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INELASTIC RESPONSE OF REINFORCED
CONCRETE ELENIENTS TO DYNANIIC LOADS

By

Jongsung Sim

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Civil and Environmental Engineering

1987

ABSTRACT

INELALSTIC RESPONSE OF REINFOCED
CONCRETE ELENIENTS TO DYNAMC LOADS

By

Jongsung Sim

Strain rate effects and shear deformations are two factors with major effects
on the dynamic response characteristics of reinforced concrete (R/C) structures,
which are generally neglected in dynamic analysis. In this investigation, the
analytical techniques capable of accounting for these effects of these two factors
were developed for dynamic analysis of R/ C elements (and structures). These
techniques were used in numerical studies on the sensitivity of R/C elements
behavior to the variations in the rate of straining under monotonic loading and to

the effects of shear deformations under cyclic loadng.

The strain rate-sensitive element model was constructed based on the strain
rate-dependent constitutive models of steel and concrete which was empirically
developed in this study. A refined (layer) element modeling technique was used
for axial-flexural analysis of R/ C beam-columns at different rates of load applica-
tion. The element model was used in numerical studies to assess the strain rate
effects on the axial compressive monotonic load-deformation behavior of R/C

columns and flexural behavior of R/ C beams.

In order to account for the effects of shear deformations on the cyclic
behavior of R/ C beams, an element model which physically separates the flexural
and shear deformations was developed. This element model considers the
differences between the shear and flexural hysteretic characteristics of R/C beams
(marked by the highly deteriorating nature of shear resisting mechanisms under
cyclic loads). The tangent stiffness matrix of the element model was also
developed, and the experimentally-observed tendency of shear deformations to
dominate the element load-deformation behavior under repeated inelastic load

cycles was verified analytically.

To my Parents

ii

ACKNOWLED GEMENT

I would like to express appreciation to those who have helped and contri-
buted to making this work a reality. This includes my dissertation advisor, Dr.
Parviz Soroushian for his guidance and assistance; all of my guidance committee
members, Dr. W. A. Bradley, Dr. R. K. Wen, and Dr. W. E. Kuan for their
interest and help during preparation of this work; and all members of the Struc-
tures Research Laboratory for their assistance in the experimental portions of this

work.

The numerical results and most of the figures presented in this dissertation
were produced using the computational and plotting facilities of the Albert H.
Case Center for Computer Aided Design at Michigan State University. This
study was financially supported by General Electric Corperation, and MSU(All

University Research Initiation Grants). This support is gratefully acknowledged.

Finally, sincere thanks are extended to my parents and parents-in-law, and
my wife Yoonhi, my son Chungwook, my brothers, my brothers-in-law, and

sisters-in-law for their encouragement.

iii

TABLE OF CONTENTS

Chapter
LIST OF TABLES .................................................................................................................

LIST OF FIGURES ................................................................................................................
LIST OF NOTATIONS ..........................................................................................................
1 INTRODUCTION ...........................................................................................................

2 STRAIN RATE EFFECTS ON THE DYNAMIC RESPONSE OF REINFORCED
CONCRETE STRUCTURES: A REVIEW OF THE LITERATURE ..............................

2.1 INTRODUCTION .....................................................................................................
2.2 CONCRETE IN COMPRESSION .............................................................................
2.3 CONCRETE IN TENSION .......................................................................................
2.4 STEEL .......... - ............................................................................................................
2.5 BOND BETWEEN STEEL AND CONCRETE .........................................................
2.6 AXIAL-FLEXURAL BEHAVIOR OF TIE R/C ELEMENTS ...................................
2.7 SHEAR BEHAVIOR OF TI-E R / C ELEMENTS ......................................................

3 A STRAIN RATE-SENSITIVE CONSTITUTIVE MODEL FOR CONCRETE
‘ UNDER COMPRESSION ...............................................................................................

3.1 INTRODUCTION .....................................................................................................
3.2 COMPRESSIVE STRENGTH ...................................................................................
3.3 STRAIN AT MAXIMUM STRESS ............................................................................
3.4 MODULUS OF ELASITICITY .............
3.5 TIE CONSTITUTIVE MODEL FOR CONCRETE ..................................................
3.6 COMPARISON WITH TEST RESULTS AND PARAMETRIC STUDES ................
3.7 SUWARY AND CONCLUSIONS ............................................................................

4 A STRAIN RATE-SENSITIVE CONSTITUTIVE MODEL FOR STEEL
UNDER TENSION .........................................................................................................

4.1 INTRODUCTION .....................................................................................................
4.2 TEST RESULTS .......................................................................................................
4.3 FACTORS INFLUENCING TI-E STRAIN RATE EFFECTS ...................................
4.4 TIE CONSTITUTIVE MODEL FOR STEEL ...........................................................
4.5 SUMMARY AND CONCLUSIONS ............................................................................

5 A STRAIN RATE-SENSITIVE MODEL FOR AXIAL/FLEXURAL ANALYSIS

OF REINFORCED CONCRETE ELEMENTS UNDER DYNAMIC LOADS ...................
5.1 INTRODUCTION .....................................................................................................

iv

Page

vi

vii

4
4
7
12
12
16
16
17

20
20
21
26
27
29
31
36

37
37
37
38
45
47

50
50

5.2 SECTION MODEL .................................................................................................... 52
5.3 FACTORS INFLUENCING TIE AXIAL /FLEXURAL BEHAVIOR OF R/ C

SECTIONS ............................................................................................................... 54
5.3.1 Eflect of Longitudinal Steel Yield Strength ....................................................... 57
5.3.2 Effect of The Concrte Compressive Strength ..................................................... 57
5.3.3 Effect of Longitudinal Steel Ratio ..................................................................... 60
5.3.4 Effect of Confinement ....................................................................................... 60
5.3.5 Effect of Loading Rate ...................................................................................... 63
5.4 ELEMENT MODEL .................................................................................................. 63
5.5 COMPARISON WITH TEST RESULTS ................................................................... 67
5.6 PARAMETRIC STUDIES ON ELEMENT BEHAVIOR ............................................ 71
5.6.1 Axial Behavior of Columns ............................................................................... 71
5.6.2 Flexural Behavior of Beams .............................................................................. 74
5.7 PRACTICAL DESIGN PROCEDURES ..................................................................... 77
5.7.1 Dynamic Axial Strength .................................................................................... 77
5.7.2 Dynamic Flexural Strength ............................................................................... 78
5.8 SUMMARY AND CONCLUSIONS ............................................................................ 80
6 INELASTIC MODELING OF REINFORCED CONCRETE ELENENTS
UNDER CY CLIC F LEXURAL AND SIEAR FORCES .................................................. 83
6.1 INTRODUCTION ..................................................................................................... 83
6.2 BACKGROUND ........................................................................................................ 85
6.2.1 Test Results ...................................................................................................... 85
6.2.2 Analytical Models ............................................................................................. 86
6.3 TIE PROPOSED ELEMENT MODEL ..................................................................... 96
6.3.1 Physical Idealization ......................................................................................... 96
6.3.2 Hysteretic Rules of Flexure and Shear ............................................................... 97
6.3.3 Selection of Hysteretic Parameters .................................................................... 103
6.4 FORMULATION OF TIE ELEMENT TANGENT STIFFNESS MATRDC ............... 105
6.5 COMPARISON WITH TEST RESULTS ................................................................... 112
6.6 SUMMARY AND CONCLUSIONS ............................................................................ 126
7 SUMMARY AND CONCLUSIONS .................................................................................. 127
LIST OF REFERENCES ....................................................................................................... 133

LIST OF TABLES

Table Page
5.1 Summary of The Loading Rate Effects on The Axial Behavior of R/ C Columns ........... 73
5.2 Summary of The Loading Rate Effects on The Flexural Behavior of R/ C Beams .......... 76
6.1 Empirical Values of The Strain Hardening Ratio and Hysteretic Parameters

of The Flexural and Shear Springs ............................................................................... 105
6.2 Properties of R/ C Beams Used in Comparison Between The Tests and Theory ............. 112

LIST OF FIGURES

Figure Page
2.1 Strain Rate Effect on The R/C Element under Seismic Forces ....................................... 6
2.2 Strain Rate-Sensitivity of The Concrete Mechanical Properties ..................................... 8
2.3 Strain Rate-Sensitivity of The Concrete Tensile Strength .............................................. 11
2.4 Strain Rate-Sensitivity of The Steel Mechanical Properties ........................................... 13
2.5 Strain Rate-Sensitivity of The Steel Yield Strength ....................................................... 15
2.6 Slip Rate-Sensitivity of The Bond between Steel and Concrete ...................................... 16
2.7 Strain Rate-Sensitivity of The Axial-Flexural Behavior of The R/ C Elements ............... 18
2.8 Strain RateSensitivity of An R / C Beam Shear Hysteresis ............................................ 19
3.1 Strain Rate Effects on The Compressive Strength of Concrete ....................................... 23
3.2 Strain Rate Efl’ects on The Concrete Strain at The Maximum Compressive Stress ......... 25
3.3 Strain Rate Efl‘ects on The Concrete Modulus of Elasticity ........................................... 28
3.4 The Constitutive Model for Concrete under Compression .............................................. 30
3.5 Comparison of The Proposed Constitutive Model with Test Results ............................. 32
3.6 Theoretical and Experimental Constitutive Diagrams for Air-Dried

Unconfined Specimens ................................................................................................... 34
3.7 Strain Rate Efl‘ects on Air-Dried and Saturated Concrete .............................................. 34
3.8 Strain Rate Effects on Confined and Unconfined Concrete ............................................. 35

Strain Rate Effects on The Steel Mechanical Properties ................................................. 39
4.2 Strain Rate Effects on The Lower Yield Strength of Steels with Different

Yield Strengths ............................................................................................................. 42
4.3 Strain Rate Effects on The Upper Yield Strength of Steels with Different

Yield Strengths ............................................................................................................. 43
4.4 Strain Rate Effects on The Ultimate Strength of Steels with Different

Yield Strengths ............................................................................................................. 43
4.5 Strain Rate Efi'ets on Strain at Initiation of Strain Hardening of Steels

with Different Yield Strength ........................................................................................ 44
4.6 Srain Rate Effects on The Ultimate Strain of Steels with Different

Yield Strengths ............................................................................................................. . 44
4.7 Comparison of The Proposed Monotonic Constitutive Models of Steel with

Test Results .................................................................................................................. 48
5.1 Loading Rate Effects on The Axial and Flexural Behavior of R/ C Elements ................. 51
5.2 Layer Model for R / C Sections ....................................................................................... 53
5.3 Typical Comparison of The Analytical Predictions with Test Results ........................... 55
5.4 Standard Beam and Column Cross Sections .................................................................. 56
5.5 Effect of Longitudunal Steel Yield Strength ................................................................... 58
5.6 Effect ,of Concrete Compressive Strength ....................................................................... 59

vii

5.7
5.8
5.9
5.10
5.11

5.12

5.13
5.14
5.15

6.1

6.2

6.3
6.4

6.5
6.6
6.7
6.8

6.9

6.10
6.11
6.12
6.13
6.14
6.15

6.16
6.17

6.18

6.19
6.20
6.21
6.22

6.23

Effect of Longitudinal Steel Area ...................................................................................
Effect of Confinement ....................................................................................................
Effect of Loading Rate ..................................................................................................
Layer Model for R/ C Elements .....................................................................................

Experimental and Analytical Axial Performances of R/ C Columns at Different
Loading Rates ..............................................................................................................

Experimental and Analytical Flexural Performances of R/ C Beams at Different
Loading Rates ..............................................................................................................

Loading Rate Effects on The Axial Behavior of R/C Columns ......................................
Loading Rate Effects on The Flexural Behavior of R / C Beams .....................................
Simplified Flexural Strain and Stress Distributions at Failure .......................................

Lateral Displacement Components of R/ C Beams, and Their Hysteretic
Characteristics ..............................................................................................................

Contribution of Shear Deformations to The Overall Element Displacements under
Cyclic Loads .................................................................................................................

Typical Crack Patterns in Critical Regions of R/ C Elements ........................................

Efi’ect of The Shear Stress Level on The Overall Hysteretic Characteristics
of R/ C Beams ...............................................................................................................

Physical Models for Simulating The Flexural Behavior of R/ C Beams ..........................
Physical Models for Simulating The Shear Behavior of R/ C Beams ..............................
Examples of Hysteretic Models ......................................................................................

Comparison of Experimental and Analytical Cyclic Load-Deflection Relationships
for R/ C Beams .............................................................................................................

Proposed Physical Idealization of Element ....................................................................
Skeleton Curve and Unloading from The Skelton Curve ...............................................
Reloading towards The Skelton Curve ..........................................................................
The Bauschinger and Pinching Effect on The Reloading Path .......................................
Small-Amplitude Reloading towards The Skelton Curve ...............................................
Small—Amplitude Reloading in A Direction with Previous Incompete Cycle ..................

Different Hysteresis Diagrams Formed by Varing Their Parameters of
The Proposed Hysteretic Model .....................................................................................

Simplified Flexural Strain and Stress Distributions at Yielding .....................................

Cracked Transformed Cross Section for Calculating The Initial F lexural and
Shear Stiffnesses ............................................................................................................

Empirical Values of The Strain Hardening Ratio and Hysteretic Parameters
of The F lexural and Shear Springs ...............................................................................

Mid-Length Inflection Point under Seismic Forces .........................................................
Cantilever Element .......................................................................................................
Displacement Degrees of Freedom in A Genaral Beam Element .....................................

Experimental and Theoretical Cyclic Load-Deformation Realationships
of Beam No. 1 .............................................................................................................

Experimental and Theoretical Cyclic Load-Deformation Realationships
of Beam No. 2 .............................................................................................................

Experimental and Theoretical Cyclic Load-Deformation Realationships

viii

61
62
64
65

68

70
72
75
79

84

84
87

87
88
89
91

93
98
98
100
100
101
101

102
104

104

106
108
108
109

114 '

117

of Beam No. 3 ............................................................................................................. 120
6.25 Experimantal and Theoretical Total Cyclic Load-Displacement Relationships

of Beam No. 4 ............................................................................................................. 123
6.26 Experimental and Theoretical Total Cyclic Load-Displacement Relationships

of Beam No. 5 ............................................................................................................. 124
6.27 Experimental and Theoretical Total Cyclic Load-Displacement Relationships

of Beam No. 6 ............................................................................................................. 125

in:

LIST OF NOTATIONS

IA, - gross area of the concrete cross section
A, =- area of tension steel
A, - area of compression steel
A“ - total area of longitudinal steel
a~ - depth of equivalent rectangular block representing compressive
resistance of concrete in beams
b a width of cross section
c - depth of neutral axis of cross section
d - effective depth of cross section
(1’ =- distance from centroid of compression steel
to the extreme compression fiber
dM = incremental bending moment of cross section
dM,- =- incremental bending moment at end i
m.

J
dM, an incremental bending moment of element

=- incremental bending moment at end j

dP = incremental axial load at cross section
dPe an incremental axial load on element
dV = incremental lateral load
(IV, = incremental lateral load of element
dt = incremental loading time
d A, a incremental free end lateral displacement of element
d6 = incremental displacement at free end of the cantilever beam
d 5, = shear spring incremental deformation
d e = strain increment
d6,- = strain increment at the i’th layer
d is,- = strain rate increment at the i’th layer
d 6? = strain increment at plastic centroid
d7, = incremental axial deformation

(1 45 =- incremental curvature

d9, = fiexural spring incremental deformation

(1 9, =- shear spring incremental deformation

E,, - dynamic secant modulus of elasticity

E,, - static secant modulus of elasticity

E, = static strain hardening modulus of steel
E): =- dynamic strain hardening modulus of steel
E, = tangent stiffness modulus of the i’th layer
E, - modulus of elasticity of steel

EM - dynamic tangent modulus of elasticity

E", - static tangent modulus of elasticity
F, =- yield force of the proposed hysteretic model
f , - concrete stress

I, - standard (quasi-static) compressive strength of concrete
I I

fc

[,4 - dynamic compressive strength of concrete

- dynamic compressive strength of concrete

fc, :- static compressive strength of concrete
I, - static upper yield strength of steel
1'; -- dynamic upper yield strength of steel
1', :1: steel stress
f,, a: compression steel stress
I, = static ultimate strength of steel
[,1 = dynamic ultimate strength of steel
f, = standard (quasi-static) yield strength of steel
f; = dynamic yield strength of steel
[1”, = yield strength of transverse reinforcement
h' = width of concrete core measured from outside of transverse reinforcement
K11, K12, K 21, and K 22 = components of the section tangent stiffness matrix
K I = tangent stiffness of fiexural spring
K15 = tangent stiffness of fiexural spring at end i
K f,- = tangent stiffness of flexural spring at end j
KG = element tangent stiffness matrix
K, = initial stiffness of the proposed model
Kh = strain hardening stiffness of the proposed model
K, = tangent stiffness of shear spring
K,,- = tangent stiffness of shear spring at end i

K,,- = tangent stiffness of shear spring at end j

K, a tangent stiffness of cantilever element
K, :- unloading stifiness of the proposed model
7 - quasi-static secant axial stiffness
= dynamic secant axial stiffness
L - total element length
l a cantilever element length
M, = static flexural strength
M,’ =- dynamic fiexural strength
P an axial force on element
P, - nominal axial strength
P,’ = dynamic axial strength
8 - center to center spacing of transverse reinforcement
V =- lateral force of cantilever element
V, =1 quasi-static lateral strength
,I - dynamic lateral strength
y,- 8 distance from centroid of the i’th layer to the plastic centroid
R = a point on the skeleton curve in the element model
R/ C a reinforced concrete
X - spring deformation in the element model
X, - yield deformation
A =- lateral displacement of element
a - a coefiicient used in the element model
3 =- a coefficient used in the element model
6 = strain
6 = strain rate (1 /sec) 2 10'5
,- = strain rate in the i’th layer
6 = strain rate at plastic centroid
c, = static strain at initiation of strain hardening
6;, = dynamic strain at initiation of strain hardening
6,, = dynamic strain at maximum stress
6,, =-= static strain at maximum stress
6, == steel strain
6, = quasi-static ultimate strain of steel
,1 = dynamic ultimate strain of steel
’7 = a coefficient used in element model

43 = section curvature

(i3 = section curvature rate
1b = element orientation angle
p = tensile steel ratio
p’ =3 compressive steel ratio
p, =- volumetric ratio of transverse reinforcement to concrete core
0 a rotation of element
0, a rotation of element at end i
9,- - rotation of element at end j
g' =- a coefficient used in the element model

2: iii

CHAPTER 1

INTRODUCTION

Experimental data have indicated that the rate of loading and the shear
deformations are two major variables influencing the inelastic response of rein-
forced concrete (R/ C) elements and structures to dynamic loads. These two fac-
tors, however, have been generally neglected in the development of analytical

models for predicting the inelastic dynamic response of R/ C structures.

Dynamic (e.g., seismic and impulsive) loadings induce relatively high strain
rates in structures as compared to the quasi-static laboratory loads. Increased
strain rates result in higher strengths and moduli of elasticity of concrete and
reinforcing steel, and consequently increased the strengths and stiffnesses of R/ C
elements[1-5]'. These modifications in element behavior at higher rates of strain-
ing might adversely influence the dynamic response characteristics of R/ C struc-
tures, resulting in higher element forces and stronger tendencies towards the brit-
tle modes of failure. Hence, the disregard for the strain rate effects in dynamic

analysis of R/ C structures might actually lead to unsafe structural designs.

Under repeated inelastic load reversals, shear deformations play an increas-
ingly important role in the inelastic load-deformation behavior of R/C

elements[6-8]. This is due to the deteriorating nature (under cyclic loads) of the

 

* Numbers in square brakets refer to the list of references.

shear resisting mechanisms when compared with the flexural ones. Many analyti-
cal predictions of dynamic response of R/ C structures, however, have neglected
the efl'ects of shear deformations, disregarding the fact that the relatively small
and negligible shear deformations (compared with the flexural ones) under mono-
tonic loads tend to grow and dominate the behavior under repeated inelastic load
cycles. Dynamic response characteristics of R/ C structures might also be
adversely influenced by the deteriorations in the shear resisting mechnisms of ele-
ments. Hence, the'disregard for shear deformations in dynamic analysis of R/C

structures can also damage the accuracy and safety of final designs.

