Development of Plastic Bending in Beams

Part A: Analytical solution;

Part B: Finite element modelling;

You are expected to demonstrate technical writing skills and address the requirements of the assignment in a clear and concise form within the available space. The recommended structure of report is: title, author details, aim and objectives, key theory background, main body of report (based on the outline of CW), concluding remarks, and references.

The Problem Outline

Beams subjected to bending moments are a very common component in the majority of engineering structures; from a simple static beam in a building construction, to a car main frame (chassis) or a component in the wing structure of an aeroplane.

Figure 1(a) shows a beam that is simply supported near its two ends and is subjected to equal transverse loads at equal distances from the supports. This arrangement represents a so called "Four-Point Bend" configuration. Using principles of static equilibrium and obtaining the bending moment distribution it is easy to show that under such loading conditions between the load points the beam will be subjected to pure bending. The beam section details are shown in Figure 1(b).

Geometry: L=L1=1000 mm, h=t=25 mm, b=W=50 mm, L2=200 mm

Material: Steel; Young's Modulus = 200GPa; Yield stress = 400 MPa

Figure 1. Schematic view of a "4-point bend" beam configuration (a); and details of shape and dimensions of its cross section (b)

This assignment aims to study the effects of a range of load levels on the beam subjected to this loading configuration. Figure 2 in the Appendix demonstrates a view of the distribution of "normal to cross section" or "axial" bending stress component (σ) in the beam for a typical load level. The model shown in Figure 2 has been produced using ABAQUS/CAE, a state of the art FEA software although you may use any FEA tool of your choice to analyse the problem.

Layout of coursework part A (analytical solution)
1. Assuming elastic material behaviour, use simple bending theory to calculate the load corresponding to a safety factor of 1.5 based on von Mises failure criterion. Also find the maximum deflection and maximum slope in the beam under this load level and specify their corresponding loci on the beam. Discuss and comment on results. Comment on distributions of tensile and compressive bending stresses. Repeat the calculation to find the load corresponding to the elastic limit (onset of plasticity!). You will need to locate the neutral plane for bending stress and work out the second moment of area for the cross section with respect to the neutral axis on the beam section.

2. Assuming elastic-perfectly plastic material behaviour find the load that will result in the full extension of plasticity in the beam section anywhere between the load points. Describe what happens to the beam at this stage. Find the ratio of the load (or bending moment) causing plastic collapse to the load (or bending moment) corresponding to the elastic limit (onset of plasticity). This is called shape factor. How does this shape factor compare with the shape factor for a beam with rectangular cross section? Discuss on the basis of potential capacity of beams to withstand overloads without collapse! To answer this section you will need to identify the initial neutral plane (i.e. up to the elastic limit) and also locate the final position of the neutral plane, i.e. when the whole section has become plastic (collapse).

3. Clearly describe the stages of progression of plasticity as the load is increasing from the elastic limit (the onset of plasticity) load, MY, to the plastic limit (fully extended plasticity; collapse) load or limit load, ML. Present graphically the distribution of bending stress in beam section (axial stress, normal to beam section) at all distinct stages of plasticity progression across the beam section, including 1) the elastic limit, 2) plasticity just initiated at the at the bottom edge (M2), and finally 3) at the limit (collapse) state. Calculate stress profile and the position of the neutral axis (e.g. its distance from bottom edge) for the above stages throughout the progression of plasticity.

Layout of coursework, Part B (finite element analysis)

1. Clearly and concisely introduce the problem, the aim and objectives. Discuss the potential modelling approaches and justify your chosen approach. Briefly describe the procedures you followed to produce the finite element model and provide reasons for decisions made. Comment on the refinement approach in order to achieve sufficient accuracy (sensitivity analysis scheme). Run a trial analyses and obtain a working model, discuss the results and comment on how the model may be improved.

2. From your trial model create and introduce details of an appropriate FE model and run a full analysis using elastic material properties. Identify the load and its corresponding moment, Md that represents a safe design by a safety factor of 1.5. Also identify the load that takes the beam to the onset of plasticity and work out its corresponding moment, MY, that represents the elastic limit state. Extract axial and von Mises stress data and plot their distributions (across the height, h) on the beam cross section at the mid span (that is under pure bending!) at the elastic limit load, MY. compare your findings with the analytical solution of part A to verify the results.

- Submit your reports online in word format using the link provided on module site
- Your model file used for part B should also be submitted.
- Maximum word count is 2000 (excluding cover page, list of contents, and references)
- Use A4, Times New Roman font, size 12, justified, with single line spacing.

Attachment:- Appendix.pdf