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This study covers optimization of I-sectional flange beams. Scope of this study is limited to medium weight flange beams of Table 1 of IS 808:1983 but it can be further extended for the other sections of this code. Best possible geometric shape of the cross-section is found for maximum performance of the beam with minimum material consumption. All possible loading conditions are considered in the study for which a beam in flexure undergoes in its life. ANSYS software program is used for the analysis and optimizing the sections. It is found that sections MB 125, MB 300 and MB 400 of Table 1 of IS 808 are not the optimum sections but other alternative of these cross-sections is available which within the same material consumption performs better than these sections of IS code.

Indian standard code of practice IS 808: 1989 was last incorporated for its amendments in 2002. This standard covers the nominal dimensions, mass and sectional properties of hot rolled sloping flange beam and column sections, sloping and parallel flange channel sections and equal and unequal leg angle sections. These sections are used by the designers and manufactures countrywide. Scope of this study covers studying the performance of medium weight I-section flange beams specified in

Designation | Mass, M | Sectional Area, A | Dimensions | Sectional Properties | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D | B | t | T | Flange Slope, Max α, | R_{1} | R_{2} | I_{x} | I_{y} | r_{x} | r_{y} | Z_{x} | Z_{y} | |||

Kg/m | cm^{2} | mm | mm | mm | mm | deg | mm | mm | cm^{4} | cm^{4} | cm | cm | cm^{3} | cm^{3} | |

(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14) | (15) | (16) |

MB 100 | 8.9 | 11.4 | 100 | 50 | 4.7 | 7/0 | 98.0 | 9.0 | 4.5 | 183 | 12.9 | 4.00 | 1.05 | 36.6 | 5.16 |

MB 125 | 13.3 | 17.0 | 125 | 70 | 5.0 | 8.0 | 98.0 | 9.0 | 4.5 | 445 | 38.5 | 5.16 | 1.51 | 71.2 | 11.2 |

MB 150 | 15.0 | 19.1 | 150 | 75 | 5.0 | 8.0 | 98.0 | 9.0 | 4.5 | 718 | 46.8 | 6.13 | 1.57 | 95.7 | 12.5 |

MB 175 | 19.6 | 25.0 | 175 | 85 | 5.8 | 9.0 | 98.0 | 10.0 | 5.0 | 1260 | 76.7 | 7.13 | 1.75 | 144 | 18.0 |

MB 200 | 24.2 | 30.8 | 200 | 100 | 5.7 | 10.0 | 98.0 | 11.0 | 5.5 | 2120 | 137 | 8.29 | 2.11 | 212 | 27.4 |

MB 225 | 31.1 | 39.7 | 225 | 110 | 6.5 | 11.8 | 98.0 | 12.0 | 6.0 | 3440 | 218 | 9.31 | 2.34 | 306 | 39.7 |

MB 250 | 37.3 | 47.5 | 250 | 125 | 6.9 | 12.5 | 98.0 | 13.0 | 6.5 | 5130 | 335 | 10.4 | 2.65 | 410 | 53.5 |

MB 300 | 46.0 | 58.6 | 300 | 140 | 7.7 | 13.1 | 98.0 | 14.0 | 7.0 | 8990 | 486 | 12.4 | 2.86 | 599 | 69.5 |

MB 350 | 52.4 | 66.7 | 350 | 140 | 8.1 | 14.2 | 98.0 | 14.0 | 7.0 | 13,600 | 538 | 14.3 | 2.84 | 779 | 76.8 |

MB 400 | 61.5 | 78.4 | 400 | 140 | 8.9 | 16.0 | 98.0 | 14.0 | 7.0 | 20,500 | 622 | 16.2 | 2.82 | 1020 | 88.9 |

MB 450 | 72.4 | 92.2 | 450 | 150 | 9.4 | 17.4 | 98.0 | 15.0 | 7.5 | 30,400 | 834 | 18.2 | 3.01 | 1350 | 111 |

MB 500 | 86.9 | 111 | 500 | 180 | 10.2 | 17.2 | 98.0 | 17.0 | 8.5 | 45,200 | 1370 | 20.2 | 3.52 | 1810 | 152 |

MB 550 | 104 | 132 | 550 | 190 | 11.2 | 19.3 | 98.0 | 18.0 | 9.0 | 64,900 | 1830 | 22.2 | 3.73 | 2360 | 193 |

MB 600 | 123 | 156 | 600 | 210 | 12.0 | 20.3 | 98.0 | 20.0 | 10.0 | 91,800 | 2650 | 24.2 | 4.12 | 3060 | 252 |

this code.

