CIVL 1112
Strength of Reinforced Concrete Beams
Reinforced Concrete Beams
Reinforced Concrete Beams
Mathematical modeling of reinforced concrete is essential to
civil engineering
Mathematical modeling of reinforced concrete is essential to
civil engineering
Concrete as a material
Concrete in a structure
Stress distribution in a reinforced concrete beam
Reinforced Concrete Beams
Reinforced Concrete Beams
Mathematical modeling of reinforced concrete is essential to
civil engineering
Mathematical modeling of reinforced concrete is essential to
civil engineering
Blast failure of a reinforced concrete wall
Geometric model a reinforced concrete bridge
Reinforced Concrete Beams
Mathematical modeling of reinforced concrete is essential to
civil engineering
Blast failure of a reinforced concrete wall
Reinforced Concrete Beams
Mathematical model for failure in an unreinforced concrete
beam
1/11
CIVL 1112
Strength of Reinforced Concrete Beams
Reinforced Concrete Beams
2/11
Reinforced Concrete Beams
Mathematical model for failure in an reinforced concrete beam
In the reinforced concrete beam project, there are three
different failure mode we need to investigate
P
Reinforced Concrete Beams
Reinforced Concrete Beams
First, lets consider the loading of the beam
The purpose of RC is the reinforcement of areas in concrete
that are weak in tension
P
P/2
P
P/2
P/2
Reinforced Concrete Beams
P/2
Reinforced Concrete Beams
Let’s look at the internal forces acting on the beam and
locate the tension zones
The shear between the applied load and the support is
constant V = P/2
P
P
 F  2 V
P/2
V is the shear force
P/2
 V
P
2
P
 F  2 V
P/2
V
P/2
 V 
P
2
CIVL 1112
Strength of Reinforced Concrete Beams
Reinforced Concrete Beams
The shear between the applied load and the support is
constant V = P/2
3/11
Reinforced Concrete Beams
The shear between the applied load and the support is
constant V = P/2
The shear force V = P/2 is constant between the applied load
and the support
P/2
P/2
P/2
P/2
Reinforced Concrete Beams
Reinforced Concrete Beams
Let’s look at the internal moment at section between the
supports and applied load
 Let’s look at the internal moment at section between the
supports and applied load
P
M  2 x
P
 The bending moment is the internal reaction to forces which
cause a beam to bend.
X max = 8 in.
M is the bending moment
x
P/2
M  P/2
4P
P/2
 Bending moment can also be referred to as torque
(lb.-in.)
Reinforced Concrete Beams
The top of the beam is in compression and the bottom of the
beam is in tension
Reinforced Concrete Beams
To model the behavior of a reinforced concrete beam we will
need to understand three distinct regions in the beam.
Two are illustrated below; the third is called shear.
Compression force on the upper
Bending
distributed
part ofmoment
the concrete
beamon
the cut surface
Bending moment distributed on
surface
Compression
MCthe cut
MC
T
P
2
T
Tension force on the lower
part of the concrete beam
P
2
Tension
CIVL 1112
Strength of Reinforced Concrete Beams
Reinforced Concrete Beams
We need models to help us with compression, tension, and
shear failures in concrete
4/11
Reinforced Concrete Beams
We need models to help us with compression, tension, and
shear failures in concrete
P
P
Compression
Tension
Reinforced Concrete Beams
We need models to help us with compression, tension, and
shear failures in concrete
Reinforced Concrete Beams
We need models to help us with compression,
tension, and shear failures in concrete
P
Shear
P
Shear
Shear
Compression
Shear
Tension
Reinforced Concrete Beams
Reinforced Concrete Beams
Compression and tension failures in a reinforced
concrete beam
Compression and tension failures in a reinforced
concrete beam
CIVL 1112
Strength of Reinforced Concrete Beams
Reinforced Concrete Beams
Shear failure in a reinforced concrete beam
Reinforced Concrete Beams
Let’s focus on how to model the ultimate tensile load in a
reinforced concrete beam
Reinforced Concrete Beams
Shear failure in a reinforced concrete beam
Reinforced Concrete Beams
Typical rebar configuration to handle tension and
shear loads
P
Tension
Reinforced Concrete Beams
Reinforced Concrete Beams
Typical rebar configuration to handle tension and
shear loads
Typical rebar configuration to handle tension and
shear loads
5/11
CIVL 1112
Strength of Reinforced Concrete Beams
Whitney Rectangular Stress Distribution
6/11
Whitney Rectangular Stress Distribution
In the 1930s, Whitney proposed the use of a rectangular
compressive stress distribution
In the 1930s, Whitney proposed the use of a rectangular
compressive stress distribution
k3f’c
0.85f’c
k 2x
b
C
c
0.5a
C
a
d
h
As
T
Whitney Rectangular Stress Distribution
Whitney Rectangular Stress Distribution
Assume that the concrete contributes nothing to the tensile
strength of the beam
k3f’c
0.85f’c
k 2x
b
C
c
Assume that the complex distribution of compressive
stress in the concrete can be approximated by a rectangle
0.5a
C
a
d
As
T
T
T
Whitney Rectangular Stress Distribution
0.85f’c
T
Whitney Rectangular Stress Distribution
The height of the stress box, a, is defined as a percentage
of the depth to the neural axis
a  1c
C
c
h
As
0.85f’c
k 2x
b
C
d
h
k3f’c
0.5a
a
T
The height of the stress box, a, is defined as a percentage
of the depth to the neural axis
f 'c  4000 psi

