Second Moment of Areas

In terms of the dimensions, a, b and t for the section, several area properties can be found about the x-x and y-y axes, such as the second moment of area, 4, and the product moment of area, 4y. However, because the section has no axes of symmetry, unsymmetrical bending theory must be applied and it is required to find the principal axes, u-u and v-v, about which the second moments of area are a maximum and minimum respectively (Urry and Turner, 1986). The principal axes are again perpendicular and pass through the centre of gravity, but are a displaced angle, a, from x-x as shown in Figure 4.63. The objective is to find the plane in which the principal axes lie and calculate the second moments of area about these axes. The following formulae will be used in the development of the problem. 

Solution The first step in analysing the foamed sandwich type structure is to calculate the second moment of area of the cross-section. This is done by converting the cross section to an equivalent section of solid plastic. This is shown in Fig. 2.18. 

From the equivalent section the second moment of area can then be calculated as... 

In any particular material, the flexural stiffness will be defined by the second moment of area, /, for the cross-section. As with a property such as area, the second moment of area is independent of the material - it is purely a function of geometry. If we consider a variety of cross-sections as follows, we can easily see the benefits of choosing carefully the cross-sectional geometry of a moulded plastic component. 

An extruded T-section beam in polypropylene has a cross-sectional area of 225 mm and a second moment of area, I, of 12.3 x lO mm. If it is to be built-in at both ends and its maximum deflection is not to exceed 4 mm after 1 week, estimate a suitable length for the beam. The central deflection, S, is given by... 

The ratio of sd esses will be equal to the ratio of second moment of area, bd 12x8 4... 

Stiffness is the ability to resist bending and buckling. It is a function of the elastic modulus of the material and the shape of the cross-section of the member (the second moment of area). 

In calculating the second moment of area of the ring some allowance is normally made for the vessel wall the use of Ir calculated for the ring alone will give an added factor of safety. 

For a rectangular section, the second moment of area is given by... 

First, if the specimen is subjected to axial stress a as well as the twist then as Biot215 has shown, Eq. (2.1) needs to be modified so that M = M — al, where M is the applied torque and I the second moment of area of the cross section with respect to the twist axis. Second, the St. Venant principle needs to be modified, as shown above, if the material is highly anisotropic, and very high length to diameter ratios may be necessary (Folkes and Arridge125). 

Second moment of area (axial) Ia (polar) Ip Inch to the fourth... 

This is for a thin column with one fixed and one free support (a cantilever). Here E is the modulus of elasticity or stiffness, I is the second moment of area and L is the length of the... 

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