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Prob. 2.3-12. A thin sheet of rubber in the form of a square (Fig. P2.3-12a) is uniformly deformed into the parallelogram shape shown in Fig. P2.3-12b. All edges remain the same length, b, as the sheet deforms. (a) Compute the extensional strain e, of diagonal AC. (b) Compute the extensional strain e of diagonal BD. -b/10 B b №(²) (1) b b (a) b D P2.3-12 b (2) (1) b (b) а

Prob. 2.3-12. A thin sheet of rubber in the form of a square (Fig. P2.3-12a) is uniformly deformed into the parallelogram shape shown in Fig. P2.3-12b. All edges remain the same length, b, as the sheet deforms. (a) Compute the extensional strain e, of diagonal AC. (b) Compute the extensional strain e of diagonal BD. -b/10 B b №(²) (1) b b (a) b D P2.3-12 b (2) (1) b (b) а

Structural Analysis
Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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**Problem 2.3-12: Analysis of Deformation in a Rubber Sheet** **Objective:** To compute the extensional strains of specific diagonals in a deforming rubber sheet. **Problem Statement:** A thin sheet of rubber initially in the form of a square (as shown in Figure P2.3-12a) is uniformly deformed into a parallelogram shape (as shown in Figure P2.3-12b). All edges of the sheet maintain the same length, \( b \), throughout the deformation process. The tasks are: (a) Compute the extensional strain \( \epsilon_1 \) of diagonal \( AC \). (b) Compute the extensional strain \( \epsilon_2 \) of diagonal \( BD \). **Figures Explanation:** - **Figure P2.3-12a: Initial Configuration** - This figure depicts the square sheet in its original, undeformed state. - The vertices are labeled \( A \), \( B \), \( C \), and \( D \). - The side lengths of the square are all \( b \). - Diagonals \( AC \) and \( BD \) are drawn as dashed lines. - **Figure P2.3-12b: Deformed Configuration** - This figure shows the deformed state where the square has been transformed into a parallelogram. - The new positions of the vertices are indicated with prime notation (e.g., \( A' \), \( B' \), \( C' \), \( D' \)). - The distances between these vertices are still \( b \), indicating that the edge lengths remain unchanged. - The initial straight angles (right angles) between edges have been altered due to deformation. - The shift from square to parallelogram introduces a small horizontal offset \( \frac{b}{10} \), as indicated in the top margin. **Computational Tasks:** 1. **Extensional Strain \( \epsilon_1 \) of Diagonal \( AC \):** - The extensional strain is calculated as the change in length of diagonal \( AC \) divided by its original length. 2. **Extensional Strain \( \epsilon_2 \) of Diagonal \( BD \):** - Similar to \( \epsilon_1 \), this strain is determined by the change in length of diagonal \( BD \) over its original length. By analyzing
Transcribed Image Text:**Problem 2.3-12: Analysis of Deformation in a Rubber Sheet** **Objective:** To compute the extensional strains of specific diagonals in a deforming rubber sheet. **Problem Statement:** A thin sheet of rubber initially in the form of a square (as shown in Figure P2.3-12a) is uniformly deformed into a parallelogram shape (as shown in Figure P2.3-12b). All edges of the sheet maintain the same length, \( b \), throughout the deformation process. The tasks are: (a) Compute the extensional strain \( \epsilon_1 \) of diagonal \( AC \). (b) Compute the extensional strain \( \epsilon_2 \) of diagonal \( BD \). **Figures Explanation:** - **Figure P2.3-12a: Initial Configuration** - This figure depicts the square sheet in its original, undeformed state. - The vertices are labeled \( A \), \( B \), \( C \), and \( D \). - The side lengths of the square are all \( b \). - Diagonals \( AC \) and \( BD \) are drawn as dashed lines. - **Figure P2.3-12b: Deformed Configuration** - This figure shows the deformed state where the square has been transformed into a parallelogram. - The new positions of the vertices are indicated with prime notation (e.g., \( A' \), \( B' \), \( C' \), \( D' \)). - The distances between these vertices are still \( b \), indicating that the edge lengths remain unchanged. - The initial straight angles (right angles) between edges have been altered due to deformation. - The shift from square to parallelogram introduces a small horizontal offset \( \frac{b}{10} \), as indicated in the top margin. **Computational Tasks:** 1. **Extensional Strain \( \epsilon_1 \) of Diagonal \( AC \):** - The extensional strain is calculated as the change in length of diagonal \( AC \) divided by its original length. 2. **Extensional Strain \( \epsilon_2 \) of Diagonal \( BD \):** - Similar to \( \epsilon_1 \), this strain is determined by the change in length of diagonal \( BD \) over its original length. By analyzing
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