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Football goal posts are sometimes torn down in the excitement of a win, which for winning home teams causes needless expenses. For the goal post and dimensions shown below, the fans have managed to put on a load W (representing their weights) in the negative y -direction at the location shown. From past goal post tear downs you have noticed that typically the base fails at the ground shown by the red dot, or at the back cross bar by the green dot. Assume the post is made of steel with a modulus E, radius R, area A, moment of inertia I, and polar moment of inertia J. INCLUDE the effects of shear from bending for this problem.

Football goal posts are sometimes torn down in the excitement of a win, which for winning home teams causes needless expenses. For the goal post and dimensions shown below, the fans have managed to put on a load W (representing their weights) in the negative y -direction at the location shown. From past goal post tear downs you have noticed that typically the base fails at the ground shown by the red dot, or at the back cross bar by the green dot. Assume the post is made of steel with a modulus E, radius R, area A, moment of inertia I, and polar moment of inertia J. INCLUDE the effects of shear from bending for this problem.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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### Educational Content Transcription This image provides a detailed look at stress analysis on elements of a structural system. The page seems to be from an educational material focusing on engineering or physics, typically related to mechanics of materials or statics. Below is a transcription and explanation suitable for an educational website. #### Homework Problem: Stress Analysis **(b) Stress Element A:** - **Objective:** Write expressions for all stresses on element A and label their directions. - **Description:** A is the red element at the base of the pole on the positive x-face. - **Stress Expressions:** - \(\sigma_{x} = \frac{M_{x} \times R}{I}\) - \(\sigma_{y} = \frac{6W \times R}{I}\) - \(\sigma_{c} = \frac{W}{A}\) **Diagram:** A box labeled "A" with arrows indicating stress directions and "Cut side at the base of the post." --- **(c) Free Body Diagram:** - **Task:** Draw a free body diagram with a cut on the post between the vertical member secured to the ground and the crossbar for element B. - **Conditions:** - \( \Sigma F_{y} = 0 \): \( R_{b} - 4G = 0 \) leading to \( R_{b} = 4G \) - \( \Sigma F_{t} = 0 \): \( W \times x - T_{b} = 0 \) resulting in \( T_{b} = 9W \) - \( \Sigma M_{B} = 0 \): \( W \times L - M_{DE} = 0 \) and \( M_{ze} = 6W \) **Diagram Description:** A labeled diagram with arrows depicting all applied and reaction forces and moments. --- **(d) Stress Element B:** - **Objective:** Write expressions for all stresses on element B and label their directions. - **Description:** B is the green element on the crossbar on the positive z-face. - **Stress Expressions:** - \(\sigma_{z} = 0\) - \(\sigma_{33} = \frac{V \times R}{I}\) - \(\sigma_{x} = \frac{4W \times A}{I}\)
Transcribed Image Text:### Educational Content Transcription This image provides a detailed look at stress analysis on elements of a structural system. The page seems to be from an educational material focusing on engineering or physics, typically related to mechanics of materials or statics. Below is a transcription and explanation suitable for an educational website. #### Homework Problem: Stress Analysis **(b) Stress Element A:** - **Objective:** Write expressions for all stresses on element A and label their directions. - **Description:** A is the red element at the base of the pole on the positive x-face. - **Stress Expressions:** - \(\sigma_{x} = \frac{M_{x} \times R}{I}\) - \(\sigma_{y} = \frac{6W \times R}{I}\) - \(\sigma_{c} = \frac{W}{A}\) **Diagram:** A box labeled "A" with arrows indicating stress directions and "Cut side at the base of the post." --- **(c) Free Body Diagram:** - **Task:** Draw a free body diagram with a cut on the post between the vertical member secured to the ground and the crossbar for element B. - **Conditions:** - \( \Sigma F_{y} = 0 \): \( R_{b} - 4G = 0 \) leading to \( R_{b} = 4G \) - \( \Sigma F_{t} = 0 \): \( W \times x - T_{b} = 0 \) resulting in \( T_{b} = 9W \) - \( \Sigma M_{B} = 0 \): \( W \times L - M_{DE} = 0 \) and \( M_{ze} = 6W \) **Diagram Description:** A labeled diagram with arrows depicting all applied and reaction forces and moments. --- **(d) Stress Element B:** - **Objective:** Write expressions for all stresses on element B and label their directions. - **Description:** B is the green element on the crossbar on the positive z-face. - **Stress Expressions:** - \(\sigma_{z} = 0\) - \(\sigma_{33} = \frac{V \times R}{I}\) - \(\sigma_{x} = \frac{4W \times A}{I}\)
**Transcription for Educational Website:** --- **Title: Structural Analysis of Football Goal Posts** **Overview:** Football goal posts are sometimes torn down in the excitement of a win, causing needless expenses. For the goal post and dimensions shown below, fans have managed to place a load \( W \) (representing their weight) in the negative \( y \)-direction at the location shown. From past goal post tear downs, it is noted that typically the base fails at the ground, shown by the red dot, or at the back crossbar by the green dot. Assume the post is made of steel with a modulus \( E \), radius \( R \), area \( A \), moment of inertia \( I \), and polar moment of inertia \( J \). **Include** the effects of shear from bending for this problem. **Diagram Explanation:** - **Figure 1 (Left):** A schematic diagram of the goal post is shown, with the following dimensions: - Vertical post heights: 20 ft - Horizontal crossbar: 18 ft - Base section \( A \): 10 ft high - Inclined section \( B \): 6 ft wide - Load \( W \) acting downward - **Figures 2 & 3 (Center):** Images depict broken goal posts due to the force exerted by fans climbing them. - **Figure 4 (Right):** Illustrates probable failure points: - Red dot at the base indicates typical ground failure. - Green dot at the back cross bar indicates failure due to stress. **Problem Statement:** (a) Draw a free body diagram with a cut at the base of the post by element \( A \), showing all applied and reaction forces and moments. Write expressions for the reaction forces and moments. --- This educational exercise aims to guide students in understanding the structural challenges and calculations involved in engineering scenarios like those experienced by football goal posts under unexpected loads.
Transcribed Image Text:**Transcription for Educational Website:** --- **Title: Structural Analysis of Football Goal Posts** **Overview:** Football goal posts are sometimes torn down in the excitement of a win, causing needless expenses. For the goal post and dimensions shown below, fans have managed to place a load \( W \) (representing their weight) in the negative \( y \)-direction at the location shown. From past goal post tear downs, it is noted that typically the base fails at the ground, shown by the red dot, or at the back crossbar by the green dot. Assume the post is made of steel with a modulus \( E \), radius \( R \), area \( A \), moment of inertia \( I \), and polar moment of inertia \( J \). **Include** the effects of shear from bending for this problem. **Diagram Explanation:** - **Figure 1 (Left):** A schematic diagram of the goal post is shown, with the following dimensions: - Vertical post heights: 20 ft - Horizontal crossbar: 18 ft - Base section \( A \): 10 ft high - Inclined section \( B \): 6 ft wide - Load \( W \) acting downward - **Figures 2 & 3 (Center):** Images depict broken goal posts due to the force exerted by fans climbing them. - **Figure 4 (Right):** Illustrates probable failure points: - Red dot at the base indicates typical ground failure. - Green dot at the back cross bar indicates failure due to stress. **Problem Statement:** (a) Draw a free body diagram with a cut at the base of the post by element \( A \), showing all applied and reaction forces and moments. Write expressions for the reaction forces and moments. --- This educational exercise aims to guide students in understanding the structural challenges and calculations involved in engineering scenarios like those experienced by football goal posts under unexpected loads.
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