentrant corner.

Table 5.2.7—Analysis and design quantities for reentrant corner

Design


Ignore twist 1 .00 0.47 0.30 1 .00 0.77 0.76 0.00 No No

Wood and Armer

1 .00 0.47 0.30 1 .00 1 .00 1 .00 0.00 No Yes

Torsion as bending

1 .00 0.47 0.30 1 .00 1 .1 3 1 .1 3 0.00 No Cons.

Torsion as shear

1 .00 0.47 0.30 1 .00 0.77 0.76 0.00 No No

Sandwich model

1 .00 0.47 0.30 1 .00 N/a 1 .23 0.00 No Ex. Cons

Twist-free analysis

0.87 0.09 0.00 1 .38 0.66 0.66 0.00 No Serv?



a) Ignore twist: Ignoring twist neither provides the correct quantity of reinforcement or a mechanism to resist the twisting moments, so it is considered not applicable for this confguration.b) Wood and Armer: The Wood and Armer method is the

most efcient applicable method for this case.c) Torsion-as-bending: Neither Condition 1 or 2 discussed

in 4.4.3 applies here, so torsion-as-bending is expected to apply. Treating torsion as bending is applicable for this case, although it requires 1 3 percent excess reinforcement. Torsion is a conservative proxy for twist.d) Torsion-as-shear: Treating torsion as shear neither

provides the correct quantity of reinforcement nor a mecha-nism to resist the twisting moments, so it is considered not applicable for this confguration. While eccentric shear causes signifcant torsion, twisting moments are still large.e) Sandwich model: This approach is applicable but


f) Twist-free analysis: The twist-free analysis approach for this slab requires the least reinforcement but results in defection predictions that are 38 percent larger than a regular (isotropic) analysis. This increase is somewhat large, so there are some concerns of noticeable cracking to achieve the twist-free load path. That makes twist-free anal-ysis questionable for this confguration. Adding diagonal reinforcement in the twist-critical reentrant corner might be adequate to ease the serviceability concerns while still requiring less reinforcement than the other methods. This, however, requires engineering judgment beyond the scope of this guide.5.2.8 Summary of typical design conditions—Table 5.2.8

displays a summary of the applicability of design methods for each condition discussed in 5.2. Note that no method is optimal for all cases, but that some conclusions can be drawn.Although the Wood and Armer method is always appli-

cable, it can be more conservative than required.The sandwich model is always applicable but is very

conservative for thin slabs such as the one investigated. For signifcantly thicker slabs, the sandwich model will have results similar to the Wood and Armer method. The strengths of the sandwich model are generally not relevant for typical slab design.Twist-free analysis, where applicable, is usually among

the most efcient methods. Twist-free analysis is applicable to most confgurations; if the twist-free defection is less

than 25 percent greater than the isotropic defection, twist-free analysis is generally applicable.The other methods, while appropriate in some circum-

stances, are either inappropriate or excessively conservative in too many cases to be used indiscriminately.While not investigated in detail in this chapter, applying

the torsion-as-shear design approach to a twist-free analysis is applicable whenever the twist-free analysis approach is applicable. This combination approach has the potential for improving vertical shear design while retaining the advan-tages of the twist-free analysis approach.

CHAPTER 6—TWO-WAY WALLS

6.1—General considerations

The discussion of twisting moments to this point has focused on two-way slabs supported by discrete columns. Twisting moments in two-way walls can also impact the out-of-plane design moments. Generally, walls are fundamen-tally diferent than slabs because: 1 ) out-of-plane fexure in the wall results from horizontal loads rather than vertical loads; and 2) boundary conditions are typically continuous rather than discrete. The former diference is largely irrel-evant to this discussion; however, the later diference can have an impact on the importance of twisting moment on the magnitude of design moments.Multiple combinations of boundary and loading condi-

tions acting on walls can result in twisting moments. Many infrastructure projects involve wall elements with contin-uous boundary conditions along two adjacent edges and out-of-plane loading along one face, resulting in a behavior similar to that shown in Fig. 3 .1 . This geometry is common in dam structures involving piers, as shown in Fig. 6.1 a.

Tanks often exhibit walls that are continuously supported on three sides, as shown in Fig. 6.1 b. This geometry can also create the conditions that result in high twisting moments occurring at locations of high bending moment.

