Beams : A section geometry study

INTRODUCTION

Beam is the long strutural member with comparatively smaller cross sectional dimensions generally designed to take up lateral loads acting on the section. It finds application by supporting floors and ceilings of the buildings. However, lathe beds, shaft supported on bearings can also be treated as the beams.

Whenever an external load acts on the struture, a resistance is developed to the change in shape of the body, this intensity of internally distributed forces is known as the stress. The externally applied load would cause change in the geometry, known as the deformation, when this deformation is taken against the original geometry, it is referred to as the Strain in the struture. Stress and strain are linerally related within the proportionality limits. This is termed as the elastic zone, in this zone, material can recover without undergoing permanent deformation. The maximum stress, beyond which the permanent deformation sets in is referred to as the yield stress. The proportionality limit is a function of the material of the structure. Similarly, stress is a function of magnitude of load, nature of loading, geometrical dimensions. Hence, the stress behaviour in the material and its design is deided by the material, load magnitude, nature of loading and the geometry of the structure.

Various failure theories such as Rankine theory, Guest theory, Strain energy theory etc are used to determine the failure stress, if the material is designed such that the maximum stress in the structure is less than the failure stress, material is said to be safe. However, material variabilities, loading variabilities, loading and application demand some amount of safety margin. Factor of safety is hence considered to limit the maximum stress even more. However, this would mean maximum material condition, hence material cannot be used as the cost and weight might pitch in, so designing is essentially optimisation between choosing factor of safety and right material weight and associated cost. Both underdesign and overdesign is detrimental.

With the above theory, in this study, beams geometry is varied and the maximum stress in the structure is mapped.

In this study, Load magnitude, Nature of loading, Material is kept constant. Study is done across Rectangular beam , I section, U section

FEA SIMULATION

Geometry: The length of the beam is maintained at 400mm for all the studies, cross section is varied as shown

Case 1: Rectangular Beam

Case 2: I Section

Case 3: U Section

MESHING AND INITIALISATION

Standard Meshing is used in all the three cases.

As shown in the figure above, one side of the structure is fixed, ( A fixed structure is one which has no vertical, horizontal or rotation) In two separate studies, 1500N of load is applied as UDL and as point load, stress response is noted in all the three cases. Loading is done on the minimum area

RESULTS

Case 1:

Case 2: I Section

CASE 3: U section

DISCUSSION ON RESULTS

1) The summary of the maximum stress in MPa for two kinds of loading in three cases is as below

Loading\Section	Rectangle	I Section	U Section
UDL	7.531	21.551	38.886
Pointed load	23.552	75.685	73.173

2) The material chosen is alloy steel whose yield stress is 620.422 MPa, for ductile materials Yield stress is a failure stress, hence the factor of safety for the above table is as below

Loading\Section	Rectangle	I Section	U Section
UDL	82.38	28.788	15.95
Pointed load	26.34	8.1974	8.47

3) From the above table, following inferences are drawn,

In applications that requires very high FOS where the loading is absolutely unknown, like in the buildings, Rectangular section could be used, however, it falls in the boundary of Overdesign

In applications where the Cost of the material is important, mass is vital, U Section is ideal

In applications where the forces are relatively well known, material cost is also important, like in Connecting rod of an automobile application, I Section is ideal

However, In general sense, if the material is alloy steel, load is 1500N, Rectangular section seems over design,

Comparing Approx areas of section

Rectangle = 3200mm2

I section = 1116 mm2

U Section = 1070mm2

So from the above consideration, I section seems to be ideal, ( The other reason being the stiffess, it induces lesser strain with minimum material condition)

In more technical terms, for the bending load, Section modulus is calculated which is the Area moment of inertia to distance to extreme fiber, and from this Moment of resistance is higher for the I section, hence I section is used to resist bending loads.