1  0.85
0.85f’c
0.5a
C
a
f 'c  4000 psi
0.5a
C
a
 f 'c  4000 
  0.65
 1000 
1  0.85  0.05 
T
T
CIVL 1112
Strength of Reinforced Concrete Beams
Whitney Rectangular Stress Distribution
Whitney Rectangular Stress Distribution
The values of the tension and compression forces are:
If the tension force capacity of the steel is too high, than
the value of a is large
C  0.85f 'c ba
T  As fy
0.85f’c
F  0  T C
a
7/11
0.5a
C
a
a
0.85f’c
As fy
0.85f 'c b
0.5a
C
a
d
If a > d, then you have too much steel
As fy
0.85f 'c b
T
T
Whitney Rectangular Stress Distribution
Whitney Rectangular Stress Distribution
The internal moment is the value of either the tension or
compression force multiplied the distance between them.
If the tension force capacity of the steel is too high, than
the value of a is large

a
 M  T  d  2 
a

M  As fy  d  
2

0.85f’c
0.5a
C
a
0.85f’c
Substitute the value for a
T
Whitney Rectangular Stress Distribution
T
Reinforced Concrete Beams
Let’s focus on how to model the ultimate shear load in a
reinforced concrete beam
P
We know that the moment in our reinforced
concrete beans is
M  4P
Ptension
C
d
M  4P
Af 

M  As fy  d  0.59 s y 
f
'c b 

Af 
Af 
= s y  d - 0.59 s y 
4 
f' c b 
0.5a
a
Af 

M  As fy  d  0.59 s y 
f 'c b 

d
The internal moment is the value of either the tension or
compression force multiplied the distance between them
a

M  As fy  d  
2

Shear
Shear
CIVL 1112
Strength of Reinforced Concrete Beams
Reinforced Concrete Beams
We can approximate the shear failure in unreinforced concrete
as:
8/11
Reinforced Concrete Beams
Lets consider shear failure in reinforced concrete
Vc  2 f 'c bd
If we include some reinforcing for shear the total shear
capacity of a reinforce concrete bean would be approximated
as:
Vn  Vc  Vs
Vs 
Av fy d
Vn 
P
2
s
Af d

Pshear  2  v y  2 f 'c bd 
 s

Reinforced Concrete Beams
Let’s focus on how to model the ultimate compression load in
a reinforced concrete beam
Reinforced Concrete Beams
There is a “balanced” condition where the stress in the steel
reinforcement and the stress in the concrete are both at their
yield points
The amount of steel required to reach the balanced strain
condition is defined in terms of the reinforcement ratio:
P
P
Compression
Reinforced Concrete Beams
The limits of the reinforcement ratio are established
as:
Reinforcement ratio definition
A
  s
bd
Reinforced Concrete Beams
The limits of the reinforcement ratio are established
as:
c
 0.600
d
 as function of c/d
c f 'c
d fy
As
bd
Compression
0.375 
  0.85 1

Beam failure is controlled by
compression
c
 0.600 Transition between tension
d
and compression control
c
 0.375
d
Beam failure is controlled by
tension
CIVL 1112
Strength of Reinforced Concrete Beams
9/11
Reinforced Concrete Beams
Reinforced Concrete Beams
Lets consider compression failure in over reinforced concrete.
First, let define an equation that given the stress in the tensile
steel when concrete reaches its ultimate strain.
Lets consider compression failure in over reinforced concrete
First, let define an equation that given the stress in the tensile
steel when concrete reaches its ultimate strain
d c 
fsteel  87,000 psi 

 c 
M  4P
only if fs  f y
c
If fsteel < fy then or
 0.600
d
Pcompression 
a
 d  c 
Mcompression  As 
  d  2  87,000 psi
 c 

