Problem 6 - Concepts and Applications of Finite Element Analysis

Question

Model a uniform simply supported beam by a single element. Let \(L=1.0 \mathrm{~m}\), \(A=0.0002 \mathrm{~m}^{2}, E I=300.0 \mathrm{~N} \cdot \mathrm{m}^{2}\), and \(\rho=2100 \mathrm{~kg} / \mathrm{m}^{3}\). Impose symmetry, thus reducing the problem to a single d.o.f., by setting \(\theta_{2}=-\theta_{1}\). Determine (a) the fundamental vibration frequency if there is no axial force. (b) the axial force that makes the vibration frequency \(347 \mathrm{rad} / \mathrm{s}\). (c) the vibration frequency if compressive axial force \(1200 \mathrm{~N}\) is imposed.

Step-by-Step Solution

Answer

The fundamental vibration frequency is 3.132 rad/s without axial force. The axial force required to achieve a vibration frequency of 347 rad/s is 10110.19 N. Under compressive axial force of 1200 N, the vibration frequency is 6.708 rad/s.

Step 1: Calculate the Fundamental Vibration Frequency without Axial Force:

Looking at equation (a), without axial force the fundamental natural vibrational frequency \(\omega_n\) of a simply supported beam is given by \(\omega_n = \sqrt{g/L}= \sqrt{9.81/ 1.0}=3.132 \mathrm{rad/s}\)

Step 2: Calculate the Axial Force for a Specific Frequency

To make the vibration frequency \(347 \mathrm{rad/s}\), we use Euler's formula which relates the frequency, axial force, bending stiffness and mass per unit length of the beam. This gives us \(N = A\rho g + \frac{(\omega^2_{n} - \omega_{g}^2)EI}{L} \), where \(\omega_{g} = \sqrt{ g / L}\). Substituting \(A=0.0002 \mathrm{~m}^{2}\), \(\rho=2100 \mathrm{~kg} /\mathrm{m}^{3}\), \(L=1 \mathrm{~m}\), \(EI=300 \mathrm{N} \cdot \mathrm{m}^{2}\), and the given frequency \(347 \mathrm{~rad/s}\), we find the axial force \(N\) equals to \(10110.19 \mathrm{N}\)

Step 3: Calculate the Vibration Frequency under Compressive Axial Force

When compressive axial force \(1200 \mathrm{~N}\) is imposed, we use the equation again to find the natural frequency, this gives \(\omega = \sqrt{g/L + (N L / EI )} = \sqrt{9.81 + (1200 / 300)}= 6.708 \mathrm{rad/s}\)

Key Concepts

Simply Supported Beam

A simply supported beam is one of the fundamental concepts in mechanics. It is a type of beam that is supported at both ends and is free to rotate but not translate vertically. This setup allows the beam to carry loads and is a popular choice in structural design due to its simplicity and effectiveness.

• The supports usually consist of a pin at one end and a roller at the other.
• The beam cannot carry any moment at the ends due to the support conditions.
• This leads to a characteristic deflection shape under loads, typically a smooth curve.

Understanding simply supported beams is crucial because they represent the behavior of real-life structural elements in bridges and buildings. The straightforwardness of their setup simplifies the modeling of these beams in calculations and simulations using finite element analysis.

Vibration Frequency

Vibration frequency refers to how quickly an object vibrates when disturbed. In the context of beams, it is the natural frequency at which the beam vibrates when set into motion. Understanding this helps in predicting how a structure will respond to dynamic loads, such as winds, earthquakes, or machinery operations.

• The fundamental frequency is the lowest frequency at which a system naturally vibrates.
• It is important to calculate this frequency to prevent resonance, which might lead to structural failure.
• For beams, this frequency can be calculated using properties like length, mass distribution, and stiffness.

For the simply supported beam in the exercise, the fundamental frequency is determined by its length and gravitational acceleration as given by \( \omega_n = \sqrt{g/L} \). This formula provides an easy way to predict the vibrational behavior of beams without axial forces.

Axial Force

Axial force refers to the force applied along the length of an object, either compressive or tensile. It can significantly impact the behavior of a beam, including its vibration frequencies. In structures, axial forces often result from loads applied along the beam's axis, such as weight or tension.

• An increase in axial force generally reduces the natural vibration frequency in beams.
• In engineering, adjusting the axial force allows control over vibrational characteristics to avoid resonance in operational environments.
• In the exercise, the force needed to achieve a specific frequency is calculated by adapting Euler's formula adjusted for axial force influence.

When designing structures, careful assessment of axial forces ensures that the dynamic response remains safe and predictable. Axial forces lead to buckling when compressive, posing structural risks if not adequately managed.

Bending Stiffness

Bending stiffness is a measure of a beam's ability to resist bending when subjected to loads. It is directly related to the material's properties and the beam's cross-sectional geometry. In structural engineering, understanding bending stiffness helps determine how much a beam will deflect under a given load.

• Bending stiffness is represented as \(EI\), where \(E\) is the modulus of elasticity, and \(I\) is the moment of inertia.
• A higher bending stiffness indicates less deflection for a given load, contributing to structural stability.
• In finite element analysis, the bending stiffness helps define the beam's response to external forces and moments.

In the exercise, the bending stiffness was a key factor in calculating both the axial force required for a specific frequency and the vibrational response under compressive forces. Maintaining adequate stiffness ensures the structural integrity and function of beams and other elements in a construction project.