CE72.52 - Lecture 3a - Section Behavior - Flexure

1
CE 72.52 Advanced Concrete
Lecture 3a:
Section Behavior
Flexure
Naveed Anwar
Executive Director, AIT Consulting
Director, ACECOMS
Affiliate Faculty, Structural Engineering, AIT
August - 2015

Capacity of RC Section
subjected to combined Flexural
Moment and Axial Force
2

Loads and Stress Resultants
3
Obtained from
analysis
Depends on
Stiffness
Dependson
SectionsandRebars
FOS
Loads Actions Deformation
Strains
Stress
Resultants
Stresses
(Sections & Readers)
Advanced Concrete l Dr. Naveed Anwar

The Response and Design
4
Applied Loads
Building Analysis
Member Actions
Cross-Section Actions
Material Stress/Strain
Material Response
Section Response
Member Response
Building Response
Load Capacity
FromLoadstoMaterials
FromMaterialstoLoadCapacity
Advanced Concrete l Dr. Naveed Anwar

Frame/ Linear Member Sections
7

Basic Section Types - Proportions
• Slender
• Buckling of section parts before reaching material
yielding
• Cols formed, thin walled metal sections
• Compact
• Material yielding first, followed by bucking of
section parts
• Most hot rolled and built-up metal sections
• Some thin concrete sections
• Plastic
• Material failure (yielding, rupture, but no buckling)
• Most concrete sections
9

Section Types – Member Usage
• Beams
• Primarily bending, shear and torsion
• Trusses
• Primarily tension and compression
• Columns
• Primarily compression, bending
• Shear and torsion also important
10

Cross-section classification based on
primary material composition
11

12
Some of the shapes used for Reinforced and Pre-stressed concrete
sections defined in CSI ETABS Section Designer.

Some common cross-sectional types
based on materials and geometry
13

14
(a)
C WF S,M H Ell Tee Tube Pipe
WF H Ell Tee Tube Pipe
(b)
(c)
I, H Circular Rectangular PipeSquare
(d)
Tee I Single Tee Double Tee Hollow case Box
Some typical standard cross-section shapes used (a) in AISC database, (b) in BS
Database, (c) in pre-cast, pre-stressed girders and slabs, (d) in pre-cast concrete
piles

Some typical parametrically defined
cross-section shapes
15
Square
b
b
a
a
Db
h
bf
bw
tf
bf
bw
h h
tf
tf
Do
Di
Rectangle Circle Tee I Pipe

16
(a)
(b)
(c)
Some typical built-up shapes and sections (a) made from standard shapes,
(b) made from standard shapes and plates, (c)made from plates

17
(a)
(b)
Some typical composite sections. (a) Concrete-Steel composite, (b)
Concrete-Concrete composite

Unified Theory for Concrete Design
• It is possible to develop a single theory for
determining the axial flexural stress
resultants of most types of concrete
members for all design methods and for
most design codes
• Unifying Beams and Columns
• Unifying Reinforced and Pre-stressed Concrete
• Unifying WSD and USD Methods
• Unifying different Cross-section Types
• Incorporating various stress-strain models
18Advanced Concrete l August-2014

Unifying Beams and Columns
19
Actions Sections
Beam Mx or My Rectangular, T, L, Box
Column P, Mx and/or My
Circular, Polygonal,
General Shape
Advanced Concrete l August-2014

Unifying Reinforced and Pre-stressed
20
Reinforced Steel Pre-stressing Steel
Un-reinforced No No
Reinforced Yes No
Partially Pre-stressed Yes Yes
Fully Pre-stressed No Yes

Unifying Reinforced and Composite
21
Reinforced Steel Pre-stressing Steel Steel Section
Reinforced Yes No No
Reinforced-Composite Yes No Yes
Partially Pre-stressed -
Composite
Yes Yes Yes
Fully Pre-stressed -
Composite
No Yes Yes

Unifying Material Models
22
Strain
Stress
Linear Whitney PCA BS-8110
Parabolic Unconfined Mander-1 Mander-2
Concrete Stress-Strain Relationships

Unifying Material Models
23
Strain
Stress
Linear - Elastic Elasto-Plastic
Strain Hardening - Simple Strain Hardening Park
Steel Stress-Strain Relationships

