Mechanics 1O

# Mechanics 1O - 2.001 MECHANICS AND MATERIALS I Lecture#12...

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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
dF ( i ) = dF x ( i ) ˆ i + dFyx ( 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 T OT ALF ACE = σ xx dA x A x V T OT ALF ACE y = σ xu dA x A x N T OT ALF ACE z = σ xz dA x A x 3

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To find the stresses on the y face take cut on x-z plane. z face take cut on x-y plane.
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