Page 174 - Marine Structural Design

150 Part II UItimate Strength

8.3.2 Plastic Hinge Load
Let’s consider a fully clamped beam under laterally uniform pressure p, the work done by
external load p may be calculated as,

We = [pdy=2p[ A &&=--B PI2 (8.57)
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where 1 is the beam length and B denotes the rotational angle at two ends where plastic hinges
occurred. The work done by the plastic hinges at two ends and the center is

= Mp8(1 + 2 + 1) = 4MpB (8.58)
Equating the work done by lateral pressure and the internal work due to hinging, we may get,
M =g (8.59)
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The collapse load P = pl may be given as,
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P = -Mp (8.60)
1
For a beam under simply support in its two end, the plastic collapse load P may be derived as,
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P=-Mp (8.61)
1
In design codes, a mean value of the collapse load P for these two extreme conditions of
boundary is used to determine the required plastic section module:
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P=--M (8.62)
I P
The required section module Z is

W=- PI (8.63)
120,

8.3.3 Plastic Interaction Under Combined Axial Force and Bending
This sub-section derives the plastic interaction equation for a beam-column due to the action
of combined moment and axial load, for two most used types of cross-sections.
Rectangular Section
The rectangular section is characterized by its width b and height h. When it is in filly plastic
status, the stress in its middle will form the reduced axial load N. The stress in upper and
lower parts will contribute to the reduced plastic moment M . Assuming the height of the
middle part that forms reduced axial load N is e, we may derive,
bh2 be2 (8.64)
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