The first part of this dissertation (Chapters 2 to 5) deals with strain rate
effects on the dynamic properties of R/ C materials and structural elements under
monotonic loading conditions. The second part (Chapter 6) examines the eflects

of shear deformations on the cyclic load-deformation relationships of the elements.

In order to assess the strain rate effects on the dynamic response of R/ C ele-
ments, strain rate-sensitive constitutive models were developed for concrete and
steel (Chapter 3 and 4, respectively), and they were incorporated into a refined
nonlinear analysis procedure for predicting the axial-flexural response of R/ C sec-
tons and beam-column elements to monotonic dynamic loads (Chapter 5). The
analysis techniques were verified using the results of a relatively large number of
quasi-static and dynamic tests on R/ C elements. The developed strain-dependent
element model was then used for numerical studies on the sensitivity of the
axial / flexural behavior of R/ C beams and columns to the variations in the rate of

straining.

In studies on shear deformations (Chapter 6), a beam element model capable
of distinguishing between the hysteretic characteristics of the shear and flexural
actions was developed. This element model consists of a physical simulation

which treats the axial and flexural actions separately, and two distinct sets of

empirical hysteretic rules for cyclic shear and flexural deformations. These hys-
teretic rules account for the higher deteriorating nature of the shear behavior
which leads to the growing dominance of shear deformations under repeated ine-
lastic load reversals. The tangent stiffness matrix of the developed element model
was also derived, and the model was verified using the results of a relatively large
number of cyclic tests on R/ C beams with wide ranges of material and geometric

characteristics.

CHAPTER 2

STRAIN RATE EFFECTS ON THE DYNANIIC RESPONSE OF
REINFORCED CONCRETE STRUCTURES:
A REVIEW OF THE LITERATURE

2.1 INTRODUCTION

The current reinforced concrete seismic analysis and design procedures[9] are
generally based on the results of quasi-static tests on R/ C materials and elements.
The rate of straining in such tests is of the order of 10’5/sec[1-4]. Typical
earthquake- and impulse-induced strain rates are, however, of the order of 10’2
and 10‘1/sec, repectively[4,5], and dynamic test results on the reinforced concrete
materials and elements[10-26] have revealed considerable dependence of the results
on the rate of straining. For example, at the high earthquake—induced loading
rates, the axial and flexural strengths of the reinforced concrete (R/C) beam-
columns have been observed to increase by about 20% over the quasi-static

values[4,5,8,12,15].

The potential problems caused by disregarding the strain rate effects in the
current methods for dynamic analysis and design of the R/ C structures are dis-

cussed below:

(a) Maximum seismic shear forces in the structural elements are proportional to

their flexural strength[9] (Figure 2.1a), and thus they increase proportionally

(b)

with the flexural strength as the rate of straining increases. The correspond-
ing increase in shear strength is, however, smaller[12,24]. Hence, a beam with
a ductile flexural failure under quasi-static loading might fail suddenly in

shear when the load is applied dynamically.

The maximum bond stress between steel and concrete as well as the max-
imum shear force in the R/ C beam-column joints under seismic loads increase
proportionally with the bar strength as the rate of straining increases[9,27]
(Figure 2.1b). If this is not accompanied by comparable increases in the
bond and shear strengths in the joints, then a beam with flexural failure
under quasi-static loads might fail by bar pull-out or joint shear cracking if
the load is applied dynamically. This will adversely influence the ductility

and energy dissipation capacity of the element.

In the present philosophy of seismic design, it is assumed that the flexural
yielding and hysteretic energy dissipation in the critical regions of the struc-
tures reduce the seismic forces in the structural elements. At higher strain
rates, however, the increased element yield strength postpones the formation
of the plastic hinges[3,8,l2,24,28]. This can increase the earthquake-induced

internal forces in the structural elements.

((1) At higher strain rates, the stifl'nesses of the R/C materials and elements

increase and thus the overall structural stiffness also increases[5,8,10,28].

This modifies the seismic response characteristics of the system.

In the coupled wall systems, the maximum axial forces in the walls are pro-
portional to the maximum shear forces (and consequently to the flexural
strengths) of the coupling girders (Figure 2.1c), and hence they increase with
increasing strain rates. The increased wall axial forces might substantially
modify the seismic response characteristics of the R/ C structures [29-32]. The

same discussion also applies to the axial forces in the columns of the R/C

 

 

D.

 

 

 

(a) Beam Shear Forces

 

mas—w

 

\J

bar force

4

.d

 

bond stress i/

 

 

 

 

——Ib—-q— -

 

 

uji— joint shear

(b) Joint Shear and Bond Stresses

 

 

 

 

 

 

 

 

 

7“w 1X11—
_. 1 no 100. 1

1X5 1)( ”L”,

 

 

(c) Wall Axial Forces

Figure 2.1 Strain Rate Efiect on The R/C' Element under Seismic Forces

frames that will be modified due to the strain rate-sensitivity of the beams.

O

(f) In the lower stories of the R/ C tall buildings, the column reinforcing bars are
usually of higher strength than the beam reinforcement. Considering that
the higher strength steel is less strain rate-sensitive[12,17,25,26], then a
strong column - weak beam design[9] under quasi-static loads might turn out

to be weak column - strong beam under dynamic loads.

(g) At higher strain rates, the increase in the element strength and the subse-
quent increase in the element seismic forces might increase. the tendency

towards the buckling failure modes in the structure.

The remainder of this chapter presents a critical review of the literature on
the strain rate-sensitivity of the R/ C materials and elements. No analytical or
experimental studies regarding the strain rate effects on the overall dynamic

response characteristics of the R / C structures have been reported in the literature.

2.2 CONCRETE IN COL/IPRESSION

Typical static and dynamic monotonic stress-strain diagrams of concrete
loaded in compression are compared in Figure 2.2(a). As shown in this figure,
with increasing strain rate, the compressive strength and modulus of elasticity of
both confined and unconfined concrete increase, and the maximum strain at

failure in unconfined concrete decreases.

Figure 2.2(b) presents an accumulation of the available test data[4,5,10-15] as
well as some empirical expressions[10,12] regarding the strain rate-sensitivity of
the concrete compressive strength. The considerable scatter of the test results
observed in Figure 2.2(b) has been attributed to the variations in the moisture
content, compressive strength and confinement of the tested concrete

specimens[4,10- 14] .

 

 

 

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2000-4

 

-- seem Rate - anodes/m
—- Strain Rate - 0.05/00:

 

 

 

 

 

 

 

j fir 1 r
0.000 0.002 0.004 0.006 0.008 0.01 0
Strain
(a) Constitutive Behavior
2.54-
5 o WMnUs] —i
- eow1[14] '
a 3110:1433] 1
j 1
o l
:r: 2.04 5
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e ‘ a A t . i
e:
1 0 a “ .S . J
' -5 151 33 33 I1 3 i 3

Strain Rate (1 /sec) in Log Scole
(b) Compressive Strength

Figure 2.2 Strain Rate-Sensitivity of The Concrete Mechanical Properties

Figures 2.2(c) and (d) show the strain rate eflect on the secant and initial
tangent moduli of concrete, respectively[4,10,12,13,33]. The initial tangent
modulus is seen to be less strain rate-sensitive than the secant modulus. The
strain rate-sensitivity of the concrete strain at maximum stress is demonstrated in
Figure 2.2(e). The model of Reference 10 does not agree well with the test results

given in this figure.

Strain rate-sensitive constitutive models for concrete are presented in Refer-
ences 10 and 33. The inaccuracies of the typical model given in Reference 10 in
predicting the test results are shown in Figure 2.2(f), where the result of a
dynamic test[15] on an R/C column with a compressive strength of 6,000 psi is

compared with the stress-strain diagram predicted by this model.

 

 

 

 

2.5 .
a WabteinUIS]
I Comll[14]
0
2:? 2.0.1
0
nil-J
U)
\
.9.
E
2
as 1.54 .’
O
-_ '
I o
I I
'n I '
1-0 l T "‘ 1 T I r
-5 -4- -3 -2 -1 0 1 2

Strain Rate (1/sec) in Log Scale
(0) Secant Modulus

Figure 2.2 Strain Rate-Sensitivity of The Concrete Mechanical Properties (cont’d)

10

2.0-

 

 

 

 

 

 

 

 

 

T o Wotatein[13] '1
' - Cowall[14] ‘
l 1
0
27.3 1.5-4
.3
m
E
.- .
é ' .
c: ‘ O I
5‘ ”H I~ ' i .I'
0'5 T T T— T T T
-5 -4- -3 -2 -1 0 1 2
Strain Rate (1 /sec) in Log Scale
(d) Tangent Modulus
2.0 _
-- Myer-[10] j
e Habit-IE] '
u Dllger[10]
o Woe-tainUs]
o - Comll[14] .
tr: 1.5-4 l
a .
H i
m
\ O
«‘3 a
E
3
> 1.0-4
a
0‘5 T T 1— I T— T
-5 -4 -3 -2 -1 0 1 2

Strain Rate (1/sec) in Log Scale

(e) Strain at Maximum Stress

Figure 2.2 Strain Rate-Sensitivity of The Concrete Mechanical Properties(cont’d)

11

 

-- Tut Fl
,.._\ -— Oils—11:10]

0---.-..O ~ --

Stress (psi)

 

 

 

 

0.005 0.010 0.015 0.020
Strain
(f) Test versus Theory

F ifure 2.2 Strain Rate-Sensitivity of The Concrete Mechanical Properties(cont’d)

 

 

 

2.5
e eomn[14] 1.
a Komlae[1 8]
I Wright[20]
l a Seabold[24]
o :1 50011433]
2:: 2.0-4 I
a9
O
E e-
(E) a
g e
g. 1.5-1 0‘3
E] 5 .
C
C
. a
:=- D q
1‘0 I I I W I
-5 -4- -3 -2 -1 0 1 2

 

Strain Rate '(1/sec) in Log Scale
Figure 2.3 Strain Rate-Sensitivity of The Concrete Tensile Strength

12

Under seismic excitations, the concrete in the R/ C structures is subjected to
repeated compression at high strain rates. The available dynamic test results are,
however, mainly concerned with the monotonic behavior of concrete, and studies

on the strain rate-sensitivity of the concrete hysteretic behavior are scarce[15].
2.3 CONCRETE IN TENSION

The available test data on concrete tensile strength [12,14,17-20,24,33] are
accumulated in Figure 2.3. Concrete tensile strength increases more rapidly than

its compressive strength with increasing strain rate (compare Figures 2.2b and

2.3).

2.4 STEEL

As typically shown in Figure 2.4(a), the steel yield and ultimate stresses as
well as its maximum yield plateau and ultimate strains increase with increasing
strain rate[5,l2,21-26]. The modulus of elasticity of steel is not, however, strain
rate-sensitive. Figures 2.4(b) through 2.4(d) present accumulations of the test
data available in the literature on the strain rate-sensitivity of the steel yield and
ultimate stresses as well as its ultimate strain. The steel yield strength is seen to
be more strain rate-sensitive than its ultimate strength. The strain rate—
sensitivity of the steel yield strength has been found[12,22,25,26] to increase with
decreasing steel strength as shown in Figure 2.5. This figure also presents the
empirical expression derived in Reference 12 that does not compare well with the
test data. Results of cyclic tests performed on steel at high strain rates are scarce,
but the available data[5] show similar tendencies in the steel strain rate-sensitivity

under monotonic and cyclic loads.

STRESS

Dynamic/Static

13

 

 

 

 

 

 

 

 

 

 

 

—- DYNAMIC
0 -- STATIC
U
STRAIN
(a) Constitutive Diagram
2.0
a Gowell[14]
. Mahin[5]
A Winlock[22]
A Staffier[26]
u Seabold[12] CI
1.5-
D D
U 00 g
8 .0 C53
& t a ,0 a a C3)
1.0— o 8 O O O
A
0.5 T
-5 -4 13 :2 31 0 1

Strain rate(1/sec) in Log Scale
(b) Yield Strength

Figure 2.4 Strain Rate-Sensitivity of The Steel Mechanical Properties

l4

 

 

 

 

 

 

 

 

2.0
o Cowell[14]
a Seabold[12]
a Winlock[22]
O
'5' 1.5...
a
«H
in
F
"i ° 051.5535
a
g 1.0- 8 ‘8 g a E) (Q
a
0‘5 I j I I f
-5 -4 ~55 -2 -1 0 1
Strain rate(1/sec) in Log Scale
(c) Ultimate Strength
2.0
a Winlockf22]
0
223 1.5-
.3
m
\
.2 A
E A A
a 3 A
E. 1.0- A 8 A
c:
0.5 , , j T 1
-5 -4 -3 -2 -1 o 1

Strain rate(1/sec) in Log Scale
(d) Ultimate Strain

Figure 2.4 Strain Rate-Sensitivity of The Steel Mechanical Properties(cont’d)

15

 

 

 

 

 

 

 

 

 

 

2.5
--- Theory[12]
a SeaboldUZ]
n Wlnlock[221
:3 2.0-4
3 /
m /
\
.9.
E
2
> 1.5-1
‘3 E E
B 8
1°C T B T n 7 1 l
-5 -4 -3 -2 -1 0 1 2
Strain Rate (1 /sec) in Log Scale
(a) f, = 30,000 psi
25. .,.._ ...._ ___._..._.,_..-
I -- 111M121 i
l . Oawell[25] 3
l
i s
i l
o i
2.01 g
.u 1 ,
(n I I,
\ g 1
u ; l
E i
2 15-i i
; /""""
c /
3,—- _/"’ll . :‘..
1.0 Wfigf-H—wrrf—w T
—s -4 —-3 -2 -1 0 i 2

Figure 2.5 Strain Rate-Sensitivity of The Steel Yield Strength

Strain Rate (1 /sec) in Log Scale

(b) I, = 90,000 psi

16
2.5 BOND BETWEEN STEEL AND CONCRETE
Monotonic test results have indicated that the characteristic bond stress

values increase with increasing slip rate[26,34-36] (Figure 2.6). Cyclic test results

on bond performed at high loading rates have not become available in the litera-

 

 

 

 

 

ture.
.3000 -—--~-——
1: Slip Rate - 0.1 ln/aec
-- 811p Rate - .001 111/000
.1 -- 3le Rate - 0.00002 In/sec
C.‘
e)
a.
V
0:
m
0
.3:
m I
i
l
l
0 j j T T J1
0.0 0.1 0.2 0.3 0.4- 0.5

Slip (inches)

F iure 2.6 Slip Rate-Sensitivity of Bond between Steel and Concrete

2.6 AXIAL-FLEXURAL BEHAVIOR OF THE R/ C ELEMENTS

As shown in Figure 2.7(a), the axial-flexural strength and stiffness of the R/ C
beam-columns increased by about 20% as the rate of straining was raised from a
quasi-static value to the seismic level in monotonic tests reported in Reference 4.

The failure of the dynamically loaded elements has been also observed to be more

l7
brittle[4,5,12,24].

As typically shown in Figure 2.7(b), in the available cyclic tests on R/C
beams[5,8,28,37] also the flexural yield strength was observed to increase by about
20% as the quasi-static loading rate was increased to the seismic level. The strain
rate effects become smaller after repetition of some large-amplitude inelastic cycles.
A relatively small increase in the beam hysteretic energy absorption was also

observed at higher strain rates.

In one of the very few analytical studies available in this area, Reference 45
has used layer models of the R/ C sections to analytically produce their interaction
diagrams at different strain rates. The results compared well with the available
test data and it was confirmed that the ACI[9] equations give quasi-static values
of the axial-flexural strength that are considerably smaller than the corresponding
dynamic strengths (Figure 2.7c). The strain rates shown in Figure 2.7 (c) belong

to the extreme compressive concrete fiber.

2.7 SHEAR BEHAVIOR OF THE R/ C ELENIENTS

Test results on simply supported R/ C beams subjected to monotonic
loads[12,24] have revealed,that with increasing loading rate, the maximum shear
force increases proportionally with the flexural strength, but the corresponding
increase in shear strength is less pronounced. Hence, in some tests the beams had
suflicient web reinforcement to force flexural yielding prior to shear failure when
loaded statically, but not enough to cause that sequence of failure modes in the
dynamic loading case. It was also observed that the share of the web reinforce-
ment in resisting shear increases and the shear failure becomes more brittle as the

loading rate increases.

18

 

 

 

:2..."
3 --~._...
3
ES 03
match

(a) Monotonic Constitutive Behavior

 

 

 

 

—“

”-
III-5
A l
W
1
4
«OJ
1
w
«4

ME 41m

 

 

 

 

(b) Hysteretic Behavior

1.1.... III}

I- new.
-.eeoeee weave-e

  
 

 

 

(c) Interaction Diagrams

Figure 2. 7 Strain Rate-Sensitivity of The Axial-Flexural Behavior
of The R/C Elements

19

Reference 33 has reported experimentally derived static and dynamic shear
load-deformation diagrams of cyclically loaded cantilever beams with shear span-
to-depth ratios of about 3 (Figure 2.8). Shear stifl‘noss degradation in these tests

was found to be smaller at higher loading rates.

Comprehensive analytical studies on the strain rate-sensitivity of the shear
strength and deformation characteristics of the R/ C elements have not been

reported in the literature.

ZOO

 

—- Dynamic l

 

Shear Force (KN)

 

 

 

 

 

.J
v n
4.0

~40 320 0 2b
Shear Deflection (mm)

Figure 2.8 Strain Rate-Sensitivity of An R / 0 Beam Shear Hysteresis

CHAPTER 3

A STRAIN RATE-SENSITIVE CONSTITUTIVE MODEL
FOR CONCRETE UNDER CONIPRESSION

3.1 INTRODUCTION

Dynamic compression tests on confined and unconfined concrete have shown

that as the strain rate increases

(1) the compressive strength, secant modulus of elasticity, and slope of the

descending branch in the concrete stress-strain diagram increase, and

(2) the maximum strain at failure decreases while the strain at maximum
stress might decrease[10] or increase[4] depending on the rate of strain-

ing.

The strain rate effects are found to decrease with evaporation of the moisture
in concrete. Some investigators have also suggested that the strain rate effects are
more significant on high strength concrete, while others have concluded from their
test results that the concrete compressive strength has no influence on the strain

rate efi'ects[11,12].

In this chapter, a constitutive model for confined and unconfined concrete
subjected to dynamic compression is proposed. This model is based on numerous
measurements of the maximum stress and strain at maximum stress in tests per-

formed by diflerent investigators at different strain rates. The proposed model

20

21

accounts for the confinement effects and distinguishes between the strain rate
effects on air-dried and saturated concrete. It also compares well with the avail-

able test results.

