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Unformatted text preview: ECCENTRIC LOADING In developing the bending formula M I =- σ y = E R , (1) we assumed that the beam was transmitting a bending moment M , but that the axial force F was zero. In fact, the axial force transmitted by the cross-section can be written as an integral (sum) of the contribution of the stress distribution over each element of area δ A as F = integraldisplay integraldisplay A σ dA . (2) Taken in combination with the results ε =- y R and Hooke’s law, the condition F = 0 leads to F =- integraldisplay integraldisplay A Ey R dA =- E R integraldisplay integraldisplay A ydA = . (3) This is the reason that in the bending theory, we must measure y from centroid of the section, since the centroid is defined by the conditions integraldisplay integraldisplay A xdA = 0 ; integraldisplay integraldisplay A ydA = in centroidal coordinates. In many practical problems, the axial force will not be zero. However, instead of reworking the bending theory from the beginning, we treat problems involv- ing bending and axial forces by superposing the stress fields due to the bending moment and the axial force considered separately. Pure axial loading If there is only an axial force F , the normal stress on the cross-section is uniform and given by σ = F A . (4) This uniform distribution adds up to (has as resultant) as force F whose line of action passes through the centroid. Thus, equation (4) applies only if the line of 1 action of the axial force passes through the centroid. For all other cases, the sameaction of the axial force passes through the centroid....
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