Unifying Service and Ultimate State
• Service State Calculations
• Neutral axis depth controlled by limit on
concrete (or steel) stresses directly
• Ultimate State Calculations
• Neutral axis depth controlled by limit on strain
in concrete (or in steel) and indirect control on
material stresses
• General
• Section Capacity based on location of neutral
axis, strain compatibility and equilibrium of
stress resultants and actions

General Procedure for Computing Capacity
• Assume Strain Profile
• Assume a specific angle of neutral axis
• Assume a specific depth of neutral axis
• Assume maximum strain and determine the strain in
concrete, re-bars, strands, and steel from the strain
diagram
• Determine the stress in each component from
the corresponding stress-strain Relationship
• Calculate stress-resultant of each component
• Calculate the total stress resultant of the
section by summation of stress resultant of
individual components

The General Cross-section
26
y
h
c
fc
Strain
Stresses for
concrete and
R/F
Stresses for
Steel
f1
f2
fn
fs NA
CL
Horizontal
Comprehensive Case

The General Stress Resultants
27
 
 
 




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
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



 



...),(
1
....,
1
...),(
1
....,
1
...),(
1
...,
1
121
3
121
2
121
1
i
n
i
ii
x y
y
i
n
i
ii
x y
x
x y
n
i
iiz
xyxAxdydxyxM
yyxAydydxyxM
yxAdydxyxN















The Comprehensive Case

Flexural Theory: Stress Resultants
28
The Most Comprehensive Case
The Most Simple Case
M f A d
a
n y st 






2
0.003 fc()
C
Strain Stress and Force
N.A.
OR
0
C
0
0.85 fc
'
jd
C
T
b
d
Section
M
 
 
 











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


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




 



...),(
1
....,
1
...),(
1
....,
1
...),(
1
...,
1
121
3
121
2
121
1
i
n
i
ii
x y
y
i
n
i
ii
x y
x
x y
n
i
iiz
xyxAxdydxyxM
yyxAydydxyxM
yxAdydxyxN















y
h
c
fc
Strain
Stresses for
concrete and
R/F
Stresses for
Steel
f1
f2
fn
fs NA
CL
Horizontal

Example: Cross-Section Response
• The Section Geometry
• Elastic Stresses
• Load Point
• Neutral Axis
• Ultimate Stresses
• Cracked Section Stresses
• Section Capacity
• Moment Curvature Curve

The Governing Equations
30
 
 
 





























 



...),(
1
....,
1
...),(
1
....,
1
...),(
1
...,
1
121
3
121
2
121
1
i
n
i
ii
x y
y
i
n
i
ii
x y
x
x y
n
i
iiz
xyxAxdydxyxM
yyxAydydxyxM
yxAdydxyxN



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
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







Nz
MxMy
y
h
c
fc
Strain
Stresses for
concrete and
R/F
Stresses for
Steel
f1
f2
fn
fs NA
CL
Horizontal

Axial-Flexural Capacity
31
Nz
Mx
My
The Stress-Resultants for Bi-Axial Bending

Load Point and Eccentricity
32

Biaxial Elastic Stress Distribution

Neutral Axis and Strain Plane

Ultimate Stress – Rectangular Block

Stresses in Rebars

Cracked Section Stresses

Axial-Flexural Capacity
Nz
Mx
My
+

The Fiber Model and Implementation
• In this approach, the section is sub-divided
into a mesh, each element called a Fiber.
A particular material model is attached to
each Fiber and then solved to compute
the response.
39
X
Y
y
xx
y
Origin of
Local Axis
Origin of
Global Axis
Rebars
Prestressed
StrandsOpening
Abi
Api
Shape of different
material/properties
BendingAxis
Plastic
Centroid
S1
S2
Sn
θ
Mx
xi
Ai, fi
yi
My
x
y

Fiber Model - Equations
40
Equilibrium equation based on
Integration
Equilibrium equation based on
Summation
Expanded Summation for Complex
Models
 
A
iiy
A
iix
A
iz dAxfMdAyfMdAfN
__
;;
_
1
_
11
;; xAfMyAfMAfN
n
i
iiy
n
i
iix
n
i
iiz  








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



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

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

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
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

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









































  
  
  
  
  
  
q
p
l
k
n
j
jjj
m
i
yi
p
y
q
p
l
k
n
j
jjj
m
i
xi
p
x
q
p
l
k
n
j
jj
m
i
zi
p
z
xAfMM
yAfMM
AfNN
1 1 11
3
1 1 11
2
1 1 11
1
1
1
1






