Question

Consider the free flexural vibrations of a simply supported uniform elastic beamcolumn of length \(L\), bending stiffness \(E I\), and mass per unit length \(m\) that is subjected to a constant compressive end load \(P_{0}\). (See Example 9.12.) Determine the first three natural frequencies and modal functions for the case where the applied load is half the static buckling load. Plot the modal functions.

Short answer

Question: Determine the first three natural frequencies and modal functions for a simply supported uniform elastic beam-column subjected to a compressive end load, \(P_0\), where the applied load is half the static buckling load.

Step-by-step solution

Calculate the static buckling load

The static buckling load, \(P_{cr}\), can be calculated using the Euler formula for a simply supported beam: \(P_{cr} = \dfrac{\pi^2 EI}{(L)^2}\)

Determine the compressive end load, \(P_0\)

Since the applied load is half the static buckling load, we can calculate the compressive end load as: \(P_0 = \dfrac{1}{2} P_{cr}\)

Write down the governing equation

The governing equation for free vibrations of a beam-column can be written as: \(E I \dfrac{\text{d} ^4 w}{\text{d} x^4} - P_0 \dfrac{\text{d} ^2 w}{\text{d} x^2} + m \dfrac{\text{d} ^2 w}{\text{d} t^2} = 0\)

Apply boundary conditions and derive the equation for natural frequencies

For a simply supported beam, the boundary conditions are: 1. \(w(0) = 0\) 2. \(\dfrac{\text{d}^2 w}{\text{d}x^2}(0) = 0\) 3. \(w(L) = 0\) 4. \(\dfrac{\text{d}^2 w}{\text{d}x^2}(L) = 0\) By applying these boundary conditions, we can transform the governing equation into an eigenvalue problem and derive the equation for natural frequencies: \(\tan(\dfrac{\alpha L}{2}) = \dfrac{\alpha^2 L^2}{4P} - \dfrac{P}{\alpha^2 L^2}\) where \(\alpha=\dfrac{\omega L}{\sqrt{EI/m}}\), \(P = \dfrac{2P_0}{\pi^2} \ge 1\), and \(\omega=\dfrac{2 \pi}{T}\) is the circular natural frequency with period \(T\).

Find the first three natural frequencies

By solving the equation for natural frequencies \(\tan(\dfrac{\alpha L}{2}) = \dfrac{\alpha^2 L^2}{4P} - \dfrac{P}{\alpha^2 L^2}\), numerically or graphically, for the first three positive roots \(\alpha_s\), \(s=1,2,3\), we can obtain the corresponding natural frequencies as follows: \(\omega_s = \dfrac{\alpha_s \sqrt{EI/m}}{L}\), \(s=1,2,3\)

Calculate the modal functions

Using the values of \(\alpha_s\) obtained in the previous step, the modal functions \(w_s(x)\) can be calculated as follows: \(w_s(x) = C_1 \cosh(\alpha_s x) + C_2 \sinh(\alpha_s x) + C_3 \cos(\alpha_s x) + C_4 \sin(\alpha_s x)\), where \(C_1\), \(C_2\), \(C_3\), and \(C_4\) are constants that can be found using the boundary conditions.

Plot the modal functions

Finally, plot the modal functions \(w_1(x)\), \(w_2(x)\), and \(w_3(x)\). This can be done by using any mathematical software or by drawing them manually, keeping in mind the boundary conditions and the shape of the trigonometric functions involved.

Key concepts

Natural Frequencies

Understanding natural frequencies is essential in analyzing vibrations. They represent the frequencies at which a system oscillates when disturbed and then left to vibrate freely. For structures like beams, these frequencies are critical, as they depict the inherent vibration properties.
Each natural frequency corresponds to a specific mode of vibration, described by a modal function. For a simply supported beam-column, these modes illustrate how the beam will deform at each frequency.

• Higher modes represent more complex deformations.
• The first mode is usually the simplest, resembling a sine wave.

For any beam, calculating these frequencies involves solving a boundary value problem. As in the exercise, the approach combines mathematical tools to find solutions that satisfy the governing equation.

Euler Buckling

Euler buckling refers to the critical load at which a slender, straight column under axial compression becomes unstable and exhibits lateral deflections. It's a vital concept in structural engineering, crucial for ensuring stability.
The Euler formula for buckling load, \(P_{cr} = \dfrac{\pi^2 EI}{(L)^2}\), shows how the load depends on length, rigidity, and support conditions. It serves as a benchmark to estimate how a beam may behave under specific loads.
When a compressive load reaches half of this critical value, as in the exercise, the beam gets evaluated in a near-buckling state. This condition influences the natural frequencies and modal shapes due to the increased sensitivity to applied loads.

Beam-Column Vibrations

The combined action of load and vibration in beams, known as beam-column vibrations, challenges engineers to account for interactions between bending and compressive forces.
The governing equation for these vibrations integrates both structural stiffness and applied pre-load: \[EI \dfrac{\text{d} ^4 w}{\text{d} x^4} - P_0 \dfrac{\text{d} ^2 w}{\text{d} x^2} + m \dfrac{\text{d} ^2 w}{\text{d} t^2} = 0\]
This equation showcases a balance between the bending moment, compressive load, and inertia.
Key points to remember:

• Beam-column interaction can lower natural frequencies.
• Effects are more pronounced near the critical buckling load.

These phenomena must be assessed carefully to ensure safety in design.

Flexural Vibrations

Flexural vibrations describe oscillations where bending is the primary deformation mode. In beams, they are a significant aspect due to their influence on structural dynamics.
These vibrations can be represented by trigonometric or hyperbolic functions, as seen in the modal functions: \[w_s(x) = C_1 \cosh(\alpha_s x) + C_2 \sinh(\alpha_s x) + C_3 \cos(\alpha_s x) + C_4 \sin(\alpha_s x)\]
The constants \(C_1, C_2, C_3,\) and \(C_4\) depend on boundary conditions and the specific mode shape.
Flexural modes are crucial in design as they dictate how a structure responds to dynamic loads. Understanding their impact helps predict performance and prevent failure under resonant conditions.