3.2 CONIPRESSIVE STRENGTH

Test results on the dynamic compressive strength of concrete have been
reported in References 4, 5, 10, ll, 13, 14, 15 and 33. These experimental studies
have covered wide ranges of compressive strength (1,000 to 7,500 psi), moisture
content (air-dried and saturated), confinement condition (plain, tied, and spirally
reinforced), and age at test (3 to 49 days). In most of the tests the strain rate has
been measured, but some studies have reported only stress rate measurements. In .
these cases, the strain rates were derived by dividing the stress rates into the con-
crete modulus of elasticity. The errors of this transformation were judged to be
acceptable, because the strain rate effects are not sensitive even to errors of the
order of 50 percent in the strain rate measurements (strain rate effects become
apparent after changes of an order of magnitude in strain rate). Figure 3.1(a)
presents a plot of the ratios of the dynamic to static compressive strengths versus
the rate of straining during the test. Each point in this figure is an average of
about three tests. This figure also shows three curves, two of which have been
based on the expressions proposed by Dilger[10] and Seabold[12], for the strain

rate effect on the compressive strength of concrete:

 

 

 

Dilger : fed = 1.38 + 0.0810g10é (3.1)
[CO
fed . .
Seabold: f = 1.17 + 0.173e + 0.06log106 (3.2)
08
where, 1 _<_ fed S 2

C8

22

The third curve, proposed in this study, has resulted from least-square curve
fitting of a second-degree polynominal (in terms of logloé) to the test results

reported by different investigators. For all tests,

fed
[cs

 

1.48 + 0.16010g10é + 0.0127(log10é)2 (3.3)
where,

f Cd:- dynamic compressive strength;
I“- static compressive strength (at a atrain rate of 10'5/sec); and
is - strain rate (1 /sec) 2 10"5
A wide scatter of the tests results is observed in Figure 3.1(a). The main
source of the scatter was found to be the variation in the moisture content of con-
crete. Figures 3.1(b) and (c) show the results of tests on air—dried and saturated
concrete, respectively. It can be concluded from these figures that the strain rate
effect on increasing the concrete compressive strength becomes more significant as
the concrete moisture content increases. The following expressions (also shown in
Figures 3.1b and c) were derived by least square curve fitting to the test results on

air-dried and saturated specimens:

For air-dried concrete,

fed

68

 

= 1.48 + 0.206log10é + 0.0221(log10é)2 (3.4)

For saturated concrete,

fed

68

 

= 2.54 + 0.58010g10é + 0.0543(log10é)2 (3.5)

The available test results did not show any considerable influence from the
compressive strength on the strain rate effects. For example, Figures 3.1(d) and
(e) compare the strain rate efiets on low strength ([63 £4,000psi) and high

strength (I ,8 > 4,000psi) air-dried concrete. These figures show that the results

23

 

   
 

 

 

 

 

 

2.5
— Proposed
-- We'rUO]
-- MHZ]
e Wat-hinUJ]
o - Oemll[14]
E 2.0-4 ; Suadztaal
Q a
o x
E a
O 0
g 1.5-1 ,..-
D
1.0-1
.5 .1!
Strain Rate (1/sec) in Log Scale
(a) General Concrete Tests
2.57
‘ ——- Proposed
-- Ollgerffll]
-- Seaboldt‘iZ]
e Wat-unna]
6 I Cowll[14]
:a 2.0.1 A Suarlz[33]
a l a DflgerUO] .
\ I A Kaplonfit]
O l X
.E .
3
5‘ 1.5.. --
1
l
l :
1.0-4:?

 

 

 

 

 

Strain Rate (1/sec) in Log Scale
(b) Air-Dried Concrete Tests

Figure 3.1 Strain Rate Eflects on The Compressive Strength of Concrete

24

 

 

 

 

 

 

 

 

 

 

2.5 - »~----,
3 -— Due-I110} / i
-- Seabcld[12] ,/ l
A Suariz[33] / i
o a Shah[15] g
z: 2.04 / l
3 l
(D
\
.9. .
E l
2
5. 1.5-1
1.0-1
-5 2
Strain Rate (1 /sec) in Log Scale
(c) Saturated Concrete Tests
2... "'
x Scott[4]
e Watstein[13]
n Dflger[10]
o - Comll[14]
:3 2.0.1 A Suarlz[33]
U
4.0
(I) O
E -/
CE! /’// I
O /‘
g. 1.5-1i A4
, I
29/
A A
1 O U a T T
. r
-5 .34 33 :2 -1 o 1 2

Strain Rate (1 /sec) in Log Scale
((1) Air-Dried Concrete with f“ S 4,000 psi

Figure 3.1 Strain Rate Eflects on The Compressive Strength of Concrete(cont’d)

25

 

 

 

 

 

2'5 T *1
! -—— Propoacd l
t Mahin[5] l
A Kaplan[11]
- Gawall[14] i
o o WatsteinUJ] i
’33 2.0%
.3
(I) O
\
.9.
E
S
> 1.5-4
(3
1-0 T T T IT 1 r
--5 -4- -3 -2 -1 0 1 2

Strain Rate (1/sec) in Log Scale
(e) Air-Dried Concrete with f“ > 4,000 psi

Figure 3.1 Strain Rate Eflects on The Compressive Strength of Concrete(cont’d)

 

 

 

 

2.0
—- Prapaaod
-- Dilgor[10] w/ fc-5 ks!
a Mahin[5]
a Diigar[1 O]
o o Watstein[13]
:3 154 I Cow-"[14] ;
.3 l
Q a l
.9. / i
E / g
g ..I/. ;
as 1.0- u t I i
i
i
Q ~-—
1
0‘5 T I T —T T r T
-5 -4- -3 -2 -1 O 1 2

Strain Rate (1/sec) in Log Scale

Figure 3.2 Strain Rate Effects on The Concrete Strain
at The Maximum Compressive Stress

26

of curve fitting to the test data are very similar for different concrete strengths.
The strain rate effect on concrete compressive strength was also found to be

independent of the age of the specimens if their moisture contents are similar.

As far as the confinement effects are concerned, very few of the test programs
have studied confined concrete, and thus no final conclusion can be drawn at this
point. The few available confined test results, however, do not show any

significant influence from confinement on the strain rate effects.

3.3 STRAIN AT MAXIIVIUM STRESS

As mentioned earlier, in some tests[4] when the strain rate was increased the
value of the concrete strain at maximum compressive stress increased while in
some other tests[lO] it decreased. Figure 3.2 shows the strain rate effect on the
strain at maximum stress in the tests reported in References 5, 10, 13 and 14.
Each point in this figure represents the average of about 3 test results. In this
figure, 60“ is the dynamic strain at maximum stress and 603 is the static strain at

maximum stress.

According to the available test results shown in Figure 3.2, increasing the
strain rate from the static value of 10'5/sec to values as high as 10'1/sec results
in reduced values of strain at maximum stress. At strain rates higher than
10‘1/sec, however, the concrete strain at maximum stress becomes larger than the

static value. A considerable scatter of test results is evident in this figure.

Figure 3.2 also presents the equation proposed by Dilger[10] as well as the
one derived in this study. Dilger’s equation is a function of the concrete compres-
sive strength and is based on the assumption that the concrete strain at maximum

compressive stress decreases linearly with the increasing strain rate:

27

60d

6 = (1.3 — 0.06logloé + 0.00007 [6' )/(1.6 + 0.00007 fc') (3.6)

 

08

where,

f c’ = 28-day quasi—static compressive strength of concrete in psi,

The proposed equation given below was derived by least square curve fitting,

using all the experimental data shown in Figure 3.2:

6ad

 

=- 1.08 + 0.11210g10é + 0.0193(log10é)2 (3.7)

608

As far as the efl'ects of concrete moisture content, compressive strength, and
confinement on the dynamic strain at maximum stress are concerned, the avail-

able test results are inadequate to derive meaningful conclusions.

3.4 MODULUS OF ELASTICITY

Figures 3.3(a) and (b) present the strain rate effects on the compressive secant
and tangent moduli of elasticity of concrete as observed in the test results
reported in References l3, l4, and 33. The secant modulus was found at a
compressive stress equal to 0.45 1;, and the tangent modulus was measured at
the origin. Compared to the secant modulus of elasticity, the tangent modulus is
found to be less influenced by the rate of straining. The following expressions
were derived by least square curve fitting to the test results presented in Figures

3.3(a) and (b):

Ecd . . 2
= 1.241 + 0.111 logloe + 0.127(log106) (3.8)

 

C8
Etd

 

=- 1.061 + 0.464 logloé + 0.00683(10g10é)2 (3.9)

ts

28

 

 

 

 

 

 

 

 

 

2.5 —~—- -—-~
T:— Prapoeod i
l o mumps]
; - Cawell[14]
O
:13 2.0-4
O
uI-J
(I)
\ 1
.2
E
8
a 1.5-4 .
C
/
a I .
I I g
1-0 T T‘T’“: T I T T
-5 -4- -3 -Z -i O 1 2
Strain Rate (1/sec) in Log Scale
(a) Secant Modulus
2.0 - .._.._._._..
o WabteinUJ] i
e e coweu[14] |
l
o .
tr: 1.5-4 l
O l
4.: l
m l
5 l
.- . I
E e i
g M i
a 1.0 . § :
0.5 #. ”EH..."
-5 I; 3; 32 31 Ta 1 2

Strain Rate (1/sec) in Log Scale
(b) Tangent Modulus

Figure 3.3 Strain Rate Eflects on The Concrete Modulus of Elasticity

29

where,

Ecd= dynamic secant modulus of elasticity;
E“: static secant modulus of elasticity;
Eta: dynamic tangent modulus of elasticity; and

Eta= static tangent modulus of elasticity.

No conclusion could be derived from the available test results regarding the
effects of moisture content, compressive strength, and confinement on the dynamic

moduli of elasticity.

3.5 THE CONSTITUTIVE MODEL FOR CONCRETE

The following constitutive model (Figure 3.4) is based on Equations (3.8) and
(3.9), which were derived from a relatively large number of dynamic compressive

test results reported by different investigators:

 

 

I 26 e
K K —
1 2]” [0.002K1K3 [0.002K1K3 n

for 6 $0.002K1K3
f a: (3.10)
K1K2fcl [l _ 2(6 — 0.002K1K3)] .>_ 0.2I(11(2fcl

for 6 > 0.002 K1K3

 

k

 

where,
f as concrete compressive stress;
€= concrete compressive strain;
p3 fyh
[(1= 1 + f I ;
C

p8= volumetric ratio of transverse reinforcement to concrete core;

f c, = 28-day compressive strength of concrete;

30

I3”, a yield strength of transverse reinforcement;
0.5

' , /2
3 + 0.002
, f‘ + -3—p, [L - 000219143
f, — 1000 4 s

h’ = width of concrete core measured to outside of the transverse

z=

 

reinforcement;

3:- center to center spacing of transverse reinforcement;

K 2.54 + 0.58010g10é + 0.0543(log10é)2 for saturated concrete
2: 1.48 + 0.16010g10é + 0.0127(logwé)2 for air-dried concrete

K3:- 1.08 + 0.112 logloé + 0.0193(log10é)2
(Note : For é<10'5/sec, K2 = K3 = 1.0)

 

 

—

0.002K1K3 ' Strain (c)
Figure 3.4 The Constitutive Model for Concrete under Compression

In this model, K 2 represents the strain rate effect on the compressive strength
of concrete [Equations (3.3), (3.4) and (3.5)], and K 3 takes care of the strain rate

effect on the concrete strain at maximum stress [Equation (3.7)]. In deriving this

31

model, it was assumed that the strain rate effect on the slope of the descending
branch of the stress-strain diagram is similar to the strain rate effect on the
compressive strength of concrete. Test results presented in Reference 10 support
this assumption. The basic static stress-strain relationship in this model is taken

from Reference 4.

The formulation does not specify a maximum strain at failure. Test results
have shown that, especially for unconfined concrete, the failure strain tends to
decrease as the strain rate increases[10l, but the available experimental data are
insufficient for a quantitative assessment of this strain rate effect. The constitu-
tive model given above closely predicts the strain rate effect on the secant modulus
of elasticity, while it overestimates the strain rate effect on the initial tangent

modulus (which is not usually considered as an important property of concrete).

3.6 CONIPARISON WITH TEST RESULTS AND PARANIETRIC STU-
DIES

The advantages of the strain rate-dependent constitutive model given in

Equation (3.10) over the one proposed in Reference 10 are:
(1) its simplicity and stability,
(2) its generality, in the sense that it was derived using more test results
reported by different investigators,

(3) its capability to distinguish between the strain rate effects on air-dried

and saturated concretes, and
(4) its more accurate consideration of the confinement effects.

Some of these advantages are shown in Figure 3.5(a), which compares the predic-

tions of the model given by Equation (3.10) and the constitutive model of Refer-

32‘

 

 

 

 

 

 

 

 

 

 

 

 
  

 

 

 

Tent
origami ;
A -.
ea ‘\‘ -
a. 3 ,
V
-‘ o ;
a ~~ 015/8 l
3: "xx,
(0
c-
0.000 0.55 0.3io 0.015 0.020
Strain
(a) Spirally Reinforced Saturated Concrete
6000
A 4000-4
'35
3
a)
8
.3
W 2000
O 1 1 I 1
0.000 0.005 0.010 0.01 5 0.020

 

Strain
(b) Tied Air-Dried Concrete

Figure 3.5 Comparison of The Proposed Constitutive Model with Test Results

33

ence 16 with a test result performed on a spirally reinfoced saturated concrete sub-
jected to a dynamic compressive strain rate of 0.0015 /sec[15]. The static compres-
sive strength of the 3 in. by 6 in. cylindrical specimen tested in Reference 9 was
6,000 psi, and it was spirally reinfoced having 1/8 in. diameter wires with a yield
strength of 60,000 psi and a pitch of 1 in. This specimen had practically no
cover. It is clear that the Dilger’s model[10] underestimates the strain rate effect
on the compressive strength of saturated concrete and also overestimates the
confinement effect on reducing the slope of the descending branch of the stress-
strain diagram. Figure 3.5(a) also shows the quasi-static constitutive diagram of
concrete as predicted by Equation (3.10), which is observed to compare with test
results far better than the other available model. The proposed model [Equation
(3.10)] also compares well in Figure 3.5(b) with concrete constitutive diagrams
obtained in quasi-static and dynamic tests on columns with a 17.7 in. square cross
section reported in Reference 4. The air-dried concrete in these clolumns had a
compressive strength of 3,600 psi and was confined by 0.39 in. ties with a yield
strength of 44,800 psi and a volumetric ratio of 0.0182. The cover thickness in

the tested specimens was 1.6 in.

Figure 3.6 compares the predictions of the proposed model given by Equation
(3.10) with the test results on air-dried, unconfined specimens tested at two
different strain rates[5]. The theoretical and experimental results are close. The
discrepancy between the theoretical and experimental values of strain at max-
imum stress in Figures 3.5(a) and 3.6 seems to have resulted from the considerable

scatter of the test data in Figure 3.2.

The more significant effect of strain rate on saturated concrete when com-
pared with air-dried concrete is shown in Figure 3.7. This figure compares the
static and dynamic constitutive diagrams predicted by Equation (3.10) for air-

dried and saturated concretes with a compressive strength of 3,670 psi. The

34

 

 

 

 

6000
~— Proposed
-- Tact
’ .‘\ ~~ 0,5
l:” ‘: ~‘\~\

c 4000-1 ’/:’ 6b\ \\‘6‘.\ °§°

03 1” 0. ‘\~ \ 0 9..

O- , I 0" 04119 x, 05 e.
V I Q cf \ /

I” ‘Q\ 49 (fee a\

a 1 ~34?

0 I” . \.\

L. I Q \ \

+1 I. 4 \

V1 20004' .!~ \\\

I \\\ l
I - \
.l
0 T T T T
0.000 0.001 0.002 0.003 0.004 0.005

Strain

Figure 3.6 Theoretical and Experimental Constitutive Diagrams
for Air-Dried Unconfined Specimens

 

   

 

 

 

1E+O4~——~»-4— —4
--- Saturated '
1 -- Air—Dried [
8000-4 -
‘\\g§—_ O 05/,“
r: l \\
3. 6000-1 r
V 6 § 0 0
g ‘[ T" ‘ _'_'§_/36_c___
_______ 6:0. “‘"---—-——..-_
g 4000-1 ------ r--Bg(:9fé’:9 M“ -----
m .4 6 = 0.00005/scc -------- “2:“ 3:73:24“
20004
0 IT 1 TT 1
0.000 0.005 0.010 0.015 0.020
Strain

Figure 3.7 Strain Rate Effects on Air-Dried and Saturated Concrete

35

column studied had an 18 in. square section confined with No. 3 stirrups having a

yield strength of 45,000 psi and a volumetric ratio of 0.0182. The cover thickness

in these columns was approximated at 0.75 in.

Figure 3.8 compares the static and dynamic constitutive models of the

confined air-dried specimen presented in the previous diagram with the static and

dynamic constitutive models of an unconfined air-dried specimen with the same
compressive strength. This figure shows that confinement considerably reduces
the slope of the descending branch in the stress-strain diagram, while a higher

strain rate slightly increases this slope. Both confinement and higher strain rates

tend to increase the concrete strength.

 

 

 

 

 

. " ‘l— Confined '7.
6 t 005/,“ -- Unconflned
4 . “N‘s
ll" 6 3292995/0ec
\ *‘-—.
A 4000"[ I, \\\ _ MN“
.65 I I \ \P\\
3 / ~. \ 0
I t" \ \' '
g l 0. "4, l
2 ”000;. s
4" J /0 \\ \ [
U’ 2000 6. ~\ \ i
\ 1
.l \‘B; l
g
o l
“I T f r I
0.000 0.002 0.004 0.006 0.008 0.010
Strain

Figure 3.8 Strain Rate Eflects on Confined and Unconfined Concrete

36

3.7 SUNIIVIARY AND CONCLUSIONS

On the basis of many test results reported by different investigators, empiri-
cal models were developed for the strain rate effects on the concrete compressive

strength, strain at maximum stress, and modulus of elasiticity. It was found that

(1) the concrete compressive strength increases at higher strain rates, and

this increase is more significant for saturated concrete,

(2) the strain at maximum stress first decreases and then increases as the

strain rate increases over the static values, and

(3) at higher strain rates, the secant modulus of elasticity of concrete
increases. The initial tangent modulus also increases at higher strain
rates but at a much slower rate when compared with the secant

modulus.

The preceding idealizations of the strain rate effects were used to develop a
practical constitutive model for confined and unconfined concrete subjected to

different compressive strain rates. This model compared well with test results.

The strain rate-dependent constitutive model developed in this chapter for
confined and unconfined concrete under compression can be used in refined (e.g.,
layer) modeling of reinforced concrete sections and elements. Such models can be
applied for dynamic analysis of reinforced concrete structures subjected to seismic,

impulsive, or wind loads.

CHAPTER 4

A STRAIN RATE-SENSITIVE CONSTITUTIVE MODEL
FOR STEEL UNDER TENSION

4.1 INTRODUCTION

The steel yield stress and ultimate strength will increase and the strains
corresponding to these stresses, as well as the strain at the beginning of the strain
hardenig range, will either increase or remain constant with increasing strain rate.
However, the steel modulus of elasticity will not be significantly influenced by the
rate of straining[5,12,22,25,26]. Dynamic tests[22,25,26] have revealed that steels
with lower yield strengths are relatively more sensitive to strain rate variations
than steels with higher strengths. Other factors, such as the chemical properties
and the cold work history of the steel, are also suggested[26] to influence the

strain rate eflects.

In this chapter, the strain rate effects on the mechanical properties of struc-
tural steel, deformed reinforcing bars and deformed wires have been compared.
Empirical strain rate-dependent constitutive models for steel have been derived on

the basis of these comparisons.

4.2 TEST RESULTS

37

38

Dynamic test results on structural steel , reinforcing bars and deformed wires
have been reported in References 12, 24, 25 and 41. The yield strength in these
tests ranged from 26,000 to 99,100 psi, and they included different types of steel
with diflerent chemical properties. The tests were all performed in tension, and

the specimens were either round or rectangular.

Figure 4.1(a) shows a typical comparison of the static and dynamic stress-
strain diagrams. As shown in this figure, the steel modulus of elasticity is not
influenced by the strain rate variations, but all the load-sustaining properties and

the percentage elongations increase with increasing strain rate.

Figure 4.1(b) shows the ratios of dynamic to static (lower) yield strengths
versus the rate of straining in test results reported by different investigators. This
figure shows that, in spite of the wide scatter in the test results, the steel yield
strength increases with increasing strain rate. The same conclusion is valid for
test results on the steel upper yield strength, as shown in Figure 4.1(c). The steel
upper yield strength seems to be more strain rate-sensitive than the lower yield
strength. The ultimate strength of steel, as shown in Figure 4.1(d), shows an
increase with increasing rate of straining, but at a slower rate when compared

with the increase in yield strength.

Figure 4.1(e) presents the ratio of the measured dynamic to static strain at
the initiation of the steel strain hardening region versus the rate of straining in
dynamic tests. A rapid increase in this strain is observed with increasing strain
rate. The ultimate strain is shown in Figure 4.1(f) to increase rather slowly with

the rate of straining. Note that the scatter in test results is rather large.

