In this article, we are going to learn about Muller Breslau Principle. This principle is too simple, interesting and that much important also. Before moving towards Muller Breslau’s principle, let’s know something about Muller Breslau.
The full and real name of Muller Breslau was “Heinrich Franz Bernhard Muller” which was a little bit difficult but Muller changed his name to distinguish his identity from another Muller.
He joined his name with his city’s name. It means the name of his city was “Breslau” in which Muller was born and that is why he identified himself as he is Muller from Breslau and then known as Muller Breslau.
Ok, this is all about his name. Muller Breslau was a German civil engineer and also a high school teacher. The interesting thing is that he provided a significant contribution to the theory of structural analysis and one of them is the “Muller Breslau Principle” which we are going to discuss in this article.
What is Muller Breslau Principle?
it state that “if we consider that there is an internal reaction component or internal stress component acting to a small distance on the structure due to that action it will give displacement or deflection to the structure that curve of deflection will show you the influence lines of that structure to some scale”
Now we will discuss this principle with the help of graphical presentation and derivation so that your all the doubts get clear over here.
For that, let’s consider a two-span continuous beam
Now, what do you mean by a two-span continuous beam? For that consider here support A support B and support C.
So you can see here, span AB is the 1st span and BC is the second span that’s why we can say it a two-span continuous beam.
Now let’s consider a unit load at a point “P” (capital P) where unit load is acting and this unit load is at a distance of “x” (small x) from support A. So currently this unit load is at point X at a distance of “x” from support A.
Now, what Muller Breslau’s principle says further? if we have to find out the reaction at support B then we have to remove support B.
What happens after removing support B?
Here, support A and C are in the same situation as it is, we just removed B. While we remove B the beam will deflect like the figure below.
Now how much it will deflect. The sign of deflection is generally denoted by “delta”
So, δB-P means deflection at B will be due to (P). Since, there is no load over B, whatever the deflection at B will be due to the load at P only.
So, it will be deflection at B due to “P” i.e (δB-P)
Now let’s talk about point “P”. That how much deflection be will over here?
here the deflection will deflect at P due to P i.e (δP-P) because the load is on itself causing deflection.
Here we saw at B & P.
Now, because it’s a moving load as we consider in influence diagram that this unit load as a moving load. and then we draw the influence line diagrams
So, because it’s a moving load, recently it was at P. Now consider that it is at B it’s just above B. Then how will be the deflection shape. It will be like the figure below ,
Here, the new deflected shape will be like this, and now, what will be this deflection? so, this deflection will be deflection at B means δB. Deflection at “B” due to B i.e (δB-B). Because the load is just over B right now.
Now if we talk about “P”.
So, how much deflection at P will be?
Deflection at P will be Deflection at p due to B. i.e (δP-B).
So, here we plotted two diagrams in a simple way and I tried my best to explain to you in the simplest method as far as possible. Now let’s see the derivation according to this. Here we just saw the deflection, Now we have to consider the “Consistent Deformation”.
What Consistent Deformation says?
it says that, this deflection (δBP) will be equal to this deflection (δBB).
Consistent deformation is trying to say that, If we talk about δBB. There will be reaction RB that we considered. There will be reaction,
Or, (RB * δBB) = δBP
One more thing you have to consider here “Maxwell’s Reciprocal Theorem”. It says to you that
this deflection (δBP) will be equal to δPB.
Or, δBP = δPB
Now we know δBP is equaled to RB into δBB, which equals to δPB. This is according to Maxwell’s Reciprocal Theorem.
Now if you want to find out reaction here (at B), then you have to make RB as Subject. It means
you will get RB = δPB/δBB. This is how you will get this whole relation.
Assume that, if δBB=1, then what happen?
RB will be equals to δPB. Which means that if we justify it, whatever the deflection at P due to be is
will be the value of reaction RB or we can say that Reaction at B will be equals to Deflection at P
or we can also say that RB is directly proportional to the deflection due to B.
At last, the important thing is that this final diagram or this deflection curve which you see here. We can say it ILD for RB, means influence lines for reaction B.
This is what I told you in Muller Breslau Principle in the beginning and here at the end, we found that this deflected curve will show you the influence line.
You can easily find out the influence line at any distance.
I hope this article on “Muller Breslau Principle” remains helpful for you.
Happy Learning – Civil Concept
Civil Engineer – Ranjeet Sahani