6.2—Impact of twisting moment on walls

exhibiting two adjacent fxed edges

The pier shown in Fig. 6.1 a demonstrates a geometry and boundary conditions common in water control struc-tures such as dam spillways. In this section, the impacts of twisting moment will be explored for wall-like spillway piers. Three diferent aspect ratios, as shown in Fig. 6.2a, are considered to quantify the impact of geometry. Walls with a

Table 5.2.8—Summary of method applicability

Design approach

Regular slab

interior Regular slab edge

Regular slab

corner

Unaligned

supports

Two-way

cantilever Reentrant corner

Ignore twist ? ? ? No ? No

Wood and Armer Cons. Cons Ex. Cons Yes Yes Yes

Torsion as bending ? Ex. Cons Ex. Cons No ? Cons.

Torsion as shear ? ? ? No ? No

Sandwich model Ex. Cons Ex. Cons Ex. Cons Ex. Cons Ex. Cons Ex. Cons

Twist-free analysis Yes Yes Yes Cons Yes Serv?



ratio close to 1 :1 generally exhibit two-way behavior. As the aspect ratio increases, the wall starts behaving more like a one-way element near the free edge.The walls in this analysis are subject to a uniform lateral

load. To provide simpler behavior that can lead to better understanding, Poisson’s ratio has been set to 0 to elimi-nate anticlastic bending. Results are produced similar to

the results of the examples in Chapter 5. However, only the following design approaches are considered: 1 ) ignore twist; 2) Wood and Armer; and 3) twist-free. For Chapter 5, the assumed design strip is the width of a slab cross section and for Chapter 6, the point values are used to produce the design moment. Results of these analyses for the three diferent aspect ratios are shown in Fig. 6.2b. Location 1 exhibits the largest twist and Location 2 exhibits the largest bending moment. Defection and required steel areas are determined using the forces obtained in these analyses. The contours shown in Fig. 6.2b are similar to the assumptions outlined in Fig. 5.2.1 , with the exception that 0 is centered on the scale allowing for negative values in this chapter.The forces at Locations 1 and 2 changed signifcantly

between the 1 :1 and 2:1 aspect ratios for the twist analysis. However, they change little between the 2:1 and 3 :1 aspect ratios.The defections and required area of steel for the design

moments are presented in Table 6.2 for the three walls and three design approaches. Whereas the largest twist occurs at Location 1 for each aspect ratio, the controlling design moment is still largely controlled by moment at the fxed

Fig. 6. 1a—Dam pier and resulting defected shape of the twist-sensitive element.

Fig. 6. 1b—Rectangular tank and resulting defected shape of the twist-sensitive element.

Fig. 6. 2a—Twist-sensitive wall examples.



boundary condition at Location 2. Therefore, the values that populate Table 6.2 are at Location 2.

The following is a summary of each of the design approaches considering all three aspect ratios:

(1 ) Ignore twist: For the examples of this study, ignoring twist was only unconservative by approximately 5 percent. However, other wall geometries and boundary conditions can create locations of higher twist and moment acting at the same location. Therefore, accounting for twist in the element stifness formulation and then ignoring the results is not recommended.

(2) Wood and Armer : This approach was applicable for all aspect ratios. However, the resulting design for the ignore twist method starts to approach the Wood and Armer method for larger aspect ratios. Twisting stifness enables a more efcient load distribution because the twist is largely concen-trated at locations of lower moment for uniformly reinforced walls. Therefore, this approach is generally recommended as a safe and efcient design method.

(3) Twist-free analysis : This approach is excessively conservative until you approach the 3 :1 ratio, where values of twist are small and the moment increases less where

Fig. 6. 2b—Twist-sensitive wall study showing twisting moment trends.



twisting stifness does not contribute to redistribution of the force. Although it could be used for large aspect ratios, it is recommended to avoid this method. The large defection multiple for the 1 :1 ratio wall and, to a lesser extent, the 3 :1 wall, is indicative of a large alteration of the load path. This alteration in load path can lead to service cracking.For the 1 :1 aspect ratio, behavior observed is close to a

pure twist (Fig. 3 .1 ) in which the fow of force is most ef-cient at a 45-degree angle to the edges and assumed rein-forcement direction. By eliminating twisting stifness, the fow of force is confned to the far more inefcient load path, which is orthogonal to the edges and, basically, creating two cantilevers. For larger aspect ratios, the most efcient fow of force is naturally more closely aligned to the edges and assumed reinforcement direction. In these situations, the ignore twist analysis starts to produce designs similar to the Wood and Armer design method.In conclusion, the Wood and Armer design approach is

generally recommended for all wall analysis with two adja-cent fxed boundary conditions and orthogonally placed reinforcement. Twist can be ignored for larger aspect ratios, but this is generally not recommended.