4.3 FACTORS INFLUENCING THE STRAIN RATE EFFECTS

In order to find the sources of the considerable scatter in the observed strain

STRESS

'N.
V ‘\
BI

‘s

a

Dynamic / Static

39

 

 

‘l
\
I
I
I
I
I
I
I
I
I
I
I
l

I

I

I

 

 

 

 

 

 

 

 

 

' I
[ 1
I
l 1
l I l
, 1 1
: : Ce: 6;
1
i [ —— DYNAMIC
[ I -- STATIC
6‘ 6;
STRAIN
(a) Static and Dynamic Constitutive Diagrams
2.0
a Cowoll[14]
o Mahin[5]
a Winlack[22]
A Staffier[26]
1:1 Seabald[12] CI
15.0
D U
El 00 g
o I “D 0°
C
A 8 °t 32° ° 9.
1.0~ a o o O
A
0.5 T l 1 T F
-5 —4 -3 -2 -1 0 1

Strain rate(1/sec) in Log Scale
(b) Lower Yield Strength

Figure 4.1 Strain Rate Eflects on The Steel Mechnical Properties

 

Dynamic / Static

Dynamic / Static

40

 

 

 

 

 

 

 

 

2.5
a Cowall[14]
111 3.0001424]
a Winlack[22]
A Staffier[26]
a Seabald[12]
2.0.1 I Wcod[41l *
at
*2?"
0 A
1.5-‘ a): A o O A
A o
2% ° . "
a
o 2 O AO*8?I§#O 0
. . O
A [a 0‘ '3. a 0 0°
1‘0 I I I fil— I
-5 —4 —3 -2 -1 0 1
Strain rate(1/sec) in Log Scale
(c) Upper Yield Strength
2.0
a Cowoll[14]
a Seabold[12]
A Mnlack[22]
1.5-1
4 ° so Dig. gas-3E8
1.04 8 ‘8 0 Ce?)
0.5
—5 34 33 -'2 31 0 1

Strain rate(1/sec) in Log Scale
((1) Ultimate Strength

Figure 4.1 Strain Rate Efiects on The Steel Mechnical Properties(cont’d)

41

2.5

 

 

 

 

 

 

 

A Winlock[22] A
A
A
.9.
46 2.0-l A
+1
(I)
E
OE A A
o
A
5. 1.5-1 A
Q A
A A
A
1'0 r I T I r
-5 ~4- -3 -2 -1 D 1
Strain rate(1/sec) in Log Scale
(e) Strain at The Initiation of Strain Hardening
2.0
A Winlock[22]
o
:r: 1.5-4
0
u...)
U)
\
.9. A
E A A
O a A
E. 1.0-4 A A Q
Q
0‘5 I I j I I
—5 -4 -:5 —2 -1 o 1

 

Strain rate(1/sec) in Log Scale
(f) Ultimate Strain

Figure 4.1 Strain Rate Effects on The Steel Mechnical Properties(cont’d)

42

rate eflects (Figure 4.1), the influences of the steel yield strength, the steel type
(structural, bar, wire), and the specimen shape (round, rectangular) were studied.
It was found that the main source of scatter is the steel yield strength, and the
other factors do not noticeably influence the strain rate eflects. For example, the
scatter in strain rate effects on the (lower) yield strength shown in Figure 4.1(b) is
considerably reduced when the test results are presented separately for different
ranges of yield strength in Figure 4.2. It is clear in this figure that the strain rate
efl'ect becomes less significant as the yield strength increases. Similar influences of
yield strength are observed in Figures 4.3, 4.4, 4.5 and 4.6 for the strain rate
effects on the upper yield strength, ultimate strength, strain at the beginning of

the strain hardening region, and the ultimate strain of steel, respectively.

 

2.0

+ Cau1:26ksi<fy<38kai

e cou2:45kai<ry<55kai

t Caa3:81ksl<fy<100kai

—- Theorybq. 4.1) for Can 1

-- Theory(eq. 4.1) for Can. 2 4"
1.5.. - Thong/(ea. 4.1) for Case 3

 

 

Dynamic/ Static

.3
O
I

 

 

 

0.5

 

I 1 I T

-5 It -3 -2 —1 o 1
Strain rate(1/sec) in Log Scale

Figure 4.2 Strain Rate Efleets on The Lower Yield Strength of Steels
with Difi‘erenl Yield Strengths

43-

 

 
    

 

 

2.0
+ Cou1:26ksi<fy<38loi +-
II Cau2:45kal<fy<55kni ++
e cau3:a1kai<fy<100kai 1'
— Thoary(eq. 4.2) for Case 1
o -- Theory(eq. 4.2) for Case 2
3;: 1.5.1 —- Theary(oq. 4.2) for Case 3
.3 + i‘ as
Q as "“ ”ZS-~-
.2 $1
E W
o
a
g; ‘LOq ¢ ¢ ¢ *
0‘5 I I I I I
~5 -4 -3 -2 -1 0 1|

 

Strain rate(1/sec) in Log Scale

Figure 4.3 Srain Rate Efi’ects on The Upper Yield Strength of Steels

2.0

1.5

Dynamic / Static
E;

0.5

with Diflerent Yield Strengths

 

l:|ex+

Cau1:26l¢sl<fy<38kol
Cau2:4-5kal<fy<55kal
00:03:51 ksl<fy<100ksl
Theary(eq. 4.3) for Can 1
111.cry(eq. 4.3) for Case 2
moary(eq. 4.3) for Case 3

 

 

 

 

 

-5

I I I I I

-4 -3 -2 -1 o 1
Strain rate(1/sec) in Log Scale

Figure 4.4 Strain Rate Eflects on The Ultimate Strength of Steels

with Difl'erent Yield Strengths

43-

 

  
 
   

 

 

 

2.0 .
+ Cou1:28kal<fy<38lal +
111 Cau2:4-5kal<fy<33kai ++
e Caa3:81ksi<fy<100ksl 1'
-— Theary(eq. 4.2) for Case 1
o -- Theory(eq. 4.2) for Case 2
:53 1.5.. —- Theory/(ea. 4.2) for Case 3 an
E +5 3‘ a“ 95“"
.2 fix?”
E W
s ' . 1*
>~ 1.0-1 ”I t ¢ ¢
O
0.5 T
-5 -4 :3 -'2 -r1 6 1

Strain rate(1/sec) in Log Scale

Figure 4.3 Srain Rate Effects on The Upper Yield Strength of Steels
with Diflerent Yield Strengths

2.0

 

 

 

+ Case1:26ksl<fy<38lal
In Cau2:43kal<fy<53kal
e: oaseszm ksl<fy<100ksl
— Thoary(eq. 4.3) for Can 1

o -- ThooMeq. 4.3) for Case 2

:3 1.5.4 -- 'l'haary(eq. 4.3) for Can 3

3

U)

\

.9.

E

2

> 1.0

O

0.5 I I I I I

 

 

 

-5 -4 -3 -2 -1 o 1

Strain rate(1/sec) in Log Scale

Figure 4.4 Strain Rate Eflects on The Ultimate Strength of Steels
with Different Yield Strengths

Dynamic / Static

Dynamic / Static

2.5

2.0

.3
U!

44

 

 

 
 

 

 

'31:;
+ ecu1:2eka1<fy<sak.1/ /
111 Cau2:45kal<fy<33kal I/
—— Theary(eq. 4.4) for Case 1 IX)!
-- “mound-q. 4.4) for Case 2 I/
-l y’
as
an
i I 7 I I
-5 -4 -.'5 -2 -1 0

Figure 4.5 Strain Rate Efl'ects on Strain at The Initiation of Strain

2.5

2.0

1.5

1.0

Figure 4.6 Strain Rate Effects on The Ultimate Strain of Steels

Strain rate(1/sec) in Log Scale

Hardening of Steels with Difierent Yield Strengths

 

.l

.4

 

+ Casa1:26kai<ty<38kai
111 Cau2:43kai<fy<53kal
—- Theary(eq. 4.5) for Case 1
-- 1h.ary(eq. 4.5) for Case 2

 

 

 

 

Strain rate(1/sec) in Log Scale

with Different Yield Strengths

 

45

4.4 THE CONSTITUTIVE MODEL FOR STEEL

Considering the influence of yield strength on the strain rate-sensitivity of
the steel mechanical properties, the following expressions were derived in terms of
yield strength for the dynamic to static ratios of the lower and upper yield and

ultimate strengths of steel and its strain hardening initiation and ultimate strains:

I

—— = (—4.51x10-7 1, + 1.46) + (—9.20><10-7 f, + 0.0927) logloé (4.1)

__ -_-. (.633x10-6 /, + 1.72) + (---1.37x10-6 1, + 0.144) logloé (4.2)

 

:1-=(—7.71><10-7 f, + 1.15) + (—2.44x10—-7 /, + 0.04969) logloé (4.3)

T .. (—4.21x10‘5 [3, + 4.46) + (—8.41X10"6 f, + 0.693) logloé (4.4)
II

I

E
.3. = (_g,gsx1o-6 f, + 1.40) + (-1.79><10-6f, + 0.0827) logloiE (4-5)

u

where,

fr’frl == static and dynamic lower yield strengths of steel, respectively (psi) ;
f? ,f,’ a static and dynamic upper yield strengths of steel, respectively;
In , f d = static and dynamic ultimate strengths of steel, respectively;
6,, ,€,',== static and dynamic strain hardening initiation strains of steel,
respectively; and

cu ,6;= static and dynamic ultimate strains of steel, respectively.

These expressions are shown to compare well with test results in Figures 4.2 to
4.6. In each figure, the value of I, used in the corresponding expression is the
average of the maximum and minimum yield strengths of the test results used in
compression.

The above expressions together with the static values of the (lower) yield'and

ultimate strengths, and strain hardening initiation and ultimate strains can be

used in the following expressions for constructing the steel monotonic stress-strain

46

diagram under the action of dynamic loads. It should be mentioned that since the
upper yield stress is a local phenomenon, it is generally excluded from the steel

constitutive models [8,26].

f I
E818 for 68 < fy /E’,I
I

r

I

ii
I. = 4 (4.6)

, 1126 -6' +2 6 —e' f’
(‘ t) +7 ’t(",—1.7)
60(6, —e,,) +2 6, —e,, f,

for

 

_<_e.<el

 

 

 

for 6,1 S 6, < e;
where,
f,- steel stress (psi);

e,= steel strain; and

E8= steel elastic modulus ( psi)

The above constitutive model requires practically all the quasi-static charac-
teristic stress and strain values of steel (fy, fa, eh, and cu) which might not be
readily available. For this reasons, a simpler bilinear dynamic constitutive model
was developed for steel subjected to monotonic loading. Selection of the bilinear
form was mainly due to its popularity in static constitutive modeling of steel.
The values required for constructing this model are the static yield stress and ulti-
mate strain, and the static strain hardening modulus of steel (Eh ), defined as the
average slope of the static stress-strain diagram in the strain hardening region,

which is about 100,000 psi:

 

 

E868 for 68 < fyI/Es
fs = I fl (4.7)
[y +E,I 6,3 — E: for E: < 63 < 6,,

 

 

where,

47

E,’ = E, [2><10"5 + 0.0770 + (000‘6 f, — 0.185) logloé] ;
f y =- static steel yield strength (psi).

The above formulation for the dynamic strain hardening modulus was derived by
curve fitting to the experimentally obtained slopes of the line connecting the yield

and ultimate points on the stress-strain diagrams of steel.

Figures 4.7(a) and (b) compare the constitutive models derived using Equa-
tions (4.6) and (4.7) with the experimental monotonic stress— strain diagrams
presented in Reference 22 at two diflerent strain rates for steel specimens with
yield strengths of 34,000 and 44,500 psi, respectively. As shown in these figures,
both models compare satisfactorily with test results. The best comparison is
obtained with Equation (4.6) which requires more experimental data on steel
static properties. Equation (4.7), which is in the simple and popular bilinear
form, also seems to be capable of predicting test results with relatively small

errors .

4.5 SUNINIARY AND CONCLUSIONS

Dynamic test results reported by different investigators on structural steel,
deformed bars, and deformed wires subjected to monotonic tension were used for
deriving empirical expressions for the ratios of static to dynamic values of the
yield and ultimate strengths, yield and ultimate strains, and the strain at the end
of the yield plateau. The principal variables in the expressions are the rate of
straining and the steel static yield strength. These expressions were then used for

developing dynamic constitutive models for monotonically loaded steel.
The following conclusions can be drawn from this chapter:

(1) All the characteristic stress and strain values on the steel monotonic con-

stitutive diagram increase with increasing strain rate. The yield

48

 

 

 

 

 

 

 

    
 

 

 

 

100
-— Test by Wlnlock[22]
-- Thoary(eq. 4.6)
-- Theary(eq. 4.7)
80-4 _ is = 0.025/sec
1 /
C i I“
g °°'* 'WV
3,, ) /,. ..
g 40.) i/ é = 0.000033/aec
20
O I I T I I —r
0.0 0.1 0.2 0.3
Strain
(a) fy -- 34,000 psi
120
— Test by Wlnlock[22]
- Thoory(eq. 4.6)
100.4 -- Theary(eq. 4.7)
4
A 80-1
"(73
3 d
m 60- é = 0.025/aec
ED)- 4 é = 0.000033/sec
4.)
(I) 40..
20-
0 Ir T Iii f 1'
0.0 0.1 . 0.2 0.3

Strain
(b) [3, = 44,500 psi

Figure 4.7 Comparison of The Proposed Monotonic Constitutive Models
of Steel with Test Results

(2)

(3)

(5)

49

strength is more strain rate-sensitive than the ultimate strength. The

steel modulus of elasticity is independent of the rate of straining.

The main factor influencing the strain rate effects is the static yield
strength of steel. The mechanical properties of steel with lower yield

strength are more strain rate-sensitive.

No considerable difl'erence was observed between the strain rate effects

on structural steel, deformed bars, and deformed wires.

The empirical strain rate-dependent constitutive models proposed in this

chapter for steel subjected to monotonic loading compare well with test

results.

The proposed dynamic constitutive models of steel can be used for
developing strain rate-sensitive analytical models of reinforced concrete
and steel elements under the action of dynamic loads. These models can

be used in dynamic analysis of structures subjected to seismic and explo-

sive loadings.

 

 

CHAPTER 5

A STRAIN RATE-SENSITIVE MODEL FOR AXIAL / FLEX'URAL
ANALYSIS OF REINFORCED CONCRETE ELENIENTS
UNDER DYNAIVIIC LOADS

5.1 INTRODUCTION

As a result of the increases in material strength and stiffness, the axial-
flexural and shear strengths and stiflnesses of reinforced concrete (R/ C) elements
also increase at higher rates of loading (Figure 5.1). Hence, reliable information
on strength and stiffness of R/C elements at higher strain rates is needed to
improve the techniques currently used in dynamic analysis and design of R/C

structures.

The increased strength and stiflness of R/ C elements at higher loading rates
might adversely influence the structural response to dynamic loads. ' This
phenomenon can result in higher element forces induced by' the application of
impulsive loads on R/ C structures. It might also encourage the tendency towards
the brittle modes of failure, for example in shear or by pull-out of anchorage bars

in joints, under dynamic loads[12].

In this chapter, a refined strain rate~sensitive analytical model for R/ C ele-
ments is developed, and it is used for studying the loading rate effects on the

dynamic response of R/ C beams and columns. Practical axial and flexural design

50

51

   
  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.000-
'3 1
Q- o
3 .. y - 0.79 in/sec.
hi fi . .
a 1.500_. 7 0.0002 in/sec.
C
A
32' « P
:2 q i
“ 1.000- 5 7..“
. °f 1&5 A“ . 5.86 sq in
h
‘ ‘1' ’5- f, - 62,930 psi
500‘ / ‘ fyh "' 44,300 P81
“ I
q : a re - 3.670 psi
‘ F“ \ 1'
417.7
0 I I l ' I f r I I I r I r I j
0 0 5 1.0 1 5
AXIAL Daromrmuun)
(a) Axial Behavior
20 -
+
15 - Y\_
E Z \ A - 5 in/sec.
.8 e
" I A - 0.1 in/sec.
m 10—
0 c1
‘’5 - V
II. -l '
3 a ) As - As - 1.2 sq in
g s— ‘ [Ea ry-smoo ps1
ii 2 l ‘yh' 52,450 psi
. fé - 4,250 psi
0 IIIIITIIrl’IfIITj—IIIITII'[ITII]
0.00 0.25 0.50 0.75 1.00 1.25 1.50

LATERAL DISPLACEMENT (in)

(b) Flexural Behavior

Figure 5.1 Loading Rate Eflects on The Axial and Flexural Behavior of R/C Elements

52

techniques are also presented.

5.2 SECTION MODEL

The axial/flexural tangent stiffness matrix of R/ C cross section was derived
using the layer modeling technique[42]. In this technique, the cross section is
divided into a number of concrete and steel layers (see Figure 5.2). The strain
increments and strain rates at these layers are calculated from the values of axial
strain increment and strain rate of the section plastic centroid, and the curvature
increment and curvature rate of the section (assuming that plain section remains

plain):

d6; = dc? + d¢-y,~ (5.1)
8'3; = ép + 431/; (5-2)

where,

dei= strain increment in the i’th layer;
é,= strain rate in the i’th layer;
dcp = strain increment at plastic centroid;
ép= strain rate at plastic centroid;
d ¢= section curvature increment;
¢= section curvature rate;

yi= distance from the centroid of the i’th layer to the plastic centroid

Once the layer strains and strain rates are obtained, tangent stiffness moduli
(E5) of various layers can be calculated using the strain rate-dependent concrete
and steel constitutive models presented in chapters 2 and 3, respectively:

= df (Eiréi)

El d e,-

(5.3)

where,

53

 

A!
plastic P
centroid
(a) Applied Forces
confined
concrete layer
steel
layers unconfined

concrete layer

 

(b) Cross Section Subdivision

curvature (¢)

\

A plastic centroid
_ anal strain (ep)

\

plastic
centroid

 

 

(c) Strain Distribution

Figure 5.2 Layer Model for R/C Sections

54'

f (chef) - constitutive equation expressing the value of stress at the i’th

layer in terms of strain and strain rate.

The section stiffness matrix (K,) can then be constructed by proper summa-

tion of the layer tangent stiffness moduli:

{14}- K« {.4}- lfi: 2:194}

dM - bending moment increment at the cross section;

where,

dP - axial load increment at the cross section;

11
K11-2A;Egy;2;

i-l

fl
K12'K21 ' ZAI'EI Vii

i-l

n
K22=2AiEii
i-l

A; =- area of the i’th layer.

5.3 FACTORS INFLUENCING THE AXIAL/FLEXURAL BEHAVIOR
OF R/ C SECTIONS

The reliability of the layer modeling technique introduced in the previous sec-
tion has been confirmed through comparisons of its predictions with test results.

The typical comparisons are shown in Figure 5.3[43].

The analysis procedure described in section 5.2 was used to study the effects
of variations in section properties on its response to axial/flexural forces. The
typical beam and column cross sections shown in Figures 5.4(a) and (b), respec-
tively, were chosen as the "standard" sections in this study, and the influence of
variations in concrete and steel strengths, steel ratio, confinement and the rate of

loading on the axial / flexural response characteristics of these sections were investi-

55

 

3670 psi
C 4

1 ' i - —rl-17 5.84 sq. i
d

9 ps

010

I ” .0 2
d

 

 

 

p8

       

 

  
    
  

 

 

 

3 CI 0.0000033/806.
a, q
1... .
5 c - 3.0167/sec.
3 1 000
8
4
¢
H
X
‘

'1' ‘\

500 —J theory

" ----- test

q

.1

0
I I I I I I I I I l I I I I T I I I I l I I I Ij

0.00 0.01 0.02 0.03 0.04 0.05
AXIAL STRAIN

(a) Axial Behavior of Column

E- 0.020/uc.
10.000— é- 0.0003/uc.

 

Materiel Proper: in

fc II 4.3 to!
f I 51.8 kl!
Y

fyh - 52.5 hi

3 tie Q 3.25"
2'7

mm (kip-1n)

l-—;.1-«l

   

th
Legend: e0.“

 

l I I I I I I I _I I I I I I I l W I
0.00 0.0005 0.0010 0.0015 0.0020
CURVA‘I'URB

(b) F lexural Behavior of Beam

Figure 5.3 Typical Comparison of The Predictions with Test Results

56

Material Properties

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fv I 60 ksi
fyh I 50 ksi
f' I 4 ksi
c
1%
“fl: W 2 :7
In H 3%.. . "r— ‘3 tie
H H ‘fi: @ 6 1n
1— w --2 #7
lp—Nl_
l 9 l
' I
(a) Beam
Material Properties
fy I 60 ksi
fyh I 50 ksi
f' I 6 ksi
c
1""
< ~-—— 3 #7
0“
vi
3 “4 II— 2 #7
v; __ (‘3 tie @ 6 in
.._.3 #7
Pl

 

 

j 15

 

1

(b) Column

Figure 5.4 Standard Beam and Column Cross Section

57

gated.

