Design for Stiffness

In structural applications for plastics, which generally include those in which the part has to resist substantial static and dynamic loads, one of the problem design areas is the low modulus of elasticity of polymeric materials. Even when such rigid polymers as the ladder types of polyesters and polyamides are considered, the elastic moduli of unfilled polymers are under one million psi as compared to metals where the range is usually 10 to 40 million psi. Ceramic materials also have high moduli. Since shape integrity under load is a major consideration for structural parts, plastics parts must be designed for efficient use of material to afford maximum stiffness. 

In the previous chapters we covered the use of material modification such as orientation and the use of fillers to increase the modulus of elasticity of plastics. This chapter is concerned with geometrical design which makes the best use of materials to improve stiffness. Structural shapes which are applicable to all materials are discussed such as sandwich structures, shells, and folded plate structures. In the case of plastics, emphasis is on the way plastics can be used in these structures and why they are preferred over other materials. In many cases plastics can lend themselves to a particular field of application only in the form of a sophisticated lightweight stiff structure and the requirements are such that the structure must be of plastics, e.g., in a radome. In other instances, the economics of fabrication and erection of a plastics lightweight structure and the intrinsic appearance and other desirable properties make it preferable to other materials.  

One of the most widely used lightweight structural concepts is the 

The above are requirements for a sandwich panel to perform its improved function in bending in the direction perpendicular to the plane of the panel. A sandwich panel exhibits no improvement in performance in other directions such as parallel to the plane of the sandwich. It is, in fact, subject to failure under lower load conditions 

Another type of stiffener is shown in Figs. 8-7 and 8-8. In this case the basic sheet of the part is converted to a series of connected I or T beams. While this construction is not as efficient as the sandwich panel, it does have the advantage that it can be molded or extruded directly in the required configuration and the relative proportions of the legs and sheet can be designed to meet the flexural requirements. One of the limitations is that it imparts increased stiffness in one direction much more than in the other. 

---

Equation (8.22) for a(0) is also special because, due to symmetry, there is only one adjacent point, a(l). The overall set may be solved by any desired method. Euler s method is discussed below and is illustrated in Example 8.5. There are a great variety of commercial and freeware packages available for solving simultaneous ODEs. Most of them even work. Packages designed for stiff equations are best. The stiffness arises from the fact that VJJ) becomes very small near the tube waU. There are also software packages that will handle the discretization automatically. 

The approximating scheme converts the system of partial differential equations to a set of ordinary differential equations in the axial spatial coordinate. The detailed equations are contained in Yu et al. (2). The advantage of reduction in this manner is that the transient location of the combustion zone does not have to be known a priori, but can be found in the course of the integrations. Gear integration, which is designed for stiff systems, is used to solve the two point boundary value problem in the axial direction. 

This material is a cyclic methylated high solids resin, designed for stiffness on Nylon and synthetics. This material is viscous in nature, and has extremely good storage stability. Catalysts are required. 

A very important gain in volume can be achieved if one uses tuned mass dampers. An example, with parametric analysis, shows clearly that one needs to consider a design for stiffness approach when dynamic loads are important. 

The numerical solution of the equation was achieved by using a library routine which selected the order and step size automatically and was designed for stiff differential equations (IMSL-DGEAR). The routine was applied in the following iterative scheme ... 

There are examples where control of deflection or deformation during service may be required. Such structural elements are designed for stiffness to control deflection but must be checked to assure that strength criteria are reached. A product can be viewed as a collection of individual elements interconnected to achieve an overall systems... 

The question of optimal design for stiffness and friction of seismic isolation systems has been addressed previously in the literature (e.g., Constantinou and Tadjbakhsh (1983), lemura et al. (2007), Jangid (2005)). The method of equivalent linearization in conjunction with a random process model for the earthquake excitation tuned to the El Centro record was used in Jangid (1996). Here, a full nonlinear dynamic analysis will be used as a basis for the computation of the structural response. The ground motion is modeled as a nonstationary (i.e., evolutionary) random process hence, the response should to be characterized in suitable probabilistic terms. 

---