Double overhanging beam deflection

[Mohmmad Maaitah]

Homework Statement

Derive an expression for the slope and deflection for the beam

Relevant Equations

Deflection of beams

Please help me finding the max slope and deflection im stuck on this problem for two weeks.
I can't get anywhere with the boundry and continutiy conditions
The problem and what I've d one:

 

[erobz]

The shear $V(x)$ can be represented with the singularity function:

$$V(x) = -P+P\langle x - a \rangle^0 +P\langle x - (L-a) \rangle^0 $$

Then you can integrate that to get $M(x)$

Then apply $EI \frac{d^2y}{dx^2} = M(x)$.

The boundary conditions on $y$ at $x=a$ and $x = L-a$ give you two equations and two unknowns to determine the integration constants $C_\theta, C_y$

 

[Mohmmad Maaitah]

The shear $V(x)$ can be represented with the singularity function:

$$V(x) = -P+P\langle x - a \rangle^0 +P\langle x - (L-a) \rangle^0 $$

Then you can integrate that to get $M(x)$

Then apply $EI \frac{d^2y}{dx^2} = M(x)$.

The boundary conditions on $y$ at $x=a$ and $x = L-a$ give you two equations and two unknowns to determine the integration constants $C_\theta, C_y$

Thanks for the answer, how is it possible with the normal integration as I'm not familiar with the singularity method.

 

[erobz]

Thanks for the answer, how is it possible with the normal integration as I'm not familiar with the singularity method.

Its messy, but the idea is the same. You will have a bunch of integration constants to figure out. The is a distinct moment curve for each region of the beam between point loads with its own constants of integration in the deflection integration. They have to be all matched up with the boundary conditions on $y$ at a particular point and use the fact that we are not to have discontinuous slope between the segments.

I suggest you see this:

https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Introduction_to_Aerospace_Structures_and_Materials_(Alderliesten)/02:_Analysis_of_Statically_Determinate_Structures/07:_Deflection_of_Beams-_Geometric_Methods/7.04:_Deflection_by_Method_of_Singularity_Function

 

[erobz]

Homework Statement: Derive an expression for the slope and deflection for the beam
Relevant Equations: Deflection of beams

Please help me finding the max slope and deflection im stuck on this problem for two weeks.
I can't get anywhere with the boundry and continutiy conditions
The problem and what I've d one:
View attachment 354449
View attachment 354450View attachment 354451
View attachment 354449

I think you are going wrong in evaluating the slope at the first support. You apparently have $x_2 = a$ in the equation? $x_2$ should not be referenced from the free end ( where $x_1$ is referenced), it should be referenced from the beginning of the segment under consideration. So when you are equating $EI \theta_1 = EI \theta_2$ be wary of that (pay attention to the value $x_2$ should take in the $\theta_2$ equation at the first support).