5.3.1 Efl'ect of the Longitudinal Steel Yield Strength

Figures 5.5(a), (b) and (c) present the quasi-static (is S 10'6/sec) moment-
curvature and axial load-axial strain relationships and the interaction diagrams,
respectively, for sections similar to the standard ones, except for their longitudinal
steel yield strengths. It may be concluded from Figure 5.5(a) that the beam
flexural strength is strongly sensitive to the variations in longitudinal steel yield
strength (:1; 30% change in yield strength causes about :1: 30% in the flexural
strength of the standard section). The efl'ect of longitudinal steel yield strength
on the column compressive strength is shown in Figure 5.5(b) to be important,
though not as much as the corresponding effect on the beam flexural strength.
Changing the longitudinal steel yield strength by :1: 30% changes the standard
column compressive strength by roughly 4; 10%. Neither the axial nor the
flexural initial stiflnesees of the standard beam and column cross sections were
sensitive to the variations in the longitudinal steel yield strength. Figure 5.5(c),
which presents the column interaction diagrams obtained at a maximum concrete
strain of 0.003, confirms that the sensitivity of section strength to the changes in

longitudinal steel yield strength increases as the flexural action becomes dominant.

5.3.2 Effect of the Concrete Compressive Strength

Figure 5.6 shows the quasi-static axial/fiexural response characteristics of
R/C beam and column cross sections similar to the standard ones, but with
difl'erent concrete compressive strengths. It can be seen in Figure 5.6(a) that the
changes in concrete compressive strength have relatively small effects on the beam
flexural stiffness and strength. There are, however, major eflects on the column
compressive stiffness and strength resulting from the variations in concrete

strength as shown in Figure 5.6(b). Changing the compressive strength by :l: 30%

58

 

 

 

 

 

 

 

 

2.0007
1.500‘ I - 'eI-Iouoooeo
1; — —- _-
? r -------—
:3- 1.000- .-.------‘.::.—..—Oo-oo
V ..—..-oo-..
1
Y
E —" 80 hi
I 500. 7o
3 ....- 60
----- 50
.e-- 60
3 I T I I 1
090 002 00‘ 0.6 008 1.0
CUIVATUIE (x 10‘3)
(a) M - (0 curves
1.250-
\
on“ 5‘\‘\
1.000- “‘1:
5 t, Silk-121:2;
o. ooooo 70
g .0-.- 6°
-' 500- ..-.. so
i .00. 60
H
S 250-
' I I . j
0.000 0.001 0.002 0.003 0.006
AXIAL 5mm
(b) P — 6? curves

AXIAL LOAD (hips)

 

 

 

1000
MT (kip- in)

1.500 2.000 2.500 3.000

(c) Interaction Curves

Figure 5.5 Efi'ect of Longitudinal Steel Yield Strength

59

 

 

 

 

1.500-
3
L
3
g
common 3 10‘3
(a) M — <15 curves
‘3
Q
3
3
3
3
1
0.000 0.001 0.002 0.003 0.00:.
AXIAL STRAIN
(b) P — 6? curves
1.500‘W
1.000-
’3
3'
5
3
3 5004
3
i
0

 

 

 

0 500 1000 1500 2000 2500 3000

mean (kip-1n)

(0) Interaction Curves

Figure 5.6 Effect of Concrete Compressive Strength

60

results in about :l: 3% and :1: 25% variations, respectively, in the flexural and

compressive strengths ( and stiflnesses).

5.3.3 Efi'ect of the Longitudinal Steel Ratio

As shown in Figure 5.7(a), the changes in tension steel ratio have major
eflects on the beam quasi-static flexural behavior (:I: 30% changes in the ratio of
tension steel changes the flexural stiflnese and strength of the standard beam sec-
tion by about :1: 30%). The quasi-static flexural response is not, however, much
sensitive to the changes in the ratio of compression steel, as shown in Figure
5.7(b). The reduction in compression steel, however, tends to reduce the ductility
of the beam flexural behavior (that is the flexural resistance at larger deforma-
tions). Figures 5.7(c) and (d) show the axial load-axial strain relationships and
interaction diagrams of sections similar to the standard column, except for their
total steel areas. It is observed that the column axial strength and, to some
extent, its axial stifl'ness are influenced by the changes in the total area of the
longitudinal steel. The column compressive strength changes by about :1: 15% as
the total longitudinal steel ratio changes by :1: 30%. Figure 5.7(d) indicates that
the column compressive strength tends to be more sensitive to the variations in

the total steel area as the flexural forces increase.

5.3.4 Efl‘ect of Confinement

Better confinement of concrete can be achieved by increasing the transverse
steel ratio (e.g. by reducing the spacing of lateral steel), and to some extent by
raising the yield strength of transverse steel. Improved confinement increases the
compressive ductility and to a lesser extent the compressive strength of concrete.
Figure 5.8(a)'shows that the changes in concrete confinement produced by reduc-
ing the spacing of transverse'steel does not influence the beam quasi-static flexural

behavior, noticing that the analytical model used in this study does not account

mm (kip-tn)

AXIAL LOAD (klpl)

 

61'

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

q -
2500 A. 1500
*— 3.8 1112 . ._._
2°°° ‘ [er-”unnane- 3.0 '~.._'-.‘.=
T 1000*
150° ‘ 3 — 2.53 1.12
'.—- fl = 1.8 3 o ...... 2.0
'1000- .......----—-------------- 1.32 g - u 1.20
- o
5005 um
—. 0.32 5 °'”
500- --- 0.61
0 r I I I fi 0 I l I I
0.0 0.1 0.5 0.6 o a 1.0 0.0 0.2 0.4 0.6 0.8 IIfl
mum: at 10"3 mum: x 10'3
(a) M - ()5 curves (b) M — (0 curves
1500 - 2000 s
‘0: 1;
10 12 3'
' 3 1500 q
. A
1000 - 3 ° 3 ., -"“--.,s u
1.0 3 ,.v',.—-.\ s... 10.12 11:2
J / .’;"'\§.“\. ....‘“.00000.- 8 o
3.52 5 1000- ., V ~\\‘\.\ 4.0
2.1:. 5 '\~,~j§.:;:.°: 3252
500 " "" 2 u.
500 -
0 1 0 I I I 1
0 1000 2000 3000 4000 0.000 0.001 0.002 0.003 0.000
AXIAL smm

MOMENT (kip-1n)

curves

(c) P —6p

(.1) Interaction Curves

Figure 5. 7 Effect of Longitudinal Steel Area

 

62

 

 

 

 

   

 

 

1500 -
a p”
d -
5 100° —- 0.0092
5 ------ - 0.0073
‘2 --- 0.0061
g 500 - ---- 0.0052
----— 0.0046
0 I T l I “I
0.0 0.2 0.1 0.6 0.8 1.0
cunvxruu x 10-3
(a) M - d5 curves
1250 -
‘3
g- 1000-
5
3 - ' 0" “m“
3 750
i —— 0.0092
5 5001- """ ° 0.0073
---- 0.0061
250 a ---- 0.0052
-«— 0.0046
0 I l I ‘1
0.000 0.001 0.002 0.003 0.00:.
Axnu. 6mm
(b) P — 6,, curves
12501
1000
’3
.S'
:5 750-1
0'
3 __ 0.0092
-‘ 500- ....... . 0.0073
'1 —--- 0.0061
g ----- 0.0052
250- ---- 0.0066
0 I I I l ‘1
0 500' 1000 1500 2000 2500

WENT (kip-1n)

(c) Interaction Curves

Figure 5.8 Effect of Confinement

63

for the possibility of compression steel buckling. It is shown in Figure 5.8(b) that
the column quasi-static compressive strength and ductility (represented by
strength at large deformations) are slightly improved by increasing the
confinement. Figure 5.8(c) confirms that the confinement effects become slightly

larger as the compression action tends to dominate the column behavior.

5.3.5 Effect of the Loading Rate

The loading rate variations are shown in Figure 5.9 to have major effects on
the axial/flexural behavior of R/C beams and columns. The data presented in
Figures 5.4 to 5.8 were all preduced at quasi-static loading rates (maximum strain
rate in the order of 10 ‘o/sec). From Figure 5.9(a) it may be concluded that the
flexural strength of the standard beam cross section increases by about 20% as the
maximum compressive strain rate of concrete increases from the quasi-static value
to 10 ’l/sec (expected under earthquake and impulsive loads). The initial flexural
stifl'ness of the beam is not, however, significantly influenced by the changes in
loading rate. Figures 5.9(b) and (c) indicate that the element axial behavior is
more sensitive to the variations in loading rate than its flexural behavior. The
compressive strength and secant stifl'ness are increased by about 30% as the max-
imum compressive strain rate of concrete in the section is raised from a quasi-

static value of 10‘6/sec to a dynamic value of 10‘1/sec.

5.4 ELENIENT MODEL

The section stiflness matrix derived in section 5.2 was incorporated into an
algorithm for nonlinear axial/flexural analysis of R/ C cantilever elements. In
flexural analysis, the inputs to the element analysis are the increments in lateral
displacement, time, bending moment, and axial load at the free end of the cantil—

ever element. In axial analysis, the inputs are the increments in axial deformation

(i4

 

 

 

2000 T
1500 q .“OOCOOOOO’::.=:;.o.-.:.-e:o:;
A 5--‘.- 0:00—00:70.-
= “’00—..—
fl .
l. o
a 1000 '1 _ 1.5-1
"""" 105-2
E 500 J --- 1.2-3
----- 1.2-6
“‘- 1.3—6
0 I I I I 1
0.0 0.2 0.5 0.6 0.8 1.0
cunvArunr x 10'3
(a) M — (t curves
2000 q

1500 -

500

AXIAL LOAD (Ripe)
E
1

 

1250

1000

750

500.

250 -

AxIAL LOAD (Ripe)

   
 

o.ooo. 1 . E-2
-0-.. 1 . E-3
"""' 1 . 5"

-..- 1 o 5-6
I , I I j

0.001 0.002 0.003 0.006
AXIAL STRAIN

(b) P -— 6? curves

-...- 1.5-3
"'"°' 1.EI‘
-"- 1.EI5

  

 

 

I I I I I fi
500 1000 1500 2000 2500 3000
MOMENT (kip-1n)

Figure 5.9 Eflect of Loading Rate

65-

(of the plastic centroid), time, bending moment and lateral load at the free end.

The model for axial/fiexural analysis of R/ C cantilever elements was constructed

by: (a) dividing the element into a number of slices along its length as shown in

Figure 5.10(a) (the devisions being more refined near the fixed end that is gen-

erally more critically stessed), and (b) subdiving each slice to a number of steel

and concrete layers as discussed in the previous section (Figure 5.10b).

 

 

 

 

7.?

A l“

I r—__-’T:'““——:-—30‘“’1 A,
cl ; g : : 7 \5

i: g | I I \I

E. : : I i \l Pa
2' : g I l ________..J:

1| L ___1___——r—' ————‘_i'--— Me

 

 

 

T—

(a) Division of Element by Slices

,1; unconfined
concrete layer

steel
layers

confined
concrete layer

 

section A —A

(b) Subdivision of Slices by Layers

Figure 5.10 Layer Model for R/C Elements

 

66

The numerical algorithm for inelastic flexural [or axial] analysis of R/C can-

tilever elements is presented below. This method accounts for the loading rate

effects as well as the material and geometric nonlinearities.

(1)

(3)

(4)

Input the incremetal values of free end lateral displacement ((1 A,), time (dt),
bending moment (dMe), and axial load (dPe) for flexural analysis [or lateral
load (dV, ), time (dt ), bending moment (dMe ), and axial displacement (d 7,)
for axial analysis]; add these increments to the values at the end of the previ-

ous time step to get the total amounts.

Estimate the total value of lateral load (Ve) at the free end for flexural
analysis [or the total value of axial force (Pa) for axial analysis] at the com-

pletion of the current time step.
For each slice:

(a) calculate the values of axial force and bending moment at the end of
current time step resulting from the application of the input (and
estimated) values of end flexural, axial and lateral forces. The contribu-
tion of axial load to the bending moment (P-A effect) should be con-

sidered.

(b) using the strain rate-sensitive section analysis formulations presented
earlier, obtain the values of curvature and plastic centroid axial strain
corresponding to the values of applied bending moment and axial load

at a specified loading rate.

Calculate the end lateral displacement in flexural analysis [or axial displace-
ment in axial analysis] of the element corresponding to the estimated values
of end forces by proper integration of the curvatures in flexural analysis [or
plastic centroid axial strains in axial analysis] of slices along the element

length.

-67.

(5) Compare the calulated value of end lateral displacement in flexural analysis
[or end axial displacement in axial analysis] with the corresponding input
value. If the error is unacceptable, revise the estimated value of lateral force
[or axial load], and repeat the procedure from step (3). Otherwise, the
analysis for this time step is complete, and the end values of rotation and
axial displacement in flexural analysis [or ratation and lateral displacement in
axial analysis] can be obtained using the curvatures and plastic centroid axial
strains of slices along the element length. The analysis can then be continued

for the next time step starting from step (1).

5.5 CONIPARISON WITH TEST RESULTS

The developed strain rate-sensitve methods for axial / fiexural analysis of R/ C
elements was used to predict the results of some tests, performed at different load-

ing rates, on R/ C columns[4,44] and beams[28].

Reference 14 has reported the results of column tests in which a concentric
axial load was applied at a constant displacement rate. The rate of displacement
increase varied in different tests. Figures 5.11(a) to (d) compare the experimental
axial load-displacement relationships reported in References 4 and 44 with the
predictions of the axial analysis procedure introduced in section 5.4. It can be
seen in these figures that the loading rate effects on axial behavior of R/C
columns are significant, and the developed theoretical approach is capable of

closely predicting the test results.

Experimental lateral load-deflection envelope curves for R/ C beams loaded at
different displacement rates are reported in Reference 28. Figures 5.12(a) and (b)
compare these test results with the predictions of the suggested strain rate-

sensitive flexural analysis technique. The proposed analytical approach is

68

 

 

 

 

 

 

 

 

 

 

 

  

2500-»——~ ~—-— ~-—~——~--—- ,, —.. - --- _. ._. . _ _ - _._,
n]
-]
fi
1 /’—~‘\
2000-4 I ‘~...~~
/"\ " \ ~~~~~~~
s. i - \~~-~~-\: ~~~~~~~~~~
:32 : ’7 ~ 0.79 in/sec “‘\ T‘-‘.__.
v 1500-1 N
D ‘.
g l
.4 +
__J 1000.: 1.17
E 4 12 020 \ —-1 C11."
"C j 5 r’ V ‘T
5004] E \ I V 010
“ 17.7 in.
4 -- TEST (l=47.2 in) L———l
‘4 —— THEORY A
0 I T fi m
0.0 0.5 1.0 1.5 2.0
AXIAL DEFORMATION (in)
(a) Test Results from Reference 4
l
l.
’«T g
.9 3
x 2
V «\ ‘
c3 \ “~\ g
S “"\‘—,\
.—J \\ \ l
-’ ‘ :
<( 1.17 i
;<— .1 D c 3670 psi ‘
< 71.”- It 5.61 eqdn.

 

 

 

 

 

57130 psi 3

/ mo ;

l

-- TEST (l=47.2 in) I 17.7 111.] l

—— THEORY 1

0‘! T T I *1
0.0 0.5 1.0 1.5 2.0

AXIAL DEFORMATION (in)
(b) Test Results from Reference 4

Figure 5.11 Experimental and Analytical Axial Performances of R/ C Columns
at Dzflerent Loading Rates

69

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7.
l
l
~‘\\‘ !
‘~~- s
1; -\ “~\-- I
‘9- /JC '\\‘ “ ‘u‘ :
é \\- ‘ l
o \\
<
O
_l
2" are
'>'<'
4: I &
zoo-ll} .21]:
] -- TEST (l=-4-0 in)
—- THEORY
01 T I r
0.0 0.5 1.0 1.5 2.0
AXIAL DEFORMATION (in)
(c) Test Results from Reference 44
1000 fi
‘ 1.0 ]
i ’\‘\\ 8“\ 1min f; ‘900 pet ]
800-1 " ‘\\ é . ' A“ 3.33 001.10. E
A I" j T N f, 71000 p91 ‘
a _] ]: 7§\0 \ \ —' a. 0.0121 g
"" .l mOO;‘ ,\ . ’1. “000 pet ]
5 5004 l,’ % .\_ 3
Q I. \\\ ilk \ I
g i I ‘\ \\\ I
_.I -] " \\\ \“‘~\ !
2" 400 :l \‘\‘ \\¥_‘ ]
Q i .' \‘\ l
< I ‘\\‘ ]
2004/.' xx .
’l \
,l

T r r
0.0 0.5 1.0 1.5

AXIAL DEFORMATI ON (in)
(d) Test Results from Reference 44

Figure 5.11 Experimental and Analytical Axial Performances of R/ C' Columns
at Diflerent Loading Rates(cont’d)

7O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.30 1'
4 —- THEORY
d -- TEST (A- 6.0 3.1/sec)
1,? 4 ‘1
.9 4 - /""‘T‘,
x A -“~ In ‘ ‘
v 204 ”w , .2] ”4:: _____________
u.I LAB 1.. ‘ r 9 I .
g -I f 1...] ,.”l ;
-I ‘ I ;
j i /' i” l
a: IO-l «III Huerta] Pronruu
Lu 4 ///’ H I' - AJ u: i
'— // 207 c g
S _] ;,’ 5” . .‘nb—I3t1006” (y - SI... b1 ]
.4 /’/ 157 I". ' $2.5 hi ;
1 1—‘ i
O J
T T T f W
0.0 0.1 0.2 0.3 0.4- . 0.5
LATERAL DISPLACEMENT (In)
(a) Test Results from Reference 28
.30
g — THEORY.
4 -- TEST ( A- 0.1 in/soc)
"a? I
Q. .l I
'2 ~ @321 «r =
v
LIJ .I 9 C~_~
o ,I ...................
95 J x
“- J . I
2% l I
0: 104 I/‘L/ Iurerul Proportion
E -] 4:.” $1 2” f; - L7 m ;
5 ul .,// ls” ' . lib—'3 £1386" f, - 51.8 ”I. ]
l” -' f - 52.5 ks! I
-l ;5’ 1 no: 2'7 Yb 1
.I / 1.9.5.1 5
o I T r n s.
0.0 0.1 0.2 0.3 0.4- 0.5

LATERAL DISPLACEMENT (In)
(b) Test Results from Reference 28

Figure 5.12 Experimantal and Analytical Flexural Performances of 13/0 Beams
at Different Loading Rates

71

observed to compare satisfactorily with test results.

5.6 PARAIVIETRIC STUDIES ON ELENIENT BEHAVIOR

In this section, the effects of the variations in different material and geometric
properties of R/C columns and beams on the sensitivity of their axial/flexural
response to strain rate variations are discussed. The following discussion first con-
centrates on the column axial behavior and then covers the beam flexural perfor-

m ance.

5.6.1 Axial Behavior of Columns

The typical column shown in Figure 5.l3(a) was chosen as the "standard"
column in this study, and the effects of the variations in different properties of
this column on its strain rate-sensitivity were investigated by the developed
analytical technique. Figure 5.13(b) presents the axial load-displacement relation-
ships of the standard column derived analytically at four different displacement
rates (covering the range from quasi-staic to impulsive loading rates, noticing that
even higher loading ratio might occur under impulsive loads). It is obvious in this
figure that axial strength and stiffness of the standard column increases with
increasing rate of load application. At the highest loading rate, an increase of
30% in column axial strength over the quasi-staic value is observed. The
corresponding increase in column secant stiffness (which is measured at 45% of its
peak axial strength) is also about 30%. The other observation in this figure is
that with increasing loading rate, the deformation at peak axial load is slightly

increased.

