L28_ PlaneStrain-fall 10

L28_ PlaneStrain-fall 10 - EAS 209-Fall 2010 Instructors...

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EAS 209-Fall 2010 Instructors: Christine Human 11/1/2010 1 Lecture 28- Strain Transformation Lecture 28 Transformation of Plane Strain We will see that the transformation of strain is very similar to the transformation of stress. Like stress transformation, we can use Mohr’s circle to solve strain transformation graphically. Today’s Objectives: Develop strain transformation equations for plane strain Use Mohr’s circle of strain to determine strain at any orientation Discuss strain gages Today’s Homework: EAS 209-Fall 2010 Instructors: Christine Human 11/1/2010 2 Lecture 28- Strain Transformation Plane Strain Deformations of the material take place within || el planes. Plane strain occurs in a plate subjected to a uniformly distributed load along its edges and restrained from expanding or contracting laterally by smooth, rigid and fixed supports. These restraints produce stresses in the out-of-plane directions due to the “ Poisson Effect” Components of plane strain: ) 0 ( , , yz xz z xy y x But yz xz z , , are not necessarily zero

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EAS 209-Fall 2010 Instructors: Christine Human 11/1/2010 3 Lecture 28- Strain Transformation The figure below shows an element in the xy plane of side Δ s before and after a load is applied which results in stains xy y x and , , . This next figure shows the same element with respect to a new set of axes x’y’ rotated by an angle about the z -axis Our task is to determine the strains in the transformed plane ' ' ' ' , , y x y x in terms of known strains xy y x , , and rotation . EAS 209-Fall 2010 Instructors: Christine Human 11/1/2010 4 Lecture 28- Strain Transformation Consider the undeformed triangle ABC below The deformed length A’B’ is ) 90 cos( ) 1 ( cos ) 1 ( )) ( 1 ( xy y x y x s which becomes ) 90 cos( sin ) 1 ( cos ) 1 ( )) ( 1 ( 2 xy y x from trig ) 1 cos , sin small, is (since sin cos sin cos cos sin sin ) 90 sin( cos ) 90 cos( ) ) 90 cos(( sin sin cos cos ) cos( xy xy xy xy xy xy xy xy xy xy y x y x y x ) sin cos ( sin ) 1 ( cos ) 1 ( )) ( 1 ( 2 xy y x where sin cos s y s x After deformations xy y x , , the triangle becomes A’B’C’ We first want to determine the strain on side AB oriented at an angle
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