Hi I was reading about deformation on the book Mechanics of

Hi !

I was reading about deformation on the book \"Mechanics of materials\" of \"Beer & Jhonston\" and I found myself in troubles to understand this statements:

I understand the mathematical part of the explanation; if you divide by 0, your rho becomes infinite at the end of the beam (where the force is applied). What I cant understand is the physical part of what is happening with de curvature radius (rho). How can I visualice what is happening whith the curvature radius a the end of the beam (A) where it is infinite and at the clamped portion (B).

I understand that if the curvature radius is small, the deformation is big and if the curvature radius is big, the deformation is smallm so I cant visualice this concepts geometrically in the beam... could you please help me to understand??

Best regards!

PB Fig. 15.3 (a) Cantilever beam with concentrated load. (b) Deformed beam showing curvature at ends.

Solution

See,in this case,firstly visualise how deflection is produced in a beam,when a oad is applied at one end of the cantilever beam(where the load is applied), So,Basically, what is happening in a beam is that, when a load is applied at one end of an cantilever beam,the load causes an deflection on the end of the beam, since at the other end of the beam--fixed support is present,which has infinite stiffness,that is,no deflection is produced at the support...

Let me Summarise this,in cantilever beam,the maximum deflection will be produced at the free end and no deflection will be produced at the fixed support end.

Now,coming to your point, the equation which you has shown in the image above, is a deflection equation---where (MOMENT)/(FLEXURAL RIGIDITY) =d²y/dx²,where y=deflection of te beam....if you will solve the given equation using Proper end conditions, you will an defection value( in case of the above example,the deflection is given as: (PL³/3EI)).

Now,talking about the curvature of the beam,the curvature of the beam will be such that,it will be maximum at the end where the load is being applied and will become zero at end where the fixed support is present, this is because at the free end,the moment will be minimum and consecuetively the deflection will be maximum and hence curvature will be maximum, but as we will walk towards the support,the moment will start increasing,since the moment will start on increasing,the curvature will start decreasing as the beam will have the ability to resist the force being applied.