Analysis of a Simple Structure - Steel Wire

Abstract

The report presents an analysis of an experiment conducted on both a straight wire and a uniquely shaped wire. Calculations were performed using numerical and analytical methods to determine the maximum displacement of the wire. The results were compared to identify any differences in the maximum displacement between the two wire configurations.

Introduction

Beams are horizontal structural members with significantly greater length than width or breadth. Various types of beams exist, including Simply Supported Beam, Fixed Beam, Cantilever Beam, and Continuously Supported Beam.

In this report, we focus on the deflection of a steel wire with one end fixed and the other free, resembling a cantilever beam. The deflection of the wire is analyzed for both a straight wire and a uniquely shaped wire. The wire dimensions are as follows: Diameter = 0.9 mm, Length = 300 mm, and a weight of 0.00605 kg is attached to the free end.

The experiment employs both numerical and analytical procedures. Abaqus software is used for numerical analysis, while analytical methods include the Deflection of Cantilever Beam with End Load formula and Castigliano’s Theorem.

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Young’s modulus of the material is determined when the wire is in its straight configuration.

Formulas

The deflection of the beam (δ_max) is calculated using the equation below:

^δ_max = ^PL / _4EI (1)

Where:

• P = Load (N)
• L = Length of wire or beam (mm)
• E = Young's Modulus (N/mm²)
• I = Moment of Inertia (mm⁴)

The load (P) is calculated as:

P = F = ma (2)

With the wire being circular, the area (A) and moment of inertia (I) are determined as:

A = πd²/4 (3)

I = πd⁴/64 (4)

Castigliano’s Theorem states that when forces act on elastic systems, tiny displacements are linearly related to the force.

It can be expressed as:

^δ_max = ^∂U/∂F (5)

Where:

• U = Strain energy

Strain energy (U) represents the external work done on an elastic member during deformation, which is converted into strain or potential energy:

U = Fδ (6)

Nomenclature

• E = Young's Modulus (N/mm²)
• P = Load (N)
• I = Moment of Inertia (mm⁴)
• L = Length of wire or beam (mm)
• δ_max = Deflection of beam (mm)
• A = Area (mm²)
• U = Strain energy
• M = Moment (Nmm)
• F = Force (N)
• m = Mass or weight of the object (kg)
• a = Acceleration (m/s²) - 9.81 m/s²

Procedure

A straight circular wire with a length of 300 mm and a diameter of 0.9 mm is fixed at one end on the wall. A load of 0.05935 N is placed on the other end of the wire.

After conducting the experiment, the deflection of the wire (δ_max) is measured as 82 mm. Using equations (2) and (4), the load and moment of inertia are calculated as 0.05935 N and 0.0322 mm⁴, respectively. Substituting these values into equation (1), Young’s Modulus is calculated as 201105.893 N/mm².

Following the determination of Young’s Modulus for the straight wire, the wire is bent into a unique shape. The experiment is repeated with the only difference being the shape of the wire.

The maximum displacement of the uniquely shaped wire is calculated using Castigliano’s Theorem, employing equations (5) and (6). Numerical analysis is also conducted using Abaqus software.

The experiment is completed, and the calculations of maximum displacement under applied load are summarized in Table 1:

Conclusion

Based on both analytical calculations and numerical simulations conducted using Abaqus, it is observed that the maximum displacement of the uniquely shaped wire does not significantly differ between the two methods. However, it is worth noting that analytical procedures are time-consuming and prone to human errors, whereas numerical procedures are faster and provide accurate results.

References

• Shigley, J. E., Mischke, C. R. & Budynas, R. G., 2003. Deflection and Stiffness. In: Mechanical Engineering Design Seventh Edition. New York: McGraw-Hill, pp. 185 - 247.