Finite Element Analysis and its application to Geotechnical problems

Dr. Seungwoo Lee has talked about some fundamental differences between linear and nonlinear analysis in structural engineering in the past. In linear analysis, the relationship between the stress and strain of a model is held constant, and the stiffness matrix of the model stays the same throughout the analysis. For a nonlinear analysis, there can be various factors that contribute to its nonlinearities, for example, material yielding, nonlinearities in the boundary conditions, and various forms of geometric nonlinearities. In this article, Dr. Lee will elaborate more on geometric nonlinearities and show how different approaches to approximate geometric nonlinearities can vary the structural analysis results.   

In the article "strut-and-tie modeling for pier caps", we have discussed the definition of strut-and-tie analysis and how to construct a strut-and-tie model using the example of pier cap. After creating the geometry of a strut-and-tie model, the next step usually is calculating dead and live loads from the superstructure. This article discusses how to determine the boundary loads for a pier cap with a superstructure that has irregular geometries. 

Bridge load posting – identifying just how much weight a bridge can bear – is a matter of public safety and a way to safeguard vital transportation infrastructure. However, load posting is less straightforward than it may seem. Not posting a bridge can create safety issues for the motoring public, while posting makes transportation more difficult for large, heavy vehicles and the industries that use them. 

How to Model the Girder and Deck Connection in Composite Bridge?

The way to simulate the connection between the girder and the deck will depend on how we construct the model. In the 2D all-frame composite bridge model shown in Figure 1, all the elements are connected in the grid within the same plane. Because it was modeled with the "all frame" model type in the midas Civil composite bridge wizard, it only consists of a 2-D grid frame composed of beam elements. In this scenario, the software considers the composite section as a lump section that incorporates both the girder section and the deck section. This means that the composite action is transformed into equivalent section properties in midas Civil. 

What is Convergence? What is a Convergence Study? 

It is easy to obtain the result from bridge finite element analysis, but to get more accurate results requires extra effort. Even the most robust finite element analysis solvers adopt the method that approximates the structural behavior, by minimizing the associated error function compared with the complex function that represents the realistic structural behavior. 

What is Strut-and-Tie Analysis? What are "Struts" and "Ties" in Strut-and-Tie Analysis?

What is the Support of Excavation System?

Iteration and Optimization

Iteration is just a repeated calculation.

If we want to solve an equation x + 1 = 5 , we can assume x and check whether x + 1 = 5 or not. If not, we can try another value of x and repeat this calculation until x + 1 = 5.
 

In this simple equation, we can solve x = 5 - 1 = 4 easily. But some engineering problems are more complicated and iteration is more efficient and/or sometimes iteration is the only way to find the solution.
 

If we want to solve another equation xy = 5, this is rather complicated because there are infinite combinations of x and y’s. However, if we can add some “constraints” like we want to minimize x + y, we can find the solution and this is called the “optimization problem”.

(If another condition is something like, x + y = 4, this is not an optimization problem because there is only one set of solutions.)

 
In our pile cap problems, there are many combinations of pile cap dimensions that satisfy all requirements. In this case, we can try to find the pile cap dimensions that satisfy all requirements and corresponds to the minimum volume, and this is a good example of both “Iteration” and “Optimization”.

(There can be an argument that the minimum volume can not be optimum. Someone can insist that the optimum shall be minimum cost and we have to consider reinforcements, forwork, etc. The author has no objection to this, but still believes minimum volume is a good choice as the target.)

 

“Optimization” does need “Iteration” and the purpose of “Iteration” is “Optimization”, so these two terminologies are somewhat mixed and have different meanings for each engineer. It is not a universal/correct definition, but the author accepts “Iteration” as checks for all possible scenarios, and “Optimization” as finding only the optimum in the fastest way.
 

Example

Creep Analysis 6

Creep Analysis 7

Think about a two-span continuous bridge, as shown in Fig 1. Let's calculate the secondary creep moments. Detailed dimensions and creep factors, etc., are not important, and the MIDAS file "Creep 2ndary Check"is attached for the reader's reference.

Creep Analysis 1 

Creep Analysis 3

The blog article of Creep Analysis 3 showed how Dr. Ghali et al. explained the creep analysis by flexibility methods.


To better understand the creep behavior, solve the previous example in a less efficient way. Here, different sign conventions will be applied.

Creep Analysis 3

We have gone through two different approaches by Dr. El-Badry so far. You can find the previous articles via these two links: Creep Analysis 1, Creep Analysis 2.

For creep analysis, the most common problem in the real-world design is continuous girders built as span by span. This example is very well explained by Dr. Ghali et al. (Concrete structures, Stresses, and deformations, 4th ed., CRC press, Example 4-2). Dr. Ghali et al. explained this problem by flexibility methods. The author will solve this same problem by stiffness methods. The programs do the matrix formulation and equation solve, and only the load matrix formulation and post-processing are our concern in the stiffness methods. Two MIDAS files are attached.


First, the author wants to define the sign convention for member forces clearly.