Since 2002, there have been tremendous improvements in the computation capabilities of software programs with high performance hardware system. Looking close to the sections of

Engineers have always tried to optimize the structural design for material saving perspective and cost reduction. Optimization can be done with respect to size, shape and topology of the structure. The most effective and efficient tool for optimization is topological optimization if loading and boundary conditions are fixed [

In 2009, K. Ghabraie [

In this paper, shape optimization is performed on I-sections flange beams. Performance of the beam is studied by changing any particular sectional dimension (

For a particular beam section, there should definitely be a unique combination of sectional parameters (_{1}), thickness of web (t), flange thickness (T) and flange slope (α) for which section gives maximum performance for applied loadings with minimum material consumption. It may also possible that for more than one combination of these parameters, section performs optimum.

For studying these realistically, let us discuss first the variation of stresses within the section as the beam is subjected to bending. For designing optimum section in flexure, beam should also be considered for torsion in case of any eccentric loading.

ing stresses across the cross-section is also shown in this figure. With the strength of material practice, it is found that magnitude of the maximum bending stress either at top or bottom layer of the cross-section is 8 to 11 times higher than that of shearing stress induces at the neutral axis [

Talking about the torsional stresses, these are zero at the centre of the cross-section then varies linearly and finally becoming maximum at the outer layer. Variation of bending stress along the cross-section is linear whereas that of shearing stresses is in a curved parabolic path as shown in

Studying the numerical values of stresses along the cross section, most robust section can be decided on the basis of that best fits in compensating these stresses.

In the alternative section, radius of fillet (R_{1}) is increased, depth is decreased and flange width is increased a bit. These all changes are made with an understanding of variation of stresses within the section. Finalizing any one of them as the best is not possible until unless performance of the beam is not known as each of these input parameter changes within a section.

ANSYS is the best tool for resolving this kind of optimization problems. Beam performance is studied by varying each of these input parameters and conclusions are made for the sections of

Geometry shown in _{2} (

Strategies are used for selecting best output parameter that optimizes the section while consuming minimum analysis time. Different simulations are tried using output parameters such as area of cross-section, different stress, different strains, material volume and strain energy etc. Three output parameters mass of the beam, safety factor and total deformation are found to be capable enough for optimizing these sections. Numerically, program

calculates safety factor from stresses induced in the structure. So, theoretically, stresses and strains (deformation) are enough to measure the performance of the beam. Mass of the beam is also opted as output parameter as it has to be minimized. Each model is subjected to all possible loading conditions which it can undergo in its lifetime and are depicted in

With several trials of meshing, element shape and size is selected which converges with the theoretical results. Shape and size of the element is adopted such that within minimum analysis time it best converges to theoretical results.

Geometry of the structure in the present study possesses curved boundaries. Hence tetrahedral shape of the elements is adopted for the analysis to avoid generation of distorted and badly shaped elements by warping or skewing [

Statistics shown in this figure are found to be in the approximate match for the other sections as well using these types of elements.

In this figure, aspect ratio of elements does not increase beyond 8.86 and the average value is 2.163. For the same element size and same geometric section if hexahedral elements were used than aspect ratio goes till 1136 with an average of 7.69. So with these statistics, using tetrahedral elements for analysis is preferable.

Another reason for adopting tetrahedral elements is observed in the convergence results of

figure, a smooth convergence of tetrahedral elements after 0.02 m element size is found which is not same in case of hexahedral elements. Smooth convergence again supports its use in the analysis.

Three dimensional, 10-node solid tetrahedral shape element that exhibits quadratic displacement behaviour is used for the analysis (

Although, using hexahedral elements when used with small elements size gives reasonable results but it in turn increases the running time of analysis because of increased degrees of freedom.