CHAPTER 7—SHELL STRUCTURES

7.1—General considerations

While shells, which are curved thin structures, behave very diferently than fat slabs, they do exhibit twisting moments similar to fat slabs. Shell behavior is typically dominated by membrane actions, although fexural behavior (including twisting moments) can be signifcant at boundary conditions, near loading points, and at openings and other discontinuities. These twisting moments should be consid-ered in design, as required in Section 6.1 .1 of ACI 31 8.2-1 4. This chapter investigates twisting moment considerations in hemispherical domes, which are commonly shell-shaped.

The conclusions may or may not be relevant for other shell shapes, in which twisting may be more or less important in the load paths.

7.2—Typical bulk material storage hemisphere

Hemispherical shells are commonly used as an efcient structural form for bulk granular material storage. This chapter considers the efects of twisting moments in the design of a typical bulk storage hemisphere. The hemisphere studied has a thickness to radius ratio of 0.01 .

7.2.1 Finite element modeling—The fnite element model of the hemisphere that was studied uses quadrilateral shell elements (Fig. 7.2.1 a). Isotropic elements are used to evaluate the ignore twist, and the Wood and Armer design approaches. Orthotropic elements are used to evaluate the twist-free design approach. The analyses performed are all linear-elastic.One signifcant diference between shells and slabs is the

varied orientation of the shell local axes from location-to-location; therefore, it is not parallel to any global axes. To account for this, a clear axes convention needs to be estab-lished. For the hemispheric shell case investigated, the local axes are defned in Fig. 7.2.1 b. Except in the apex region, reinforcement is placed parallel to the local axes shown in the fgure.The base of the hemisphere is constrained against vertical

translation (local x-axis direction) and against translation tangential to the radius from the center vertical axis of the hemisphere (local y-axis direction). The base is allowed to translate radially (local z-axis direction). Figure 7.2.1 c illus-trates constraints at the base of the dome near the opening. The fgure shows the undeformed and deformed shape of the model. Arrows at the base of the hemisphere depict the constraint against translation used in the model. The cylin-drical coordinate system is provided to clarify the boundary conditions. The model is constrained against translation in

Table 6.2—Analysis and design quantities for various aspect ratios

1:1 ratio design approach M Twist Def. Md As Applicability

Ignore twist 1 .0 0.08 1 .00 0.92 0.92 No

Wood and Armer 1 .0 0.08 1 .00 1 .00 1 .00 Yes

Twist-free analysis 1 .6 0.00 2.26 1 .44 1 .43 Ex. Cons


Ignore twist 1 .0 0.02 1 .00 0.98 0.98 No


Twist-free analysis 1 .4 0.00 1 .52 1 .36 1 .41 Ex. Cons


Ignore twist 1 .0 0.005 1 .00 1 .00 0.995 Yes


Twist-free analysis 1 .1 0.000 1 .1 2 1 .09 1 .087 Cons



the θ- and Z-directions, as shown in the fgure. The fgure illustrates how the model is only allowed to translate radially. While the region of the opening is shown, the constraints are typical all around the base. The ring foundation continues under the opening to provide the constraint in the θ direction.

7.3—Typical loading conditions

Three common loadings are considered for the hemi-sphere: 1 ) self-weight; 2) pseudo-hydrostatic; and 3) a point load at the apex.

7.3.1 Self-weight loading—The self-weight of the hemi-sphere due to gravity is a signifcant load for this type of structure. The efects of this loading are considered in 7.4.1 and 7.4.2.

7.3.2 Pseudo-hydrostatic loading—Bulk granular mate-rials stored in the hemisphere apply a load to the shell similar to a hydrostatic load. This loading is diferent from a hydro-static pressure in that it acts horizontally and not perpendicu-larly to the shell. Figure 7.3 .2 illustrates the load considered. The efects of this loading are considered in 7.4.1 and 7.4.2.

