reinforced concrete beams

Reinforced Concrete Beams

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



Geometric model a reinforced concrete bridge



Blast failure of a reinforced concrete wall



Blast failure of a reinforced concrete wall


Mathematical model for failure in an unreinforced concrete beam

CIVL 1112 Strength of Reinforced Concrete Beams 1/11


P

In the reinforced concrete beam project, there are three different failure mode we need to investigate


P

P/2 P/2

First, lets consider the loading of the beam


P

P/2 P/2

The purpose of RC is the reinforcement of areas in concrete that are weak in tension


P

P/2 P/2

Let’s look at the internal forces acting on the beam and locate the tension zones

2

PF V

V

2

PV

V


P/2

The shear between the applied load and the support is constant V = P/2

P/2

2 2

P PF V V


P/2


P/2



P/2


P/2

The shear force V = P/2 is constant between the applied load and the support


P

P/2 P/2

Let’s look at the internal moment at section between the supports and applied load

P/2

M2

PM x

x

X max = 8 in.

(in. lb.)4M P


Let’s look at the internal moment at section between the supports and applied load

The bending moment is the internal reaction to forces which cause a beam to bend.

Bending moment can also be referred to as torque M

2

P


The top of the beam is in compression and the bottom of thebeam is in tension

Bending moment distributed on the cut surface

C

T

Compression force on the upper part of the concrete beam

Tension force on the lower part of the concrete beam


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.

2

P

MBending moment distributed on the cut surface

C

T

Compression

Tension


P

Tension

We need models to help us with compression, tension, and shear failures in concrete



P

Compression



P

Shear Shear


P

Tension

CompressionShear Shear


Reinforced Concrete Beams Reinforced Concrete Beams

Compression and tension failures in a reinforced concrete beam

Compression and tension failures in a reinforced concrete beam

Reinforced Concrete Beams Reinforced Concrete Beams Shear failure in a reinforced concrete beam


Shear failure in a reinforced concrete beam

Reinforced Concrete Beams Reinforced Concrete Beams

P

Tension

Let’s focus on how to model the ultimate tensile load in a reinforced concrete beam

Typical rebar configuration to handle tension and shear loads

Reinforced Concrete BeamsTypical rebar configuration to handle tension and

shear loads


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

b

h d

As

T

Cc

k3f’c k2x

T

0.85f’c

a C

0.5a



Assume that the concrete contributes nothing to the tensile strength of the beam

b

h d

As

T

Cc

k3f’c k2x

T

0.85f’c

a C

0.5a


Assume that the complex distribution of compressive stress in the concrete can be approximated by a rectangle

b

h d

As

T

Cc

k3f’c k2x

T

0.85f’c

a C

0.5a


The height of the stress box, a, is defined as a percentage of the depth to the neural axis

T

0.85f’c

a C

0.5a

1a c


The height of the stress box, a, is defined as a percentage of the depth to the neural axis

1' 4000 0.85cf psi

' 4000cf psi

1

' 40000.85 0.05 0.65

1000cf

T

0.85f’c

a C

0.5a


The values of the tension and compression forces are:

0.85 'cC f ba

s yT A f

0.85 's y

c

A fa

f b

0F T C

T

0.85f’c

a C

0.5a


If the tension force capacity of the steel is too high, than the value of a is large

0.85 's y

c

A fa

f b

If a > d, then you have too much steel

d

T

0.85f’c

a C

0.5a



If the tension force capacity of the steel is too high, than the value of a is large

2

aM T d

2s y

aM A f d

d

T

0.85f’c

a C

0.5a


The internal moment is the value of either the tension or compression force multiplied the distance between them.

2s y

aM A f d

Substitute the value for a

0.59's y

s yc

A fM A f d

f b

4M PT

0.85f’c

a C

0.5a

d

We know that the moment in our reinforced concrete beans is

0.59's y

s yc

A fM A f d

f b


The internal moment is the value of either the tension or compression force multiplied the distance between them

4M P

0.59s y s ytension

c

A f A fP = d -

4 f' b

P

Shear Shear

Let’s focus on how to model the ultimate shear load in a reinforced concrete beam


n c sV V V


We can approximate the shear failure in unreinforced concrete as:

2 'c cV f bd If we include some reinforcing for shear the total shear

capacity of a reinforce concrete bean would be approximated as:

v ys

A f dV

s

2n

PV

2 2 'v yshear c

A f dP f bd

s


Lets consider shear failure in reinforced concrete



P

Compression

Let’s focus on how to model the ultimate compression load in a reinforced concrete beam