Furthermore, results of error estimation [

Total deflection of the beam is calculated considering both effects bending as well as shear. As for all sections, beam length is chosen to be 1 m. So it is also possible that for increasing sectional dimensions, in case of deep sections shear deformations become high. So, for validation of results, following Timoshenko beam equation [

where M and q is moment and shear force at any cross-section, E is modulus of elasticity of material, G is shearing modules, A and I are the area and moment of inertia of the cross-section and k is the shear constant which depends upon the shape of beam cross-section. For I-section shear constant is calculated as under [

In this equation, all other parameters are as shown in

Solving the equation for a cantilever beam loaded with uniformly distributed load, deflection at the free end comes out to be.

where w is the magnitude of uniformly distributed load and l is the length of the beam. Putting all the numerical values of these parameters, total deflection of beam at free end was calculated for validation of result.

This section describes the behavior of the beam as the sectional dimensions vary in a section. Each parameter i.e. depth, radius of fillet, thickness of web, flange thickness, flange slope and width of the section, is varied by keeping all other parameters constant to understand the behavior of section thoroughly.

Studying these variations within a section helps in deciding a particular value of it for maximum performance hence optimum design.

Following results (Figures 9-11) were obtained with a section of flange width 140 mm. All other parameters were kept constant and depth is varied from 100 mm to 650 mm.

With the results of

Important observations can be seen with the graph of

In the performance graph, it can be observed that the limit of this depth is around 300 mm which is approximately double the width of the section.

Safely factor increases linearly as the depth of the section increases (

If web thickness of the section is increased mass of the section increases linearly (

So, suitably selecting web thickness optimizes the section for maximum performance.

Increasing radius of fillet increases the mass of the section in a curved path with the increasing gradients (

Using very small radius of fillet is not suitable for designing the section (

In the following Figures 18-20, on the horizontal axis, numerical values are for B/4 (

Performance of the section with both the parameters, flange width and flange thickness, is almost same and can also be compared with the graphs of Figures 21-23.

In the graphs of Figures 19-23, it can be observed that with increasing the width and depth of the flange improves the performance of the section with increasing gradient whereby increasing the mass linearly (

So increasing the flange width or thickness of the section is the effective way of increasing the performance of the section. It increases the mass in a linear path whereby increases the performance of the section with higher gradients.

Astonishing results are found for any section when flange slope changes within the geometric limits. In

This is the only parameter which is found like increasing its value decreases the mass whereby improving the performance of the section with high gradients. It improves the performance of the section without increasing the mass proportionally.

With the results of safety factor of

For any arbitrary loading, following results are found out for IS sections as well as for optimally suggested sections. It can be observed in the this

S.N. | Sections | D (mm) | B (mm) | t (mm) | α (deg) | T (mm) | R_{1} (mm) | R_{2} (mm) | Mass (kg/m) | S.F | T.M.D. (mm) |
---|---|---|---|---|---|---|---|---|---|---|---|

1. | MB 125 | 125 | 70 | 5 | 98 | 8 | 9 | 4.5 | 13.086 | 3.0541 | 1.934 |

Optimal Section | 125 | 60 | 6 | 98 | 8.5 | 9 | 6.4 | 13.045 | 3.7034 | 1.812 | |

2. | MB 300 | 300 | 140 | 7.7 | 98 | 13.1 | 14 | 7 | 45.3 | 2.5854 | 2.1137 |

Optimal Section | 300 | 140 | 7 | 97 | 13.75 | 15 | 10.7 | 45.296 | 2.6381 | 2.0023 | |

3. | MB 400 | 400 | 140 | 8.9 | 98 | 16 | 14 | 7 | 60.396 | 2.1315 | 2.6412 |

Optimal Section | 400 | 148 | 8.8 | 95 | 15 | 15 | 12.9 | 60.323 | 2.3168 | 2.3861 |

For better validation of these results, following structural errors are found in discretization of the model geometry (

・ Section MB 125 should be changed for the parameters, preferably flange width (B) and flange thickness (T) so that the safety factor of the section can be increased.

・ Section MB 400 should be amended for the sectional dimension, preferably for flange width (B), web thickness (t) and flange slope (α) so that the total deformation of the section can be reduced.

・ Section MB 300 should be amended such that deformation as well as the safety factor can be improved and the suggestions are web thickness (t), flange thickness (T) and flange slope (α).

Except these three sections of IS 808,

Further, this study can be performed for the other sections of IS 808 for the better performance of the sections.

Himanshu Gaur,Krishna Murari,Biswajit Acharya, (2016) Optimization of Sectional Dimensions of I-Section Flange Beams and Recommendations for IS 808: 1989. Open Journal of Civil Engineering,06,295-313. doi: 10.4236/ojce.2016.62025