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Propped cantilever boundary conditions

Short answer

A propped cantilever has one fixed end and one simply supported end. At the fixed end, deflection y=0 and slope dy/dx=0; at the propped end, deflection y=0 and bending moment M=0 (no moment resistance).

Draw a propped cantilever and write down its boundary conditions.

A propped cantilever is a beam that is fixed at one end and simply supported (propped by a roller or hinge) at the other end. It is statically indeterminate to the first degree.

  fixed |============================== o  (roller/prop)
   A    x = 0                      x = L   B

Boundary conditions (A fixed at x=0x=0, B propped at x=Lx=L)

EndConditionMeaning
A (fixed)y(0)=0y(0)=0no deflection
A (fixed)$\dfrac{dy}{dx}\Big_{x=0}=0$
B (prop)y(L)=0y(L)=0the prop prevents deflection
B (prop)$M(L)=EI\dfrac{d^2y}{dx^2}\Big_{x=L}=0$

Rotation at B is free and the moment at A is a reaction that is not zero. These four conditions fix the four constants of integration of EI d4y/dx4=w(x)EI\,d^4y/dx^4=w(x). In the double-integration method (EI d2y/dx2=M(x)EI\,d^2y/dx^2=M(x)) the condition M(L)=0M(L)=0 is already built into M(x)M(x), and the other three conditions give the two constants of integration and the redundant prop reaction RBR_B. For a UDL ww they lead to RB=3wL8R_B=\dfrac{3wL}{8} and MA=wL28M_A=\dfrac{wL^2}{8} (hogging).

✓ Answer checked by Claude Opus 5.5 (high effort)

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