Problem 16 - Concepts and Applications of Finite Element Analysis

Question

17,16 The horizontal bar in the sketch is perfectly rigid and is constrained to remain horizontal as load \(P\) and displacement \(v\) increase. The three vertical bars are elastic-perfectly plastic with \(A=1, E=1\) and \(L=2\). These bars have the respective yield point loads \(F_{1}=2, F_{2}=4\), and \(F_{3}=6\). Use the tangent-stiffness method to generate the \(P\) versus \(p\) relation. Use three steps. Scale each step so that one bar begins to yield at the end of the step. Problem \(17.16\)

Step-by-Step Solution

Answer

The P-p curve starts linearly with a slope equal to the combined stiffness of the bars. It then deviates from the linear path when the first bar yields (at P=2). The curve has a smaller slope for the next step when the second bar yields (at P=6), finally, the slope reduces further when the last bar yields (at P=12). The displacement 'p' increases at each step, corresponding to each yield point.

Step 1 - Calculate initial stiffness

First, calculate the initial stiffness (\(K\)) of each bar using the formula \(K = E \cdot A / L\), where \(A=1\), \(E=1\), and \(L=2\). The initial stiffness will be the same for all bars.

Step 2 - Determine first yielding

Next, find the first bar to yield by comparing the yield loads (\(F_1=2, F_2=4, F_3=6\)). The bar with the smallest yield load will be the first to yield.

Step 3 - Calculate displacement at yield

Calculate the displacement (\(p\)) at the point of yielding using the formula \(p = F / K\), where \(F\) is the yield load of the first bar to yield.

Step 4 - Repeat for other bars

Repeat steps 2 and 3 for the remaining bars. Calculate the displacement at yield for each bar. The overall load \(P\) at each step will be the sum of the yield loads up to that point.

Key Concepts

Elastic-perfectly plastic analysis

Elastic-perfectly plastic analysis is a method that simplifies the material behavior into two distinct phases: elastic and plastic. In this method, a material initially behaves elastically, which means it deforms proportionally to the applied load and returns to its original shape when the load is removed.
After reaching a specific point, known as the yield point, the material behaves plastically. This means it will endure permanent deformation even if the load is removed. These characteristics help in predicting the response of structures under excessive loads.
Understanding elastic-perfectly plastic analysis is crucial because it helps engineers design structures that can handle high loads without abrupt failure. The method involves:

• Elastic Phase: This phase is governed by Hooke's Law, where stress is directly proportional to strain and the material behaves predictably.
• Plastic Phase: After yielding, the material does not follow Hooke's Law. It has a constant stress level, which is the yield stress. Load increases do not increase stress significantly but cause more permanent deformation.

This analysis helps ensure safety and reliability in structural engineering projects by providing clear points where the material will begin to yield and take permanent shape changes.

Yield point loads

Yield point loads are the specific loads at which a material transitions from elastic to plastic behavior. This point is crucial as it designates the maximum stress that the material can withstand while still being able to return to its original shape once the load is removed.
Each yield point load is unique for different materials or sections within a structure due to varying material properties or geometric configurations. In structural design, recognizing the yield point helps in predicting when a part of the structure might begin to fail permanently.
Here's how yield point loads come into play:

• Material Specific: Different materials, like steel or aluminum, have different yield strengths. Yield point loads need to consider these material properties to ensure safety.
• Consistent Performance: Using yield point loads, engineers can design structures that behave predictably under increasing loads, preventing surprises that could lead to structural failures.
• Design Criterion: Structural elements are often designed to function under loads below their yield point to maintain longevity and safety.

Being aware of these loads is crucial for the longevity and integrity of constructions, ensuring they remain safe under anticipated service conditions.

Structural analysis

Structural analysis is the process used by engineers to determine the effects of loads and forces on physical structures and their components. It plays a critical role in designing buildings, bridges, and other structures to ensure they withstand anticipated loads without excessive deformation or failure.
The analysis allows engineers to identify how different forces affect the structure’s response and helps in making informed decisions about necessary changes or reinforcements.
Here’s a simplified breakdown of structural analysis:

• Load Analysis: Identify all types of forces and loads that may act on the structure, including live loads (like people and furniture), dead loads (the structure's own weight), and environmental loads (such as wind and earthquakes).
• Stress and Strain Analysis: Determine how these loads cause stress and strain within structural elements and ensure they do not exceed material capacities.
• Deflection Analysis: Calculate how much elements will deflect under loads, aiming for deflections that are minimal to maintain structural functionality and aesthetics.

By performing structural analysis, engineers can evaluate safety, functionality, and economy, ensuring the design will be capable of withstanding the loads it is expected to encounter during its lifespan. This methodical approach helps in avoiding costly failures and ensuring public safety.