Plastic Analysis and Ultimate Load · Introduction
Linear-elastic analysis uses allowable stress as the design criterion — any local yielding is treated as failure. But for plastic materials (especially statically indeterminate structures), yield ≠ failure: the structure can continue carrying load until enough sections yield and the system becomes a mechanism. Plastic analysis uses this ultimate state as the design basis — making better use of the material.
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Elastic design and its limitations
MotivationPrevious chapters dealt with linear-elastic analysis: the material stays in the elastic range under load, and the structure fully recovers upon unloading, with no residual deformation. Design by cross-sectional stress requires the maximum stress $\sigma_{\max}$ not to exceed the allowable stress:
This "allowable-stress" approach is elastic design.
For plastic-material, statically indeterminate structures — local material failure ≠ loss of structural capacity. Elastic design cannot capture the strength reserves of the structure. Even after local yielding, the structure can carry more load. This motivates plastic analysis: use the ultimate state (mechanism) as the design basis — only then can the material's full capacity be used.
Plastic development in an indeterminate truss (see Animation 12.1.2)
Truss ExampleUsing the indeterminate truss of ($EA$ constant), we see how a structure continues carrying load after yield:
As $F_{\mathrm P}$ increases, bar $CD$'s stress first reaches $\sigma_{\mathrm s}$. $CD$ then continues to stretch but its axial force stays constant — can the truss still be loaded?
Answer
Although $CD$ has yielded, its effect can be replaced by a pair of external forces of magnitude $A\sigma_{\mathrm s}$ on the rest of the structure. The structure remains geometrically stable and can carry more load.
Further loading continues until bars $AC$ and $BC$ also reach yield — the system becomes a mechanism, and only then does real failure happen.
Determinate truss Determinate
- One bar failing forms a mechanism — no strength reserve;
- Ultimate load = smallest yield load of a single bar.
Indeterminate truss Indeterminate
- After one bar yields, there is still strength reserve (no material strain-hardening);
- Multiple bars must yield for the system to become a mechanism;
- The extra capacity unlocked by plastic analysis comes from this.
Plastic development in a continuous beam (see Animation 12.1.3)
Continuous BeamFor flexural members, strength reserve comes not only from redundancy but also from the cross-section itself — fiber yielding is progressive, from the outermost fibers to the full section, giving a substantial margin.
$M_{\max} < M_{\mathrm s}$
By elastic analysis, the maximum moment (at section $D$) has not yet reached $\sigma_{\mathrm s}$. The whole beam is elastic.
$M_{\mathrm s} \le M_{D} < M_{\mathrm u}$
Edge fibers of section $D$ yield first; inner fibers remain elastic. Stress is no longer linear, but the structure still carries more load — cross-sectional strength reserve.
$M_{D} = M_{\mathrm u}$
All fibers at section $D$ yield ($\sigma = \pm\sigma_{\mathrm s}$), reaching the ultimate moment $M_{\mathrm u}$. $D$ loses its rotational restraint — it becomes a hinge.
- Cross-sectional reserve: from first yield of the outermost fiber to full yield of the section — $M_{\mathrm s} \to M_{\mathrm u}$, ~1.5× for a rectangular section;
- Structural reserve: from the first plastic hinge to the formation of a mechanism — multiple hinges form in sequence;
- The product of the two gives the total strength reserve of an indeterminate flexural structure — the additional capacity unlocked by plastic design.
Key features of the plastic hinge
- A plastic hinge forms at the section of maximum moment (location depends on $M$ diagram, not on the structure itself);
- The two sides always carry a constant couple of $M_{\mathrm u}$;
- A plastic hinge can rotate, but in one direction only — only in the direction of $M_{\mathrm u}$; reverse rotation restores the rigid connection (unlike a regular hinge);
- Each new plastic hinge reduces the degree of redundancy by 1.
As loading continues, the next-highest-moment section (e.g. $B$) also reaches $M_{\mathrm u}$ and forms another plastic hinge — hinges form one by one, until the system becomes a mechanism (a bending collapse mechanism) and the structure loses capacity. The corresponding load is the ultimate load $F_{\mathrm{Pu}}$. Since the 1930s–40s, this methodology of designing by ultimate load — i.e. plastic analysis — has been well established.
Ideal elasto-plastic model · Load-path dependence
Idealized σ-εTo build a plastic-analysis theory, we first make reasonable simplifications of the material's stress-strain behavior. is a typical tensile stress-strain curve of mild steel; we idealize it as an ideal elasto-plastic material (see Animation 12.1.4), assumed to behave identically under tension and compression.
Three rules of ideal elasto-plasticity
- Elastic range (segments $OA$, $OD$): stress and strain are linearly related, $\sigma = E\varepsilon$.
- Plastic flow (segments $AC$, $DE$): once $\sigma_{\mathrm s}$ is reached, the material enters the plastic-flow regime — stress stays constant while strain grows indefinitely.
- Unloading (segment $BF$): after plastic deformation, unloading follows a line parallel to $OA$. During unloading $\Delta\sigma/\Delta\varepsilon = E$ still holds — the material is ideal elasto-plastic on loading but linear elastic on unloading.
After unloading, the elastic part $\varepsilon_{\mathrm e}$ vanishes with the stress, while the plastic part $\varepsilon_{\mathrm p}$ remains — the residual strain.
Strain and stress are no longer one-to-one — the same stress $\sigma_{\mathrm s}$ can correspond to different strains (the three paths $OA'$, $OABB'$, $OACC'$ in give $\varepsilon_{1}, \varepsilon_{2}, \varepsilon_{3}$ all different).
Equivalently: the final state depends on the load path. Plastic analysis cannot rely on the instantaneous state alone; we must trace the whole stress-deformation history — which sections yielded first, which unloaded, which reloaded, …. This makes plastic analysis substantially more complex than elastic analysis. Fortunately, under proportional loading a set of beautiful theorems (§12-4) restores much of the simplicity.
Some materials (e.g. aluminum alloys) exhibit strain hardening after the proportional limit — stress continues to rise with strain, with no obvious flow plateau (a/b). This chapter restricts attention to the ideal elasto-plastic model — the simplest and most engineering-relevant case.
- Elastic design uses allowable stress — and underestimates the capacity of redundant structures;
- Plastic design uses the ultimate state (mechanism) — unlocking both cross-section and system strength reserves;
- The plastic hinge forms at the section of maximum moment, maintains $\pm M_{\mathrm u}$ couples, and rotates in one direction;
- Ideal elasto-plastic material: on loading, stress caps at $\sigma_{\mathrm s}$; on unloading $\Delta \sigma/\Delta \varepsilon = E$;
- Plastic analysis is path-dependent — the same stress may correspond to many strains;
- Next, §12-2 derives the ultimate moment $M_{\mathrm u}$ of a rectangular-section pure-bending beam.