7.3.3 Apex point loading—Point loading at the apex of the hemisphere is typical for bulk material storage facilities. This load comes from the equipment to fll the hemisphere. The efects of this loading are considered in 7.4.3 .

7.4—Typical design regions

Three regions of the hemisphere are considered: 1 ) the apex; 2) an opening at the base; and 3) a typical region at the base. Figure 7.4 identifes the three regions.For each region the ignore-twist, Wood and Armer, and

twist-free analysis approaches are evaluated. All results are normalized by the Wood and Armer results; bending and twisting analysis results are normalized together so the rela-tive magnitude of twist and bending is apparent. The reported defections are the maximum defection in any direction. Note that scales are intentionally omitted from the plots. The importance of the fgure is to show qualitative behaviors. Refer to Fig. 5.2.1 for scale information.

7.4.1 Typical base region—At the typical base section away from the opening, the results are observed to be similar to those of a pure hemisphere with no discontinuities. As the loadings are all radially symmetric, no signifcant twist is observed. Figure 7.4.1 shows this region subjected to the self-weight and hydrostatic loadings; the contours shown in Fig. 7.4.1 (b) are all nearly zero and are primarily a result of

numerical rounding in the FEA analysis. The section cut of the area analyzed is shown in the fgure. The length of the analyzed strip is approximately 20 percent of the radius of curvature of the shell. The analyzed strip location was chosen at the location of highest Mx moment. This is diferent than previous chapters because the twisting moment is very small.The calculation of the values in Table 7.4.1 is calculated

the same way as in previous chapters. Note the following regarding design methods:

Fig. 7. 2. 1a—Finite element model (geometry and mesh).

Fig. 7. 2. 1b—Local shell axes.

Fig. 7. 2. 1c—Hemisphere base boundary conditions and

cylindrical coordinate system.



a) Ignore twist: Ignoring twist results in the correct quan-tity of reinforcement, although it is questionable as it does not guarantee a mechanism exists to resist the twist.b) Wood and Armer: This approach is applicable,

although it results in a 4 percent excess reinforcement penalty.c) Twist-free analysis: This approach results in defec-

tions that are similar to a regular (isotropic) analysis. There are no concerns of signifcant cracking required to achieve the twist-free load path. Twist-free analysis is the most ef-cient applicable method for this case.

7.4.2 Opening base region subject to self-weight

loading—Figure 7.4.2 shows this region subjected to the self-weight loading. The section cut of the area analyzed is shown in the fgure. The length of the analyzed strip is approximately 5 percent of the radius of curvature of the shell. The analyzed strip location was chosen at the loca-tion of highest twisting moment.

The area around the opening of the curved shell shows a large amount of twisting moment. This is illustrated in Table 7.4.2.a) Ignore twist: This approach is not applicable in this

area, as the twisting moment is signifcant and the isotropic load path requires twisting moments. The ignore twist design moment is only 44 percent of the most efcient applicable design method and, hence, could lead to unsafe designs.b) Wood and Armer: This method is excessively conser-

vative because it results in a 47 percent excess reinforcement penalty.c) Twist-free analysis: This approach results in defec-

tion predictions that are 1 5 percent larger than a regular (isotropic) analysis. This increase is relatively small, so there are no concerns of signifcant cracking required to achieve the twist-free load path. Twist-free analysis is the most ef-cient applicable method for this case.

7.4.3 Opening base region subject to pseudo-hydro-

static loading—Figure 7.4.3 shows this region subjected to pseudo-hydrostatic loading. The section cut of the area analyzed is shown in the fgure. The length of the analyzed strip is approximately 5 percent of the radius of curvature of the shell. The analyzed strip location was chosen at the loca-tion of highest twisting moment.The area around the opening of the curved shell shows a

noticeable amount of twisting moment. This is illustrated in Table 7.4.3 .a) Ignore twist: This approach is not applicable in these

areas, as the twisting moment is signifcant and the isotropic load path requires twisting moments. The ignore twist design moment is only 58 percent of the most efcient applicable design moment and, therefore, could lead to unsafe designs.b) Wood and Armer: This method is excessively conser-

vative because it results in a 75 percent excess reinforcement penalty.

Fig. 7. 3. 2—Pseudo-hydrostatic loading.

Fig. 7. 4—Design regions.

