Deriving Moment & Elastic Curve Equations for Incomplete Triangular Load on Beam

[Quantumsatire]

I was wondering how you would derive the moment and elastic curve equations for an incomplete triangular load. Say you have a pin at the left end of the beam and a roller at L/2 from the left, and a triangular load that goes from the pin and ends at the roller. I know you have to do some kind of extension, but how do you come up with the formula.

 

[timthereaper]

When you say "triangular load", you mean a distributed load that's zero at the left end and increases linearly to the roller at L/2?

 

[paisiello2]

If you know Δ(x) and θ2 for a simply supported beam as a function of L' = L/2 with a triangular load then the deflection would be:

Δ = Δ(x) for x=<L/2
= θ2*(x-L/2) for x>=L/2

 

[Quantumsatire]

@timthereaper yes

 

[alphy]

To derive the moment and elastic curve equations for an incomplete triangular load on a beam, we can use the principles of statics and mechanics of materials. The first step would be to draw a free body diagram of the beam, including all external forces and reactions. In this case, we have a pin support at the left end, a roller support at L/2 from the left, and a triangular load acting on the beam between these two supports.

Next, we can apply the equations of static equilibrium to determine the reactions at the supports. For a pin support, we know that it can only exert a vertical reaction force, while a roller support can exert both vertical and horizontal reaction forces. By summing the forces in the vertical and horizontal directions, we can solve for the reactions at the supports.

Once we have the reactions, we can use the principle of superposition to determine the internal forces and moments at any point along the beam. This means that we can break down the triangular load into smaller, simpler loads, such as point loads or distributed loads, and analyze each one separately. Then, we can sum the individual results to get the overall response of the beam.

To determine the moment equation, we can use the moment equation M = Fd, where M is the moment, F is the force, and d is the perpendicular distance from the point of interest to the line of action of the force. By applying this equation to each individual load, we can determine the moment at any point along the beam.

To determine the elastic curve equation, we can use the equation EIy'' = M, where E is the modulus of elasticity, I is the moment of inertia, y'' is the second derivative of the deflection curve, and M is the moment. By integrating this equation twice and applying boundary conditions, we can solve for the deflection of the beam at any point.

In summary, to derive the moment and elastic curve equations for an incomplete triangular load on a beam, we need to apply the principles of statics and mechanics of materials, use the equations of equilibrium and superposition, and solve for the reactions and internal forces using appropriate equations.