Design of a Beam Structure for Failure Prevention at Critical Loading Conditions

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1 International Academic Institute for Science and Technology International Academic Journal of Innovative Research Vol. 3, No. 10, 2016, pp ISSN X International Academic Journal of Innovative Research Design of a Beam Structure for Failure Prevention at Critical Loading Conditions Owunna Ikechukwu, Ikpe Aniekan E. *, Satope Paul Mechanical Engineering Department, Coventry University, UK. Abstract To avoid failure in most engineering designs, von-mises failure criterion must be met. This criterion requires that the stress a body is subjected to is less than the yield strength of the material of the body. In this analysis, a beam structure has been designed with CATIA software to ensure that the von-mises criteria is met. This was done with critical evaluation of the mass requirement and the pivot constraints on the body. The first load case considers a total force of 18kN while the second load case considers a force of 9kN. The final weight of the initial model was kg. The maximum von-mises stress obtained from the initial analysis for load case 1 was 1560MPa whereas, the maximum von-mises stress obtained from the initial analysis for load case 2 was 530MPa. For the final design analysis, weight of the final model was kg. The maximum von-mises stress obtained from the final analysis for load case 1 was 102MPa whereas, the maximum von-mises stress obtained from the final analysis for load case 2 was 101MPa. Hence, one of the methods that failure can be avoided in beam design is to ensure that the maximum allowable stress is not exceeded. Keywords: Beam Structure, Design, Deformation, Weight, Stress, Applied Load. 32

2 Introduction: A beam is a structural member that is horizontally positioned between vertically supporting member (column) in such a manner that carries load acting at right angle to the length of the beam. A beam is generally used for resisting loading reactions resulting from shear forces, vertical loads, and bending moments (Sankararaj, 2013). A typical beam has a small cross-section compared to its span, for example, the width and depth of a given beam is less than span/10. In practice, a beam is generally known to be subjected to two types of external forces which includes external loads acting on the beam as well as reactions to the load from supporting members. Moreover, a beam is also known to be subjected to two types of internal forces such as bending moments and shear forces, and both the internal shear forces as well as the internal bending moment can be considered as pairs of forces (Nilantha, 2015). Generally, applied load is usually responsible for bending moment which results in bending stresses on the beam structure and consequently tensile and compressive stresses. Under positive moment as oftentimes the case, tensile stresses are produced underneath the beam whereas, compressive stresses are found on top of the beam and it is important for beam structures to resist both tensile and compressive stress in beam related applications (Frederick and Dominic, 2009; Bansal, 2010). The classification of beam structures can be done based on three major criteria which includes geometry such as beam straight profile, curved profile, tapered cross section, shape of cross section (for example, beam with I cross section, T cross section, C cross section etc.). The second criteria in which the classification of a beam structure is based includes equilibrium conditions such as statically determinate beam and statically indeterminate beam whereas, the third criteria for beam classification includes the type of support such as Simply supported beam, Cantilever beam, Overhanging beam, Continuous beam and Fixed beam respectively (Erasmo et al., 2011; Chennu, 2015). Solving engineering problems could be time consuming basically because there is always need to meet different design conditions. To achieve such solutions engineers use best design practices. In cases where weight is important practices like cut outs are fundamental in removing wastage. The dilemma of finding engineering solutions is not the unavailability of innovative designs, the major challenge is making designs that can be easily manufactured. In doing this, engineers use mostly standard parts obtainable from the market and then remove weight where possible. The customer requirement for the part contained in this analysis requires that the product meets two different load cases. To ensure that this is achieved in this paper, the analysis of the load cases was carried out in CATIA software with special emphasis on the mesh used in performing the analysis. The mesh size that can provide accuracy with small simulation time was employed in the analysis. By using 2D meshing, the critical points in the design would be determined easily and reinforcements provided where possible. Different kinds of weight reduction techniques will be utilized in ensuring that the weight limit for this product is not exceeded or the minimum possible. Analysis of Initial Load Case For components with thickness of less than or equal to 10mm, the mid surface of the component is often created and meshed (Bastien, 2013), and such principle is applied in this analysis. The mid surface of the beam was designed in CATIA. The mid surfaces created for both the lug and the hollow beam was meshed with 2D elements of maximum element length of 8mm as shown in Figure 1. To reduce stress concentrations within regions in the model, the same mesh size was applied on all parts of 33

The beam is provided with a pivot at the holes. The pivot at the holes retrains the body in the Z and Y axes but not the X-axes, some form of rotation in the Y axes is also permitted.