Mx
xi
Ai, fi
yi
My
x
y

Procedure for Computing Stress Resultants
• Define the material models in terms of basic
stress-strain functions. Convert these functions
to discretized curves in their respective local
axes;
• Model the geometry of the cross-section using
polygon shapes and points, called “fibers”
• Assign the material models to various fibers
• Locate the reference strain plane based on
the failure criterion. The failure criterion is a
strain in concrete defined in corresponding
material model and design code;

• Compute the basic stress profiles for all
materials, using the reference strain profile;
• Modify the stress profiles for each material
based on appropriate material functions,
and special factors;
• For each material stress profile compute
the corresponding stress resultant for the
resulting triangles and points in the
descretized cross-section. The detailed
procedure for determining the resultants is
discussed in the next section of this note;

• Modify the stress resultants using the
appropriate material specific and strain-
dependent capacity reduction factors as
defined in design codes; and,
• Compute the total stress resultants for all
material stress profiles.
• Steps 5 to 9 are repeated for other
locations of the reference strain plane. The
computed sets of Nz, Mx, and My are used
to define the capacity surface.

44
Plain concrete shape Reinforced concrete section Compact Hot-rolled steel shape
Compact Built-up steel section
Reinforced concrete,
composite section
Composite section
Application of General Equations

Cross-section Stiffness and Cross-
section Properties
• As described earlier, the action along each degree of
freedom is related to the corresponding deformation by the
member stiffness, which in turn, depends on the cross-
section stiffness. So there is a particular cross-section
property corresponding to member stiffness for each degree
of freedom. Therefore, for the seven degrees of freedom
defined earlier, the related cross-section properties are:
•
• uz  Cross-section area, Az
• ux  Shear Area along x, SAx
• uy  Shear Area along y, SAy
• rz  Torsional Constant, J
• rx  Moment of Inertia, Ix
• ry  Moment of Inertia, Iy
• wz Warping Constant, Wz or Cw
46

Basic and Derived Properties
• Difference between Geometric and Section
Properties
• Geometric properties – No regard to material stiffness
• Cross-section Properties: Due regard to material stiffness
• Cross-sectional properties can be categorized in
many ways. From the computational point of
view, we can look at the properties in terms of;
• Basic or Intrinsic Properties
• Derived Properties
• Specific Properties for Reinforced Concrete Sections
• Specific Properties for Pre-stressed Concrete Sections
• Specific Properties for Steel Sections
47

Basic or the Intrinsic Properties
• The area of the cross-section, Ax
• The first moment of area about a given axis, (A.y
or A.x etc.)
• The second moment of area about a given axis,
(A.y2 or A.x2 etc.)
• The moment of inertia about a given axis, I
• The shear area along a given axis, SA
• The torsional constant about an axis, J
• The warping constant about an axis, Wz or Cw
• The plastic section modulus about a given axis, ZP
• The shear center, SC
49

Derived Properties
• The geometric center with reference to the given
axis, x0 , y0
• The plastic center with reference to the given axis,
xp , yp
• The elastic section modulus with reference to the
given axis, sx , sy
• The radius of gyration with
reference to the given axis, rx , ry
• Moment of inertia about the principle axis of
bending, I11 , I22
• The orientation of the principal axis of bending, J
50

Section Modulus
51
Elastic Plastic
y
I
S xx
x 
4
2
bh
ZPx 

Centroids
52
CG – Center of Gravity
SC – Shear Center
PC – Plastic Center

The significance of geometric and
plastic centroid in columns
53
Pu
Pu
Pne
b
h
b
h
h/2 h/2h/2 h/2
Pn
GC
GC
PC
Mu = Pu . e
(a) (b)
(a) Symmetric rebar arrangement, (b) un-symmetric rebar arrangement

Torsional Constant, J
57
Circle
Square
A finite element solution is need for general sections

Warping Constant, Cw
58
A finite element solution is need for general sections

Cracked Section Properties – RC Section
60
Icr = Moment of inertia of cracked section transformed to concrete, mm4
Ie = Effective moment of inertia for computation of deflection, mm4
Ig = Moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement,
mm4
Mcr = Cracking Moment, N-mm
Ma = Applied Moment, N-mm
fc’ = Compressive strength of concrete, Mpa
fr = Modulus of rupture of concrete, Mpa
λ = Factor for lightweight aggregate concrete
yt = Distance from centroidal axis of gross section, neglecting reinforcement, to tension face, mm

