Chapter 12 · Section 4

Theorems for Judging the Ultimate Load under Proportional Loading

Complex indeterminate structures may admit multiple collapse mechanisms, and the true failure form is often hard to predict. Under proportional loading, three theorems — the minimum theorem (upper bound), the maximum theorem (lower bound), and the uniqueness theorem — provide the rigorous mathematical foundation and an enumeration verification method for the ultimate-load computation.

How to use

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12.4.1

Proportional loading · Three conditions at the ultimate state

Three Conditions

Definition of proportional loading

Proportional loading means all loads grow in a fixed ratio — the whole load group can be represented by a single load parameter $F_{\mathrm P}$ that increases monotonically without unloading. In this section we assume:

Material is ideal elasto-plastic;
Positive and negative ultimate moments are equal in magnitude ($M_{\mathrm u}^{+} = M_{\mathrm u}^{-} = M_{\mathrm u}$);
Ignore axial and shear effects on the ultimate moment.

Three conditions to be simultaneously satisfied at the ultimate state

① Equilibrium

Equilibrium

At the ultimate state, the whole structure or any part remains in equilibrium.

② Yield (capacity)

Yield / Strength

The moment magnitude at every section does not exceed the section ultimate moment: $|M(x)| \le M_{\mathrm u}$.

③ Mechanism

Mechanism

The structure has enough sections at their ultimate moment that it becomes a mechanism — capable of moving in the direction that the load does positive work.

12.4.2

Three load types · Basic theorem

Base Theorem

Three load types

Name	Symbol	Conditions satisfied
Collapse load	$F_{\mathrm P}^{+}$	Equilibrium + Mechanism (not necessarily Yield)
Safe load	$F_{\mathrm P}^{-}$	Equilibrium + Yield (not necessarily Mechanism)
Ultimate load	$F_{\mathrm{Pu}}$	All three at once — both a collapse load and a safe load

Basic theorem

Any collapse load is never less than any safe load:

$$ F_{\mathrm P}^{+} \;\ge\; F_{\mathrm P}^{-} $$

(12.4-1)

Proof (virtual work)

① Let the structure under $F_{\mathrm P}^{+}$ have become a mechanism with $n$ plastic hinges. By the virtual-work principle on the mechanism motion:

$$ F_{\mathrm P}^{+}\cdot \Delta \;=\; \sum_{i=1}^{n}\, |M_{\mathrm u i}|\cdot |\theta_{i}| \qquad \text{(a)} $$

(12.4-2)

$\theta_{i}$ rotates in the direction of $M_{\mathrm u i}$, so $M_{\mathrm u i}\theta_{i}$ is always positive

② Let a safe load $F_{\mathrm P}^{-}$ act with moment diagram $M^{-}$. Subjecting this equilibrium system to the same mechanism motion:

$$ F_{\mathrm P}^{-}\cdot \Delta \;=\; \sum_{i=1}^{n}\, M_{i}^{-}\, \theta_{i} \qquad \text{(b)} $$

(12.4-3)

$M_{i}^{-}$ is the moment in $M^{-}$ at the $i$-th plastic hinge

③ By the yield condition $|M_{i}^{-}| \le M_{\mathrm u}$:

$$ \sum_{i} M_{i}^{-}\, \theta_{i} \;\le\; \sum_{i} |M_{\mathrm u i}|\cdot |\theta_{i}| $$

(12.4-4)

Substituting (a) and (b) with $\Delta > 0$:

$$ F_{\mathrm P}^{+}\, \Delta \;\ge\; F_{\mathrm P}^{-}\, \Delta \quad \Rightarrow \quad F_{\mathrm P}^{+} \ge F_{\mathrm P}^{-} \qquad \blacksquare $$

(12.4-5)

12.4.3

Three theorems · Upper / lower / uniqueness

Bounds & Uniqueness

The basic theorem yields three interrelated important theorems:

Anim. 12.4.3

① Minimum theorem Upper-Bound Theorem

The smallest collapse load is the ultimate load.
Any collapse load is an upper bound: $F_{\mathrm{Pu}} \le F_{\mathrm P}^{+}$.

Proof: since $F_{\mathrm{Pu}}$ is itself a safe load $F_{\mathrm P}^{-}$, the basic theorem gives $F_{\mathrm P}^{+} \ge F_{\mathrm{Pu}}$.

② Maximum theorem Lower-Bound Theorem

The largest safe load is the ultimate load.
Any safe load is a lower bound: $F_{\mathrm{Pu}} \ge F_{\mathrm P}^{-}$.

Proof: since $F_{\mathrm{Pu}}$ is itself a collapse load $F_{\mathrm P}^{+}$, the basic theorem gives $F_{\mathrm{Pu}} \ge F_{\mathrm P}^{-}$.

③ Uniqueness theorem (single-value theorem)

The ultimate-load value is unique. If a load is both a collapse load and a safe load, then it is the ultimate load.

