Running head: STRUCTURAL DESIGN SAMPLE WORK
Structural Design Design of a Shaft (STATIC ANALYSIS)
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Table of Contents Aims....................................................................................................................................................... 3 Problem................................................................................................................................................. 3 Figure 1. Layshaft (Marshall Brian) Requirements of the project ........................................................... 3 Methodology......................................................................................................................................... 4 Analysis.................................................................................................................................................. 5 Conclusions.......................................................................................................................................... 11 References........................................................................................................................................... 12
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Aims The main goal of this project is to design a shaft which undergoes torsional loading. Problem The issue is to design a shaft of any mechanical system where the torque and rotation are to be transmitted with the most appropriate cross-section shape (hollow or solid) with the minimum weight. A picture of a layshaft with three gears which transmit the torque to the shaft is shown below:
Figure 1. Layshaft (Marshall Brian) Requirements of the project The current project has some basic requirements: - The values of a torque which is transmitted by gears on the shaft are as follows: T1=5kN*m; T2=7kN*m; T3 should be found.
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- The material of a shaft should be steel with these material properties: allowable stress: Ďƒy=160MPa; the maximum torsional deflection of the shaft: θ=0,8degree/m. - The cross-section should be round (hollow or solid). - The dimensions and shape of the cross-section should be chosen with relation to the lowest weight. Methodology The next steps should be completed in order to design a shaft: 1. Evaluation of inner forces; 2. Calculation of the allowable tangent stress in accordance with the third theory of strength. 3. Evaluation of the minimum values of the moment of inertia and the moment of resistance. 4. Calculation of the cross-section diameter for solid and hollow sections from the strength point of view. 5. Calculation of the cross-section diameter for solid and hollow sections from the stiffness point of view. 6. Selection of the diameter dimension from the standard shapes. 7. Evaluation of torsional deflection of the shaft for solid and hollow sections. 8. Diagrams of torque and torsional deflection of the shaft with the hollow and solid cross-sections.
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9. Comparison of the solid and hollow cross-sections with respect to weight. Analysis Step 1. Evaluation of inner forces The free body diagram is presented in the picture below:
Figure 2. Free-body diagram of a shaft. The shaft is considered to be in equilibrium. Therefore, the value of the outer torque (T3) can be found from the equation of equilibrium below:
In order to accomplish the static analysis of the most loaded sections, these sections should be found. The torque diagram of the torque distribution through the shaft length is presented in Figure 3 below:
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Figure 3. Diagram of the torque distribution along the shafts can be seen from Figure 3. Above the most loaded part of the shaft is the second one where the torque equals to 12 kN*m. Step 2. Calculation of the allowable tangent stress in accordance with the third theory of strength In accordance with the third theory of strength, the value of allowable tangent stress is connected with the value of a normal allowable stress as follows:
Step 3. Evaluation of the minimum values of moment of inertia and moment of resistance
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STRUCTURAL DESIGN SAMPLE WORK
The minimum values of the moment of inertia and moment of resistance can be found from the strength and stiffness conditions.
Step 4. Calculation of cross-section diameter for solid and hollow sections from the strength point of view The subsequent formulas of the moment of resistance for hollow and solid circle crosssections will be used to determine the diameters of cross-sections:
;
; Step 5. Calculation of cross-section diameter for solid and hollow sections from the stiffness point of view The following formulas of the moment of inertia can be used in order to calculate the crosssection diameters:
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STRUCTURAL DESIGN SAMPLE WORK
Step 6. Selection of the diameter dimension from the standard shapes From steps 4 and 5, above the minimum allowable diameter of a solid cross-section is:
From steps 4 and 5, above the minimum allowable diameter of a hollow cross-section is:
From the list of standard cross-sections, the closest diameter dimensions are:
- for solid cross-section:
- for hollow cross-section:
;
.
Step 7. Evaluation of torsional deflection of the shaft for solid and hollow sections In order to determine the torsional deflection of the shaft, the moments of inertia should be calculated for diameters, which were obtained in the previous step.
Stiffness of both cross-sections can be calculated as below:
STRUCTURAL DESIGN SAMPLE WORK
The torsional deflection of each part of the shaft: for a solid cross-section:
for a hollow cross-section:
The torsional deflection of the specific cross-sections of the shaft:
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STRUCTURAL DESIGN SAMPLE WORK 10 Step 8. Diagrams of torque and torsional deflection of the shafts (hollow and solid)
Step 9. Comparison of the solid and hollow cross-sections with respect to weight The weight of the shaft can be obtained from the equation below:
where A is a cross-sectional area. l is a length of the shaft and Ď is the density of the material. The cross-sectional areas are:
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The density and the length are equal on both cases, therefore the weight difference in hollow and solid cross-sections are dependent only within cross-sectional areas. Therefore, the bigger the cross-section, the higher weight obtained. From this observation, the hollow cross-section would be more appropriate than the solid cross-section. Conclusions The diameters of both solid and hollow cross-sections were obtained and are: D(solid)=105 mm and D(hollow)=120mm. Deformations in each section of the shaft were found and diagrams of torque and torsional deflection of the shafts were plotted. As was obtained, stiffness of both shafts is almost equaled. The direct connection between the cross-section area and the weight of the shaft was obtained. Therefore the best cross-section shape appeared to be a hollow tube with an outer diameter of 120 mm, which meets all requirements of the project.
STRUCTURAL DESIGN SAMPLE WORK 12 References Marshall Brian. “How Manual Transmissions Work”. Howstuffworks. Internet resource: http://auto.howstuffworks.com/transmission2.htm, Web: 25.12.2015
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