Eng. 6002 Ship Structures 1 Hull Girder Response Analysis Lecture 8: Review of Beam Theory (deformations)

Overview In this lecture we will look at beam deformations caused by bending moments We will also review the differential equations of beam theory and solve them using Maple

Deformations In simple beam theory we make the assumption that only the bending moments cause distortions. So we can assume that the beam is bent into a simple circular curve over any short length.

Deformations 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

Deformations For the top fibre of the figure we see that the strain can be represented by Substituting, we have:

Deformations Integrating, we have For most sections EI is constant, so

Deflections To find the deflections, v, we have:

Family of Differential Equations Consider a beam between two supports, we describe the deflections with the variable v(x). Recall:

Family of Differential Equations Combining these equations gives: We can describe all responses as derivatives of deflection

Family of Differential Equations We can also express the relationships in integral form to give

Solving the Beam Equations To solve beam equations, we need to know the beam geometry and properties (L, E, I), the loading conditions (length, shape and magnitude of load) the boundary conditions (fixed, pinned, free, guided). The boundary conditions are described in terms of what is known. For a fixed end we know that the deflection and rotation are zero. For a guided end we know that the shear (reaction) and rotation are zero.

Solving the Beam Equations Four standard boundary conditions are shown Each of the boundary conditions results in the deflection or one of its derivatives being equal to zero.