The effect of loading rate variations on the column axial strength and secant

stiffness are summarized, among other information, in Table 5.1. In this table,

72

10 bars

Material Properties:

 

 

 

 

 

 

 

 

 

 

e
2 5y - 60,000 psi (414.0 MPa)
O -
I-d fyh 60,000 psi
fé ' 4,000 psi (27.6 HPa)
dry and normal weight
1//8910(Ast - 10.16 sq in)
r"#4 hoops @ 4 in
o 'J.‘- (as - 0.02082)
N , .
.h‘
Zefimuf's
(a) Standard Column
4, '—
000 L
I-
’a'.‘ 3,000 __
2' .
:5 b y - 50.0 in/sec.
g : I - 5.0 in/sec.
o o
a t Y ' 0.5 in/sec.
.3 _. .
5 2,000 Y . 0.0005 in/sec
3‘5 .
1,000 '—
0 l j J l 1 l l 1 I l I l I 1 l i l L L 1 i l 1 I
0 0.1 O 2 0.3 0 4 0 5

AXIAL DEFORMATIQN (in)

(b) Axial Behavior of Standard Column

Figure 5.13 Loading Rate Eflects on The Axial Behavior of R/C Columns

73

Table 5.1 Summary of The Loading Rate Eflects on The Axial Behavior
of R/ C Columns

 

I

 

 

 

 

 

 

 

 

 

 

 

 

 

,7 Pn /Pn K1 /K'1

SECTION . refined practical refined

(‘0 /sec) analysis Eqn. (5.6) analysis
0.5 1.16 1.16 1.22
standard 5.0 1.25 1.25 1.28
50.0 1.36 1.35 1.31
0.5 1.18 1.17 1.23
low 1, (I. =- 40 Im‘ ) 5.0 1.28 1.27 1.28
50.0 1.40 1.39 1.37
0.5 1.12 1.13 1.23
high f, (I. = 80 Im‘) . 5.0 1.20 1.21 1.29
50.0 1.31 1.30 Lifl
0.5 1.16 1.16 1.23
low 21,, (A. - 8.0 in”) 5.0 1.26 1.25 1.29
50.0 1.37 1.37 1.32
0.5 1.15 1.15 1.22
high A... (A. - 12.48%”) 5.0 1.24 1.24 1.27
50.0 1.35 1.34 1.30
0.5 1.14 1.16 1.23
low confinement ( p. = 0.0139 ) 5.0 1.23 1.25 1.28
50.0 1.35 1.35 1.31
0.5 1.17 1.16 1.23
high confinement ( p. - 0.4164 ) 5.0 1.25 1.24 1.26
50.0 1.37 1.35 1.31
, 0.5 1.16 1.15 1.21
low I. (I: - 3 Im‘ ) 5.0 1.25 1.24 1.26

50.0 1.36 1.34 129—,

. . I , , 0.5 1.15 1.16 1.24
11‘3“ f. U: =' 6 ’m l 5.0 1.25 1.26 1.30
50.0 1.31 1.37 1.33
0.5 1.39 1.37 1.51
saturated concrete 5.0 1.69 1.65 1.78
50.0 2.07 2.00 2.05
0.5 1.23 1.16 1.15
circular shape ( D = 22-53 in l 5.0 1.29 1.25 1.24
50.0 1.31 1.35 1.36

 

 

 

 

 

 

74

P" and P", are the quasi-static and dynamic axial strengths, and K 7 and K :1 are
the quasi-static and dynamic secant axial stiffnesses, respectively. Table 5.1 also
summarized the strain rate effects on columns similar to the standard column,
except for the change in a single variable which could be confinement, concrete
compressive strength, degree of saturation of concrete, or cross sectional shape.
From the data presented in Table 5.1, it may be concluded that the variations in
steel ratio, concrete compressive strength, confinement and cross sectional shape of
columns do not significantly influence the column loading rate-sensitivity. There
is, however, a slight increase in the loading rate-sensitivity of column axial
strength with decreasing yield strength of steel. Both the axial strength and
stiffness of the saturated concrete column are also observed to be far more loading

rate-sensitive than those of air-dried concrete columns.
5.6.2 Flexural Behavior of Beams

In the sudies on strain rate-sensitivity of the R/ C beam fiexural behavior, a
typical R/ C ‘beam shown in Figure 5.14(a) was chosen as the "standard" one. Fig-
ure 5.14(b) shows the lateral load—displacement relationships of this beam derived
analytically at four diffrent displacement rates. The fiexural strength, but not the
initial stiffness of the beam, is observed to increase at higher loading rates. The
maximum increase in fiexural strength is about 15% at the highest rate of lateral
displacement application used in this study. Table 5.2 summarizes the loading
rate effects on the lateral strength of the standard R/ C beams as well as beams
similar to the standard one except for the change in a single variable which could
be steel yield strength, concrete compressive strength, the ratio of compression or
tension steel, confinement, degree of concrete saturation, or the beam length. In
this table, Va and V", are the quasi-static and dynamic values of lateral strength,
respectively. From the data presented in Table 5.2, it can be concluded that R/ C

beams with lower yield strength steels and higher tension to compression steel

'75

#3 stirrups @ 6 in

- \
-.1/103 0.011, )¥

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 2#7 53 in

Material Properties: fy . 52,400 psi (361.3 MPa)
fyh- 50,000 psi (344.8 MP3)

   

fé - 2,660 psi (18.2 HPa)
dry and normal weight

(a) Standard Beam

 

 

20
C"
. é - 5.0 in/eee
_ 9 - 1.0 in/see.
9 - 0.1 in/sec
15 — A - 0.000001 111/5
’3
s- l
:5 r 2
fl " /
8 10 _ '
2 .
A l-
5 _
e-
j .
5 __
P
l
0 1 I i 1 l 1 l l 1 l 1 I 4L1 l l
0.0 0.5 1.0 1.5

LATERAL DISPLACEMENT (in)

(b) Flexural Behavior of Standard Beam

Figure 5.14 Loading Rate Eflects on The Flexural Behavior of R/C Beams

76

Table 5.2 Summary of The Loading Rate Eflects on The Flexural Behavior

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

of R/C Beams
A Vn /Vn Vn / Va
SECTION refined practical
(in /sec) analysis Eqn. (5.7)
0.1 1.08 1.09
standard 1.0 1.11 1.13
5.0 1.15 1-18
0.1 1.11 1.10
low I, (I. =- 40 Im') 1.0 1.14 1.15
5.0 1.17 1.21
0.1 1.08 1.08
high I, (I. - 60 (mi) 1.0 1.10 1.12
5.0 1.13 1.16
, 0.1 1.08 1.09
low A, (.41 20.61012) 1.0 1.12 1.13
5.0 1.17 1.18
0.1 1.10 1.09
high A, (A. - 2.0 in”) 1.0 1.15 1.14
' 5.0 1.18 1.18
, 0.1 1.07 1.09
high confinement( P. = 0.016 1 1.0 1.10 1.13
5.0 1.15 1.18
, , 0.1 1.07 1.09
high I. (I. =6 In") 1.0 1.12 1.13
5.0 1.15 1.17
0.1 1.09 1.09
saturated concrete 1.0 1.13 1.12
5.0 1.17 1.16
( flange width = 0.1 1.08 1.07
T - beam 18 in ) 1.0 1.11 1.12
5.0 L15 1.16
0.1 1.09 1.10
short beam (1 = 40 in ) 1.0 1.12 1.14
5.0 1.15 1.19

 

 

77

ratios are more sensitive to the variations in loading rate. Beams with saturated
concrete are also more loading rate-sensitive than those with air-dried concrete.
The variations in concrete compressive strength and confinement, element length,
and cross sectional shape do not seem to influence the loading rate-sensitivity of

the fiexural behavior of R/ C beams.

5.7 PRACTICAL DESIGN PROCEDURES

The layer modeling technique developed in this chapter for loading rate-
sensitive axial / flexural analysis of R/ C elements is not convenient for every day
use by designers. This section illustrates two simple design techniqes developed
for computing the axial and fiexural strengths of R/C cross sections as functions
of the applied deformation rates. The results of these simple techniques are also
compared with those obtained from the more refined layer analysis procedures

described in section 5.2.
5.7.1 Dynamic Axial Strength
ACI code (318-83)[9] suggests the following equation for calculating the nomi-

nal axial strength (Pu) of R/ C columns:

Pa = 0'85 fol (Ag _ Ast) + Ast fy (5'5)

where,

A9 = gross cross sectional area of concrete;

A at = total area of longitudinal steel.

The above equation has been based on the results of quasi-static tests. This
equation can be modified as follows for calaulating the column axial strength at

high loading rates (Pu, ):

P,,' = 0.85 1,” (A9 —A3,) +213, f!" (5.6)

78

where,

f c” = dynamic compressive strength of concrete (psi);
f
[C] [1.48 + 0.160 logloé + 0.0127(logloé)2]

for air-dried concrete

1; [2.54 + 0.580 logloé + 0.0543(logloé)2]

for saturated concrete

 

L

f; as dynamic yield strength of steel (psi);
- /, [—4.51x10“6 1, + 1.46 + (—9.2OX10-7 j, + 0.0927) logloé];
5:: strain rate (1 /sec) 2 10"5;
fy- standard (quasi-static) yield strength of steel (psi).
Table 5.1 compares the ratios of dynamic to static compressive strengths
obtained from the above equation with those obtained from the refined (layer)
analysis procedure described earlier. The suggested simple approach is observed

to compare well with the test results of the refined analysis.
5.7 .2 Dynamic Flexural Strength

The ACI code (318-83)[9] fiexural design procedure is based on simplified ulti-
mate flexural strain and stress distributions at cross section shown in Figure 5.15.
It should be noticed that the tension steel area in this approach is assumed to be
small enough (i.e. below balanced area) such that the tension steel yields at the
ultimate fiexural condition. In order to develop a simple procedure for computing
the dynamic fiexural strength of R/C beams, it was assumed that the absolute
values of strain rate in concrete under compression and steel under compression
and tension are equal to the rate of straining at the extreme compression fiber.
This assumption leads to the following procedure for calculating the loading rate-

sensitive fiexural capacity of a doubly reinforced beam cross section:

(1) Assume a value for the depth of neutral axis (c in Figure 5.15).

 

79

(2) Find steel stresses:

I

d

In compression, f“ = 87,000£;c—— _<_ f I

y
, I
In tenswn, f y

(3) Find the new value of c for the above steel stresses:

0 = As [3; -Asl fee

 

0.85 f,” b
c = .1.
:81
where,
0.05(],’ ’ - 4,000)
3,: 0.85 - but, 0.65 g s, _<_ 0.85

 

1,000

(4) Repeat from step (2) with the new value of 0 until the change in c is real-

tively small.

(5) Find the strain rate-dependent fiexural strength of the section:

M;= A8 fy'(d - d’) + 0.85 f,” ab(-;— — d’) (5.7)

b . 0.003 0.85 f."

‘1
‘1

 

f...

A.’ ’° '
d ___.".",-—--——_-—-l—— - -_4L -—-—1V.A.

 

 

  
 

 

 

 

 

 

 

«2'
A.
il— 0 o o - - — - f.’
' Strain Distribution Stress Distribution
At Failure At Failure

Figure 5.15 Simplified F lexural Strain and Stress Distributions at Ivauure

80

The value of strain rate at the extreme compressive fiber (6) used for calcu-
lating I; and fen in the above equation should be obtained from the given rates

of displacement application to the beam, using approximate analytical techniques.

The suggested procedure for calculating the loading rate-dependent flexural
strength of R/ C sections was applied to all the beams used in the numerical study
discussed earlier. The results are compared with the predictions of the refined
(layer) modeling technique in Table 5.2, and the comparison is observed to be

satisfactory.

5.8. SUWARY AND CONCLUSIONS

Empirical strain rate-sensitive constitutive models of steel and concrete were
incorporated into an analytical procedure for axial/flexural analysis of R/ C
beam-columns at diflerent loading rates. The analytical predictions compared well
with quasi-static and dynamic test results. Simple practical analysis procedures
were also developed for calculating the axial and flexural strengths of R/C

columns and beams as functions of the applied loading rates.

The refined (layer) analytical method was used to perform a numerical study
on the sensitivity of the axial and flexural performances of R/ C sections and ele-
ments to the changes in steel and concrete strengths, concrete confinement and the

rate of load application. The results of this numerical study indicated that: .

(a) Variations in longitudinal steel yield strength have major effects on the
flexural strength, and some important effects on the axial strength of the typ-
ical beam and column cross sections considered in this study (i 30% varia-
tions in the longitudinal steel yield strength causes changes of the order of :1:
30% in flexural strength and :l: 10% in axial compressive strength). The

axial and flexural stiffnesses, however, are not much influenced by the varia-

(d)

(f)

(8)

-81

tions in yield strength.

Variations in the concrete compressive strength have relatively small effects
on the beam flexural behavior, but major eflects on the column compressive
stiffness and strength. For the typical beam and column cross sections stu-
died here, :I: 30% changes in concrete compressive strength resulted in varia-
tions of about :1: 3% and i 25%, respectively, in the flexural and compressive

strengths (and stiffnesses).

The flexural behavior of reinforced concrete beam sections is strongly
influenced by the variations in the tension steel‘ratio, but not much by the
changes in compression steel ratio. A :1: 30% change in tension steel ratio
results in about :1: 30% change of a typical beam flexural stiffness and
strength. The axial strength and stiflness of R/ C column cross sections are
also found to be sensitive to the variations in their total steel area. Changes
of the order of :1: 15% were observed in strength of a typicaal column as a

result of :l: 30% changes in the total steel area.

Increased confinement does not have major effects on the beam flexural
behavior (neglecting the possibility of compression steel buckling). It, how-

ever, slightly increases the compression strength and ductility of the column.

Under typical impulsive loading rates, the axial strength and stiffness of typi-
cal R/ C columns increase by about 30% over the corresponding quasi-static

values.

The flexural strengths of typical R/ C beams increase by about 15% at typi-
cal impulsive loading rates over the quasi-static values. The flexural
stiflnesses of R/C beams is not much influenced by the loading rate varia-

tions.

The loading rate-sensitivity of R/C column axial behavior increases with

decreasing yielding strength and increasing degree of saturation of concrete.

(h)

82

The variations in steel ratio, concrete compressive strength, confinement and
cross sectional shape of columns do not significantly influence the column

loading rate-sensitivity.

The loading rate-sensitivity of R/C beam flexural behavior increases with
decreasing steel yield strength and increasing steel ratio and concrete degree
of saturation. The variations in concrete compressive strength, confinement,
and element length and cross sectional shape do not significantly influence

the loading rate-sensitivity of R/C beams in flexure.

CHAPTER 6

INELASTIC MODELING OF REINFORED CONCRETE ELEMENTS
UNDER CY CLIC FLEXURAL AND SHEAR FORCES

6.1 INTRODUCTION

Total lateral displacements in reinforced concrete beams result from the
flexural and shear deformations. Each of these displacement components has dis-
tinct hysteretic characteristic (Figure 6.1) [6,28]. The shear hysteresis is dis-
tinguished from the flexural one by severe stiflness and strength deteriorations and
lower energy dissipation. Shear deformations, due to their deteriorating nature,
tend to gradually dominate the lateral response of R/ C beams to repeated inelas-

tic load reversals.

The available analytical models for predicting the hysteretic behavior of R/ C
beams are generally based on the assumption that flexure fully dominates the ele—
ment behavior[32]. These models can not properly idealize the inelastic shear
deformation, and this generally leads to poor comparisons between the analytical
and experimental hysteretic curves, especially after large inelastic cyclic deforma-

tions in elements subjected to relatively high shear stresses.

A practical model for predicting the hysteretic behavior of R/ C beams has
been developed in this chapter. The model accounts for the distinct hysteretic
characteristics of shear and flexural deformations, and its predictions compare well
with test results performed on R/C elements with a variety of geometric and

material properties.

83

84

Force Force

 

Displ. Displ.

 

 

Figure 6.1 Lateral Displacement Components of R/ C Beams, and
Their Hysteretic Characteristics

 

 

I]: Deflection

\ \\\\ \KLXX

 

Total
Shear

 

C—ycle

 

 

 

 

Figure 6.2 Contrtibution of Shear Deformations to The Overall Element
Displacements under Cyclic Loads

85

6.2 BACKGROUND

In this section, first the hysteretic characteristics of R/ C beams observed in
test results will be discussed, and then the available models for these characteris-
tics will be critically reviewed. The section will conclude with a brief discussion
on the influence of element models on the overall seismic response characteristics

of R/ C structures.

6.2.1 Test Results

Reference 33 has compared the histories of shear and flexural displacements
in a beam under cyclic loads (Figure 6.2). The results confirmed that the shear
deformations, due to their deteriorating nature, tend to dominate the behavior at

later cycles.

The beam inelastic deformations are generally concentrated in the critical
regions near the beam-column connections, where the flexural and shear forces are
both relatively large. The performance in these regions is marked by the appear-
ance of flexural and shear cracks (Figure 6.3a) which intersect the opposite cracks

under load reversals (Figure 6.3b) and from vertical slip lines[45].

The relatively small fiexural degradations observed under cyclic loads are
usually attributed to the reduction in stiffness during the closure of cracks (that
have opened under loads in the opposite direction), the Bauschinger eflect in steel
bars, and buckling of the compression reinforcement[45,46]. The relatively large
shear deteriorations are caused by closure of the diagonal cracks that remain open
when the loading in the opposite direction is removed, sliding shear along the
vertical slip lines shown in Figure 6.3(b), and the loss of shear transfer by truss

action resulting from spalling of the concrete in critical regions[6,8,28,46].

The R/C beam hysteretic characteristics are influenced by the element shear
stress level (or shear span-to—depth ratio), longitudinal steel percentage and

configuration, and transverse steel ratio. In general, the stiffness and strength

 

86

 

Ml Tl
W ll” l
E. . Li

(a) Under Monotonic Load (b) Under Cyclic Load

(ii if .

 

 

 

 

Figure 6.3 Typical Crack Patterns in Critical Regions of R/C Elements

Load Load

2%
Displ. flV Displ.

(a) Low Shear Stress Level (b) High Shear Stress Level

 

 

Figure 6.4 Effect of The Shear Stress Level on The Overall Hysteretic
Characteristics of R/C Beams

87

deteriorations are more pronounced in elements with higher shear stress
levels[7,45,46](see Figure 6.4). An increase in the percentage of longitudinal steel
increases the flexural stength and consequently the maximum shear stresses that
can be applied in R/ C beams. This increase in shear strength seems to be the
main reason for the inferior hysteretic behavior of elements with higher longitudi-
nal steel ratios[45,46]. The configuration of longitudinal steel also influences the
element hysteretic characteristics. With the ratio of top to bottom steel areas
approaching unity, the flexural hysteresis is observed to become more stable[46].
Finally, the increase in transverse steel ratio generally improves the element hys-
teretic behavior by increasing the shear-friction resistance, confinement of the con-
crete core, and by providing support against buckling of the longitudinal

bars [7,45] .

6.2.2 Analytical Models

This section briefly reviews the common physical idealizations for simulating
the flexural and shear behaviors of R/ C beams, and then presents the hysteretic

models commonly applied to R/ C elements.

The flexural behavior of R/C beams has been physically idealized by the
layer modeling technique[16] (Figure 6.5a), the multi—spring models[32] (Figure
6.5b), the single component simulation[8] (Figure 6.5c), the parallel multi-
component models[47] (Figure 6.5d), and the single component models with distri-
buted inelasticities[32] (Figure 6.5e). The physical idealizations of shear behavior
in R/C beams have generally been based on a rotational simulation[48] (Figure
6.6a), a truss simulation[8] (Figure 6.6b), or a single spring simulation[8] (Figure
6.6c) of the performance. Many investgators have neglected the shear deforma-
tion, and some have assumed that the ratio of shear to flexural stifinesses in the

inelastic range is constant and equal to this ratio in the elastic range[28,32].

88

 

is—rs—e—c—sl

steel layers

(a) Fiber Model (b) Multiple Spring Model

 

elasto-plastic component

is 4: ff 2

P

 

 

elastic component

(c) Single Component Model ((1) Multi-Component Model

inelastic zone

elastic zone

/ l
'—_L_ J—_'l

 

 

“‘ “

 

 

\X‘fl

(e) Single Component Model with
Distributed Inelasticities

Figure 6.5 Physical Models for Simulating The F lexural Behavior of R/ C Beams

89

assumed hinge

/

m\\xxxx\xxxxxm

 

‘ shear displ.
F-

 

 

(a) Rotational Idealization

plastic hinge length
IL :1k

1L

"I.

