Modeling the plate elements as part of the framing system is not a good idea
01 /11/2014
The title of this article may
surprise many, so let me start explaining from the very foundation of
structural analysis.
Before we learned to use
computers for matrix analysis of structures, we learned about moment
distribution method in which the stresses of adjacent beams (continuous beams)
are distributed based on the relative stiffness of the adjacent beams. This
means that if the left beam has 75% of the total stiffness at the junction, it
will carry 75% of the stresses, thus the other beam carries the remaining 25%.
Suppose the stiffness ratio of the two beams is so high, say 1,000,000/1. Then slight
changes to the number of decimal point precision on the stiffness value of the
smaller one, changes significantly the stiffness ratio at the junction. On the
contrary, small changes in precision on the larger stiffness beam do not have
significant impact to the stiffness ratio.
Now, let us go back to modelling of plate elements. Plate elements being significantly smaller in cross-sections compared to beam elements, have much smaller stiffness especially in bending. This will lead to the
condition I explained earlier. This is called “ill-conditioned” stiffness matrix. If you search the term “ill-conditioned” stiffness matrix in the internet, you will find many explanation and discussion about this topic. I will not explain this topic further since there are many literatures available from the internet. Some might say that plates or slabs maybe thin but they are wide thus the cross sectional area is large so the stiffness is large as well. My answer to that is, in finite element analysis of slabs modelled as moment resisting plates, it requires that the slab be sub-divided into small plate elements thus the stiffness of each small plate element is much smaller than the stiffness of the beams which may lead to ill-conditioned stiffness matrix.
One safer way of modelling slabs is to use “plane strain elements” which is only resisting planar loads. However, I don’t bother using this method because I can model this effect using a more efficient way. Plate elements are computationally expensive (I mean time consuming).
This is why I don’t like including plate elements in FrameCE. If I want to consider the effect of slabs to the frame structure, I would either use the master and slave method or model the slab stiffness contribution using compression elements. In the early days of computers, master and slave method was not used primarily to model plate effects, it was used to make the calculation much faster because this method significantly reduces the size of the stiffness matrix. I remember when I was doing my masters, my thesis supervisor who was a proud graduate of UC Berkeley in the US, let me use an analysis program to analysis a 10-storey building. In just a few seconds, the analysis was completed. I was not using a powerful computer back then, instead I was only using Pentium 90 MHz with 128 MB RAM. It was all quick because the analysis used a master-slave node method for each floor.
Master and slave method imposes a condition that displacements of the slave node will equate to the displacement of the master node. However, I am not a big fan of this method either since slabs usually deflects and contracts thus under-estimating stresses in beam members. That leaves me more inclined to model the slab contribution conservatively using compression elements or not to model the effect at all.
Now, let us go back to modelling of plate elements. Plate elements being significantly smaller in cross-sections compared to beam elements, have much smaller stiffness especially in bending. This will lead to the
condition I explained earlier. This is called “ill-conditioned” stiffness matrix. If you search the term “ill-conditioned” stiffness matrix in the internet, you will find many explanation and discussion about this topic. I will not explain this topic further since there are many literatures available from the internet. Some might say that plates or slabs maybe thin but they are wide thus the cross sectional area is large so the stiffness is large as well. My answer to that is, in finite element analysis of slabs modelled as moment resisting plates, it requires that the slab be sub-divided into small plate elements thus the stiffness of each small plate element is much smaller than the stiffness of the beams which may lead to ill-conditioned stiffness matrix.
One safer way of modelling slabs is to use “plane strain elements” which is only resisting planar loads. However, I don’t bother using this method because I can model this effect using a more efficient way. Plate elements are computationally expensive (I mean time consuming).
This is why I don’t like including plate elements in FrameCE. If I want to consider the effect of slabs to the frame structure, I would either use the master and slave method or model the slab stiffness contribution using compression elements. In the early days of computers, master and slave method was not used primarily to model plate effects, it was used to make the calculation much faster because this method significantly reduces the size of the stiffness matrix. I remember when I was doing my masters, my thesis supervisor who was a proud graduate of UC Berkeley in the US, let me use an analysis program to analysis a 10-storey building. In just a few seconds, the analysis was completed. I was not using a powerful computer back then, instead I was only using Pentium 90 MHz with 128 MB RAM. It was all quick because the analysis used a master-slave node method for each floor.
Master and slave method imposes a condition that displacements of the slave node will equate to the displacement of the master node. However, I am not a big fan of this method either since slabs usually deflects and contracts thus under-estimating stresses in beam members. That leaves me more inclined to model the slab contribution conservatively using compression elements or not to model the effect at all.
The beauty and relevance of dynamic analysis
02/11/2014
Whenever I use FrameCE, before starting the real work I have this habit of entertaining myself for a few seconds by watching models dancing to different modes of my choice. I am referring to performing dynamic analysis of structural models using FrameCE. FrameCEs dynamic analysis is second to none in terms of ease in using. Dynamic or frequency analysis is usually given low priority by many structural engineers in less earthquake-prone countries. In all projects in Australia where I was involved in the structural design, I’ve never been asked to do a frequency analysis except for a footbridge of a hospital project where vibration was a concern, which was unrelated to the strength of the structure.
Dynamic analysis has many applications even in low seismicity countries like Australia. One example is determining the best location for a bracing system. I observed that the common practice here is that senior engineers designate where the bracing locations are, before designing the sizes of the bracing rods. It is my opinion that efficiency of bracing systems will be significantly improved if structural engineers do a dynamic analysis and see where the bracing is most effective. For symmetrical and regular shaped steel structures, the engineer can easily point out the best locations for a bracing system. However, with irregular shaped structures, it may not be easy to pinpoint the best location.
What makes frequency an attractive first step for determining bracing location is that it with FrameCE features, the process is done in a few seconds and that the user is given the option to view the mode shape of interest. Moreover, FrameCE allows the user to eliminate unnecessary degrees of freedom to facilitate the calculation.
In dynamic analysis using FrameCE, the user is given the easy option of viewing the first 20 mode shapes of the structure. This number of mode shapes is more than enough to know the structure’s critical modes for determining efficient bracing locations.
Another application of frequency analysis which many structural engineers fail to realise it that the procedure may give the structural engineer a hint if there are mistakes in the models. For example, if a symmetrical structure will show irregular mode shapes, most likely points to error in modelling.