Engineering Mechanics & Design of Steel Structures : Plastic Analysis

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Plastic Sections Modulus

• Plastic Section Modulus

Shape Factor (∝)

Where Z_y is elastic section modulus

Load factors 

Load Factor = Factor of safety x Shape factor

λ = FS × α

Shape factors for different shapes

Reserve Strength (Ψ)

• It is the ratio of the ultimate load W_u to the load at first yield W_y of the structure. 

Length of a Plastic hinge (L_p)

• It is the length of the beam over which the moment is greater than the yield moment (M_y).

(a) For simply supported beam carrying a concentrated load, length of plastic hinge is given by
 (for rectangular section)

 (for I section) 
(b) For simply support beam carrying UDL length of plastic hinge is given by 

Where ∝ is shape factor and L is the total length of the beam.

• The length of plastic hinge depends on loading and geometry.

Collapse Load Formulae

• Simply supported beam with the concentrated load at the center

• Simply supported beam with the uniformly distributed load

• Propped cantilever with a concentrated load at the center

• Propped cantilever with a uniformly distributed load

• Fixed beam with a concentrated load at the center

• Fixed beam with eccentric loading

• Fixed beam with the uniformly distributed load

• Fixed beam with hydrostatic loading

• Continuous beam with a uniformly distributed load

The positions of the plastic hinges are one at the support B and one on each side of the central support at a distance of 0.414L from A & C.

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