Slope Equation
Let's try and solve the Bernoulli-Euler Equation. If we integrate it once, then:
\(\frac{d\Delta}{dx}=\theta=\int\frac{M}{EI}dx+C_1\)
Recall that \(\frac{d\Delta}{dx}\) is the slope of the deflected curve at point \(x\). It tells us the rotation \(\theta\) at said position. Using this equation, we can find the angular deflection at any point along the beam. We call this the slope (or rotation) equation.
Deflection Equation
Now, let's integrate the slope equation once more to get:
\(\Delta=\iint\frac{M}{EI}dx+C_1x+C_2\)
In this equation, \(\Delta\) represents linear deflection at point \(x\) or its translation. This equation helps us find such quantity at any point along the beam. We call this the deflection (or translation) equation.
Slope and Deflection Diagrams
Now that we integrated the Bernoulli-Euler equation. We have two equations we can use to describe the beam's deflection. It is enough to find the deflection at any point along the model.
We can do more. We can use these equations to create the beam's slope and deflection diagrams. It's a graphical representation of a structure's deflections (similar to axial, shear, moment, and torsion diagrams).
Creating such diagrams is as simple as plotting the function in a graph.
Two Approaches
Let's move on to the two approaches to this method. Generally, there are two approaches: a general equation or a segmented approach. It's best to explain these using examples.
Remember - this method of solving movements and rotation is only possible if we obey the Bernoulli-Euler Beam model.
Summary
Let's summarize:
The Double Integration Method is an analytical procedure for solving beam deflections.
This method aims to find an expression for the structure's deflected shape through a function.
If we integrate the Bernoulli-Euler equation once, we get the slope equation: \(\frac{d\Delta}{dx}=\theta=\int\frac{M}{EI}dx+C_1\)
If we integrate the Bernoulli-Euler equation twice, we get the deflection equation: \(\Delta=\iint\frac{M}{EI}dx+C_1x+C_2\)
Generally, there are two approaches: a general equation or a segmented approach.
Remember - this method of solving deflections is only possible if we obey the Bernoulli-Euler Beam model.