 

 

 

Ir
_J

 

 

 

l

I

I
.L.

I
I""'"

 

 

(b) Truss Idealization

flexural spring

///

Wt

 

382%

(0) Spring Idealization

Figure 6.6 Physical Models for Simulating The Shear Behavior of R/C Beams

90

The physical models presented above generally require certain hysteresis rules
for predicting the element behavior under cyclic loads. A wide variety of hys-
teresis models (or rules) are available in the literature, a number of which are

presented in Figure 6.7 [8,32,37,48-54].

The avialable analytical models for predicting the hysteretic behavior of R/C
beams, usually consisting of a physical model (Figures 6.5 and 6.6) and some hys-
teretic rules (Figure 6.7), generally disregard the significant increase in shear
deformations caused by repeated inelastic load reversals. A very popular (and
typical) model proposed in Reference 55 has been based on a single component
physical idealization of flexural behavior (Figure 6.5c) and the Takeda hysteresis
rules (Figure 6.7c). The shear deformations are accounted for in this model by
assuming that the ratio of the shear to flexural deformations is constant in both
the elastic and inelastic ranges. This ratio is, however, relatively small in the elas-
tic range and it has been observed to increase (due to the more severe deteriora-
tions of the shear resisting mechanisms compared with the flexural ones) under
inelastic load reversals. Disregard for this increase in shear deformations is shown
in Figures 6.8(a), (b), and (c) to lead to major discrepancies between the analytical
and experimental results. These discrepancies are observed to be more pronounced
for beams with smaller shear span-to-depth ratios (and consequently higher shear

stress levels).

The analytical predictions of the inelastic seismic response characteristics of
R/ C frames have generally been based on element models incapable of accurately
predicting inelastic shear deformations. Large deviations from test results, similar
to the one shown in Figure 6.8, are expected in the analytical predictions of these
element models. This shortcoming can produce large discrepancies in the
predicted response of frame structures (especially those with relatively deep ele-

ments) under severe seismic excitations. The disregard for the deteriorations in

91

 

 

 

’7 /
Z f U/x

(a) Bilinear Model (b) Clough Model[49] (c) Takeda Model [50]

 

 

 

 

 

 

(e) Origin-Oriented Model[8]

,3!

strength decay H) '
guidelines

i /.
4 ” x

 

 

 

\N

(f) Sinha Model[53] (g) Model with Strength Decay[32]

Figure 6.7 Examples of Hysteretic Models

92

 

 

   

(h) Kustu Model for Flexure[48] (i) Kustu Model for Shear[48]

 

 

 

 

 

 

(l) Ramberg-Osgood Model [51]

 

 

 

 

 

F F
/ ’1
7/ 5(—
(1) Cubic Model[54] (m) Modified Cubic Model[37]

Figure 6.7 Examples of Hysteretic Models(cont’d)

93

 

 

 

 

 

 

 

 

 

 

 

 

 

20
-- TEST
J
A 10
3 'l
3 I
B
o O
_I
"E I
.3
3 -104
1
‘20 f n F r
-3.0 -2.0 . -1.0 0.0 1.0 2.0 . 3.0
End Displacement (in)
20
— THEORY A
A 10 //
8. * l
3 J ’
"5
3 ° I /
E l
B Z
3 -10“ {I ’
/ W
.. /%
"'20 r f T I
-:‘.l.0 ~2.0 -1.0 0.0 1.0 2.0 3.0

 

End Displacement (in)

Figure 6. 8(a) Comparison of Experimental and Analytical Cyclic Load-Deflection
Relationships for R/ C Beams with large a/d

94

 

 

 

 

Lateral Load (kips)

 

 

 

 

 

T T I
~ll.0 -2.0 -1 .0 0.0 1.0 2.0 3.0
End Displacement (in)

 

 

 

 

 

 

 

 

20
-- THEORY
A .0. //

3 ,, / I
’3 J I l
V l
.. l I

U l
.8 0 l —-—1
E 1 // l
3 // I
3 “'1 O-l {/I” l

7% ‘
‘20 _ T I T r i
-.‘.l.0 -2.0 -1 .0 0.0 1.0 2.0 3.0

End Displacement (in)

Figure 6.8(b) Comparison of Experimental and Analytical Cyclic Load-Deflection
Relationships for R/ C Beams with medium a/ d

 

—- TEST

 

 

 

\\
m.
‘l

\l

Lateral Load (kips)
O
l
I
!

 

 

 

 

 

 

 

 

 

 

 

 

r T i T
-’ .0 -2.0 -1.0 . 0.0 1.0 2.0 3.0
End Displacement (in)
20 j
—- THEORY :
1//’
A 104 //
3 //
.3
V -l /
'U l,
8 o / l i _4
_1 /
is + ”v
4.. . /
/’,,/
3 -10-l ”I
%W
4 4%
‘20 f I - 1 r
-1 .0 ~2.0 -1.0 0.0 1.0 2.0 3.0

End Displacement (in)

Figure 6.8(c) Comparison of Experimental and Analytical Cyclic Load-Deflection
Relationships for R/ C Beams with small a/d

96

shear stiffness and strength have been observed to result in increased deforma-
tions, longer periods of vibration and premature failures in actual structures when
compared with the idealized ones under earthquake ground motions. Hence, it is
important to develop element models which account for the inelastic shear defor-
mations in predicting the hysteretic behavior of R/ C beams, in order to achieve
more reliable predictions of nonlinear seismic response characteristics of R/C

structures.

6.3 THE PROPOSED ELENIENT MODEL

The element model developed in this study, which consists of a physical
idealization of the element and hysteretic rules for flexural and shear force resis-

tance mechanisms, is described below.
6.3.1 Physical Idealization

Figure 6.9 presents the physical model of R/ C beams used in this study. In
this model, all the shear and flexural elasto-plastic deformations under seismic
forces are assumed to be concentrated at dimensionless end springs. In order to
differentiate between the hysteretic rules of shear and flexure, distinct springs
representing each of these types of behavior are attached to the beam. The

behavior of each spring is defined by a skeleton curve and a set of hysteretic rules.

The model shown in Figure 6.9 is a refined version of the single-component
model. It is practical (economical), because it involves limited number of degrees
of freedom. A comprehensive review of the literatures also indicates that its accu-
racy in predicting the flexural response is comparable to that of the more complex
ones like the multiple spring model (Figure 6.5d) and the single component model
with distributed inelasticies (Figure 6.5e). The key advantage of the model pro-
posed in Figure 6.9 is, of course, its capacity to account for the distinct hysteretic

characteristics of shear and fiexural deformations which leads to major

97

improvements in its accuracy as the shear deformations tend to dominate the

behavior in later inelastic load cycles.
6.3.2 Hysteretic Rules of Flexure and Shear

In order to form the tangent stiffness matrix of the element model presented
in Figure 6.9, the tangent stifiness of each of the springs needs to be derived
(using its force-deformation relationship) as a function of the level and history of
loading. In the following, first a general set of hysteresis rules capable of simult-
ing a variety of hysteresis characteristics (by proper selection of the variables
involved) are introduced. Then the hysteretic parameters for flexural and shear
types of behavior, which have been derived empirically, are presented. In this
presentation, the spring forces (F) might be flexural or shear force, depending on
the type of the spring, and the spring deformation (X) might be fiexural rotation
or shear deformation. The cyclic force-deformation model presented below con-

sists of a skeleton curve and a set of hysteresis rules.

A bilinear curve (Figure 6.10) has been selected in this study, mainly due to
the fact that the more complex trilinear and cuvilinear ones do not significantly
improve the accuracy of the model[8,32,49]. The initial loading takes place
linearly with the stiffness K; up to the yield force Fy, at deformation Xy.

Thereafter the stillness drops to a strain hardening value of K}, .

Unloading before yielding takes place on the loading curve (with a stiffness
equal to K,). In the post-yield region, the unloading stiffness (Kn) will be decided
using the parameter a (see Figure 6.10). This parameter should be derived empir-

ically for each spring type.

The rule for reloading towards the skeleton curve in the opposite direction is
illustrated in Figure 6.11. The point on the skeleton curve (R) towards which the
reloading occurs is defined by a parameter fl that can have any value greater than

zero.

98

fiexural spring

shear spring

// rigid segment

\

mid-length inflection point

Figure 6. 9 Proposed Physical Idealization of Element

 

 

Force(F)
Fr ---- A” I
l I
l I
l I
: I
l I
l K
K.- : ' IK-
JI .‘
Xv LEE}. Deformation(X)
l Y 41

 

Figure 6.10 Skeleton Curve and Unloading from The Skeleton Curve

99

The basic reloading rules presented in Figure 6.11 have to be refined to
account for the pinching effect which is dominant in shear behavior and the
Bauschinger efi'ect displayed especially by the flexural behavior. For this purpose,
a line is specified which intersects the horizontal (deformation) axis at the same
point as the strain hardening portion of the skeleton curve, and its slope is a frac-
tion (5’) of the skeleton curve hardening stiffness (see Figure 6.12). Another hys-
teretic parameter '1, that is between -1.0 and 1.0, defines the degree to which the
actual reloading path deviates from a straight line. A positive value of ’7 (Figure
6.12a) is indicative of the Bauschinger effect, while a negative value (Figure 6.12b)

leads to the pinching of the reloading path.

The majority of load cycles sustained by a structural element during a
seismic event have deformation amplitudes within the bound of previous max-
imum and minimum deformations. Upon a small-amplitude load reversal, if the
reloading takes place towards the skeleton curve (indicating a large-amplitude
reversal from the skeleton curve in the previous half-cycle), the path shown in Fig-
ure 6.13 will be followed. If reloading is in a direction with the previous load
reversal being incomplete, the reloading would take place towards the peak point
of the incomplete cycle. This reloading path will involve pinching and bowing if
the incomplete cycle peak is above the line with slope g discussed earlier (Figure
6.14a). Otherwise, it will simply be a straight line from the horizontal axis to the

peak of the previous incomplete cycle (Figure 6.14b).

It should be noted that the small amplitude rules apply as far as the cyclic
deformations are within the previous maximum and minimum values. If any of

these limits are exceeded, the regular large-amplitude rules will be effective again.

The suggested hysteretic rules are quite versatile and, depending on the
values of their parameters, they can represent a wide variety of hysteretic charac-

teristics as shown in Figure 6.15.

100

 

 

 

 

 

 

 

 

   

 

 

F Y 4 FR fiY .
BY 1“— Y . 7
R '" Ii"
X x
bWDSflSM) amass

Figure 6.11 Reloading towards The Skeleton Curve

 

 

 

 

 

 

 

 

@NDSvglfl qungqgon

Figure 6.12 The Bauschinger and Pinching Efl'ect on The Reloading Path

101

——°—-— previous cycle

————- current cycle

 

 

 

 

 

 

Figure 6.13 Small-Amplitude Reloading towards The Skeleton Curve

_. ——-— previous incomplete cycle

——“" current cycle

 

 

 

 

(a) 0359 1 (b) Case 2

Figure 6.14 Small-Amplitude Reloading in A Direction with Previous Incompete Cycle

102

 

 

 

 

 

 

‘4
l

 

 

 

 

eta-0.0, cam) a=0.0, fi=0.0 a=0.0, B=1.0 a=0.0, B=o.0
7-1.0, $-11) 7.0.0, {-1.0 ,7330.0, §=I.O 7:0-5, (=05

 

 

 

 

 

 

 

 

 

 

 

. %_ 1..

a=0.0, Ham a=0.0, fl=0.0 a=0.0, 5=1.0 a=0.(). d=1.2
q-os, {-0.5 was—0.5, g=0.25 1=—0.5, §=O.25 7=—0.5, g=0.25

I: l3 l7 1...

 

 

 

7’ —

 

 

a=0.0, B=0.0 a=0.0, B=1.0 a=0.75. 8=1.0
era-1.0, (=00 7-3-10, g=0.0 7=0.0, g=1.o

 

Figure 6.15 Different Hysteresis Diagrams Formed By Varing
Their Parameters of The Proposed Hysteretic Alodel

103

6.3.3 Selection of The Hysteretic Parameters

In order to complete the hysteretic simulation of the fiexural and shear
springs in the proposed element model, the yield force, the initial and strain har-
dening stiffnesses and the hysteretic parameters (0:, ,8, '7, and 5') of these springs
have to be determined. The empirical values of these variables or the analytical

methods for calculating them are presented below.

Yielding of the flexural and shear springs in Figure 6.9 is assumed to occur
simultaneously when the bending moment in the fiexural spring reaches the
flexural yield strength of the R/ C cross section. The reported test data of R/ C
beams support this assumption. The yield moment in flexure can be derived using
the simplified strain and stress distributions presented in Figure 6.16. At the
yield point, the concrete stress distribution across the section is assumed to be
linear, and yielding of the tension steel is assumed to mark the yield point. The
depth of neutral axis in Figure 6.16 can be obtained from the equilibrium of axial
forces, and then the yield moment of the cross section can be derived. The force
in the shear spring at yielding of the fiexural spring is assumed to be the yield
shear force. For unsymmetric sections, the yield values of shear force and bending

moment should be calculated for both loading directions.

The initial fiexural and shear stiffnesses of the springs are based on the
moment of inertia and the area of the cracked transformed cross section (Figure

6.17), using the conventional elastic formulations.

It should be mentioned that in calculating both the yield and initial
stiflnesses of the flexural and shear springs, it is assumed that the inflection point
of the element under seismic forces is at the element mid-length (Figure 6.19).
Hence, the shear spring force is always equal to the fiexural spring bending
moment devided by half the element length, and the initial stifinesses of the

flexural and shear springs at each end represent the elastic flexural and shear

104

 

  
  

 

 

  

 

 

 

1I_—
. , O
A, _
- - —— - — NA.
A.
O O 0 yield strain yield force
,St‘iain , stress
distribution distribution

Figure 6.16 Simplified Flexural Strain and Stress Distributions at Yielding

 

 

 

 

3‘
a:

 

 

 

 

 

 

 

 

 

Figure 6.17 Cracked Transformed Cross Section for Calculating
The Initial F lexural and Shear stifinesses

105

stiflnesses of half the element (treated as a cantilever beam with a cracked tras-
formed cross section). The variations in inflection point location under seismic
forces has been observed to have relatively small effects on the element response
characteristics[39], and the assumption of a fixed inflection point at mid-length

seems to be warranted.

The strain hardening ratio (the strain hardening stiffness over the initial
stiffness) and the hysteretic parameters of the flexural and shear springs were
derived empirically using the test data presented in References 6—8, 28, 37, 45 and
46. All these references have reported the results of cyclic tests on R/ C beams
with separate measurements of the flexural and shear deformations. The test
techniques used in these refernces have eliminated, to a large extent, the deforma-
tions associated with the fixed-end rotation. Table 6.1 presents the values of

these vaiables for the flexural and shear springs (see Figure 6.18).

Table 6.1 Empirical Values of The Strain Hardening Ratio and Hysteretic
Parameters of The Flexural and Shear Springs

 

Strain
Spring Hardening a g '1 5'
Type Ratio

 

Flexural 0.021 0.16 0.97 0.42 0.75
Shear 0.043 0.11 1.22 -0.49 0.29

 

6.4 FORMULATION OF THE ELEIVIENT TANGENT STIFFNESS
MATRIX

The hysteretic models of the flexural and shear springs in the proposed ele
ment model (Figure 6.9) can be used to derive the tangent stiflnesses of these
springs at any step in the loading history. These stiffnesses can be used in con-

structing the overall tangent stiffness matrix of the element. In the following, the

FORCE (F)

FORCE (F)

106

 

-- Previous Cycle Y2
— Current Cycle 6Y2 L l

 

 

 

 

 

 

 

 

0: = 0.16
fiaam
'1 - 0.29
g a 0.71

 

 

 

 

 

 

DEFORMATION (X)
(a) Flexural Spring

 

-- Previous Cycle
— Current Cycle

 

 

 

 

 

   
 

 

 

 

 

 

 

 

DEFORMATION (X)
(b) Shear Spring

Figure 6.18 Empirical Values of The Strain Hardening Ratio and
Hysteretic Parameters of The F lezural and Shear Springs

107

tangent stiffness matrices for cantilever beams (commonly used in experimental
studies) and a general beam element (for incorporation in the overall structural

analysis programs) will be developed.

The cantilever beam (Figure 6.20) has only one degree of freedom, that is the
lateral displacement (6) at the free end of the element. This displacement results
from the shear deformation of the shear spring and rotation of the flexural spring:

d6=- d6, + l-d6f

_dV dV-l
K, K,

 

=dV (6.1)

 

 

 

where,

d6= incremental displacement at free end ;
d 6,= shear spring incremental deformation ;
d 0, - flexural spring incremental rotation ;
d V8 incremental lateral load at free end ;
la: cantilever element length ;
K,=- shear spring tangent stiffness ;
K I =- flexural spring tangent stiffness.

Hence, the tangent stillness (Kt),of cantilever element shown in Figure 6.8 can be

derived from the following expression:

dV =Kt-d6
1
1 z

n+3?

d6 (6.2)

 

 

In a general beam element, there are six displacement degrees of freedom (r,,,
r,,,-, r,,-, rzj, ryj, and raj), shown in Figure 6.21(a). These six degrees of freedom
may be condensed to the rotation at each end of the element (9,, 91-) and the

change in element length AL, as shown in Fig. 6.21(b). The beam axis will be

108

9?
<7\ /l<\o

I inflection point

Figure 6.19 Mid-Length Inflection Point under Seismic Forces

Actual a I

 

 

ldealized @— l

\

shear spring (Kc)

fiexural spring (K I )

Figure 6. 20 Cantilever Element

109

 

 

 

   

(a) Six Degrees of Freedom

 

 

 

(b) Condensed Degrees of Freedom

Figure 6. 21 Displacement Degrees of Freedom in A Genaral Beam Element

110

rotated due to the end translation by an amount 6, and it will be

9 =- [(rw- — ’yi) cost/J - (rxj — rzi) sinwl/L (6.3)

where,

1P= element orientation angle;
L =- total element length.

Hence, the total rotation of each end of the element will be

0: a 6 + r0:
6.4
61'. 0 + T" ( )
The change in element length is given by
AL- (rzj - r,,-)cos1/J + (ryj — ry,-)sin1/J (6.5)

Deformations of the element degrees of freedom are correlated with the shear
and rotational deformations in the shear and flexural springs, respectively. Incre~

mental deformations in the shear springs change the element rotation (d 68) by:

dV

d0 ..
. K,-L

 

(6.6)
where,

d V= incremental shear force at element ends

M, + m]

=3
7

 

L

dM,-= incremental moment at end i ;

 

dMJ-s incremental moment at end 1' ;

L:- total element length ;

1
K = ;
8 l/Ksi + l/Ksj

K“:- tangent stifl'ness of shear spring at end i ;

 

K814:- tangent stiffness of shear spring at end 3'.

The incremental element end rotations caused by the flexural spring rotations

(dB/,- and d 0”) are given by:

111

(6.7)

 

Adding the shear and flexural components, the total element end rotations

can be obtained as follow:

{‘16,} {49, + w,,}

 

 

 

 

 

 

 

 

 

 

l + l 1
KaL2 K]; K3L2 _ dM,-
" 1 1 . 1 dM- (6'8)
+ J
l K81.2 K,L2 Kfj .
Hence, the element tangent stiflness matrix (K) is,
_;_ + _L. __1_ "
K,L2 K;.- K,L2
K a l 1 1 (6.9)
2 2 +
L [(314 K8 L Kfj

 

 

The element tangent stiffness matrix (KG) in the global coordinates, for
incorperation into the structural stiffness matrix, can be obtained through an

appopriate transformation:

[Kc] = [TlT'lKl'lTl (6-10)

where, [T] is a transformation matrix which can be obtained from Equations (6.3)

 

 

 

 

and (6.4):
. ’n ’w’ ’w '2.» ry, To];
Sln‘l/J cost!) sinw cosw
_ L L 1 L ‘ 1.. 0
T: _sim/J cost/2 0 siml’ _cosL/J (6°11)
L L L L J

 

 

In Equation (6.10), [T]T is the transpose matrix of [T].

112

The above element tangent stiffness matrix can be used to construct the
overall tangent stiffness matrix of complete structural systems for performing

incremental nonlinear static and dynamic analysis of structures.

