Converting Rates in One Coordinate System to Another
The Importance of Structural Design Analysis
Structural design analysis of any construction project helps to determine whether the structure that is to be built will be able to perform as has been conceived by the designer. This requires certain standards of reliable and safety to be maintained and other long established standards to be followed. Structural analysis will examine the physical integrity that goes into any structure, be it a building, bridge, or even a vessel or aircraft. The process has to depend on the principles and laws of mathematics and physics so that the stability of a structure can be predicted depending on the loads it is subject to. The main idea behind structural analysis is computations of deformations, stresses, and internal forces when the structure is required to assume certain loads. The geometry of the structure, the support conditions, and the materials of which the structure is made are all relevant to any equations that ultimately decide the performance and compare them to failure criteria.
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The Approach to Structural Design
Structural analysts approach the process in three different ways. In the first, the strength and mechanics of the material used to form the structure are the primary concern. This is the simplest form of analysis and is basically a linear approach that does not require complex equations to be solved. The second method has the same linear approach, but elasticity is the main consideration. The third method is the finite element approach where the structure is assumed to be composed of a number of finite elements all interconnected with each other to form a grid. It is the strength of the grid which finally decides the safety and efficiency of the structure. This approach is quite complex and is heavily dependent on computational methods.
The Basics of Finite Element Analysis
Finite element analysis, or FEA, is a way of solving such complex engineering problems. A computational model is made of the structure in the form of finite elements that can be mathematically defined. This gives rise to a finite element mesh and other boundary conditions and material properties that can be applied to each of the finite elements. Each element has a connection with one or more other elements and these connections do play their part in the functioning of each element and its effect on the overall structure.
Software can then be used to solve the various equations that are generated when the elements are related to each other and to any other external forces that act on the elements.
Once the solutions are obtained they have to be checked for their validity and for the values of displacements and stresses.
Every structural design has to deliver the results that are required, and FEA allows even separate sections of a structure to be analyzed so that each plays its part in the overall requirement.
Coordination Systems and Their Transformation
Members of any structure are oriented in different directions with each member having its own local coordinates. These coordinates give the dimensions of the member, which again are responsible for the stiffness that each member has. It is ultimately the stiffness of each member that determines the part it will play in the stability of the structure. To determine the overall impact that such stiffness plays in the overall system that makes up the structure, it is necessary to convert the local coordinates that each member has into a common global coordinate system. When the member stiffness is determined in global coordinates common for all the members, it is then possible to determine the combined stiffness of the structure to decide on its stability. So the end forces and end displacement of each member have to be transformed to global coordinates for the required structural analysis.
Converting rates in one coordinate system to another has become a very simple affair due to the advent of computers and their abilities to perform extensive calculations that such transformations require. NASTRAN is a finite element analysis program that is in extensive use in the automobile and space industries, and is software originally commissioned by NASA which has since gone into industrial usage.
The modification of coordinate system components is a common operation that is necessary when the results of finite elements are finally processed. Material systems for defining coordinates use the material definition and angle. Materials exhibit different properties in different axes, and these properties are used when deciding the stresses and strains on any element in the NASTRAN material coordinate system. This allows all the properties of the material and its position in the structure to be fully utilized and can result in significant reduction in sizes of elements, which leads to a lot of economy in the final design.
Image source: Wikimedia
References
LS-Dyna support - Coordinate system for stress
Scribd - Nastran User’s Guide
