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Triangular cross section

Finally, we consider the torsion of a cylinder with equilateral triangle cross section, as shown in figure 9.21. The boundary of this section is defined by,

Triangular Cross Section - Civil Engineering (CE)

where a is a constant and we have simply used the product form of each boundary line equation. Assuming that the Prandtl stress function to be of the form,

Triangular Cross Section - Civil Engineering (CE)

so that the boundary condition (9.50) is satisfied. It can then be verified that the potential given in equation (9.99) satisfies (9.47) if

Triangular Cross Section - Civil Engineering (CE)

Substituting equation (9.99) in (9.57) and using (9.100) we obtain the torque to be

Triangular Cross Section - Civil Engineering (CE)

where the polar moment of inertia for the equilateral triangle section, J = 3 √ 3a4 .

The shear stresses given in equation (9.44) evaluates to

Triangular Cross Section - Civil Engineering (CE)

 Triangular Cross Section - Civil Engineering (CE)

on using equations (9.99) and (9.100). The magnitude of the shear stress at any point is given by,

Triangular Cross Section - Civil Engineering (CE)

Since, for torsion the maximum shear stress occurs at the boundary of the cross section, we investigate the same at the three boundary lines. We begin with the boundary x = a. It is evident from (9.102) that on this boundary σxz = 0. Then, it follows from (9.103) that σyz is maximum at y = 0 and this maximum value is 3aµΩ/2. It can be seen that on the other two boundaries too the maximum shear stress, τmax = 3aµΩ/2. Substituting (9.99) in equations (9.45) and (9.46) and solving the first order differential equations, we obtain the warping displacement as,

Triangular Cross Section - Civil Engineering (CE)

on using the condition that the origin of the coordinate system does not get displaced; a requirement to prevent the body from displacing as a rigid body.

 

Triangular Cross Section - Civil Engineering (CE)

Figure 9.22: Example of a multiply connected cross section. Elliptical cross section with two circular holes.

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FAQs on Triangular Cross Section - Civil Engineering (CE)

1. What is a triangular cross section?
Ans. A triangular cross section refers to a shape that is formed when a solid object is cut perpendicular to its length, resulting in a triangle-shaped slice.
2. How is the area of a triangular cross section calculated?
Ans. The area of a triangular cross section can be calculated by using the formula: Area = 1/2 * base * height, where the base is the length of the triangle's base and the height is the perpendicular distance between the base and the opposite vertex.
3. What are some real-life examples of objects with triangular cross sections?
Ans. Some examples of objects with triangular cross sections include triangular prisms, roof trusses, and pyramids. These shapes can be commonly found in architecture, engineering, and construction.
4. How does a triangular cross section affect the stability of a structure?
Ans. A triangular cross section is known for providing greater stability compared to other shapes. The triangular shape distributes forces evenly, reducing the risk of bending or collapsing. This is why triangular structures, such as trusses, are often used in bridges and buildings.
5. Can a triangular cross section be used for fluid flow applications?
Ans. Yes, a triangular cross section can be utilized for fluid flow applications. Triangular channels or pipes can help maintain a uniform flow rate and minimize pressure drop, making them suitable for certain hydraulic systems or piping designs.
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