Need help to remap the searchlight's beam

Hello,

I would like to create a very long beam for the land searchlights. The problem is that the 3D model of the beam is not properly mapped with the texture. Have a look :

 

 

So, I would like the texture to be placed correctly on each rectangle which compose the beam model.

 

Moreover, it seems that a kind of shadow is applied on the model, and I would like to keep this effect. I have added a blue color in order you can see that better :

 

About the basis (the small circle on the searchlight's head), it has to be plain white (no modifications to do) :

 

Anyone to do this little work ?