Boundary conditions on a fixed-fixed bar

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In analyzing a fixed-fixed bar subjected to a distributed axial load w(x)=CX/L, the user is seeking assistance with determining the appropriate force boundary conditions. The known conditions are U(0)=0 and U(L)=0, indicating fixed displacements at both ends of the bar. The user is specifically looking for guidance on how to define U' at a certain value, which relates to the bar's axial deformation under load. A visual aid has been provided to clarify the problem. Additional insights on deriving the force boundary conditions would be beneficial.
Bryan morrow
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I am working with a fixed fixed bar with a distributed axial load to the right as w(x)=CX/L. I am having a hard time determining the force boundary conditions. I know that U(0)=0 and U(L)=0. However, I need to come up with something in regards to U'(Value). Any help would be appreciated.
 
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