Plastic Collapse Continuous Beams: Non-Prismatic Span Guide (2026)

1. Introduction to Non-Prismatic Continuous Beam Limit Analysis

In modern structural bridge and commercial building design, continuous beams are frequently fabricated with variable cross-sections, cover plates, or tapered haunches over intermediate supports. The study of plastic collapse continuous beams establishes how moment redistribution occurs when cross-sectional plastic capacities vary along the span. Classical elastic analysis often underestimates the load capacity of non-prismatic members because it overlooks post-yield plastic capacity redistribution. Evaluating plastic collapse continuous beams allows engineers to predict the ultimate collapse load factor $\lambda_c$ across multi-span systems.

                  STEPPED CONTINUOUS BEAM CAPACITY PROFILE
    Mp_span = 1.0 M0                                         Mp_span = 1.0 M0
   |-------------------+                  +-----------------------------------|
                       |  Mp_sup = 2.0 M0 |
                       +------------------+
   ====================|==================|====================================
   ^ (A) Pin                              ^ (B) Interior Roller                ^ (C) Roller
   |<------------ Span L1 --------------->|<-------------- Span L2 ----------->|

When evaluating plastic collapse continuous beams, designers must account for localized hinge formation at points of maximum bending moment or cross-sectional capacity transitions. By applying kinematic virtual work and static equilibrium theorems, the true multi-span collapse load can be established for any non-prismatic beam configuration.

This technical guide presents the governing mechanics, virtual work formulations, and step-by-step worked examples required to assess non-uniform continuous girder collapse modes.

2. Mechanics of Plastic Hinge Formation in Continuous Spans

2.1 Sequential Hinge Formation and Moment Redistribution

In a continuous beam of degree of statical indeterminacy $D_s$, increasing the service load initiates yield at the most stressed cross-section (typically the negative hogging moment over intermediate supports). As the moment reaches the plastic capacity $M_{p,sup}$, an idealized plastic hinge develops.

The formation of continuous beam plastic hinges reduces the effective degree of indeterminacy by one for each hinge. The beam continues to carry additional load by redistributing excess moments to positive sagging in-span regions until a complete kinematic collapse mechanism forms.

2.2 Statical Indeterminacy and Hinge Count Criteria

For an indeterminate continuous beam with indeterminacy $D_s$, the minimum number of plastic hinges $N_h$ required to form a full or partial collapse mechanism is:

$$N_h = D_s + 1 \quad (\text{for a complete collapse mechanism})$$

However, in continuous multi-span beams, failure frequently occurs via a partial collapse mechanism confined to a single critical span. For a single propped or continuous span bounded by supports, only 2 or 3 hinges are required to cause span failure, regardless of the total indeterminacy of the overall multi-span bridge structure.

3. Non-Prismatic Beams: Stepped Profiles and Haunched Girders

3.1 Plastic Capacity Discontinuities

In a non-prismatic beam, the plastic moment capacity varies as a spatial function $M_p(x)$. In steel plate girders, transitions occur abruptly where flange cover plates terminate or flange thickness changes. In reinforced concrete or haunched steel beams, $M_p(x)$ varies smoothly or linearly across tapered segments.

       HAUNCHED SUPPORT REGIME           VS        STEPPED FLANGE REGIME

       Mp(x)                                       Mp(x)
         |     /\                                   |   +---------+
         |    /  \                                  |   |         |
         |   /    \                                 |   |         |
         |--/      \--                              +---+         +---+
         +---------------> x                         +-----------------> x

When evaluating a non-prismatic beam, plastic hinges can form at:
1. Support centerlines where negative moments peak,
2. Cross-sectional transition points where $M_p(x)$ drops abruptly,
3. In-span locations where positive sagging moment matches the local $M_p(x)$.

