Structural Design
When design engineers consider the structural design of a new project, an initial thought should be what materials to use. Sometimes the project may decide the type of material to use. If weight is important such as aircraft design then it might be appropriate to select aluminium or plastic. Structures requiring strength, hardness, or if cost is a consideration then selecting steel might be the most appropriate material. The design of any structure involves methodical investigation of materials proposed. This choice of materials will depend on the stability, strength, and rigidity required of the structure. Also, the selected material must be able to carry the load or withstand any stress loading without distortion or fracture. While, a well engineered design will reduce any risk of costly failure.
Stress and Strain
Stress and Strain are two words that anyone involved in engineering design work will need to understand. More importantly, they will need to know how to calculate the stress on the design structure. This is an engineering discipline that would take many webpages to explain in detail. There are already many books and websites specializing in this subject. As these texts explain the subject in great detail, I cannot improve on this material. Therefore, all I will do is provide samples of how to use stress calculations.
Finite Element Analysis
The development of Engineering computer software has seen the introduction of Finite Element Analysis and Static stress analysis. The introduction of 3D computer aided design software has also allowed this. Modern plastic material can also create curves and other complex, irregular shapes. This has made it extremely difficult to carry out manual stress analysis without the aid of finite element analysis. I have provided links below to some websites for anyone who wishes to learn more about this subject.
Image of The Forth Rail Bridge by dassel from Pixabay
Mechanical Properties
To do any type of stress analysis on materials, requires the mechanical properties of the materials. Generally, the material manufacturers can provide this type of information. This might include cross-section sizes, tensile and compressive strength or shear strength. The recognised equations for estimating the stress and strain under load use these data. To obtain a value of both stress and strain, use the following equations below: .
The following shows a stress strain curve produced when a material is subject to a tensile test. The Yield Point shows when the stress on the material causes it to deform. This identifies the end of the elastic behaviour and the beginning of plastic behaviour of a material. Some materials undergo a sudden extension without an increase in load. To calculate the yield stress at this point, divide the load at yield by the original cross-sectional area. In the elastic range, the material returns to its original length. If the material passes the yield point, it enters the plastic range. In this range, the material undergoes permanent extension. Once it passes the point of maximum tensile strength, it continues to extend with a reducing load. The stress increases up until the material fractures.
Design Codes
Scotland, like many countries, has Building Standards providing guidance to ensure good standards under the Building (Scotland) Regulations 2004. Also, the Scottish Government issues technical handbooks to assist with achieving these standards. Certainly, engineers designing structural systems in Scotland need to be familiar with these standards. The design codes in the technical handbooks reference the European Standards for structure (Structural Eurocodes). Any new structure or building designs must comply with the details set out in the Eurocodes. The table below lists the Eurocodes:
| Eurocodes | Details |
| Basis of structural design | |
| 1 | Actions on structures |
| 2 | Design of concrete structures |
| 3 | Design of steel structures |
| 4 | Design of composite steel and concrete structures |
| 5 | Design of timber structures |
| 6 | Design of masonry structures |
| 7 | Geotechnical design |
| 8 | Design of structures for earthquake resistance |
| 9 | Design of aluminium structures |
Worked Example
The following is an example of how to find the maximum bending stress on a specified beam:
It is based on a British Steel section serial size 356 x 171 x 67 Universal beam. This has a mass of 67.1 kg per metre. The drawing opposite shows the cross-section of this Universal beam (UB).
Properties of the material
Young’s modulus of elasticity indicates how easily a material such as in the UB could bend or stretch. Dividing the maximum stress by the strain, produces the Young’s modulus and indicates the stiffness of the material. For this example, the choice selected is a Young’s modulus of E = 210 GPa.
The Yield Strength indicates the elastic limit of the material. If the stress exceeds the Yield Strength, then the material will deform permanently. If the stress applied does not exceed the Yield Strength, then the material will return to its original shape. For this example, the chosen Yield Strength used is σy = 250 Mpa. A Factor of Safety (FOS) of 5 applied to the calculations as a safeguard.
The length of the beam is L = 2.0 metres long, with a support at each end. A load applied of F = 800.7 kN to the centre of the beam, as shown in the sketch:
Calculations
To obtain the maximum bending stress requires using all the equations below. In order to obtain the correct units, some equations have had the formula slightly changed.
The cross-sectional area of the universal beam allows estimation of normal stress. Also, to allow estimation of the maximum bending moment, requires estimation of the second moment of area on the x-axis. Additionally, known as the second moment of inertia and determines the shape’s ability to resist bending when loaded. Dividing the yield strength by the factor of safety determines the maximum allowable bending stress. The equations also obtain the distance of the point load from each of the supports of the universal beam. Finally, the equations also obtain the distance required between the point load and neutral axis. This will be the final equation to determine the maximum bending stress on the beam. The following equations demonstrate how to obtain the above values.
Equations
Cross-sectional area of UB section:
Second moment of area on x-x axis:
Distance of point load from neutral axis:
Bending Moments
Maximum bending moment of beam at point load:
Maximum allowable bending stress:
Maximum bending stress on the beam:
Stress and Strain on the beam
Normal stress on the beam:
Strain on the beam:
Deflection and Extension of the beam
Maximum deflection of beam:
Maximum extension of beam:
The results show that the maximum bending stress is greater than the yield strength. This suggests that the beam would fail in this situation.
Online Files
NOTE:
The blank rectangles in the sheets above are the diagrams showing the longitudinal section of the beams. To enable this feature, download the SMath Studio Desktop and install the Plot Region plugin.
Click to download file – This sheet provides how to calculate the stress, bending moment, strain, maximum deflection and maximum extension of a Circular Hollow Section. The CHS is supported at both ends and has a point load applied.
Click to download file – This sheet demonstrates how to calculate the stress, bending moment, strain, maximum deflection and maximum extension of a Universal beam. The UB is supported at both ends and has a point load applied.
Click to download file – This sheet demonstrates how to calculate the stress, bending moment, strain, maximum deflection and maximum extension of a Universal beam. The UB is supported at both ends and has two equidistant point loads applied.
Click to download file – This sheet demonstrates how to calculate the stress, bending moment, strain, maximum deflection and maximum extension of a Universal beam. The UB is supported at both ends with distributed loading applied.
NOTE:
The blank rectangles in the sheets above are the diagrams showing the longitudinal section of the beams. To enable this feature, download the SMath Studio Desktop and install the Plot Region plugin.
PDF Files
Click to open PDF file – This sheet provides how to calculate the stress, bending moment, strain, maximum deflection and maximum extension of a Circular Hollow Section. The CHS is supported at both ends and has a point load applied.
Click to open PDF file – This sheet demonstrates how to calculate the stress, bending moment, strain, maximum deflection and maximum extension of a Universal beam. The UB is supported at both ends and has a point load applied.
Click to open PDF file – This sheet demonstrates how to calculate the stress, bending moment, strain, maximum deflection and maximum extension of a Universal beam. The UB is supported at both ends and has two equidistant point loads applied.
Click to open PDF file – This sheet demonstrates how to calculate the stress, bending moment, strain, maximum deflection and maximum extension of a Universal beam. The UB is supported at both ends with distributed loading applied.
Downloadable Zip File
Click to download file – This ZIP document contains four SMath live documents for calculating a Circular Hollow Section and Universal Beams with different point loading. The sheets demonstrate how to calculate stress, bending moments, strain, maximum deflection and maximum extension when different loads are applied.