Table 7.4.1 —Comparisons for typical base section

Design

approach M |twist| Md Defection Applicable

Ignore twist 1 .00 0.03 0.97 1 .00 ?

Wood and Armer

1 .00 0.03 1 .00 1 .00 Cons

Twist-free analysis

0.99 0.00 0.96 1 .00 Yes



c) Twist-free analysis: This approach results in defec-tion predictions that are 11 percent larger than a regular (isotropic) analysis. This increase is relatively small, so there are no concerns of signifcant cracking required to achieve the twist-free load path. Twist-free analysis is the most ef-cient applicable method for this case.

7.4.4 Apex region—Due to the radial symmetry of the loadings and the structure in the region of the apex (that is, the base opening is far away), the behavior in this region is largely radially symmetric. Therefore, there are no signif-cant twisting moments about the meridional and hoop axes.

Fig. 7. 4. 1—Typical section of hemisphere at base.

Fig. 7. 4. 2—Base at opening subject to self-weight.

Table 7.4.2—Comparisons for base at opening

subject to self-weight

Design


Ignore twist 1 .00 2.33 0.30 1 .000 No

Wood and Armer

1 .00 2.33 1 .000 1 .000 Ex. cons

Twist-free analysis

2.26 0.000 0.68 1 .1 5 Yes

Table 7.4.3—Comparisons for base at opening

subject to pseudo-hydrostatic loading

Design


Ignore twist 1 .00 2.02 0.33 1 .00 No

Wood and Armer

1 .00 2.02 1 .000 1 .00 Ex. cons

Twist-free analysis

1 .72 0.000 0.57 1 .11 Yes



The reinforcement at the apex, however, cannot be placed in a radially symmetric pattern due to constructability constraints. Reinforcement in this region is placed along orthogonal axes that are approximately in a horizontal plane. When the moments about the radial axes are transformed to the reinforcement axes, signifcant twisting moments arise.Figure 7.4.4 shows the apex region. Only the apex point

loading is considered, as that is the most signifcant loading

in this region. Note that moments, normal and twisting, are theoretically infnite under a point load.The section cut of the area analyzed is shown in the

fgure. The length of the analyzed strip is approximately 1 0 percent of the radius of curvature of the shell. The analyzed strip is slightly of center of the apex. This is so that the twisting moment is not exactly zero along the length of the analysis strip.Table 7.4.4 notes the normalized key design quantities and

results for the apex region shown in Fig. 7.4.4.a) Ignore twist: This approach is not applicable in these

areas, as the twisting moment is signifcant and the isotropic load path requires twisting moments. The ignore twist design moment is only 94 percent of the most efcient applicable design moment and, therefore, could lead to unsafe designs.b) Wood and Armer: This approach is applicable,

although it results in a 1 9 percent excess reinforcement penalty.c) Twist-free analysis: This approach results in defec-

tion predictions that are 3 percent larger than a regular (isotropic) analysis. This increase is very small, so there are no concerns of signifcant cracking required to achieve the twist-free load path. Twist-free analysis is the most efcient applicable method for this case.

Fig. 7. 4. 3—Base at opening subject to pseudo-hydrostatic loading.

Fig. 7. 4. 4—Apex of hemisphere under point loading.

Table 7.4.4—Analysis and design comparisons for

apex loading

Design


Ignore twist 1 .000 0..27 0..79 1 .000 ?

Wood and Armer

1 .000 0.27 1 .00 1 .000 Cons

Twist-free analysis

1 .07 0.00 0.84 1 .03 Yes



7.4.5 Summary—A hemispherical shell is modeled with several diferent loading scenarios. The response of the shell under a point load at the apex, near an opening with a pseudo-hydrostatic force, and in a typical section with pseudo-hydrostatic forces is investigated. These loads are additional to the load caused by the self-weight of the struc-ture. Generally, the twisting moments in this shell are more infuenced by geometric discontinuities than by loading discontinuities. The following is a written summary of the behavior of the shell for twisting moments, which is also summarized in Table 7.4.5.Ignore twist: The analyses found that the ignore twist

method was not a valid methodology for all the investi-gated loading scenarios. The stresses in the shell away from openings do not exhibit large twisting moments because the membrane action dominates the behavior of the shell in these areas. However, it does not guarantee a mechanism exists to resist the twisting moments.Wood and Armer: The Wood and Armer method is an

appropriate analysis method for shells in all locations. However, it also results in excess reinforcement in areas of geometric or loading discontinuities.Twist-free analysis: The twist-free analysis is an appro-

priate analysis method. The defections are very similar to the defections produced when considering twist. This small diference in defections create no concern of signifcant cracking required to achieve the twist-free load path. Difer-ences between the shells and slabs are due to the membrane capabilities of the shells. The twist-free analysis results in diferent membrane forces in the shell that need to be consid-ered in design.

