### Determining Normal and Shear Stresses in a Pressurized PipeThe pipe shown has an outer diameter of 200 mm and a wall thickness of 10 mm. The length \(a\) is 1600 mm, and length \(b\) is 800 mm (\(b\) is from the center of the pipe to the application of the load). There is an applied load of 25 kN as pictured, and the pipe contains a static fluid pressurized at 2 MPa (this causes normal stresses in both the hoop and long directions).**Objective:** Determine the normal and shear stresses on the side of the pipe at point \(K\), which is coplanar with the yz-plane.**Pipe Cross-Sectional Properties:**\[ A = 12 \cdot 10^3 \, \text{mm}^2 \]\[ Q_{\text{max}} = 180 \cdot 10^3 \, \text{mm}^3 \]\[ J = 54 \cdot 10^6 \, \text{mm}^4 \]\[ I_x = I_y = 27 \cdot 10^6 \, \text{mm}^4 \]**Representation of Stresses:**Present your answers in the following stress matrix in the xyz-coordinate system:\[ [\sigma]_K = \begin{bmatrix} - & - & - \\ - & - & - \\ - & - & - \\\end{bmatrix} \, \text{MPa} \]#### Explanation of the DiagramThe image depicts the pipe along with a coordinate system (x, y, z). The pipe is fixed at one end, and a load of 25 kN is applied downward at the other end. The lengths of the pipe sections, \(a\) and \(b\), are marked.In summary, the problem involves calculating the stress components—both normal and shear—at a specified point \(K\) on the pipe, using the given material properties and boundary conditions. This is to be presented in matrix form for clarity in the xyz-coordinate system.

Answered: Image /qna-images/answer/928b1bb7-1ab9-4ee6-a662-de5aa360161d.jpg

The pipe shown has an outer diameter of 200 mm and a wall thickness of 10 mm. The length a is 1600 mm and length b is 800 mm (b is from the center of the pipe to the application of the load). There is an applied load of 25 kN as pictured and the pipe contains a static fluid pressurized at 2 MPa (this causes normal stresses in both the hoop and long directions). Determine the normal and shear stresses on the side of the pipe at point K which is coplanar with the yz-plane. K Use the following cross-sectional properties: A = 12· 103 mm² J = 54· 106 mm4 Qmax = 180 · 103 mm3 I = Iy = 27· 106 mm4 V 25 kN

The pipe shown has an outer diameter of 200 mm and a wall thickness of 10 mm. The length a is 1600 mm and length b is 800 mm (b is from the center of the pipe to the application of the load). There is an applied load of 25 kN as pictured and the pipe contains a static fluid pressurized at 2 MPa (this causes normal stresses in both the hoop and long directions). Determine the normal and shear stresses on the side of the pipe at point K which is coplanar with the yz-plane. K Use the following cross-sectional properties: A = 12· 103 mm² J = 54· 106 mm4 Qmax = 180 · 103 mm3 I = Iy = 27· 106 mm4 V 25 kN

Elements Of Electromagnetics

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Author:Sadiku, Matthew N. O.

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Concept explainers

Stress transformation

The term stress indicates the measure of internal forces that develop due to an external force's application on an object. When a specific external force is applied to an object, internal stress generates in the material. The value of stress can obtain by taking the ratio of the externally applied force and cross-sectional area of the object. There are different types of stress depending on the type of external force.

Question

### Determining Normal and Shear Stresses in a Pressurized Pipe

The pipe shown has an outer diameter of 200 mm and a wall thickness of 10 mm. The length \(a\) is 1600 mm, and length \(b\) is 800 mm (\(b\) is from the center of the pipe to the application of the load). There is an applied load of 25 kN as pictured, and the pipe contains a static fluid pressurized at 2 MPa (this causes normal stresses in both the hoop and long directions).

**Objective:**
Determine the normal and shear stresses on the side of the pipe at point \(K\), which is coplanar with the yz-plane.

**Pipe Cross-Sectional Properties:**
\[ A = 12 \cdot 10^3 \, \text{mm}^2 \]
\[ Q_{\text{max}} = 180 \cdot 10^3 \, \text{mm}^3 \]
\[ J = 54 \cdot 10^6 \, \text{mm}^4 \]
\[ I_x = I_y = 27 \cdot 10^6 \, \text{mm}^4 \]

**Representation of Stresses:**
Present your answers in the following stress matrix in the xyz-coordinate system:
\[ [\sigma]_K = \begin{bmatrix}
- & - & - \\
- & - & - \\
- & - & - \\
\end{bmatrix} \, \text{MPa} \]

#### Explanation of the Diagram
The image depicts the pipe along with a coordinate system (x, y, z). The pipe is fixed at one end, and a load of 25 kN is applied downward at the other end. The lengths of the pipe sections, \(a\) and \(b\), are marked.

In summary, the problem involves calculating the stress components—both normal and shear—at a specified point \(K\) on the pipe, using the given material properties and boundary conditions. This is to be presented in matrix form for clarity in the xyz-coordinate system.

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