What is Capacity?
• The axial-flexural capacity of the cross-
section is represented by three stress
resultants
• Capacity is property of the cross-section
and does not depend on the applied
actions or loads

What is Capacity?
63
• Capacity is dependent on
failure criteria, cross-section
geometry and material
properties
• Maximum strain
• Stress-strain curve
• Section shape and Rebar
arrangement etc

How to Check Capacity
• How do we check capacity when there are
three simultaneous actions and three
interaction stress resultants
• Given: Pu, Mux, Muy
• Available: Pn-Mnx-Mny Surface
• We can use the concept of Capacity Ratio,
but which ratio
• Pu/Pn or Mux/Mns or Muy/Mny or …
• Three methods for computing Capacity Ratio
• Sum of Moment Ratios at Pu
• Moment Vector Ratio at Pu
• P-M vector Ratio
67

Sum of Mx and My
• Mx-My curve is plotted at applied axial
load, Pu
• Sum of the Ratios of Moment is each
direction gives the Capacity Ratio
68

Vector Moment Capacity
• Mx-My curve is plotted at applied axial load
• Ratio of Muxy vector to Mnxy vector gives the
Capacity Ratio
69

True P-M Vector Capacity
• P-M Curve is plotted in the direction of the
resultant moment
• Ratio of PuMuxy vector to PnMuxy vector gives
the Capacity Ratio
70

Load Point and Eccentricity Vector
• The load point location depends on the
direction of the eccentricities in the x and y
directions
71

Interpretation of Capacity Surface
72
+Mx
+My

+My
-Mx
+Mx
+My
+Mx
-My
-Mx
-My
Moment Directions on the M-M Curve
-My
-Mx
Load Point
Applied
Load Vector
+Mx
+My

+My
-Mx
+Mx
+My
+Mx
-My
-Mx
-My
Moment Directions on the M-M Curve
-My
-Mx
Load Point
Applied
Load Vector

What is Capacity
73
1- Based on Sum of Moments at Pu 2- Based on Moment Vector at PU
3- Based on True Capacity Vector in 3D

P-M Interaction Curve
74
• The curve is
generated by
varying the neutral
axis depth


















zi
N
i
si
z A
cny
N
i
si
A
cnx
dAfdzdafM
AfdafN
si
b
si
b
1
1
.)(
)(


Safe
Un-safe

Mx-My Interaction
75
-Mz
Muy
(-) Mnz
(+) Mnz
+ My
- My
+ Mz
Mx-My Interaction is the basis for many approximate methods

P-Mx-My Interaction Surface
76
• The surface is
generated by
changing Angle
and Depth of
Neutral Axis
 
 
 



























 
 
  



...),(
1
....,
1
...),(
1
....,
1
...),(
1
...,
1
121
3
121
2
121
1
i
n
i
ii
x y
y
i
n
i
ii
x y
x
x y
n
i
iiz
xyxAxdydxyxM
yyxAydydxyxM
yxAdydxyxN
















What is Uni-axial Bending
• Uni-axial bending is induced when column bending
results in only one moment stress resultants about
any of the mutually orthogonal axis.
No Bending
Mx = 0, My = 0
Strain StressSection
x
y e
P
fc
P fs1
fc
fs2
P
x
y
P
ey
P
e
fs1
fs2
fc
Uni-axial Bending
Mx <> 0, My = 0

What is Bi-axial Bending
• Biaxial bending is induced when column bending
results in two moment stress resultants about two
mutually orthogonal axis
78
y
x
ey
e
ex
P
x
P
ey
ey
P
x

P-M Interaction Diagram

Effect of Compressive or Tensile
Strength on Interaction curve

Symmetrical Column Section
Unsymmetrical Column
Section
Effect of Symmetry of Column
Section

