Comparison of Bending in Beams

Abstract

In this project, the main objective is to compare the bending performance of the three types of beams(Rectangular beam, I -section beam, and C-section beam) which is extruded to a length of 400mm. After which, a baseline simulation is run on these beams at an external force of 1500N acting upon their smaller face and compare the results of stress, displacement, and strain for all the three beams. Also to suggest the best out of the three.

1. Introduction

1.1 FEA(Finite Element Analysis):

FEA is an analysis method using FEM(Finite Element Method) to solve the structural problems. The finite element method is a numerical technique for obtaining an approximate solution to a wide variety of engineering problem. The method originated in the aerospace industry as a tool to study stresses in the complex air-frame structure.

Since its birth, the finite element method has gone through various developmental stages. Started with the formulation of finite element theory, variational and Galerkin's approaches for two dimensional and three-dimensional problems were formulated. Isoparametric formulations for both linear and non-linear, static and dynamic analysis were developed. Also, computational methods required by the finite element method were established. This method includes numerical integration of element equations, a solution of the global system of algebraic equations, study of numerical stability and error analysis solution algorithms.

1.2 Stress:

When a body is acted upon by an external force or load, internal resisting force is set up, such a body is said to be in a state of stress, hence stress is the resistance offered by the body to deformation.

1.3 strain:

when a force or load acts on a body it undergoes deformation. This deformation per unit length is known as unit strain or simply strain.

1.4 Stress-Strain Curve:

To determine the relation between the stress and strain for a given material, let’s conduct an experiment. Take a test cylinder or wire and stretch it by an applied force. Record the fraction change in length (strain) and the applied force needed to cause the strain. Increase the applied force gradually, in steps, and record the readings.

Now, plot a graph between the stress (which is equal in magnitude to the applied force per unit area) and the strain produced. The graph for a typical metal looks as follows:

• Analysis of the Curve

From the figure, we can see that in the region between O and A, the curve is linear. Hence, Hooke’s Law obeys in this region. In the region from A to B, the stress and strain are not proportional. However, if we remove the load, the body returns to its original dimension.

The point B in the curve is the Yield Point or the elastic limit and the corresponding stress is the Yield Strength (Sy) of the material. Once the load is increased further, the stress starting exceeding the Yield Strength. This means that the strain increases rapidly even for a small change in the stress.

This is shown in the region from B to D in the curve. If the load is removed at, say a point C between B and D, the body does not regain its original dimension. Hence, even when the stress is zero, the strain is not zero and the deformation is called plastic deformation.

Further, point D is the ultimate tensile strength (Su) of the material. Hence, if any additional strain is produced beyond this point, a fracture can occur (point E). If,

• The ultimate strength and fracture points are close to each other (points D and E), then the material is brittle.
• The ultimate strength and fracture points are far apart (points D and E), then the material is ductile.

1.5 Bending of Beams:

In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

Fig illustrates Bending of an I-beam.

2. Geometry Of Beams(given):

2.1 Rectangular Beam

2.2 I-section Beam

2.3 C-section Beam

3. Procedure:

• Creating the geometry in SolidWorks.
  • Case 1: Rectangular Beam

• Case 2: I-Section Beam

• Case 3: C-Section Beam

• After creating the geometry, extrude it to 400mm.
• Under Simulation, perform a new study for static analysis.
• General steps in FEA problem:
  • Pre-processing:-
    • Defining material property. Here we have defined the material for each beam to be alloy steel.
  • Solving:-
    • Types of Analysis: Static Analysis
    • Beams are fixed at one end of the extrusion along the cross-section.
    • A 1500N load was applied over the top face
    • The Simulation was run using a basic mesh.
  • Post Processing:-
    • Results: Stress, strain, and displacement contours/plots.

4. Output

Case 1: Rectangular Beam

• Observation:-
  • Stress Contour: It achieves maximum yield strength at 6.204e+002N/mm.
  • Displacement Contour: It is quite evident from the contour that fixed end does not experience any changes, as we move to the other end, displacement is maximum at a distance of 3.1e-002mm.
  • Starin Contour: Maximum strain is experienced on the fixed end i.e. 2.402e-005.

Case 2: I-Section Beam

• Observation:-
  • Stress Contour: Here stress is evenly distributed. It achieves maximum yield strength at 6.204e+002N/mm.
  • Displacement Contour: It's quite similar to case 1 stress contour, but slightly higher displacement i.e. 8.537e-002mm, this is due to lose in weight as compared to a rectangular beam(case 1).
  • Starin Contour: Maximum strain is experienced on the fixed end i.e. 7.564e-005.

Case 3: C-Section Beam

• Observation:-
  • The beam experiences a torsion due to the vertical force applied onto the smaller face.
  • Stress Contour: It achieves maximum yield strength at 6.204e+002N/mm.
  • Displacement Contour: Displacement is maximum at a distance of 3.021e-001mm.
  • Starin Contour: Maximum strain is experienced on the fixed end i.e. 1.385e-004.

5 Results:-

• If we compare the stress plots, maximum stress induced is less on the rectangular beam then on the I section and is maximum at C-section beam.
• Strain figures indicate that maximum change in length takes place in the C-section beam and minimum change in length occurs in I-section beam and the displacement plots indicate that minimum displacement occurs in the rectangular beam after application of 1500N and maximum displacement occurs in the C-section beam.
• It can be seen that the C-section beam is subjected to torsion, therefore, it is the least preferred among the three beams.
• Since stress induced in both the I-section and rectangular beam is under the yield strength therefore both the beams will sustain the force of 1500N properly without failing, but if we look at the distribution of stress, it is quite evenly distributed in the I-section, which I feel will withstand larger force as compared to rectangular beam.
• So, I would prefer the I-section beam as the best of the three beams.