Motives for better Engineering
Explore horizontal earth pressure,
Coulomb's theory, and its applications.
Compare geotechnical results and
understand the trial wedge method's nuances.
Explore the technical content on vessel collision
to calculate the annual frequency of bridge component collapse.
Introducing the concept of seismic isolation design.
See more
Sometimes we need tiresome calculations, even though they are not critical nor difficult, but they are essential, and they take time if we do not have proper tools. One of these is the moment of inertia calculation for cracked, circular concrete sections. We need this calculation when we perform service stress checks for flexure and deflection checks.
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.
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.
The scary part of FEM is sometimes FEM gives wrong results without any error message. The analysis may be meaningless if an engineer cannot check or interpret the results. Let’s consider a simple example similar to the case from Dr. Gallagher (Finite Element Analysis: Fundamentals, 1975).
For the previous example, we can use high-order triangular elements. This element has six nodes per element and assumes the displacement is quadratic within an element. Also, each side edge can be curved, as shown.
Continuing on to the third part of this multi-part blog, another option is a quadrilateral element. As always, let’s start with an example.
Optimum crack angle θ
From the previous example, we can catch that there are some possible crack angle ranges for the given εx and vu/f’c. Now our question is which values of θ and β are the optimums? The previous example shows that, without considering longitudinal reinforcements, mostly (not always) the lowest crack angle results in the least number of stirrups. However, with considering longitudinal reinforcements, the optimum crack angle increases. The methodology to find out the optimum crack angle is proposed by Rahal and Collins (Background to the general method of shear design in the 1994 CSA-A23.3 standard, Canadian Journal of Civil Engineering, February 2011).
Now it’s time to solve the previous example from full iteration. For simplicity, interaction with flexure is not considered. In other words, it is assumed that the status is in a pure shear condition which rarely exists in the real world.
Finally we came to the composite section analysis. One of the best examples is from Dr. Gilbert and Dr. Ranzi (Example 5.10, Time-dependent behavious of concrete structures, CRC Press, 2010.)