6.5 CONEPARISON WITH TEST RESULTS

A large number of cyclic test data on R/C cantilever beams have been
reported in the literature. This section compares the experimental cyclic load-
deformation relationships with the predictions of the developed models. Six tested
R/ C beams with a variety of geometric and material characteristics were selected
for this comparative study. Some major properties of these elements are given in
Table 6.2. In this table, I is the element length, l/d is the shear span to depth
ratio, b is the (rectangular) cross sectional width, p and p' are the ratios of bot-
tom and top reinfocements, and p, is the volumetric ratio of transverse reinfoce-
ment at the critical location near the fixed end, and f; and [1, are the concrete

compressive strength and steel yield strength, respectively.

Table 6.2 Properties of R/ C Beams Used in Comparison
Between The Tests and Theory

Beam ’ ’ . ‘ .
No. ’0'") ”d 5 98p 9. f.(k8*) fylks')1131:f.

1 59 3.1 7.9 0.0178 0.0043 6.11 45.11 51

 

 

59 3.1 7.9 0.0178 0.0043 5.45 44.38 51
59 3.1 7.9 0.0178 0.0043 4.82 45.1 1 51
60 4.7 15.0 0.0103 0.01 1 4.75 54.4 35
60 4.7 15.0 0.0103 0.013 4.06 51 .8 35

cocaine/ow

60 4.7 15.0 0.0103 0.020 4.25 51.8 35

 

113

Figures 6.22(a) and (b) compare the total cyclic force-deformation relation-
ships obtained experimentally and theoretically, respectively, for the cantilever
beam No. 1 (see Table 6.2). The proposed model is observed to be capable of
predicting the test results with a reasonable accuracy. Figures 6.22(c) and ((1)
compare the experimental and theoretical flexural components, and Figures 6.22(e)
and (f) compare the experimantal and theoretical shear components of the cyclic
deformations. These figures (both experimental and theoretical ones) indicate that
shear deformations play an increasingly important role in the element deforma-

tions under repeated inelastic load cycles.

Another comparison between the experimental and the theoretical total,
flexural, and shear cyclic load-deformation relationships is presented in Figure
6.23 for the beam No. 2 of Table 6.2. Here also the model is observed to satisfac-
torily predict the test results. It is also interesting to notice that at the relatively
large inelastic cycles repeated with the same amplitude (see for example, the last
four cycles in Figures 6.23a and b), the flexural end displacements (Figures 6.23c
and d) tend to decrease while the shear deformation (Figures 6.23c and f) continue

to increase under repeated cycles.

Another comparison between the experimental and theoretical total, flexural
and shear load-displacement relationships is given in Figure 6.24. The theoretical
results for this beam (No. 3 of Table 6.2) are also observed to compare well with

the experimental ones.

Figures 6.25 through 6.27 present comparisons between the experimental and
theoretical total cyclic load-displacement relationships for. beam Nos. 4, 5 and 6,
respectively, of Table 6.2. In all these beams, which have relatively low shear
forces compared to beam Nos. 1 to 3, a reasonable comparison can be observed

between the experimental and theoretical results.

 

 

Lateral Force (kips)

Lateral Force (kips)

114 -

 

20-1

 

 

 

 

 

 

 

-4.0

T
-3.0

1

0.0
Total End Displacement (in)

(a) Total Deformation (Experiment)

I 1 I
1.0 2.0 3.0 4.0

 

-ANALYSl$.

 

 

 

 

 

 

 

 

-‘|r.0

T
-3.0

0.0

l f 7

1.0 2.0 3.0 4.0
Total End Displacement (in)

(b) Total Deformation (Theory)

Figure 6. 22 Experimental and Theoretical Cyclic Load-Deformation
Relationships of Beam No. 1

115

 

 

 

Lateral F orco (klps)

 

 

 

 

-4.a -§0 30 -i.o are 130 £0 35 4.0
Flexural End Displ. (in)

(c) F lexural Deformation (Experiment)

49

 

4 -MALYSB
30-4

d

2.0-4
ia-JI [III
I //

0
I I” ~ 7
_10-4
4
—2o-l
.4

 

 

 

Lateral F orce (klps)

 

~30-4
-l

 

'40 T 1T I I T T T
-4.0 ~3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

Flexural End Displ. (in)
(d) Flexural Deformation (Theory)

 

 

Figure 6. 22 Experimental and Theoretical Cyclic Load-Deformation
Relationships of Beam No. 1(cont’d)

 

 

Lateral Force (klpe)

Lateral Force (klps)

116

 

 

 

 

 

 

 

-4.0 30 -i0 -1'.0 01.0 110 in 330 4.0
Shear End Displ. (in)

(e) Shear Deformation (Experiment)

 

.I --ANALY38
304

\\

,M’

204

 

\\\
\
\

10-

 

 

 

 

 

 

 

-10.;

~20-l //
. lfl’ Ir

_304

 

 

 

 

 

\ \I‘ \\
\

 

 

 

j

W 35.0 -i0 —i .0 0.0 1T0 230 3T0 4.0

Shear End Displ. (in)
(f) Shear Deformation (Theory)

L

Figure 6. 22 Experimental and Theoretical Cyclic Load-Deformation
Relationships of Beam No. 1(cont’d)

117

 

 

 

 

Lateral F orce (kips)
D

 

 

‘40 ' I T T T T T 1 T
-ll.0 -2.0 -1 .0 0.0 ‘l .0 2.0 3.0

Total End Displacement (in)

(a) Total Deformation (Experiment)

 

j *7

 

 

 

 

 

Lateral Force (kips)

 

 

 

 

 

-.‘.l.O 32} -T0 0.0 if?) 2:0 3.0

Total End Displacement (in)
(b) Total Deformation (Theory)

Figure 6. 23 Experimental and Theoretical Cyclic Load-Deformation
Relationships of Beam No. 2

 

Lateral Force (klps)

Lateral Force (kips)

118

 

304 < =—’,
20—

101

 

 

-10.

   

 

 

 

 

-3.0 -§.0 -i .0 0.0 £0 210 3.0
Flexural End Displ. (in)

(c) F lexural Deformation (Experiment)

 

.l — ANALYSIS
30-4 .
20- - 17

/

I?"
a '”/ /
I/
0 I /
-l 7/
-10-I / I
~ , I

 

    
 
 

 

 

 

 

-20~ / /’
cl / I /
_30—l d
—40 1
-3.0 —10 —i.0 0.0 1T0 £0 3.0

Flexural End Displ. (in)
(d) Flexural Deformation (Theory)

Figure 6. 23 Experimental and Theoretical Cyclic Load-Deformation
Relationships of Beam No. 2(cont’d)

Lateral Force (kips)

Lateral Force (kips)

119

 

 

 

 

 

 

 

 

 

i r T l T
-300 -200 -1 .0 0.0 1 .0 20° 30°
Shear End Displ. (in)

(e) Shear Deformation (Experiment)

 

.l—ANALYSIS

'l /l/l/I

['1
20" //

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-3.0 3.0 -i.0 0.0 E0 £0 3.0

Shear End Displ. (in)
(f) Shear Deformation (Theory)

Figure 6. 23 Experimental and Theoretical Cyclic Load-Deformation
Relationships of Beam No. 2(cont’d)

 

120

 

 

 

 

 

 

 

 

 

 

 

4c _.
. -- TBT
30..
’33 204 (f
0' J
'3'
V 104 /(
9 .
e 0
IE . )”
2,17
aid
3 ~20- /
.. I!"
-30...
4
-—4c 1
.0 -3.0 ~10 ~i.0 030 To 210 3.0 4.0
Total End Displacement (in)
(a) Total Deformation (Experiment)
40
. -- ANALYSIS
30d /
cl
’6? 20- /
.9- .J
x
V lO-il /
o q /
0
E5 0
L‘— J I /
ӎ -lO-l
° 4
‘5 a
.30...
cl
-40
--4.0 ~10 -§.0 -i.0 010 To 210 3.0 4.0

 

Total End Displacement (in)

(b) Total Deformation (Theory)

Figure 6. 24 Experimental and Theoretical Cyclic Load-Deformation

Relationships of Beam No. 3

Lateral F orce (kips)

Lateral F orce (kips)
O

121

 

4e m
.4 — m

30" 1 f I
4 V , '

20-1 /

 

I
'8
L
“\—

4 J
/

 

 

 

fir

--4.0 -Ei -10 J .0 0.0 :0 20 3.0 4.0
Flexural End Displ. (in)

(c) F lexural Deformation (Experiment)

 

w _A
¢—ANALYSIS

 

 

 

0‘ A ,J/

-10-l

 

1

-4.0 3.0 ~i0 -i.0 0.0 1:0 25 $0 4.0

Flexural End Displ. (in)
(d) Flexural Deformation (Theory)

 

 

Figure 6. 24 Experimental and Theoretical Cyclic Load-Deformation
Relationships of Beam No. 3(cont’d)

Lateral Force (kips)

Lateral F orce (klps)

122

 

304

20-4

10-

 

 

-10.;

-204

-30-l

 

 

 

 

—4

-£.0 -T0 0.0 1T0 25 f0
Shear End Displ. (in)

(e) Shear Deformation (Experiment)

I».O -3.0 4.0

 

— ANALYSIS

 

 

 

 

 

~43

Figure

 

 

 

r I l r T
-2.0 -1 .0 0.0 1.0 2.0

Shear End Displ. (in)
(f) Shear Deformation (Theory)

r
-3.0

6. 24 Experimental and Theoretical Cyclic Load-Deformation
Relationships of Beam No. 3(cont’d)

123

 

 

 

 

L

Lateral Force (klps)
O

J.
O
i

 

I

 

 

 

 

.0fi-EIT—1f0 . 0.0 r ifofi 2E) T 3.0
Total End Displacement (in)

(a) Total Deformation (Experiment)

20

 

—- ANALYSIS

NH

 

 

 

Lateral Force (kips)
O

 

 

 

 

f

Total End Displacement (in)
(b) Total Deformation (Theory)

Figure 6. 25 Experimantal and Theoretical Total Cyclic Load-Displacement
Relationships of Beam No. 4

20

124 .

 

10-l

 

 

 

Lateral Force (kips)
O

 

 

 

 

 

 

-3.0 1 -2r.0fi -1'.0 ‘ 0.0 T 17.0 T 20 T 3.0
Total End Displacement (in)
(a) Total Deformation (Experiment)
20
-- ANALYSIS

 

 

Lateral F orce (kips)
O

.l.
O
L

 

 

 

 

 

 

T if I T
0 -2.0 ~1.0 0.0 .1.0 2.0 3.0

Total End Displacement (in)
(b) Total Deformation (Theory)

Figure 6.26 Ezperimantal and Theoretical Total Cyclic Load-Displacement

Relationships 0] Beam No. 5

125

20 _

 

10-l

 

 

 

Lateral F orce (kips)
O

 

 

 

 

 

 

 

 

 

 

 

 

 

-::.0 -{0 ' 0.0 Er W: “1 2.0
Total End Displacement (in)
(a) Total Deformation (Experiment)
20
-- ANALYSB
.1 7’.
E 104
3 .
8
‘5 O
a.
'5 .l
L-
o
4.;
3 ~10-l
//,
q 1414““, L
“’20 r “F T fi
-:z.0 -T.0 03 1l0 2.0

Total End Displacement (in)
(b) Total Deformation (Theory)

Figure 6.27 Experimantal and Theoretical Total Cyclic Load-Displacement
Relationships of Beam No. 6

 

126

6.6 SUNIMARY AND CONCLUSIONS

Under monotonic loads, shear deformations generally constitute only a minor
fraction of the overall displacements of the R/ C beam elements. Consequently,
the shear deformations have traditionally been neglected in analytical modeling of
the R/ C elements. Repeated inelastic load reversals, however, tend to deteriorate
the shear resisting mechanisms in a faster pace than the flexural ones. Hence,
shear deformations play an increasingly important role in determining the element
behavior under inelastic load cycles, and the disregard for shear deformations in
analytical modeling can lead to major discrepancies between cyclic test results and

theoretical predictions.

A practical element model was developed in this study to predict the hys-
teretic behavior of R/ C beams, accounting for the distinct hysteretic characteris-
tics of shear and flexural deformations. All the elasto-plastic deformations were
assumed to be Concentrated in dimensionless serial shear and flexural springs at
the element ends. The behavior of each spring was defined by a semi-empirical
skeleton force-deformation curve and a set of empirical hysteretic rules which were
derived from the available cyclic test results on R/ C beam (in which the shear and

flexural deformations were measured separately).

Based on the developed physical model (with concentrated shear and flexural
springs), tangent stiffness matrices were constructed for cantilever beams and gen-

eral beam elements.

The element model was checked against the results of a number of cyclic tests
performed on R/ C cantilever beams with wide ranges of geometric and material
properties. The comparison between test and theory was satisfactory, and both
the experimental and theoretical cyclic load-deformation relationships indicated

that the repetition of inelastic load cycles leads to a growing dominance of shear

deformations in the cyclic behavior of R/C beams.

 

 

CHAPTER 7

SUNINIARY AND CONCLUSIONS

This investigation dealt with two critical aspects of the dynamic response
characteristics of R/C structures which are generally neglected in developing
dynamic analysis and design procedures. These two aspects are the effects of the
relatively high strain rates induced by the dynamic (e.g., seismic and impulsive)
loads in structures, and the role of shear deformations in the cyclic load-

deformation behavior of R/ C elements and structures.

In the first phase of this investigation, summarized in chapters 2 to 5, refined
strain ratesensitive models were developed for nonlinear axial—fiexural inelastic
analysis of R/C elements under monotonic dynamic loads. These models were
used to assess the strain rate-sensitivity of the axial behavior of R/ C columns and
the flexural behavior of R/C beams. The steps involved in this phase of the

research, and conclusions at each step are presented below.

Constitutive Modeling of Concrete

On the basis of many test results reported by different investigators, empiri-
cal models were developed for the strain rate effects on the concrete compressive

strength, strain at maximum stress, and modulus of elasiticity. It was found that

(1) the concrete compressive strength increases at higher strain rates, and this

increase is more significant for saturated concrete,

 

128

(2) the strain at maximum stress first decreases and then increases as the strain

rate increases over the static values, and

(3) at higher strain rates, the secant modulus of elasticity of concrete increases.
The initial tangent modulus also increases at higher strain rates but at a

much slower rate when compared with the secant modulus.

The preceding idealizations of the strain rate effects were used to develop a
practical constitutive model for confined and unconfined concrete subjected to

different compressive strain rates. This model compared well with test results.

Constitutive Modeling of Steel

Dynamic test results reported by different investigators on structural steel,
deformed bars, and deformed wires subjected to monotonic tension were used for
deriving empirical expressions for the ratios of static to dynamic values of the
yield and ultimate strengths, yield and ultimate strains, and the strain at the end
of the yield plateau. The principal variables in the expressions are the rate of
straining and the steel static yield strength. The following conclusions was

derived:

(1) All the characteristic stress and strain values on the steel monotonic constitu-
tive diagram increase with increasing strain rate. The yield strength is more
strain rate-sensitive than the ultimate strength. The steel modulus of elasti-

city is independent of the rate of straining.

(2) The main factor influencing the strain rate effects is the static yield strength
of steel. The mechanical properties of steel with lower yield strength are

more strain rate-sensitive.

(3) No considerable difference was observed between the strain rate effects on

structural steel, deformed bars, and deformed wires.

129

(4) The empirical strain rate-dependent constitutive models proposed in this
chapter for steel subjected to monotonic loading compare well with test

results.

Element Modeling and Numerical Studies

Empirical strain rate-sensitive constitutive models of steel and concrete were
incorporated into an analytical procedure for axial/flexural analysis of R/C
beam-columns at different loading rates. The analytical predictions compared well
with quasi-static and dynamic test results. Simple practical analysis procedures
. were also developed for calculating the axial and flexural strengths of R/C

columns and beams as functions of the applied loading rates.

The refined (layer) analytical method was used to perform a numerical study
on the sensitivity of the axial and flexural performances of R/ C sections and ele-
ments to the changes in steel and concrete strengths, concrete confinement and the

rate of load application. The results of this numerical study indicated that:

(1) Variations in longitudinal steel yield strength have major effects on the
fiexural strength, and some important effects on the axial strength of the typ-
ical beam and column cross sections considered in this study (:1: 30% varia-
tions in the longitudinal steel yield strength causes changes of the order of j;
30% in fiexural strength and :1: 10% in axial compressive strength). The
axial and fiexural stifinesses, however, are not much influenced by the varia-

tions in yield strength.

(2) Variations in the concrete compressive strength have relatively small effects
on the beam flexural behavior, but major effects on the column compressive
stifi'ness and strength. For the typical beam and column cross sections stu-
died here, d: 30% changes in concrete compressive strength resulted in varia-

tions of about :I: 3% and :l; 25%, respectively, in the flexural and compressive

(3)

(4)

(5)

(7)

(8)

130

strengths (and stiflnesses).

The flexural behavior of reinforced concrete beam sections is strongly
influenced by the variations in the tension steel ratio, but not much by the
changes in compression steel ratio. A :1: 30% change in tension steel ratio
results in about :I: 30% change of a typical beam flexural stiffness and
strength. The axial strength and stiffness of R/ C column cross sections are
also found to be sensitive to the variations in their total steel area. Changes
of the order of :l: 15% were observed in strength of a typicaal column as a

result of :l: 30% changes in the total steel area.

Increased confinement does not have major effects on the beam flexural
behavior (neglecting the possibility of compression steel buckling). It, how-

ever, slightly increases the compression strength and ductility of the column.

Under typical impulsive loading rates, the axial strength and stiffness of typi-
cal R/C columns increase by about 30% over the corresponding quasi—static

values.

The flexural strengths of typical R/ C beams increase by about 15% at typi-
cal impulsive loading rates over the quasi-static values. The flexural
stifl'nesses of R/C beams 'is not much influenced by the loading rate varia-

tions.

The loading rate-sensitivity of R/C column axial behavior increases with
decreasing yielding strength and increasing degree of saturation of concrete.
The variations in steel ratio, concrete compressive strength, confinement and
cross sectional shape of columns do not significantly influence the column

loading rate—sensitivity.

The loading rate-sensitivity of R/C beam flexural behavior increases with
decreasing steel yield strength and increasing steel ratio and concrete degree

of saturation. The variations in concrete compressive strength, confinement,

131

and element length and cross sectional shape do not significantly influence

the loading rate-sensitivity of R/ C beams in flexure.

In the second phase of research, summarized in chapter 6, an element model
capable of distinguishing between the shear and flexural hysteretic characteristics
for R/ C elements was developed and verified against test results. The findings of

this phase are summarized below.

Under monotonic loads, shear deformations generally constitute only a minor
fraction of the overall displacements of the R/ C beam elements. Consequently,
the shear deformations have traditionally been neglected in analytical modeling of
the R/ C elements. Repeated inelastic load reversals, however, tend to deteriorate
the shear resisting mechanisms in a faster pace than the flexural ones. Hence,
shear deformations play an increasingly important role in determining the element
behavior under inelastic load cycles, and the disregard for shear deformations in
analytical modeling can lead to major discrepancies between cyclic test results and

theoretical predictions.

A practical element model was developed in this study to predict the hys-
teretic behavior of R/ C beams, accounting for the distinct hysteretic characteris-
tics of shear and flexural deformations. All the elasto-plastic deformations were
assumed to be concentrated in dimensionless serial shear and flexural springs at
the element ends. The behavior of each spring was defined by a semi-empirical
skeleton force-deformation curve and a set of empirical hysteretic rules which were
derived from the available cyclic test results on R/ C beam (in which the shear and

flexural deformations were measured seperately).
Based on the developed physical model (with concentrated shear and flexural
springs), tangent stifl'ness matrices were constructed for cantilever beams and gen-

eral beam elements.

132

The element model was checked against the results of a number of cyclic tests
performed on R/ C cantilever beams with wide ranges of geometric and material
properties. The comparison between test and theory was satisfactory, and both
the eXperimental and theoretical cyclic load-deformation relationships indicated
that the repetition of inelastic load cycles leads to a growing dominance of shear.

deformations in the cyclic behavior of R/ C beams.

10.

11.

12.

13.

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