Effect of Column Type on Shape of
Interaction Diagram

Effect of Reinforcement Ratio on
Moment Curvature Curve

86
Effect of
Reinforcement Ratio
Effect of Reinforcement
Spacing

Effect of ultimate concrete strain ϵcu

Flexural Design of RC Beam
Sections – A Special Case of
General Approach
88

First: The Class Project
Beam That will not fail !
89

The way to go!
• Try to generate the entire
“Load-Deformation Curve”
• Including “Residual Strength”
• Have to rely on “Ductility”, “Plastic hinges”
and “Catenary Action”
• Make sure beam does not fail in shear
93

Design Process for Class Project
• Flexural Design
• Shear Design
• Ductility and Plastic Hinges
• Catenary/Axial Capacity
94

Stress Block – Singly Reinforced
Concrete
95
N A
x
ε’cu=0.0035
εst=0.002
k1fcu
0.87fy
BS 8110
β=0.9ε’cu
k1= 0.45 fcu
βx
k1fcu
0.87fy

Balanced Condition
• For balanced condition, the concrete Crushing
and yielding of reinforcing bars take place
simultaneously
96
ε’cu
xu
d
d
E
f
x
xd
x
E
f
s
y
cu
cu
u
u
u
s
y
cu
















'
'
'



Xu=Neutral axis for
balanced condition

ε’cu
xu
d
s
y
E
f
Balanced State
Concrete and steel
reach their failure
strain simultaneously
x
s
y
E
f
ε >ε’cu
Over reinforced State
Concrete reaches failure
strain prior to Steel (x>xu)
Under reinforced State
Steel reaches failure strain
prior to concrete (x<xu)
x
s
y
E
f
ε <ε’cu

ACI - Determine Mrc
98
0.003 fc
()
C
N.A.
OR
0
C
0
0.85 fc
'
jd
C
T
b
d
Section
M
• Mrc is a measure of the capacity of
concrete in compression to resist
moment .
• It also ensures someductility by
forcing failure in tension
• It primarily depends on fc , b, d
 
  
c s
y
s
b c s b b c
f
E
c c a c f
 
  
0003. ,
( , ), ( )
M f a b d
a
b c b
b
 





(. )'
85
2
M M torc b  , . .05 075
),,,(
'
dbffM ycrc

Determine Ast for Singly Reinforced
Beam
99
0.003 fc()
C
N.A.
OR
0
C
0
0.85 fc
'
jd
C
T
b
d
Section
M









bf
M
dd
y
f
b
c
f
A
c
u
st

 22








2
a
df
M
A
y
u
st

where a c fc, . ( )'
 
M f ab d
a
n c 





 (. )'
85
2
a
A f
f b
st y
c

.85
This procedure for
Ast is iterative
b = 0.85 to 0.65 , f =0.9

Ast and Asc for Doubly Reinforced
Beam
100
0.003 fc()
C
N.A.
OR
0
C
0
0.85 fc
'
jd
C
T
b
d
Section
M
A A Ast st mrc sc ( )  
A A
M
f d a
st mrc stb
b
y b
( )
.
 



5
A
M M
f d dsc
u rc
y



( )
( )'

75.05.0 to

Reinforcement Limits for Flexure
101
• Minimum Steel
• For Rectangular Beams and
Tee beams with flange in
compression
• For Tee beams with flange in
tension
• (All values in psi and inches)
• Maximum Steel
y
w
w
y
c
s
f
db
thanlessnot
db
f
f
A
200
3 '
min, 
db
f
f
A w
y
c
s
'
min,
6

bd
A
bd
A stsc
b




'
'
max 75.0

Check for Flexural Cracking
• The cracking depends on distribution of rebars
in the tension zone and on steel stress
• Crack width w is given by
• The control of cracking is
given by
• z should be less than 175 kip/in2 for interior
• z should be less than 145 kip/in2 for exterior
• fs = 0.6 fy
• A =Area surrounding the bars
• dc = centroid of the bars
102
Adfz cs
3
Adfw cs
3076.0 

Design for Bending Moment
OK
RevisedSectionMaterial
OK
Mu, fc, fy
Section
Computer Mrc
Doubly Reinforced
Beam
Compute Ast, Asc
Singly Reinforced
Beam
Compute Ast
fMrc > Mu
Check
Ast (Max)
Check
Ast (Min)
Moment Design
Completed
Use
Ast (min)
Determine the
Layout of Rebars
Y
OK

CE72.52 - Lecture 3a - Section Behavior - Flexure

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CE72.52 - Lecture 3a - Section Behavior - Flexure