CHAPTER 8—REFERENCES

Committee documents are listed frst by document number and year of publication followed by authored documents listed alphabetically.

ACI 31 8-1 4—Building Code Requirements for Structural Concrete and CommentaryACI 31 8.2-1 4—Building Code Requirements for Concrete

Thin Shells and Commentary

CSA Group

CSA-A23.3-04—Design of Concrete Structures

European Committee for Standardization

EN 1 992-1 -1 :2004—Eurocode 2: Design of Concrete Structures, General Rules and Rules for Buildings

Authored documents

Armer, G. S. T. , 1 968, “The Reinforcement of Slabs in Accordance with a Pre-Determined Field of Moments,” by R. H. Wood, discussion of reference 1 2.1 5, Concrete (London) , V. 2, No. 8, Aug., pp. 31 9-320.Brøndum-Nielsen, T. , 1 985, “Optimization of Rein-

forcement in Shells, Folded Plates, Walls, and Slabs,” ACI Journal Proceedings , V. 82, No. 3 , May-June, pp. 304-309.Burgoyne, C., 2004, “Are Structures Being Repaired

Unnecessarily?” The Structural Engineer, V. 82, No. 1 , Jan., pp. 22-26.Comité Euro-International du Béton fb, 1 993, “CEB-FIP

Model Code 1 990,” Lausanne, Switzerland, 460 pp.Comité Euro-International du Béton fb, 2008, “Practi-

tioners’ Guide to Finite Element Modeling of Reinforced Concrete Structures,” Bulletin 45, Lausanne, Switzerland, 344 pp.Deaton, J. B. , 2005, “A Finite Element Approach to Rein-

forced Concrete Slab Design,” MS thesis, Georgia Institute of Technology, Atlanta, GA, 1 70 pp.Den Hartog, J. P. , 2014, Advanced Strength of Materials ,

Courier Corporation, Dover Publications Inc., New York, 379 pp.Kennedy, G., and Goodchild, C., 2003, Practical Yield

Line Design , British Cement Association, 1 71 pp.Lourenço, P. B. , and Figueiras, J. A., 1 995, “Solution

for the Design of Reinforced Concrete Plates and Shells,” Journal of Structural Engineering , V. 1 21 , No. 5, May, pp. 81 5-823. doi: 1 0.1 061 /(ASCE)0733-9445(1995)121 :5(81 5)Marti, P. , 1 990, “Design of Concrete Slabs for Transverse

Shear,” ACI Structural Journal, V. 87, No. 2, Mar.-Apr., pp. 1 80-1 90.May, I. M., and Lodi, S. H., 2005, “Defciencies of the

Normal Moment Yield Criterion for RC Slabs,” Proceedings of the Institution of Civil Engineers—Structures and Build-

ings, V. 1 58, No. 6, Dec., pp. 371 -380.Park, R. , and Gamble, W. L. , 2000, Reinforced Concrete

Slabs , second edition, John Wiley and Sons, New York, 71 6 pp.Shin, M.; Bommer, A.; Deaton, J. B.; and Alemdar, B. N.,

2009, “Twisting Moments in Two-Way Slabs,” Concrete International , V. 31 , No. 7, July, pp. 35-40.Timoshenko, S. , and Woinowski-Krieger, S. , 1 959, Theory

of Plates and Shells , second edition, McGraw-Hill Book Co.Wood, R. H., 1 968, “The Reinforcement of Slabs in Accor-

dance with a Pre-Determined Field of Moments,” Concrete (London) , V. 2, No. 2, pp. 69-76.

Table 7.4.5—Summary of method applicability

Design approach Typical Opening with in-plane loading Opening with out-of-plane loading Point load at apex

Ignore twist ? No No ?

Wood and Armer Cons Ex. cons Ex. cons Cons

Twist-free analysis Yes Yes Yes Yes



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