5003 Lectures - Faculty of Engineering and Applied Science

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E5003 - Ship Structures I 72 © C.G. Daley Adding Deformations So far we have differential equations for load/shear/bending relationships. Now we add deformations. The shear force and bending moments are causing stresses and strains in the beam. We make the assumption that we can ignore the shear deformations (this is part of what we call simple beam theory), so that only the bending moments cause distortions. This means that only consider the shortening of the compression side of the beam and the lengthening of the tension side. When this happens, the beam deforms from being straight to being a curve. The curve shape for any short length is an arc of a circle, with a radius R. The local radius, as we can show, turns out to depend only on the local bending moment. The figure below show a short length of a bending beam. The curved shape is also presented in differential form, meaning essential or limit shape for a very small value of dx.
E5003 - Ship Structures I 73 © C.G. Daley The neutral axis (NA) does not stretch or contract. The upper and lower parts of the beam compress and/or stretch. We can use the two ‘known’ relationships, the stress-moment equation; and 1D Hooke's law;
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5003 Lectures - Faculty of Engineering and Applied Science

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