Mechanics 1O

Mechanics 1O - 2.001 - MECHANICS AND MATERIALS I Lecture...

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2.001 - MECHANICS AND MATERIALS I Lecture # 10/23/2006 Prof. Carol Livermore MULTI-AXIAL STRESS AND STRAIN Recall: Internal Forces and Moments Axial stress from uniaxial loading P σ = A Note: σ is an average axial stress. For slender (long, thin) objects A is either uniform or slowly-varying . 1 12
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What about a more general case? Note: dF ± is not in general parallel to ˆ n . 2
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± ± ± ± dF ± ( i ) = dF x ( i ) ˆ i + dF yx ( i ) ˆ j + dF z ( i ) k ˆ n ˆ= ˆ i N totalforce = dF x ( i ) A x σ ¯= N totalforce /A x Traction: ± t Force per unit area at a point ( i )( i i ) ± t ( i ) = dF ( i ) = dF x ˆ i + dF y ˆ j + dF z k ˆ dA x dA x dA x dA x dF x ( i ) σ x x = (Normal Stress) dA x dF y ( i ) σ x y = (Shear Stress) dA x dF z ( i ) σ x z = (Shear Stress) dA x So: ± t ( i ) = σ xx ˆ i + σ xy ˆ j + σ xz k ˆ N TOTALFACE = σ xx dA x A x V y = σ xu dA x A x N z = σ xz dA x A x 3
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To fnd the stresses on the y Face take cut on x-z plane. zFace take cut on x-y plane. And Follow the same procedure. ± t ( j )
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Mechanics 1O - 2.001 - MECHANICS AND MATERIALS I Lecture...

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