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

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

Elastic Moment Capacity

Always C T from equilibrium consideration.

Elastic moment capacity

5.7.4

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

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

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

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

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

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

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

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

Now,

or,

or,

or,

or,

or,

Ans.

5.7.13

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