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 blog article intro to Time-dependent Analysis for Concrete Structures, we have touched upon the importance of construction stage analysis for concrete structures. The material time function can be plotted and inputted into analysis software like midas Civil to simulate their changing material behavior in various stages of the construction. This article will go over the process of calculating various parameters that contribute to the shape and location of the material's time functions. 

What is the Support of Excavation System?

There are a couple of ways to analyze suspension bridges. As previously discussed (Uniqueness and Difficulties of Suspension Bridge Analysis), the deflection theory proposed by Melan and solved by Moisseiff is one of the first. The second one may be trigonometric methods proposed by Timoshenko (Theory of suspension bridges, Journal of the Franklin Institute, Volume 235, Issue 4, April 1943, Pages 327-349). In this article, a brief explanation of trigonometric methods and related excel, and the example bridge dimensions will be provided.

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