3.2 Locating the Critical Intermediate In-Span Hinge

Under distributed loading $w$, the bending moment profile $M(x)$ is parabolic. In a non-prismatic span, the plastic hinge does not necessarily form at midspan. The exact location $x_0$ of the in-span hinge is determined by maximizing the collapse load parameter or locating the point of zero shear:

$$V(x_0) = \frac{dM(x)}{dx} = 0 \implies x_0 = \frac{L}{2} + \frac{M_{p,\right} – M_{p,\left}}{w L}$$

When a step in cross-section occurs near this peak, the hinge is frequently pinned directly to the step location if the capacity drop exceeds the gradient of the moment curve.

4. Kinematic Formulation of Multi-Span Collapse Mechanisms

4.1 Independent Span Mechanisms

To evaluate the multi-span collapse load of continuous girders, each span is analyzed as an independent kinematic mechanism. The governing collapse load of the entire continuous structure is the minimum load factor among all possible span mechanisms:

$$\lambda_c = \min \left( \lambda_1, \lambda_2, \dots, \lambda_n ight)$$

       SPAN 1 COLLAPSE MECHANISM                 SPAN 2 COLLAPSE MECHANISM

       (A)      (Hinge)    (B)                   (B)      (Hinge)    (C)
       ^-----------o--------^                     ^-----------o--------^
        \         /                                \         /
         \       /                                  \       /
          \  δ1 /                                    \  δ2 /
           v   v                                      v   v
       Span 2 remains rigid                       Span 1 remains rigid

4.2 Virtual Work Equilibrium for Non-Uniform Capacity Profiles

For any assumed kinematic collapse mechanism with virtual displacement field $\delta(x)$ and plastic hinge rotations $\theta_j$, the internal energy dissipation equates to external work:

$$W_{ext} = \int_{0}^{L} w(x) \delta(x) \, dx + \sum P_i \delta_i$$

$$D_{int} = \sum_{j=1}^{N_h} M_{p,j} |\theta_j|$$

Equating $W_{ext} = D_{int}$ yields the kinematic collapse load for the non-prismatic continuous span.

5. Step-by-Step Worked Calculation: Two-Span Continuous Stepped Girder

Consider a two-span continuous steel girder $A\text{–}B\text{–}C$ resting on simple supports at $A$, $B$, and $C$.

• Span 1 ($A\text{–}B$): Length $L_1 = 10\text{ m}$, supporting a concentrated point load $P_1 = 2P$ at midspan ($x = 5\text{ m}$).

• Span 2 ($B\text{–}C$): Length $L_2 = 8\text{ m}$, supporting a concentrated point load $P_2 = P$ at midspan ($x = 4\text{ m}$).

• Plastic Moment Capacities:

• Span 1 interior capacity: $M_{p1} = 300\text{ kN}\cdot\text{ m}$.
• Interior Support $B$ haunch capacity: $M_{pB} = 450\text{ kN}\cdot\text{ m}$.
• Span 2 interior capacity: $M_{p2} = 200\text{ kN}\cdot\text{ m}$.
• Supports $A$ and $C$ are simple end supports ($M_{pA} = M_{pC} = 0$).

                2P                                            P
                 v                                            v
       A         |                   B                        |         C
       ^=========o===================^========================o=========^
       |<--------- L1 = 10 m ------->|<---------- L2 = 8 m ----------->|
       Mp1 = 300 kNm                 MpB = 450 kNm            Mp2 = 200 kNm

5.1 Structural Geometry and Plastic Capacity Definition

The system has 1 degree of statical indeterminacy ($D_s = 1$). A span collapse requires 2 plastic hinges in either Span 1 or Span 2:
1. One positive sagging hinge under the concentrated load,
2. One negative hogging hinge at interior support $B$.

5.2 Span 1 Collapse Mechanism Evaluation

Postulate a collapse mechanism in Span 1 with hinges at midspan ($x = 5\text{ m}$) and support $B$:

• Virtual downward displacement at load $2P$: $\delta_1 = \delta$.

• Rotation at end support $A$: $\theta_A = \frac{\delta}{5}$.

• Rotation at interior support $B$: $\theta_{B1} = \frac{\delta}{5}$.

