OPTI_222_W8 - OPTI 222 Mechanical Design in Optical...

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OPTI 222 Mechanical Design in Optical Engineering 43 Shear Forces and Bending Moments in Beams Bending Stress: My I σ = Moment of Inertia: I x = A y 2 dA I y = A x 2 dA Parallel Axis Theorem: I x = I xc + Ad 2 I y = I yc + Ad 2 Beam Classifications: Beams are also classified according to the shape of their cross sections. Shear Force and Bending Moment: When a beam is loaded by forces or couples, internal stresses and strains are created. To determine these stresses and strains, we first must find the internal forces and couples that act on cross sections of the beam. Consider the following example.
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OPTI 222 Mechanical Design in Optical Engineering 44 It is convenient to reduce the resultant to a shear force, V, and a bending moment, M. Because shear forces and bending moments are the resultants of stresses distributed over the cross section, they are known as stress resultants and in statically determinate beams can be calculated from the equations of static equilibrium. Deformation Sign Conventions: As can be seen in the previous diagram (left hand section vs. right hand section), we recognize that the algebraic sign does not depend on its direction in space, such as upward or downward or clockwise or counterclockwise. The sign depends on the direction of the stress resultant with respect to the material against which it acts. Shear and Moment Sign Convention Deformations highly exaggerated Positive shear forces always deform right hand face downward with respect to the left hand face. Positive bending moments always elongate the lower section of the beam.
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OPTI_222_W8 - OPTI 222 Mechanical Design in Optical...

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