IN-LB Inch-Pound Units

SI International System of Units

Report on Deflection
of Nonprestressed
Concrete Structures
Reported by ACI Committee 435
ACI 435R-20
 First Printing
November 2020
ISBN: 978-1-64195-128-9

Report on Deflection of Nonprestressed Concrete Structures

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 ACI 435R-20

Report on Deflection of Nonprestressed Concrete

Structures
Reported by ACI Committee 435
Eric S. Musselman, Chair Dylan Freytag, Secretary

Peter H. Bischoff Mamdouh M. El-Badry Adam S. Lubell Debrethann R. Orsak

Allan P. Bommer Amin Ghali Faris A. Malhas Maria A. Polak
Flora A. Calabrese Mayrai Gindy Michael C. Mota Mahmoud M. Reda Taha
Eamonn F. Connolly Shawn P. Gross Hani H. Nassif Andrew Scanlon
Norbert J. Delatte Young Hak Lee Edward G. Nawy Richard H. Scott

Consulting Members
Alex Aswad Satyendra Ghosh Bernard L. Meyers Himat T. Solanki
Finley A. Charney Peter Lenkei Vilas S. Mujumdar Susanto Teng

This report presents a consolidated treatment of initial and CHAPTER 2—NOTATION AND DEFINITIONS, p. 3
time-dependent deflection of nonprestressed reinforced concrete 2.1—Notation, p. 3
members such as simple and continuous beams and one-way and 2.2—Definitions, p. 3
two-way slab systems. It presents the current state of practice of
deflection prediction as well as analytical methods for computer
CHAPTER 3—MATERIAL PROPERTIES, p. 3
use in deflection estimation. Topics include material properties,
3.1—Objective, p. 3
deflection of reinforced concrete one-way flexural members, deflec-
tion of two-way slab systems, and reducing deflection of concrete 3.2—Material properties affecting deflection, p. 4
members. 3.3—Concrete material properties, p. 4
3.4—Reinforcement material properties, p. 9
Keywords: camber; cracking; creep; curvature; deflection; modulus of
rupture; moments of inertia; serviceability; shrinkage; time-dependent CHAPTER 4—DEFLECTION OF REINFORCED
deflection.
CONCRETE ONE-WAY FLEXURAL MEMBERS, p. 9
4.1—General, p. 9
CONTENTS 4.2—Control of deflection, p. 10
4.3—Short-term deflection calculation, p. 11
CHAPTER 1—INTRODUCTION AND SCOPE, p. 2 4.4—Long-term deflection calculation, p. 18
1.1—Introduction, p. 2 4.5—Temperature-induced deflections, p. 21
1.2—Scope, p. 2
CHAPTER 5—DEFLECTION OF A TWO-WAY
SLAB SYSTEM, p. 22
5.1—Introduction, p. 22
ACI Committee Reports, Guides, and Commentaries are 5.2—Deflection calculation methods for two-way slab
intended for guidance in planning, designing, executing, and systems, p. 23
inspecting construction. This document is intended for the use 5.3—Minimum thickness requirements, p. 26
of individuals who are competent to evaluate the significance 5.4—Loads for deflection calculation, p. 28
and limitations of its content and recommendations and who 5.5—Variability of deflections, p. 31
will accept responsibility for the application of the material it
contains. The American Concrete Institute disclaims any and 5.6—Allowable deflections, p. 32
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 ACI 435R-20 supersedes ACI 435R-95(03) and became effective November 2020.
Copyright © 2020, American Concrete Institute.
documents. If items found in this document are desired by
All rights reserved including rights of reproduction and use in any form or by any
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they shall be restated in mandatory language for incorporation chanical device, printed, written, or oral, or recording for sound or visual reproduction
by the Architect/Engineer. or for use in any knowledge or retrieval system or device, unless permission in writing
is obtained from the copyright proprietors.

1
2 REPORT ON DEFLECTION OF NONPRESTRESSED CONCRETE STRUCTURES (ACI 435R-20)