• Total rotation at midspan hinge 1: $\theta_1 = \theta_A + \theta_{B1} = \frac{\delta}{5} + \frac{\delta}{5} = \frac{2\delta}{5}$.

Work and Energy Equations:
$$W_{ext,1} = (2P) \cdot \delta_1 = 2P \delta$$

$$D_{int,1} = M_{p1} \theta_1 + M_{pB} \theta_{B1} = (300) \left( \frac{2\delta}{5} ight) + (450) \left( \frac{\delta}{5} ight) = \left( \frac{600 + 450}{5} ight) \delta = \frac{1050}{5} \delta = 210 \delta\text{ (kN}\cdot\text{m)}$$

Equating $W_{ext,1} = D_{int,1}$:
$$2P \delta = 210 \delta \implies 2P = 210 \implies P_{collapse,1} = 105.0\text{ kN}$$

Total midspan load on Span 1 at collapse $= 2P = 210.0\text{ kN}$.

5.3 Span 2 Collapse Mechanism Evaluation

Postulate a collapse mechanism in Span 2 with hinges at midspan ($x = 4\text{ m}$ from $B$) and support $B$:

• Virtual downward displacement at load $P$: $\delta_2 = \delta$.

• Rotation at interior support $B$: $\theta_{B2} = \frac{\delta}{4}$.

• Rotation at end support $C$: $\theta_C = \frac{\delta}{4}$.

• Total rotation at midspan hinge 2: $\theta_2 = \theta_{B2} + \theta_C = \frac{\delta}{4} + \frac{\delta}{4} = \frac{2\delta}{4} = \frac{\delta}{2}$.

Work and Energy Equations:
$$W_{ext,2} = P \cdot \delta_2 = P \delta$$

$$D_{int,2} = M_{p2} \theta_2 + M_{pB} \theta_{B2} = (200) \left( \frac{\delta}{2} ight) + (450) \left( \frac{\delta}{4} ight) = 100 \delta + 112.5 \delta = 212.5 \delta\text{ (kN}\cdot\text{m)}$$

Equating $W_{ext,2} = D_{int,2}$:
$$P \delta = 212.5 \delta \implies P_{collapse,2} = 212.5\text{ kN}$$

5.4 Statical Admissibility and Moment Envelope Verification

Comparing the kinematic collapse load parameters:

• Span 1 mechanism governs at $P_c = 105.0\text{ kN}$ (Midspan load $= 210.0\text{ kN}$).

• Span 2 mechanism requires $P = 212.5\text{ kN}$.

Therefore, the multi-span collapse load is:

$$P_c = 105.0\text{ kN}$$

       BENDING MOMENT DIAGRAM AT GOVERNING COLLAPSE (kNm)

       Support A (0)                                Support B (-450)             Support C (0)
       ^---------------------------------------------------^---------------------------^
        \                                                 / \                         /
         \                                               /   \                       /
          \                                             /     \   Span 2 remains    /
           \                                           /       \     elastic       /
            \                                         /         \                 /
             \-------------o-------------------------/           \-------v-------/
                       Span 1 Hinge (+300 kNm)                     M2_sag = +160 kNm (< 200)

At $P = 105.0\text{ kN}$, Span 2 remains purely elastic with maximum sagging moment $M_2 = 160\text{ kN}\cdot\text{ m} < M_{p2} = 200\text{ kN}\cdot\text{ m}$. Span 1 undergoes a ductile partial collapse mechanism under the plastic collapse continuous beams limit state.

6. Engineering Failure Modes and Forensic Case Studies

Limit analysis of continuous bridge girders reveals key forensic failure modes:

1. Premature Local Flange Buckling at Haunches: Negative moment regions over interior piers experience steep stress gradients combined with high shear. If haunch flanges violate compactness ($b_f/2t_f > \lambda_p$), local buckling prevents full plastic hinge rotation.
2. Web Shear-Flexure Interaction: High vertical reaction shear at interior supports reduces the effective plastic flexural capacity $M_{p,red}$ according to:
   $$M_{V,Rd} = M_{pl,Rd} \left[ 1 – \left( \frac{2 V_{Ed}}{V_{pl,Rd}} – 1 ight)^2 ight]$$
3. Lateral-Torsional Buckling of Unsupported Bottom Flanges: In continuous beams, the bottom flange is in compression near interior supports. Inadequate lateral bracing leads to out-of-plane buckling before mechanism formation.

