Understanding Plastic Theory in Structural Analysis

Structural analysis methods help engineers understand how structures behave under load. There are different approaches, including elastic analysis and plastic analysis, often referred to as plastic theory.

Elastic theory assumes that materials behave elastically under load, meaning they return to their original shape once the load is removed. Stress is proportional to strain within the elastic limit. Design based on elastic theory typically focuses on ensuring stresses remain below the yield strength of the material, using a factor of safety.

Plastic theory, on the other hand, considers the behavior of materials beyond their elastic limit, into the plastic range. In the plastic range, a material continues to deform even if the stress remains constant (perfect plasticity model), or with increasing stress (strain hardening). Plastic theory allows engineers to determine the ultimate load-carrying capacity of a structure, considering that parts of the structure can yield and form plastic hinges, redistributing internal forces.

Applying Plastic Theory to Different Structures

The application of plastic theory is particularly beneficial for certain types of structures, especially those with redundancy. Redundancy means that if one part of the structure fails (like forming a plastic hinge), the load can be redistributed to other parts, and the structure can still carry additional load before complete collapse.

• Columns: Columns are primarily compression members. Their failure is often governed by buckling rather than material yielding across the entire cross-section to form a plastic hinge, especially in slender columns. While plasticity can influence buckling strength (inelastic buckling), plastic theory as a primary design method is not as universally applied to isolated columns as it is to frame structures.
• Beams: Simple, statically determinate beams can be analyzed using plastic theory to find the collapse load (when a single plastic hinge forms at the point of maximum moment). However, the full advantage of plastic theory, which lies in load redistribution after yielding, is most prominent in indeterminate structures. For simple beams, elastic analysis often suffices for design based on allowable stress.
• Rigid Frame Structures: Rigid frame structures are typically statically indeterminate and possess significant redundancy. In a rigid frame, moments, shear forces, and axial forces are present. As loads increase, moments at critical sections (like beam-column joints or points of maximum moment in beams/columns) can reach the plastic moment capacity of the section, leading to the formation of plastic hinges. Because the frame is indeterminate, the formation of one or more plastic hinges does not necessarily lead to immediate collapse. The internal forces are redistributed to other parts of the frame. Collapse occurs only after enough plastic hinges form to create a collapse mechanism, transforming the stable frame into a mechanism. Plastic theory is widely used for analyzing and designing rigid frames to determine their ultimate load-carrying capacity and potentially achieve more economical designs compared to purely elastic methods.
• Roofs: Roof structures can take various forms, including trusses, plates, shells, or frames. While plastic analysis concepts might be applied to specific elements within a roof structure (like beams or frames), plastic theory is not generally used as the overarching primary analysis method for all types of roof structures. For instance, truss roofs are typically designed assuming pin joints and elastic behavior of members, with failure often related to buckling or yielding of individual members under axial load, rather than the formation of widespread plastic hinges and collapse mechanisms in the same way as rigid frames.

Based on the principles of plastic analysis and structural redundancy, plastic theory is most generally and advantageously applied to rigid frame structures.

Summary of Plastic Theory Application

Therefore, the plastic theory is generally used for rigid frame structures due to their indeterminate nature and ability to redistribute forces after yielding, allowing engineers to determine the ultimate collapse load.

Revision Table: Plastic Theory Basics

Additional Information on Plastic Analysis of Frames

Plastic analysis provides a different perspective on structural safety compared to elastic design. Elastic design prevents yielding under service loads, whereas plastic design (based on plastic theory) ensures the structure can withstand a factored load (design load multiplied by load factor) before reaching collapse. The ratio of the collapse load (determined by plastic theory) to the working load is the factor of safety against collapse.

Key assumptions in basic plastic theory include:

• Material is rigid-plastic or elastic-perfectly plastic.
• Deformations are small, and equilibrium is based on the original geometry (small displacement theory).
• Buckling is prevented or accounted for separately.
• The plastic moment capacity ($M_p$) is constant once a hinge forms.
• Failure is due to bending only (shear and axial effects on plastic moment are sometimes neglected or simplified).

Plastic analysis methods include the static method (lower bound theorem), kinematic method (upper bound theorem), and mechanism method (a type of kinematic method focusing on potential collapse mechanisms). The mechanism method is commonly used for frames.

Designing structures using plastic theory can sometimes lead to more efficient use of material, as it accounts for the reserve strength available beyond the elastic limit, especially in indeterminate structures like rigid frames.