CHAPTER 6—REDUCING DEFLECTION OF of prestressed concrete are not addressed in this document,
CONCRETE MEMBERS, p. 32 although prestressing can be an effective tool for controlling
6.1—Introduction, p. 32 both short- and long-term deflections.
6.2—Design techniques, p. 32
6.3—Construction techniques, p. 34 1.2—Scope
6.4—Materials selection, p. 35 The principal causes of deflections taken into account
6.5—Summary, p. 36 in this report are those due to elastic deformation, flexural
cracking, creep, shrinkage, and temperature effects. This
CHAPTER 7—REFERENCES, p. 36 document is composed of two introductory chapters and four
Authored documents, p. 36 main chapters that provide information on calculating and
controlling deflections of members constructed using rein-
APPENDIX A—DEFLECTION DESIGN EXAMPLES, forced concrete. The organization of the report is:
p. 39 a) Chapter 1—Introduction and Scope provides back-
Example A.1—Deflection of a simply supported slab, p. 39 ground information on the document.
Example A.2—Age-adjusted deflection of simply b) Chapter 2—Notation and Definitions provides a
supported slab, p. 43 listing of the notation used throughout the document.
Example A.3—Short- and long-term deflection of a four- c) Chapter 3—Material Properties discusses material
span continuous beam, p. 44 properties that affect deflections.
Example A.4—Temperature-induced deflections, p. 48 d) Chapter 4—Deflection of Reinforced Concrete
One-Way Flexural Members discusses behavior of
APPENDIX B—TWO-WAY SLAB DEFLECTION uncracked and cracked members, and time-dependent
EXAMPLES, p. 48 effects. It also includes the relevant code procedures
Example B.1—Deflection design example for long-term and expressions for deflection computation in reinforced
deflection of a two-way slab, p. 48 concrete beams. Numerical examples are included to
Example B.2—Deflection calculation for a flat plate using illustrate the standard calculation methods for simply
the crossing beam method, p. 52 supported and continuous concrete beams.
Example B.3—Minimum thickness calculation, p. 54 e) Chapter 5—Deflection of Two-Way Slab Systems
covers the deflection behavior of reinforced two-way-
CHAPTER 1—INTRODUCTION AND SCOPE action slabs and plates. This chapter gives an overview
of classical and other methods of deflection estimation,
1.1—Introduction such as the crossing beam analogy and the finite element
Design for serviceability is central to the work of structural method for immediate deflection computation. It also
engineers and code-writing bodies. It is also essential to users discusses approaches for determining the minimum thick-
of the designed structures. Increased use of high-strength ness requirements for two-way slabs and plates and gives
concrete and higher-strength reinforcing bars, coupled with a detailed computational example for evaluating the long-
more detailed computer-aided designs, has resulted in lighter term deflection of a two-way reinforced concrete slab.
and more material-efficient and, thus, more flexible structural The chapter emphasizes the uncertainties inherent in esti-
members and systems. This in turn has necessitated better mating deflections of two-way slab systems.
prediction and control of short-term and long-term behavior f) Chapter 6—Reducing Deflection of Concrete
of concrete structures at service loads. Members gives practical and remedial guidelines for
This report presents a consolidated treatment of initial improving and controlling the deflection of reinforced
and time-dependent deflection of nonprestressed reinforced concrete members, hence enhancing their overall long-
concrete members such as simple and continuous beams and term serviceability.
one- and two-way slab systems. It presents current engineering It should be emphasized that the magnitude of actual
practice in design for control of deformation and deflection of deflection in concrete structural members, particularly in
concrete members and includes methods presented in ACI 318 buildings, which are the emphasis and the intent of this
plus selected other approaches suitable for computer-based report, can only be estimated with limited accuracy. This
use in deflection computation. Design examples are given at is because of the large variability in the properties of the
the end of one- and two-way framing chapters showing how constituent materials of these members, the quality control
to evaluate deflection and, thus, control it through adequate exercised in their construction, and the construction methods
design for serviceability. The content of the report as well used. Therefore, for practical considerations, the computed
as the step-by-step examples are intended to familiarize deflection values in the illustrative examples at the end of
practitioners with the current methods for estimating deflec- each chapter should be interpreted with this in mind.
tions in buildings as well as analytical methods suitable for In summary, this single document gives design engineers
computer-based application. The examples apply ACI 318 the key tools for estimating, and thereby controlling through
requirements and a recommended alternative approach with design, the expected deflection in nonprestressed reinforced
a lower cracking moment (to account for shrinkage restraint). concrete building structures. The material presented and the
Methods for predicting initial and time-dependent deflections design examples will help to enhance serviceability when

