LECTURE 4 W5 Strain Transformation.pptx

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1. Strain Transformation Complex stresses developed within this airplane wing are analyzed from strain gauge data. 1

2. Plane Strain State of strain at a point is described by six strain components: Three normal strains: ε x , ε y , ε z Three shear strains: γ xy , γ xz , γ yz These components depend upon the orientation of the line segments and their location in the body These components tend to deform each face of an element, just like stress. 2

3. General Equations of Plane Strain Transformation According to the established sign convention, if ε x’ is positive, the element elongates in the positive x’ direction, and if ɣ x’y’ is positive, the element deforms as shown. If the normal strain in the y’ direction is required, it can be obtained from this ε x’ Eq. by simply substituting ( θ+90) for θ. The result is

4. Example 1 An element of a material at a point is subjected to a state of plane strain ε x = 500 (10 -6 ), ε y = -300 (10 -6 ), ɣ xy = 200 (10 -6 ), which tends to distort the elements as shown. Determine the equivalent strains acting on the element oriented at the point, clockwise 30° from the original position.

5. Principal Strains Like stress, an element can be oriented at a point so that the element’s deformation is caused by normal strains with no shear strain. (Principal Strain) Expressions for determining in-plane principal directions, in-plane principal strains, and the maximum in-plane shear strain:

6. Maximum In-Plane Shear Strain Expressions for maximum in-plane shear strain and its associated average normal strain:

7. Example 2 A differential element of material at a point is subjected to a state of plane strain defined by ε x = -350 (10 -6 ), ε y = 200 (10 -6 ), ɣ xy = 80 (10 -6 ), which tends to distort the element as shown in Fig. Determine the principal strains at the point and the associated orientation of the element. Determine the maximum in-plane shear strain at the point and the associated orientation of the element.

8. Mohr’s Circle Set axis: X-axis: normal strain ε is +ve to the right Y axis: half of shear strain (γ/2) +ve downwards Find center, located at ε axis at distance ε average . Plot ref point A (ε x , γ xy /2). Connect AC. Find radius, R. Sketch circle.

9. Mohr’s Circle for Strain: Procedure

10. Example 3: Mohr’s Circle for Plane Strain The state of plane strain at a point is represented by the components ε x = 250 (10 -6 ), ε y = -150 (10 -6 ), and ɣ xy = 120 (10 -6 ). Determine the principal strains and the orientation of the element. Determine the maximum in-plane shear strains and the orientation of an element.

11. Example 4 The state of plane strain at a point is represented on an element having components ε x = -300µ , ε y = -100µ , and ɣ xy = 100µ . Determine the state of strain on an element oriented 20° clockwise from this reported position.