Deflection of Tapered Beam with Elliptic Cross Section

In summary: M(x)}{EI} \text{ and }\\&\text{The solution for the deflection at the free end diverges when equated with the solution for the deflection of a tapered beam with elliptic cross-section.}\\&\text{This problem may be due to a difference in the way the semi-axes are defined in the two cases.}\end{align*}In summary, the area moment of inertia for a tapered beam with an elliptic cross-section is a linear function of the beam length, but the solution for the deflection at the free end diverges when equated with the solution for the deflection of a tapered beam with
  • #1
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I am working on deriving expression for deflection of a tapered beam with an elliptic cross-section. Hence, area moment of inertia is a linear function of the beam length. The beam is fixed at one end, and a concentrated force F is applied on its tip at the free end. I am using the known relation between second derivative of deflection and moment. To verify my solution of the first integral (yielding the slope), I compare the result with the expression for the slope of tapered beam with circular cross-section under the same boundary conditions, by equating the minor and major semi-axes of the elliptic cross-section. The problem is: by equating the minor and major axes of the elliptic cross section, the solution diverges (resulting in 0/0). I tried integration manually and with mathematica, but the solution always diverges. Attached is my solution.
I would appreciate hints about what is wrong with this solution.
Thanks!
 

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\begin{align*} &\text{Let a tapered beam with an elliptic cross section of length } L \text{ be fixed at one end and a force F is applied to its tip at the free end.} \\& \text{The area moment of inertia } I \text{ is a linear function of the beam length }L.\\&\text{Assuming that the deflection }y(x) \text{ is non-linear, we apply the relation between second derivative of deflection and moment:}\\&\frac{d^2y}{dx^2}= \frac{M(x)}{EI}\\&\text{Integrating twice, the general solution for the deflection is given by: }\\&y(x)= C_1 x + C_2 + \frac{M_0x^2}{2EI}+ \frac{1}{EI} \int_0^x \int_0^u M(t) dt du\end{align*}To find the constants of integration, we use the boundary conditions: $y(0)=0$ and $y_x(L)=0$. Then,\begin{align*}&C_1=0 \text{ and } C_2=0 \\&\implies y(x)= \frac{M_0x^2}{2EI}+ \frac{1}{EI} \int_0^x \int_0^u M(t) dt du\end{align*}To find the slope at the free end, we differentiate the expression twice and set $x=L$.\begin{align*}y_x(L)=\frac{M_0L}{EI}+ \frac{1}{EI} \int_0^L \int_0^u M(t) dt du\end{align*}Now, let $a$ and $b$ be the major and minor semi-axes of the elliptic cross-section respectively. Then,\begin{align*}&\text{From the known expression for the slope of a tapered beam with circular cross-section under the same boundary conditions, we get }\\&y
 

FAQ: Deflection of Tapered Beam with Elliptic Cross Section

What is the significance of the deflection of a tapered beam with an elliptic cross section?

The deflection of a tapered beam with an elliptic cross section is an important factor to consider in structural engineering. It determines the strength and stability of the beam and can impact the overall performance of a structure.

How is the deflection of a tapered beam with an elliptic cross section calculated?

The deflection of a tapered beam with an elliptic cross section can be calculated using the Euler-Bernoulli beam theory, which takes into account the beam's material properties, cross-sectional geometry, and applied loads. Advanced numerical methods such as finite element analysis can also be used for more accurate results.

What factors can affect the deflection of a tapered beam with an elliptic cross section?

The deflection of a tapered beam with an elliptic cross section can be affected by various factors such as the beam's length, taper ratio, cross-sectional dimensions, material properties, and applied loads. Temperature changes and external forces such as wind or earthquakes can also influence the beam's deflection.

How does the deflection of a tapered beam with an elliptic cross section compare to that of a uniform beam?

The deflection of a tapered beam with an elliptic cross section is typically larger than that of a uniform beam of the same length and material. This is due to the varying cross-sectional dimensions along the length of the tapered beam, which result in uneven stress distribution and higher deflection.

What are some practical applications of the deflection of a tapered beam with an elliptic cross section?

The deflection of a tapered beam with an elliptic cross section is an important consideration in the design of various structures such as bridges, aircraft wings, and wind turbine blades. It is also relevant in the development of new materials and structural designs, as well as in the analysis and maintenance of existing structures.

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