Problem 5 - Concepts and Applications of Finite Element Analysis

Question

Assume that members of a truss carry only uniaxial stress, and that members in compression will buckle elastically at their critical loads without ever yielding. Assume that members in tension have elastic-perfectly plastic behavior. Outline a tangent-stiffness algorithm for computation of displacements produced by monotonically increasing loads.

Step-by-Step Solution

Answer

Develop an iterative tangent-stiffness algorithm considering the elastic-perfectly plastic behaviour of tensile members and elastic buckling of compressive members. The algorithm is applied in incremental load steps, recalculating member states (compression or tension) each time. Termination is when the total applied load is reached or when the change in displacements per iteration is less than a predetermined value.

Step 1: Review Basic Truss Theory

Truss members carry only axial forces, either in tension or compression. Tensile forces cause elongation while compressive forces cause contraction or buckling under critical loads.

Step 2: Understand Elastic-Perfectly Plastic Behaviour

In such behaviour, materials deform elastically and return to their original shape until the yield stress is reached after which the material deforms plastically and does not return to its original shape even after the applied stress is removed. In truss members, this is exhibited in tensions.

Step 3: Develop Algorithm Structure

A tangent stiffness algorithm calculates displacements based on iteration. At each subsequent loading step, an equilibrium equation is solved comprising member forces (elastic buckling for compression, elastic-perfectly plastic for tension) and loading forces. The unknowns in this equation are the nodal displacements. The displacements are computed iteratively by solving the equilibrium equation until convergence is achieved.

Step 4: Detail Loading Increments

The algorithm should be based on increments of loading. With each increment, the conditions of each member (either bend in compression or elongate in tension) are reassessed. Load increments continue until the complete load has been applied to the structure.

Step 5: Implement Convergence Criterion

The convergence criterion checks for the difference in displacements between consecutive iterations. If this difference is less than a predetermined tolerance, it signifies that the iteration process has achieved a sufficient solution and can be terminated.

Step 6: Calculate Final Displacement

Using the determined algorithm, calculate the final displacements under the applied loads in a step-by-step or monotonically increasing manner, always checking for equilibrium at each step.

Key Concepts

Truss Analysis

In engineering, a truss is a structure composed of members commonly arranged in connected triangles. Truss members are slender and direct loads through members either in tension or compression. This means that a truss structure efficiently carries loads, making it ideal for bridges, roofs, and structures needing to span large distances.
Truss analysis allows engineers to determine how these axial forces affect each member. In trusses, tensile forces elongate the members, while compressive forces can lead to either contraction or buckling, depending on the load magnitude.
The primary goal is to ensure that each member can safely carry the forces without failing. To do this, engineers use techniques such as the method of joints or sections to calculate the internal forces, ensuring they do not exceed the material's limits.

Tangent-Stiffness Algorithm

The tangent-stiffness algorithm is pivotal in finite element analysis, particularly for structures like trusses. It is used to compute displacements, which reveal how a structure deforms under loads.
This algorithm works iteratively, meaning it gradually calculates the displacements by solving a set of equilibrium equations at each step.
For trusses, these equations involve the forces within each member and the applied external loads. The approach relies on adjustments in stiffness—specifically, using the tangent to the curve of the stress-strain relationship at each load step.
By incrementing the loads step-by-step and solving the equations repeatedly until the changes between iterations are minimal (which signifies convergence), the algorithm finds the nodal displacements accurately.

Uniaxial Stress

Uniaxial stress refers to stress applied along a single axis, crucial for understanding the behavior of truss members.
In this context, truss members are assumed to only experience uniaxial loads, which allows these elements to be simplified as bars stretching or compressing along a line.
This simplification is beneficial, as it reduces the complexity of the calculations and focuses on the axial forces, ignoring more complex stress components such as shear or bending.
Understanding uniaxial stress is essential for predicting how truss elements will react under specific loads, ensuring accurate and safe design of structures.

Elastic-Perfectly Plastic Behavior

Elastic-perfectly plastic behavior describes how materials respond to stress. Initially, a material will deform elastically; this means the structure will return to its original shape if the stress is removed before reaching the yield point.
Once the yield stress is reached, the material deforms plastically, and any further stress leads to permanent deformation. In trusses, this behavior is significant, particularly for members in tension.
When such a member experiences loading, it stretches elastically until the yield point. Afterward, it begins to plastically deform, which is crucial for considering the material's performance under long-term loads.
Elastically perfectly plastic materials ensure a predictable, constant yield point, making them invaluable for designing structures that need to safely dissipate loads without a sudden failure.