Plastic Analysis & Design of Steel Notes

Last reviewed 16 Sept 2026 · 5 min read

Philosophy

Mild steel is ductile: after yielding it strains a great deal at nearly constant stress. A statically indeterminate steel structure therefore does not fail when one section yields; the yielded section becomes a plastic hinge, moments redistribute, and the load keeps increasing until enough hinges form to create a collapse mechanism. Plastic design sizes members so that the factored loads equal (or are less than) the collapse load.

Benefits: more realistic assessment of strength; economy in continuous beams and frames (often 10–20% lighter than elastic design); simpler calculation of the collapse load than of exact elastic moments.

(Theory of plastic hinges, theorems and standard mechanisms is developed in Structural Analysis — Plastic Analysis. This chapter focuses on steel design use.)

IS 800:2007 requirements for plastic analysis

Code ProvisionConditions for using plastic analysis (key points)

Members where plastic hinges form must be of plastic (Class 1) cross-section; other members at least compact.
Steel must have adequate ductility: ratio $f_{u} / f_{y} \geq 1.2$ , sufficient elongation, and yield plateau (strain at $f_{u}$ well beyond yield).
Yield stress of steel generally not more than 450 N/mm².
Members must be restrained laterally at hinge locations and within specified distances to prevent lateral–torsional buckling before the mechanism forms.
Axial force in members with hinges is limited (a column with high axial load cannot develop full rotation); stiffeners are needed where concentrated loads act at hinges.
Connections at hinge locations must develop the plastic moment and rotation.
Second-order (P-Δ) effects are considered for sway frames.

Plastic moment of rolled sections

M_{p} = Z_{p} f_{y}, M_{d} = \frac{Z _{p} f _{y}}{γ _{m 0}}

$Z_{p}$ (plastic section modulus) of I-sections is tabulated; the shape factor $Z_{p} / Z_{e}$ is about 1.12–1.15 for major-axis bending of rolled I-sections (1.5 for rectangles, 1.7 for solid circles).

For an I-section, approximately:

Z_{p} \approx b t_{f} (h - t_{f}) + \frac{t _{w} ( h - 2 t _{f} ) ^{2}}{4}

Effect of axial force and shear

Axial force $N$ occupies part of the section, reducing the moment capacity. For I-sections about the major axis (with $n = N / N_{d}$ ): $M_{p r} \approx 1.18 M_{p} (1 - n) \leq M_{p}$ (a common form; IS 800 uses $1.11 M_{d z} (1 - n)$ for design).
Shear above about 60% of the shear capacity reduces moment capacity (web partly used by shear), as in Steel Beams.

Collapse loads — summary

Member / loading	Collapse load
Simply supported, central point load	$W_{c} = 4 M_{p} / L$
Simply supported, UDL	$w_{c} = 8 M_{p} / L^{2}$
Fixed beam, central point load	$W_{c} = 8 M_{p} / L$
Fixed beam, UDL	$w_{c} = 16 M_{p} / L^{2}$
Propped cantilever, central point load	$W_{c} = 6 M_{p} / L$
Propped cantilever, UDL	$w_{c} = 11.66 M_{p} / L^{2}$
End span of continuous beam (UDL, simply supported outer end)	$w_{c} = 11.66 M_{p} / L^{2}$
Interior span of continuous beam (UDL)	$w_{c} = 16 M_{p} / L^{2}$
Portal frame, fixed bases: beam mechanism (central $V$ )	$V = 8 M_{p} / L$
Portal frame: sway mechanism (horizontal $H$ at beam level, height $h$ )	$H = 4 M_{p} / h$

For portal frames, check beam, sway and combined mechanisms; the smallest load factor governs.

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Plastic Analysis & Design of Steel

Philosophy

IS 800:2007 requirements for plastic analysis

Plastic moment of rolled sections

Effect of axial force and shear

Collapse loads — summary

This chapter is in the syllabus of

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