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 REPORT ON DEFLECTION OF NONPRESTRESSED CONCRETE STRUCTURES (ACI 435R-20) 3

used judiciously by the engineer. Designers, constructors, t = time, s

and codifying bodies can draw on the material presented w = uniformly distributed load (load per unit length),
in this document to achieve serviceable deflection of lb/in. (N/mm)
constructed facilities. wc = unit weight of normalweight concrete or equilib-
rium density of lightweight concrete, lb/ft3 (kg/m3)
CHAPTER 2—NOTATION AND DEFINITIONS yt = distance from centroidal axis of gross section,
neglecting reinforcement, to extreme fiber in
2.1—Notation tension, in. (mm)
As = area of nonprestressed tension steel, in.2 (mm2) ∆ = elastic deflection of a beam or slab, in. (mm)
As′ = area of nonprestressed steel in compression zone, ∆cr = additional deflection due to creep, in. (mm)
in.2 (mm2) ∆inc = incremental deflection that occurs after attachment
b = width of the section, in. (mm) on nonstructural elements (includes long-term
bw = web width, in. (mm) deflection ∆LT from sustained loads and immediate
Ct = creep coefficient of concrete at time t, days deflection from the remaining part of live load that
Cu = ultimate creep coefficient of concrete is not sustained, in. (mm)
c = depth of centroidal axis, in. (mm) ∆L = initial (immediate) deflection due to live load, in.
d = distance from the extreme compression fiber to (mm)
centroid of tension reinforcement, in. (mm) ∆LT = deflection from long-term effects, in. (mm)
d′ = distance from the extreme compression fiber to ∆sh = additional deflection due to shrinkage, in. (mm)
centroid of compression reinforcement, in. (mm) ∆sus = initial (immediate) deflection due to sustained load,
db = bar diameter, in. (mm) in. (mm)
E = modulus of elasticity, psi (MPa) εcf = strain due to stress in the concrete
Ec = modulus of elasticity of concrete, psi (MPa) εo = free strain, such as unrestrained shrinkage
fc = stress in concrete, psi (MPa) εsf = strain due to stress in nonprestressed steel
fc′ = specified compressive strength of concrete, psi (MPa) εsh = shrinkage strain of concrete
fcr = stress to cause cracking in concrete, psi (MPa) (εsh)t = shrinkage strain of concrete at time t, days
fr = modulus of rupture of concrete, psi (MPa) (εsh)u = ultimate shrinkage strain of concrete
fres = stress from restraint to shrinkage, psi (MPa) εt = total strain
fs = stress in nonprestressed steel, psi (MPa) ζ = distribution coefficient
fy = specified yield strength of nonprestressed rein- κ = cross section curvature, in.–1 (mm–1)
forcing steel, psi (MPa) κ sh = shrinkage curvature, in.–1 (mm–1)
h = overall thickness of a member, in. (mm) λc = creep multiplier for long-term deflection
hf = flange thickness, in. (mm) λsh = shrinkage warping multiplier for long-term deflection
I = moment of inertia, in.4 (mm4) λt = total multiplier for long-term deflection
Icr = moment of inertia of the cracked section trans- λ∆ = time-dependent multiplier for long-term deflection
formed to concrete, in.4 (mm4) ν = Poisson’s ratio
Ie = effective moment of inertia for computation of ρ = nonprestressed tension reinforcement ratio (As/bd)
deflection, in.4 (mm4) ρ′ = reinforcement ratio for nonprestressed compres-
Ig = moment of inertia for gross concrete section about sion steel (As′/bd)
centroidal axis, neglecting reinforcement, in.4 (mm4) ξ = time-dependent multiplier for deflection
k = depth of compression zone divided by d
ℓ = span length, in. (mm) 2.2—Definitions
ℓn = distance from the inside of the support to the inside Please refer to the latest version of ACI Concrete Termi-
of support, clear span, in. (mm) nology for a comprehensive list of definitions.
M = bending moment, lb-in. (N-mm)
M1 = moment at End 1 of a continuous member, lb-in. CHAPTER 3—MATERIAL PROPERTIES
(N-mm)
M2 = moment at End 2 of a continuous member, lb-in. 3.1—Objective
(N-mm) Deflections in reinforced concrete structures are affected
Ma = maximum service load moment (unfactored) at significantly by numerous material properties, including
stage deflection is computed, lb-in. (N-mm) concrete and reinforcement moduli of elasticity, concrete
Mcr = cracking moment, lb-in. (N-mm) modulus of rupture, creep, and shrinkage. The purposes
MD = moment due to dead load, lb-in. (N-mm) of this chapter are to: 1) briefly address how each of these
ML = moment due to live load, lb-in. (N-mm) material properties affects deflection; and 2) provide brief
Mm = midspan moment, lb-in. (N-mm) guidance on the most common expressions recommended
N = axial member load, lb (N) by various ACI committees for estimation of these param-
n = modular ratio Es/Ec eters during the design process. This chapter is not intended
T = temperature, °F (°C) to provide a comprehensive review of all the material prop-

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