**State of Strain in a Plane Element**The state of strain in a plane element is given by:- \( \varepsilon_x = -200 \times 10^{-6} \)- \( \varepsilon_y = 0 \)- \( \gamma_{xy} = 75 \times 10^{-6} \)Determine the equivalent state of strain which represents:(a) The principal strains(b) The maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element.**Diagram Explanation**The accompanying diagram illustrates a differential element subjected to plane strain:- The \( x \)- and \( y \)- axes define the plane.- The element has dimensions \( dx \) and \( dy \).- The deformed element is shown as a dashed line, with rotations due to shear strain.- \( \gamma_{xy} \) represents the shear strain, causing a rotation of \( \frac{\gamma_{xy}}{2} \) on each face of the element.- \( \varepsilon_x dx \) indicates the contraction or elongation in the \( x \)-direction.The diagram demonstrates the effect of normal strain \( \varepsilon_x \), which contracts the element along the \( x \)-axis, and shear strain \( \gamma_{xy} \), which causes angular distortion.**Educational Objective**Understand how to determine principal strains and maximum in-plane shear strain from a given state of strain, and how to relate these values to the orientation of the deformed element in-plane stress analysis.

(a) Write the formula for principal strains in terms of the given strains. ε1,2=εx+εy2±εx-εy22+γxy22…

The state of strain in a plane element is ex = -200 × 10-6, €y = 0, and yxy = 75 x 10-6, as shown below. Determine the equivalent state of strain which represents (a) the principal strains (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element. y Yxy 2 dy Yxy FExdx dx

The state of strain in a plane element is ex = -200 × 10-6, €y = 0, and yxy = 75 x 10-6, as shown below. Determine the equivalent state of strain which represents (a) the principal strains (b) the maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element. y Yxy 2 dy Yxy FExdx dx

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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Question

**State of Strain in a Plane Element**

The state of strain in a plane element is given by:
- \( \varepsilon_x = -200 \times 10^{-6} \)
- \( \varepsilon_y = 0 \)
- \( \gamma_{xy} = 75 \times 10^{-6} \)

Determine the equivalent state of strain which represents:
(a) The principal strains
(b) The maximum in-plane shear strain and the associated average normal strain. Specify the orientation of the corresponding elements for these states of strain with respect to the original element.

**Diagram Explanation**

The accompanying diagram illustrates a differential element subjected to plane strain:
- The \( x \)- and \( y \)- axes define the plane.
- The element has dimensions \( dx \) and \( dy \).
- The deformed element is shown as a dashed line, with rotations due to shear strain.
- \( \gamma_{xy} \) represents the shear strain, causing a rotation of \( \frac{\gamma_{xy}}{2} \) on each face of the element.
- \( \varepsilon_x dx \) indicates the contraction or elongation in the \( x \)-direction.

The diagram demonstrates the effect of normal strain \( \varepsilon_x \), which contracts the element along the \( x \)-axis, and shear strain \( \gamma_{xy} \), which causes angular distortion.

**Educational Objective**

Understand how to determine principal strains and maximum in-plane shear strain from a given state of strain, and how to relate these values to the orientation of the deformed element in-plane stress analysis.

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