3 the assembly. The mesh size was however carefully selected after a thorough mesh convergence study was carried out on the initial design. Figure 1: 2D mesh generated for initial analysis The beam is provided with clamping holes at the edges about 40mm from the holes on the axes of the beam. The beam is provided with a pivot at the holes. The pivot at the holes retrains the body in the Z and Y axes but not the X-axes, some form of rotation in the Y axes is also permitted. The design problem requires the body to withstand a 4500N force at each of the slug in the first load scenario. The second scenario is a 1250N horizontal force and a 1kN vertical force at each slugs. The first scenario requires a total of 18KN bending force while the second scenario is a total of 9KN torsional load. This forces were exerted on the beam. The first load case situation is shown in Figure 2. 34

4 Figure 2: Load case 1 for the beam analysis The material for the beam was steel which has a density of 7850 kg/m3, Young s modulus, 210GPa, Yield Strength 800MPa and Poisson s ratio of 0.3. The thickness of the pipe for 2D analysis is therefore 3.85mm and this was applied as 2D property on the pipe in the CATIA analysis uniform thickness was applied for the hollow beam, however, the lug plates had a different thickness of 10mm and thus different 2D properties were made for the lugs. The Finite Element Analysis (FEA) was prepared from CATIA and the results for the von-mises plot and the deflections were generated. The final weight of the initial model was kg. The maximum von-mises stress obtained from the analysis is 1560MPa for the initial load case and maximum displacement stood at 24.9mm. The maximum von-mises stress obtained from the analysis is 530MPa for the second load case and maximum displacement stood at 6.53mm. With a linear relationship between maximum stress and displacement reducing either the stress or displacement will reduce the other simultaneously (Borst et al., 2012). Its application in this design therefore is to use such techniques that can result in direct reduction of any of stress or displacement. Design Parameters Structural design plays a vital role in bridges, buildings as well as other complex load bearing infrastructures that require salient design considerations in the design phase in order to achieve the purpose of such design. The following design parameters were taken into consideration while carrying out the beam structure design. Moment of Inertia The moment of inertia of a body determines the resistance of any object to rotational forces or bending forces. The moment of inertia of the body however depends on the axes of reference of the calculation of the moment of inertia (Ashby, 1992). The moment of inertia is often calculated with reference to the central axes of the body in question. For beam structures, the moment of inertia is given as (1) (2) The second moment of area of a beam on the other determines the deflection of the beam under the influence of applied load, and can be expressed as shown in equation (4) Where, h = height of the beam structure b = width of the beam structure (3) (4) 35

To ensure a reduction in the deflection of the body in order to ensure conformity with stress restrictions in the design, the second moment of area must increase.

5 To ensure a reduction in the deflection of the body in order to ensure conformity with stress restrictions in the design, the second moment of area must increase. The moment of inertia of the initial design of the pipe is given in Figure 3. Figure 3: Measure of Moment of Inertia Design Requirement The design problem requires that the factor of safety for the design be 2.0. The beam is made of steel with yield strength of 250MPa. With this F.O.S requirement, the allowable stress in the new beam can be calculated. From the hand calculation, F.O.S for the allowable stress and yield strength shows that the allowable stress for the new design is 147MPa. Factor of Safety and Von-Mises Stress Failure of engineering materials are often judged by using different criterion for analysis of the design based on the type material. The design parameter shown in appendix 3 requires a factor of safety of 1.7. The factor of safety is a ratio of failure load and the design load (Budynas and Nisbett, 2008). For the beam design, to obtain a value of 1.7 for the factor of safety, the allowable stress is calculated as follows: ( ) (5) Therefore, 36

6 ( ) (6) The allowable stress for this design is therefore 147MPa. Elastic Region and Ultimate Tensile Strength. For every material there is a maximum level of stress and strain that the material can withstand, this is referred to as yield level of strain or stress. Before such level, the stress or strain in the body is elastic (the body returns to its initial state after the application of a system of forces or load. Before such level, the stress or strain in a body is elastic (the body returns to its initial state after the application of a system of forces or load). Beyond the yield point the extension in the material is plastic. The yield stress therefore defines the advent of plastic deformation in a material (Tyler, 2007; Bucciarelli, 2009). The design being analysed is required to stay within its elastic region throughout the application of the load. However, this must be considered in the design of a beam to avoid sudden failure when it is operating under extremely high load. The maximum stress a material withstands when subjected to applied load is defined as the material s ultimate tensile strength (σ u ) which is given as; Dividing the applied load at the point of failure by the original cross sectional area of the beam gives the material s ultimate tensile strength. Where, = Applied load at the point of failure = Original cross sectional area of the beam If the beam is in tension under the influence of applied load, equation 3 can be rewritten as, (7) Under this condition, the load (P) is converted to stress by calculation and A o which is the original area of the beam is given as; (8) (9) Where, d o = Original diameter of the beam Final Design Processes The final design processes adopted for the beam design is as follows Design Modification The 25mm holes remained 2200mm distance to each other and the lugs stayed 600mm to the edge of the beam. The shape of the lug were retained. The connecting edges of the bell crank retained their shapes and sizes all through the re-design. This is basically because they are connection points for 37