Proof: suppose two ultimate states exist with $F_{\mathrm{Pu1}}, F_{\mathrm{Pu2}}$. Each is both $F_{\mathrm P}^{+}$ and $F_{\mathrm P}^{-}$, so by the basic theorem $F_{\mathrm{Pu1}} \ge F_{\mathrm{Pu2}}$ (treating 1 as $+$, 2 as $-$) and $F_{\mathrm{Pu2}} \ge F_{\mathrm{Pu1}}$ (treating 2 as $+$, 1 as $-$). Hence $F_{\mathrm{Pu1}} = F_{\mathrm{Pu2}}$. $\blacksquare$

Practical use of the three theorems

Minimum theorem → enumeration method: list all possible collapse mechanisms, compute $F_{\mathrm P}^{+}$ for each — the smallest is $F_{\mathrm{Pu}}$;
Maximum theorem → trial method: assume a $M$ diagram satisfying the yield condition and use equilibrium to compute $F_{\mathrm P}^{-}$ — the largest is $F_{\mathrm{Pu}}$;
Uniqueness theorem → verification method: once a value of $F_{\mathrm P}$ is found, verify that all three conditions are satisfied — if so, it must be $F_{\mathrm{Pu}}$.

Note: a structure under the same generalized load may have more than one ultimate state, but all share the same ultimate load — the ultimate load is unique, but the ultimate state is not necessarily unique.

12.4.4

Example 12-4 · Continuous beam via enumeration

Worked Example

Find the ultimate load of the equi-section continuous beam of Animation 12.4.4. Section ultimate moment $M_{\mathrm u}$. The two loads grow proportionally: $F_{\mathrm P}$ and $1.2\, F_{\mathrm P}$.

Anim. 12.4.4

① Mechanism 1 · AB span alone (see Animation 12.4.4)

Assume positive hinge at $D$ in $AB$ and negative hinge at $B$. Virtual-work equation:

$$ F_{\mathrm P}\, \theta\, \dfrac{l}{2} \;=\; 2\, M_{\mathrm u}\, \theta \;+\; M_{\mathrm u}\,\theta $$

(12.4-6)

Collapse load:

$$ F_{\mathrm P}^{+(1)} \;=\; \dfrac{6\, M_{\mathrm u}}{l} $$

(12.4-7)

From equilibrium we can draw the $M$ diagram. Check: the midspan moment in $BC$ is $2.27\, M_{\mathrm u}$ — well above $M_{\mathrm u}$, so the yield condition fails. This value is only an upper bound.

② Mechanism 2 · BC span alone

Similarly, for a $BC$-span mechanism with positive hinge at $E$ and negative hinge at $B$:

$$ 1.2\, F_{\mathrm P}\, \theta\, \dfrac{l}{3} \;=\; M_{\mathrm u}\cdot \theta \;+\; M_{\mathrm u}\cdot \dfrac{3\theta}{2} $$

(12.4-8)

Solving:

$$ F_{\mathrm P}^{+(2)} \;=\; \dfrac{(5/2)\, M_{\mathrm u}}{1.2\, (l/3)} \;=\; \dfrac{6.25\, M_{\mathrm u}}{l} $$

(12.4-9)

Check the $M$ diagram: $AB$-span midspan moment is $2.06\, M_{\mathrm u} > M_{\mathrm u}$ — yield condition again fails; still only an upper bound.

③ Mechanism 3 · Combined

Plastic hinges at $D$ in $AB$, $E$ in $BC$, and $B$ at the middle support. Virtual-work equation:

$$ F_{\mathrm P}\, \theta\, \dfrac{l}{2} \;+\; 1.2\, F_{\mathrm P}\, \theta\, \dfrac{l}{3} \;=\; M_{\mathrm u}\cdot 2\theta \;+\; M_{\mathrm u}\cdot \theta \;+\; M_{\mathrm u}\cdot \dfrac{3\theta}{2} $$

(12.4-10)

Solving:

$$ F_{\mathrm P}^{+(3)} \;\approx\; \dfrac{(9/2)\, M_{\mathrm u}}{0.9\, l} \;=\; \dfrac{5\, M_{\mathrm u}}{l} $$

(12.4-11)

Check the $M$ diagram: $AB$-span midspan $\approx M_{\mathrm u}$, $BC$-span midspan $\approx 0.056\, M_{\mathrm u}$ — all $\le M_{\mathrm u}$, yield condition satisfied. It is also a collapse load. By the uniqueness theorem:

$$ \boxed{\; F_{\mathrm{Pu}} \;=\; \dfrac{5\, M_{\mathrm u}}{l} \;} $$

(12.4-12)

Key points of the enumeration method

List all possible collapse mechanisms — single-span, multi-span combined, etc.;
Compute $F_{\mathrm P}^{+}$ via virtual work for each;
The smallest $F_{\mathrm P}^{+}$ is the ultimate load (upper-bound theorem);
Verify the $M$ diagram satisfies the yield condition — if so, the uniqueness theorem confirms.

Section summary

Under proportional loading, the ultimate state requires simultaneous satisfaction of equilibrium · yield · mechanism;
Any collapse load $F_{\mathrm P}^{+}$ is an upper bound; any safe load $F_{\mathrm P}^{-}$ is a lower bound;
If a load is both $F_{\mathrm P}^{+}$ and $F_{\mathrm P}^{-}$, it must be $F_{\mathrm{Pu}}$ (uniqueness theorem);
Enumeration procedure: list mechanisms → virtual work to find $F_{\mathrm P}^{+}$ → pick smallest → verify yield;
Next, §12-5 extends this methodology to the ultimate load of plane frames.

§12-5 Frame Ultimate Load →