Reinforced Concrete Beams
Consider the different types of failures in reinforced concrete:
As  d  c  
a
d   87,000 psi
4  c  
2
Reinforced Concrete Beam Analysis
Let’s use the failure models to predict the ultimate strength-toweight (SWR) of one of our reinforced concrete beams from
lab
Consider a beam with the following characteristics:
Concrete strength f’c = 5,000 psi
Steel strength fy = 60,000 psi
The tension reinforcement will be 2 #3 rebars
The shear reinforcement will be #3 rebars bent in a U-shape spaced at
4 inches.
Use the minimum width to accommodate the reinforcement
Reinforced Concrete Beam Analysis
Reinforcing bars are denoted by the bar number. The
diameter and area of standard rebars are shown below.
Reinforced Concrete Beam Analysis
Based on the choice of reinforcement we can compute an
estimate of b and d
#3 rebar diameter
Bar #
3
4
5
6
7
8
9
10
11
2
Diameter (in.) As (in. )
0.375
0.11
0.500
0.20
0.625
0.31
0.750
0.44
0.875
0.60
1.000
0.79
1.128
1.00
1.270
1.27
1.410
1.56
b
Minimum cover
b  2  0.375 in  2(0.75 in ) 2(0.375 in )
0.75 in 
d
6 in.
#3
#3 rebar diameter
Space between bars
3.75in.
CIVL 1112
Strength of Reinforced Concrete Beams
Reinforced Concrete Beam Analysis
If we allow a minimum cover under the rebars were can
estimate d
Half of #3 bar
diameter
b
d 6 
#3
We now have values for b, d, and As
4.69 in.
Af 

M  As fy  d  0.59 s y 
f 'c b 

b
0.375
0.375 0.75
2
#3 rebar diameter
d
Reinforced Concrete Beam Analysis
Minimum cover
d
6 in.
The As for two #3 rebars is:
d
6 in.
As  2(0.11in.2 )  0.22 in.2
#3
Reinforced Concrete Beam Analysis
10/11
Reinforced Concrete Beam Analysis
b
Compute the moment capacity
Let’s check the shear model
Af d

Pshear  2  v y  2 f 'c bd 
s


Af 

M  As fy  d  0.59 s y 
f 'c b 

d
6 in.
#4
Area of two #3 rebars

0.22in.2 (60ksi) 
 0.22in.2 (60ksi)  4.69in.  0.59

5ksi (3.75in.) 



 2 0.11in.2  60,000psi 4.69in.

 2
 2 5,000psi  3.75in. 4.69in. 


4in.


Shear reinforcement spacing
 56.4 k  in. 
P
M
4
 14.1kips
 35,928lb.

35.9kips
Since Ptension < Pshear therefore Ptension controls
Reinforced Concrete Beam Analysis
An 1 estimate is given as:
Let’s check the reinforcement ratio
A
  s
bd
Reinforced Concrete Beam Analysis
Reinforcement ratio definition
 as function of c/d
c f 'c
  0.85 1
d fy
To compute , first we need to estimate 1
f 'c  4000 psi
1  0.85

f 'c  4000 psi
 f 'c  4000 
  0.65
 1000 
1  0.85  0.05 
 5,000  4,000 
  0.80
1,000


1  0.85  0.05 
CIVL 1112
Strength of Reinforced Concrete Beams
Reinforced Concrete Beam Analysis
Check the reinforcement ratio for the maximum steel allowed
for tension controlled behavior or c/d = 0.375
  0.85 1
Reinforced Concrete Beam Analysis
An estimate of the weight of the beam can be made as:
Size of concrete beam
c f 'c
5ksi
 0.85(0.80)0.375
d fy
60ksi
 0.021
11/11
W 
bhL
 145lb. 
3
3 
1728in. ft.  ft.3 
Unit weight of steel
Additional weight of rebars
c/d = 0.375 for tension
controlled behavior

A
0.22 in.2
 s 
 0.0125
bd 3.75in.(4.69in.)
Unit weight of concrete
As L
 490lb.  145lb. 

3
3 
1728 in. ft. 
ft.3

The amount of steel in this beam is tension-controlled
behavior.
Reinforced Concrete Beam Analysis
An estimate of the weight of the beam can be made as:
Size of concrete beam
W 
Reinforced Concrete Beam Analysis
In summary, this reinforced concrete beam will fail in tension
Unit weight of concrete
Unit weight of steel
Additional weight of rebars
4.69 in.
6 in.

(0.22in. )(30in.)  490lb.  145lb. 

1728 in.3 ft.3 
ft.3

2
 56.64 lb.  1.32 lb.  57.96 lb.
Reinforced Concrete Beam Analysis
Questions?
S  P  14.1 kips

W  57.96 lb.

3.75 in.
(3.75in.)(6in.)(30in.)  145lb. 
 ft.3 
1728in.3 ft.3


#3
SWR 
14,100 lb.

57.96 lb.
243