7 other parts of the system in which the bell crank functions. The design material remained the same throughout the process. Since the product is expected to be either machined or casted, there were no considerations taken for joining the materials. Result The final design of the beam has a weight of kg which is relatively smaller than the maximum permitted. This implies that the beam design is safe due to the significance it would have if applied on structures, as exceeding the maximum permitted weight may result in unforeseen failure. In the final design, the stress level were below the maximum allowable. This also imply that the beam design in this paper is safe, as the higher the stress on a given material, the higher the chances of failure at increasing load. Analysis of Moment of Inertial for final design is shown in Figure 4. The first load case gave a stress of 102MPa and the second load case gave a stress of 101MPa as shown in Figure 5 and 6. The final beam design is shown in Figure 7. The overall factor of safety for the new design is To further modify the beam without increasing the length, the sectional modulus of the beam may be employed. This can be used in subsequent design of the beam for further studies of the load case. Appendix 1 and 2 represents failed beam design with maximum stress exceeding allowable design stress. It is obvious that the failed design shown in appendix 1 and 2 deformed slightly due to the effect of applied load, which may have resulted from the exceedingly high maximum stresses of about 745 and 344 MPa. Also, comparing the beam design shapes in appendix 1 and 2 to load case 1 and 2 in Figure 5 and 6, it can be concluded that any design operating below the maximum allowable stress will exhibit more threats of stability and longevity compared to a given design operating above the maximum allowable stress. 38

8 Figure 4: Analysis of Moment of Inertial for final design 39

9 Figure 5: Von-Mises of Beam for load case 1 Figure 6: Von-Mises of beam for load case 2 40

10 Figure 7: Final design of the beam Costing The cost of the material for the beam can be estimated using the average pipe of a steel beam of that size. The material cost for the B-pillar assembly is estimated by referring to CES EduPack software, 2014 version. i. 7.5m of 40mm X 40mm X 3mm hollow pipe is from FH Brundle UK. ii. The modified beam is 3.3m. iii. Hence the material for the beam will cost around The overall cost may be a bit higher that this price because of the cost of producing the slugs, cost of machining the beams and cost of wastages that may be incurred. Transportation may also increase the price of the final design. iv. The total cost of the piece may therefore be estimated at GBP. Manufacturing For the new design, the geometry is simple. The design requires that a hollow square beam be cut to size and then welded with a small 5mm bar on the top surface. Rectangular pocket can then be made on the side to reduce weight of the design. Conclusion An appropriate mesh size of 8mm was used in analysing the load cases required in this paper after adequate mesh convergence studies have been carried out. After thorough analysis of the hollow cylindrical pipe in the initial design, the cross-section was changed to a rather stable cross section. For the new design, a solid rectangular beam was used to reinforce the design at the top end. This reinforcement is of same material with the initial beam and can help the beam against bending load. The pivots were also reinforced to ensure compliance with the second load case. The final design has a mass of kg. 41

All requirements for the new design have therefore been met.

11 The first load case gives a stress of 102MPa while the second load case gives a stress of 101MPa. The design was done with CATIA and the final drafts are included in this paper. All requirements for the new design have therefore been met. Appendices Appendix 1: Failed Beam Design with Maximum stress Relatively Higher Allowable Design Stress Appendix 2: Failed Beam Design with Maximum stress Exceeding Allowable Design Stress 42

Appendix 3: Beam Design Parameters References Bansal, R. K (2010) A Text Book of Strength of Materials, Laxmi Publications, 2010. ISBN: 9788131808146. Bastien C. (2013) FEA in Continuum, Lecture Note.

12 Appendix 3: Beam Design Parameters References Bansal, R. K (2010) A Text Book of Strength of Materials, Laxmi Publications, ISBN: Bastien C. (2013) FEA in Continuum, Lecture Note. Coventry University, Bucciarelli, L. L. (2009) Engineering Mechanics for Structures. Dover Publications, Budynas, R. G. and Nisbett, J. K. (2008) Shigley s Mechanical Engineering Design. Eighth Edition, McGraw-Hill, ISBN: Chennu, V. R. (2015) Different Types of Beams, (ME MECHANICAL, 2015). [online] available from < [ 4 January 2015]. 43

13 D. Borst, R. Crisfield, M. A. Remmers and C.V. Verhoosel, Non-Linear Finite Element Analysis of Solids and Structures, Erasmo, C., Gaetano, G. and Marco, P. (2011) Beam Structures: Classical and Advanced Theories. John Wiley & Sons Ltd, ISBN: Frederick, A. L. and Dominic, J. D. (2009) Strength & Stiffness of Engineering Systems. Springer, Lancaster, M. F. Ashby, M. F. (1992) Materials Selection in Mechanical Design. Oxford: Pergamon Press, Nilantha, P. (2015) Column and Beam system in construction, (Basic Civil Engineering). [online] available from < [2 January 2016]. Sankararaj, S. (2013) Beam-Definition and Types, (Mechteacher, 2013). [online] available from < [2 January 2015]. Tyler, H. G. (2007) Handbook of Civil Engineering Calculations. Mc Graw Hill,

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