5003 Lectures - Faculty of Engineering and Applied Science

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E5003 - Ship Structures I 164 © C.G. Daley dy = d + f x , d , d , d ) (why is this more 2 ( 3 5 6 complex?) For this beam element, we made use of what is called ‘beam theory’, to solve for the loads and deflections under certain loading conditions. However, in the case of most finite elements, such as 2D planar elements, plate elements, and solid elements, we will not start from some general analytical solution of a loaded membrane, plate or solid. These solutions are too complex and will not give practical results. Instead, we assume some very simple behaviors, highly idealized, but which satisfy the basic requirements for equilibrium (i.e. forces balance, energy is conserved). With this approach, the single element does not really model the behavior or a comparable real solid object of the same shape. This is ok, because the aggregate behavior of a set of these simple elements will model the behavior quite well. This is something like modeling a smooth curve as a series of straight lines (even horizontal steps). This is locally wrong, but overall quite accurate. Constant Stress Triangle To illustrate the way that finite elements are formulated, we will derive the full description of an element called the constant stress triangle (cst). This is a standard 2D element that is available in most finite element models. Consider a 2D element which is only able to take in-plane stress. The three corners of the triangle can only move in the plane. For this element the force balance is; δ e F = K 6x 1 = 6x6 6x1 { } [ ]{ } We want to determine the element stiffness matrix Ke , and we want it to be valid for any triangle;
E5003 - Ship Structures I 165 © C.G. Daley So, while we have six degrees of freedom, as we did in the beam case, we don't have any hand analytical solutions. To create a general solution that will apply to all triangles we will make some very simple assumptions which will allow us to model 2D stress problems (such as a web in shear, or stresses in plane around a cutout in a web. . We will follow the outline in Hughes (p. 245-253). Step 1 - select a suitable displacement function. Consider the movement of a general triangle. Each corner moves differently, and every point inside moves. The movement in x is defined as u and the movement is y is defined a v. Both u and v are functions of x and y ; ∆
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5003 Lectures - Faculty of Engineering and Applied Science

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