Lecture 7 Flexure

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Lecture 7 - Flexure

• January 29, 2003
• CVEN 444

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Lecture Goals

• Safety factors
• Loading and Resistance
• Balanced Beams

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Safety Provisions
Structures and structural members must always be
designed to carry some reserve load above what is
expected under normal use.
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Safety Provisions
There are three main reasons why some sort of
safety factor are necessary in structural
design. 1 Consequences of failure. 2
Variability in loading. 3 Variability in
resistance.
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Consequences of Failure
A number of subjective factors must be considered
in determining an acceptable level of safety.

• Potential loss of life.
• Cost of clearing the debris and replacement of
  the structure and its contents.
• Cost to society.
• Type of failure warning of failure, existence of
  alternative load paths.

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Variability in Loading
Frequency distribution of sustained component of
live loads in offices.
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Variability in Resistance

• Variability of the strengths of concrete and
  reinforcement.
• Differences between the as-built dimensions and
  those found in structural drawings.
• Effects of simplification made in the derivation
  of the members resistance.

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Variability in Resistance
Comparison of measured and computed failure
moments based on all data for reinforced concrete
beams with fc gt 2000 psi.
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Margin of Safety
The distributions of the resistance and the
loading are used to get a probability of failure
of the structure.
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Margin of Safety
The term Y R - S is called the safety margin.
The probability of failure is defined as and
the safety index is
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Loading

• SPECIFICATIONS
• Cities in the U.S. generally base their building
  code on one of the three model codes
• Uniform Building Code
• Basic Building Code (BOCA)
• Standard Building Code

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Loading
These codes have been consolidated in the 2000
International Building Code. Loadings in these
codes are mainly based on ASCE Minimum Design
Loads for Buildings and Other Structures has been
updated to ASCE 7-98.
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Loading
The loading variations are taken into
consideration by using a series of load factors
to determine the ultimate load, U.
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Loading
The equations come from ACI code 9.2 on loading
(4.6 in your book), D Dead Load W Wind
Load L Live Load Lr Roof Load F Fluid
Pressure R Rain Load E Earthquake Load T
Temperature Load S Snow Load H Soil Load
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Loading
The most general equation for the ultimate load,
U (Mu) that you will see is going to be
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Resistance
The load factors will generate the ultimate load,
which is used in the design and analysis of the
structural member. Mu Ultimate Moment Mn
Nominal Moment f Strength Reduction Factor
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Resistance
The strength reduction factor, f, varies from
member to member depending whether it is in
tension or compression or the type of member.
The code has been setup to determine the
reduction.
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Three possibilities in Inelastic Behavior

• Compression Failure - (over-reinforced beam)
• Tension Failure - (under-reinforced beam)
• Balanced Failure - (balanced reinforcement)

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Inelastic Behavior
Compression Failure
The concrete will crush before the steel yields.
This is a sudden failure. The beam is known as an
over-reinforced beam.
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Inelastic Behavior
Tension Failure
The reinforcement yields before the concrete
crushes. The concrete crushes is a secondary
compression failure. The beam is known as an
under-reinforced beam.
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Inelastic Behavior
Balanced Failure
The concrete crushes and the steel yields
simultaneously. The beam is known as an
balanced-reinforced beam.
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Inelastic Behavior
Which type of failure is the most desirable?
The under-reinforced beam is the most
desirable. fs fy es gtgt ey You want ductility
system deflects and still carries load.
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Balanced Reinforcement Ratio, rbal
rbal unique r value to get simultaneous ec
0.003 es ey
Use similar triangles
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Balanced Reinforcement Ratio, rbal
The equation can be rewritten to find cb
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Nominal Moment Equation
The equation can be rewritten in the form
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Nominal Moment Equation
The equation can be rewritten in the form
Use the ratio r b/d and r
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Nominal Moment Equation
Use w r fy/fc and
Use the ratio r b/d and R
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Strain Limits Method for Analysis
The strength reduction factor, f, will come into
the calculation of the strength of the beam.
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Limitations on Reinforcement Ratio, r
The selection of the steel will be determined by
the Lower Limit on r ACI 10.5.1

ACI Eqn. (10-3) fc fy are in psi
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Limitations on Reinforcement Ratio, r
Lower Limit on r ACI 10.5.1 Lower limit
used to avoid Piano Wire beams. Very small As
( Mn lt Mcr ) es is huge (large deflections) when
beam cracks ( Mn gt Mcr ) beam fails right away
because Mn lt Mcr
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Additional Requirements for Lower Limit on r
If As (provided) 4/3 As (required) based on
analysis, then As (min) is not required.
for
As (provided) See ACI 10.5.3
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Additional Requirements for Lower Limit on r
Temperature and Shrinkage reinforcement in
structural slabs and footings (ACI 7.12) place
perpendicular to direction of flexural
reinforcement. GR 40 or GR 50 Bars As (TS)
0.0020 Ag GR 60 or Welded Wire Fabric (WWF)
As (TS) 0.0018 Ag Ag - Gross area of the
concrete
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Example
Given fc 3 ksi fy 40 ksi and As 4
in2 Determine (1) (2)
Determine if the beam will satisfy ACI code. If
fc 6 ksi?
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Example
Given fc 3 ksi fy 40 ksi and As 4 in2
The minimum steel ratio is
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Example
Given fc 3 ksi fy 40 ksi and As 4 in2
The neutral axis is
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Example
The strain in the steel is
There for the beam is in the compression zone and
f would be 0.65, however c/d ratio is greater
than 0.375 so the beam will need to be redesigned.
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Example
Given fc 6 ksi fy 40 ksi and As 4 in2
The minimum steel ratio is
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Example
Given fc 6 ksi fy 40 ksi and As 4 in2
The neutral axis is at
39
Example
The strain in the steel will be
There for the beam is in the tension zone and f
will be 0.9.
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Homework (Due 2/5/03)

• Do the following problems
• Problem 5.2 from the text.
• Problem 5.3 from the text.