Marine Technology - January 2008 - Page#53

<BODY> <span id="lblContents">? Marine Technology, Vol. 45, No. 1, January 2008, pp. 9?20 On Introducing "Section Modulus for Asymmetric Bending" of Stiffeners Lyuben D. Ivanov1 A new geometric property of shipbuilding structural profiles is introduced to consider asymmetric bending. This is the so-called section modulus for asymmetric bending, which allows for determining the total bending stresses in the structural profile for any angle of the attached plate and any angle of the plane of the applied load. As for symmetric bending, the major assumption behind the developed procedure for calculating the section modulus for asymmetric bending is that the structural profile works as a unit, that is, independently. 1. Introduction THERE ARE MANY AREAS in ship hull structure where the web plates of stiffeners (or girders) are not perpendicular to the attached plates. In general, although the load may be applied in the plane of web plates, bending of stiffeners/girders may be asymmetric if the load acts in a plane that does not coincide with any of the principal axes (bending will be symmetric only in cases where the cross section of stiffeners/girders are symmetric and the web plates are perpendicular to the attached plates). In such cases the "elastic" neutral axis (NA) is not perpendicular to the plane of the bending moment. In such a case, the section modulus calculated relative to the horizontal neutral axis should be corrected. This is done in several ways: ? When the angle (see Fig. 1) is smaller than 15?, the section modulus of stiffeners/girders is calculated relative to horizontal neutral axis (NA), although the attached plate may not be horizontal (ABS 2007). ? When > 15?, the section modulus is calculated either relative to NA parallel to the attached plate or by the equation (ABS 2007): Z = 1.45 - 40.5 Z90 (1) where the subscript refers to the angle between the attached plate and the web plate; the subscript 90 indicates the attached plate is perpendicular to the web plate. ? When > 15?, the section modulus is calculated either relative to NA parallel to the attached plate or by the equation (IACS 2006): Z = Z90 sin (2) The well-developed engineering beam theory allows for analyzing the behavior of stiffeners/girders under asymmetric bending quite accurately. The first step is finding the NA for asymmetric bending. can be determined as a sum of the normal stress resulting from the two projections of the bending moment on the two planes of the principal axes (Feodosiev 1972). For a point "i" inside the bulb plate at cross section "z" with coordinates x2,i (z) and y2,i (z) the normal stress will be (see Fig. 2): i z= MX2 z y2,i z MY2 z x2,i z - IX2 IY2 (3) where: MX2(z) is the bending moment at section "z" acting in plane perpendicular to X2. MY2(z) is the bending moment at section "z" acting in plane perpendicular to Y2. x2,i(z), y2,i(z) is the coordinates of any point "i" in the principal coordinate system X2, Y2 located at section "z." IX2 is the moment of inertia relative to the principal axis X2. IY2 is the moment of inertia relative to the principal axis Y2. Note: The sign in equation (3) is minus because the assumed point "i" is located in the area of negative values of X2 (see Fig. 2). The "elastic" NA can be determined on the basis of the fact that the normal stresses in points laying on it should be equal to 0, that is: 0= MX2 z y2,0 z MY2 z x2,0 z - IX2 IY2 (4) 2. Neutral axis for asymmetric bending A convenient way to find out the location of the NA for asymmetric bending is to project the bending moment on the two principal axes X2 and Y2 (see Fig. 2). Let us assume that the distribution of these projections along the stiffener's length is known. Then, the normal stress in the elastic range American Bureau of Shipping, Houston, Texas, USA. Manuscript received at SNAME headquarters March 2007. 1 Fig. 1 Stiffener with inclined attached plate. The Neutral Axis refers to symmetric bending when the load is applied in the stiffener's web plane; = angle between the attached plate and the Neutral Axis JANUARY 2008 0025-3316/08/4501-0009$00.00/0 MARINE TECHNOLOGY 9</span> <br /> <br /> <a href="http://www.digitalwavepublishing.com/" title="Digital Magazine">Digital Wave Publishing</a> </BODY>