Semi rigid connections are widely used and studied in steel structure field, they provide a correct stiffness to the structure, but I think that the mess is still the moment‐rotation relationships of these connections. You should read the "Semi-Rigid Connections Handbook" Edited by Wai-Fah Chen, Norimitsu Kishi, and Masato Komuro

Best Regards,

As a construction aspect of view, semi-rigid connections need less work and weaker details, while they can cause increasing experienced lateral displacements under earthquake excitation and maybe concentrating damages in connections and not the beams for example in moment-resisting frames. In previous earthquakes this issue has caused local or global collapse in buildings.

On the other side, where no considerable earthquake risk exists and gravity loads may control the design, using rigid connections could increase redundancy which consequently leads to smaller sections. 

In addition to what Reza said about reduction of the elements section, flexibility of connections allows to reduce internal forces in the structure under non-mechanical loads such as thermal load. However, structural systems with semi-rigid or hinged joints require spatial stiffening to ensure global stability of the system.

The use of "semi-rigid" connection is recommended by SAFETY! Some times, the believe that a beam is rigidly or pinned connected to a column can lead to ignore a little capacity in the last or a high cost design in the former, nothing new.

But, most of all, there are conections which can behave like if it were 'almost' rigid lowing the cost and others which behavior can be dangerous, as their response becomes negligible as the connection rotation grows quickly after some point. And this is the problem.

The called 'same' connection can change the expected answers depending on the beam or the column, the bolts, the undesired column stiffners, etc. where it is placed. Some apparently little changes in a connection can also change  the answer. This was forgiven in the past but it is a matter of cost unforgotten today.

The semi-rigid connection study is a objetive of my research, which I began in my DSc dissertation at 2010. Since then, need of work (begin of professor carreer), take me away the possibility to continue it, to publish what I found (the 'new DSc answers') and most, to make experimental research to  compare the theorical and practical answers I found. There's so much to do and this 5 years run out of my desired trend.

The theorethical answers of my work did follow the ones published by CHEN and CHAN very well, as other researchers did, but there's much more to work out. I ask to continue my work but everyday seems more afar like a broken dream. Technology changes much faster then one can follow and the computing ways of grouth is not in the same direction of what -researchers- wanted they grow. Now I need a place where I can continue this research and I didn't have it. I need an 'old computer' with more resources that are not given by the new ones.

Computer grew aimed to vectorize, to parallel machines, to make direct functions: design to 'object' and hide it (cloud), lgs. Our need is to make bigger loops task. They give us numbers of 64 bits, while for us 16 still is enough. But we ask for million of numbers, thousand of loops. Everywhere the task of mounting and solve matrices' equations with millions of variables is the problem. This is time consuming task of all study today. Mine too.

Did they give us a ship to solve these mathematics' problem? No. But Windows can talk, make 'selfies' and present flowers running down the screen (lgs) and so on.

The Eurocode made a lot of work in this subject, giving you ways of approximately find the answers following the builded moment rotation curve, based on the work of several researchers, like FAELLA, PILUSO and RIZZANO showed (old Structural Steel Semirigid Connections, 2000). There's much more new today about this.

This moment-rotation curve tells the semi-rigid joint behavior and this is the base of all the following analysis. If this curve is not quite understood or stabilished, the answers are not reliable and this implies in risk. That's why, I talk about SAFETY. Hope you get a good picture.

Semi rigid connection:

advantages: more economic, less stiffening, connection design easier, fabrication easier, architecturally preferred like: flush end plate

Disadvantages: require complicated calculation when considering it in member design, allow for rotation after reaching small load.

Rigid connection: like extended end plate and splices, heavy connection weight, more stiffeners, difficult connection design, need to use slip critical bolts...

advantages: the behavior of connections is clear and meet assumptions of rigidity used while designing members.