7. Modern Design Code Standards: AISC and Eurocode Provisions

Both AISC 360 (Appendix 1) and Eurocode 3 (EN 1993-1-1 / EN 1993-1-5) establish explicit rules for plastic design of non-prismatic continuous beams:

• Compactness Limits: Only Class 1 plastic cross-sections may be used where plastic hinges form. Stepped transitions must be detailed with tapered transitions (maximum 1:4 slope) to minimize stress concentrations.

• Moment Redistribution Percentage: In continuous bridge beams analyzed by elastic methods, codes permit limited moment redistribution (up to $15\text{–}20\%$ in Eurocode 3, up to $20\%$ in AASHTO LRFD) provided section ductility is verified.

• Bracing Restraints: Negative moment plastic hinges at intermediate supports must have lateral restraints located within distance $L_m = (0.12 + 0.076 M_1/M_2)(E/f_y)r_y$ from the hinge point.

8. Synthesis and Engineering Wrap-Up

Mastering the mechanics of plastic collapse continuous beams allows structural engineers to unlock the hidden strength of indeterminate bridge and frame systems. By analyzing independent span mechanisms and accounting for capacity variations across haunched and stepped sections, designers achieve optimum material efficiency without compromising limit-state structural safety.

Balancing capacity transitions with rotational ductility guarantees that continuous multi-span structures perform with predictable resilience under extreme overloads.

References & Standards Cited

1. American Association of State Highway and Transportation Officials (AASHTO). (2020). AASHTO LRFD Bridge Design Specifications. 9th Edition, Washington, DC: AASHTO.
2. American Institute of Steel Construction (AISC). (2022). Specification for Structural Steel Buildings (AISC 360-22). Chicago, IL: AISC.
3. European Committee for Standardization (CEN). (2006). Eurocode 3: Design of steel structures – Part 1-5: Plated structural elements (EN 1993-1-5). Brussels: CEN.
4. Horne, M. R. (1979). Plastic Theory of Structures. 2nd Edition, Oxford: Pergamon Press.
5. Neal, B. G. (1977). The Plastic Methods of Structural Analysis. 3rd Edition, London: Chapman and Hall.

Frequently Asked Questions (FAQ)

How does a change in cross-section affect the plastic hinge location?

In a **non-prismatic beam**, an abrupt reduction in plastic moment capacity $M_p(x)$ often forces the plastic hinge to form exactly at the section step or cover plate cutoff, where the ratio of applied bending moment to local plastic capacity $M(x)/M_p(x)$ reaches unity.

Can a continuous beam fail by partial collapse without all spans forming hinges?

Yes. Multi-span continuous beams almost always fail through a partial collapse mechanism where plastic hinges form exclusively in the most heavily loaded or flexible span ($N_h = D_{span} + 1$). The remaining spans stay elastic.

What is the effect of support settlement on the plastic collapse load?

Support settlement introduces initial elastic self-equilibrating residual moments. However, according to the plastic collapse theorems, differential settlement does not affect the ultimate plastic collapse load factor $lambda_c$, provided the steel has sufficient rotational ductility.

How do tapered haunches improve the multi-span collapse load?

Tapered haunches increase both the elastic stiffness and the plastic moment capacity $M_{p,sup}$ over interior supports. This forces the negative support moment to absorb a higher proportion of the total load, delaying the formation of in-span mechanisms.

Why is lateral-torsional bracing critical near interior continuous supports?

Over interior continuous supports, the bottom flange is in compression. Because plastic analysis requires large plastic rotations $theta_p$ to redistribute moments to adjacent spans, unbraced bottom flanges are highly susceptible to premature lateral-torsional buckling.