reinforced concrete beams v2.ppt - · PDF fileGeometric model a reinforced concrete bridge...

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


Mathematical model for failure in an reinforced concrete beam


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 is the shear force

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

M is the bending moment

2

PM x

x

X max = 8 in.

(lb.-in.)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 surfaceCompression

Tension

C

T



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




Compression and tension failures in a reinforced concrete beam

Compression and tension failures in a reinforced concrete beam



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



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


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

0.85f’c

a C

0.5a

T

Cc

k3f’c k2x


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

0.85f’c

a C

0.5a

T

Cc

k3f’c k2x


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 bdIf 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 #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


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 0.375b in

#3 rebar diameter Minimum cover #3 rebar diameter

2(0.75 )in 2(0.375 )in

0.75 in

Space between bars

3.75in.

b

6 in.d

#3



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

6d

Half of #3 bardiameter

Minimum cover

4.69 in.d


#3 rebar diameter

0.375

20.375 0.75

b

6 in.d

#3

We now have values for b, d, and As

0.59's y

s yc

A fM A f d

f b


2 22(0.11in. ) 0.22 in.sA

The As for two #3 rebars is:

b

6 in.d

#3

Compute the moment capacity

0.59's y

s yc

A fM A f d

f b

22 0.22in. (60ksi)

0.22in. (60ksi) 4.69in. 0.595 (3.75in.)ksi

56.4 k in. 14.1kips4

MP


35.9kips 35,928lb.


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 4.69in.2 2 5,000psi 3.75in. 4.69in.

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


0.021

20.22 in.

0.01253.75in.(4.69in.)

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 tension-controlled behavior.

c/d = 0.375 for tensioncontrolled behavior

3 3 3

145lb.

1728in. ft. ft.

bhLW

3 3 3

490lb. 145lb.

1728 in. ft. ft.sA L

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

Size of concrete beam

Additional weight of rebarsUnit weight of steel

Unit weight of concrete

3 3 3

(3.75in.)(6in.)(30in.) 145lb.

1728in. ft. ft.W

2

3 3 3

(0.22in. )(30in.) 490lb. 145lb.

1728 in. ft. ft.

56.64 lb. 1.32 lb. 57.96 lb.

Reinforced Concrete Beam AnalysisAn 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

14.1kips

57.96 lb.

S P

W

14,100 lb.

24357.96 lb.

SWR


In summary, this reinforced concrete beam will fail in tension

3.75 in.

6 in.4.69 in.

#3


Questions?


reinforced concrete beams v2.ppt - · PDF fileGeometric model a reinforced concrete bridge...

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