section modulus
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5.7 PLASTIC BENDING OF STRAIGHT BEAMS
Recall the stress-strain diagram for mild steel.
5.7.1
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Example
5.7.2
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is the bending moment at which the entire section will become plastic, or the
bending moment associated with yielding that has penetrated the entire depth is called
the fully plastic moment. Note that the is the ultimate moment capacity of a
section. If , then a structure will collapse. When M reaches , we
say that a plastic hinge has been formed. If the number of plastic hinges exceeds
certain value, then the structure or a member will collapse.
COMPUTATION OF ELASTIC-PLASTIC MOMENT CAPACITIES
5.7.3
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Elastic Moment Capacity
Always C T from equilibrium consideration.
Elastic moment capacity
5.7.4
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PLASTIC MOMENT CAPACITY
C T always, from equilibrium consideration.
Plastic moment capacity (fully plastic)
U ltim ate m o m en t cap acity o f th e
sec tio n
5.7.5
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Elastic and Plastic Section Modulus
For a rectangular section
at the bottom fibre,
or,
Here called section modulus
Note that for unsymmetrical section, , and we may have two different section
moduli, one corresponding to and the other corresponding to . For a general case
5.7.6
Note that
For elastic bending we have
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Plastic Neutral Axis
Note that for a general case, plastic neutral axis will be different from the elastic neutral
axis. For a plastic section, always.
where C = Total compressive force =
T = Total tensile force =
Now
where called plastic section modulus.
Shape factor
5.7.7
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The ratio between fully plastic and fully elastic moments is called the shape factor of the
section.
Shape Factor =
Problem 5-21
Required: Determine
Elastic Moment Capacity
5.7.8
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Locate neutral axis:
or,
y 6 2 5. m m from bottom of the section
Find moment of inertia about neutral axis
and
(with respect to top)
(with respect to bottom)
Minimum section modulus
Elastic moment capacity
Ans.
5.7.9
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Fully Plastic Moment Capacity
For fully plastic moment:
Hence the neutral axis may be located by inspection 75 mm from the bottom of the
section.
Plastic section modulus
Here
,
Shape factor
5.7.10
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CALCULATION OF PARTIALLY PLASTIC MOMENTS M PP
Example: Problem 5-118, pp. 212
Because the section is partially plastic, only a portion of the section has experienced
yielding.
Note that because the section is symmetric
and
Now,
, by inspection
5.7.11
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or, Ans.
EXAMPLE: PARTIALLY PLASTIC MOMENT CALCULATION
Problem 5-19 pp. 213
This problem is very similar to problem 5-18.
Because of symmetry we have: and
or,
5.7.12
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Now,
or,
or,
or,
or,
or,
Ans.
5.7.13
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