P

Compression

sA

bd

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:

1

'0.85 c

y

fc

d f

sA

bd


The limits of the reinforcement ratio are established as:

Reinforcement ratio definition

as function of c/d


The limits of the reinforcement ratio are established as:

0.375c

d

0.600c

d Beam failure is controlled by

compression

Beam failure is controlled by tension

0.375 0.600c

d Transition between tension

and compression control

87,000steel

d cf psi

c

87,0002compression s

d c aM A d psi

c


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

If fsteel < fy then or 0.600c

d

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

87,0004 2

scompression

A d c aP d psi

c


4M P only if s yf f



Consider the different types of failures in reinforced concrete:

Reinforced Concrete Beam Analysis

Let’s use the failure models to predict the ultimate strength-to-weight (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 #4 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


Bar # Diameter (in.) As (in.2)

3 0.375 0.114 0.500 0.205 0.625 0.316 0.750 0.447 0.875 0.608 1.000 0.799 1.128 1.00

10 1.270 1.2711 1.410 1.56

Reinforcing bars are denoted by the bar number. The diameter and area of standard rebars are shown below.

Based on the choice of reinforcement we can compute an estimate of b and d

2 0.5b in

#4 rebar diameter Minimum cover #3 rebar diameter

2(0.75 )in 2(0.375 )in

0.75 in

Space between bars

4.0in.

b

6 in.d

#4


If we allow a minimum cover under the rebars were can estimate d

6d

Half of #4 bardiameter

Minimum cover

4.625 in.d


#3 rebar diameter

0.5

2 0.3750.75

b

6 in.d

#4

We now have values for b, d, and As

0.59's y

s yc

A fM A f d

f b


2 22(0.20 in. ) 0.40 in.sA

The As for two #4 rebars is:

b

6 in.d

#4


Compute the moment capacity

0.59's y

s yc

A fM A f d

f b

2

2 0.4in. (60ksi)0.4in. (60ksi) 4.625in. 0.59

5 (4in.)ksi

94.0 k in. 23.5 kips4

MP


35.76kips35,757lb.


Let’s check the shear model

Area of two #3 rebars

2 2 'v yshear c

A f dP f bd

s

22 0.11in. 60,000psi 4.625in.2 2 5,000psi 4in. 4.625in.

4in.

Shear reinforcement spacing

Since Ptension < Pshear therefore Ptension controls

b

6 in. d

#4

sA

bd


Let’s check the reinforcement ratio

1

'0.85 c

y

fc

d f

To compute , first we need to estimate 1

Reinforcement ratio definition

as function of c/d

An 1 estimate is given as:

1' 4000 0.85cf psi

' 4000cf psi

1

' 40000.85 0.05 0.65

1000cf


1

5,000 4,0000.85 0.05 0.80

1,000

0.02120.4 in.

0.0224in.(4.625in.)

sA

bd


Check the reinforcement ratio for the maximum steel allowed for tension controlled behavior or c/d = 0.375

1

' 5ksi0.85 0.85(0.80)0.375

60ksic

y

fc

d f

The amount of steel in this beam is just a bit over the allowable for tension controlled behavior.

c/d = 0.375 for tensioncontrolled behavior 0.034


However, the maximum about of steel where compression is in control is c/d = 0.600

1

' 5ksi0.85 0.85(0.80)0.600

60ksic

y

fc

d f

Therefore, the beam is in the lower part of the transition zone and for our purposes is OK.

c/d = 0.600 for compressioncontrolled behavior

20.4 in.0.022

4in.(4.625in.)sA

bd


3 3 3

(4in.)(6in.)(30in.) 145lb.

1728in. ft. ft.W

2

3 3 3

(0.4in. )(30in.) 490lb. 145lb.

1728 in. ft. ft.

60.42lb. 2.39lb. 62.81lb.

Reinforced Concrete Beam Analysis An estimate of the weight of the beam can be made as:

Size of concrete beam

Additional weight of rebars Unit weight of steel

Unit weight of concrete

23.5 kips

62.81 lb.

S P

W

23,500 lb.374.2

62.81 lb.SWR


In summary, this reinforced concrete beam will fail in tension

4 in.

6 in.4.625 in.

#4


Questions?


reinforced concrete beams

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