Concepts of Structural Analysis<\/p>\n
Structures can be classified in a variety of ways.\u00a0The casual observer might first consider classifying structures according to their respective functions: buildings, bridges, ships, aircraft, towers, and so on.<\/p>\n
This basis for structural classification is in fact fundamental;\u00a0all structures have some functional reasons for existence.\u00a0It is the need to fulfill some function that prompts the designer to give life to a structure.\u00a0Furthermore, it is the need for a safe, serviceable, feasible, and aesthetically pleasing fulfillment of a function that dictates the form, material, and manner of loading of a structure.<\/p>\n
Once the form and material have been determined, a structure may be further classified according to either its form (e.g., an arch, truss, or suspension structure) or the material out of which it is constructed (e.g., steel concrete, or timber).\u00a0The form and material of a structure in turn dictate its behavior, which in turn dictates the character of the analytical model.\u00a0Fig. 6.1 illustrates schematically the relationships among the function a structure is to fulfill, the form and material and loading on the structure, the behavior of the structure, and the analytical model of the structure.\u00a0At this point, we need to discuss some of the aspects of structural behavior indicated in Fig.6.1 and to explain their respective relationships to the form and material of the structure.\u00a0A structure is linear if its response to loading, say displacement at a point, is directly proportional to the magnitude of the applied load.If this proportionality does not exist, the structure is said to be nonlinear.Structural nonlinearities are of two types:(1) material nonlinearities that arise when stress is not proportional to strain,\u00a0and (2) geometric nonlinearitis that arise when the configuration of the structure under load is markedly changed from the unloaded configuration. 2\uff09(the presence of cables in a structure often leads to geometric nonlinearity because displacements can occur owing to a change in cable sag, which can be shown to be nonlinearly related to the force in the cable.)materials, and therefore structures built from them, may be classified as elastic, plastic, or viscoelastic.\u00a0Elastic materials rebound to their initial configuration when the load is removed, whereas plastic materials retain a permanent setThe deformations of viscoelastic materials depend on time and therefore load history, whereas the deformations of elastic and plastic materials do not.\u00a0A structural system is unconservative or conservative depending on whether or not energy is lost from the system during a cycle of loading and unloading.Energy is generally lost if a system does not recover its initial shape after unloading owing either to plastic behavior of the material or to friction forces within or between parts of the structure.<\/p>\n
All these behavioral aspects of the structure will have a significant influence on the nature of the analysis used in studying the structure.\u00a0In addition, in developing the analytical model it will be necessary to consider whether the structural material is homogeneous or nonhomogeneous and whether it is irotropic, orthortropic, or anisotropic.\u00a0(the physical properties of homogeneous materials are the same at each point; those of nonhomogeneous material are not.\u00a0The physical properties of isotropic materials are the same in all directions at a point;\u00a0those of anisotropic materials are not.An orthotropic material is a special anisotropic material whose properties are different in three principal directions but whose properties in all other direction are dependent on those in the principal directions. Other aspects of the structure, although important design considerations, will not usually have a significant impact on the analysis technique.\u00a0These include brittleness, ductility, flammability, texture, color, hardness, and machinability.<\/p>\n
Finally, the nature of the loading, which is dependent on the function of the structure, will also influence the analysis.\u00a0The only truly static loading on a structure is the dead, or gravity, loading.\u00a0However, if other loadings are applied gradually enough, they are called quasi-static loadings and may be considered static for analysis purposes.\u00a0Whether or not the rate of loading is gradual enough depends on whether or not the time it takes to apply the load is longer than the fundamental period of vibration of the structure being analyzed.\u00a0Loads usually need to be treated as dynamic only if they are periodic in nature or if they are applied very suddenly.\u00a0Even then, sometimes an \u201cimpact factor\u201d is applied to an analysis with a static-loading result to account for the effect of a suddenly applied load.Loads can also be categorized as either external applied forces or internal initial distortions.\u00a0Thermal loading is an example of an internal initial distortion (or initial strain) loading.<\/p>\n
Unfortunately, the picture of structural behavior is generally not so clear as that just painted.\u00a0That is, materials are not either \u201clinear\u201d or \u201cnonlinear\u201d and \u201celastic\u201d or \u201cplastic\u201d; instead, their behavior depends on circumstances such as environment and rate of loading.\u00a0The picture is further clouded in that the type of behavior that must be considered in an analysis may depend on the type of response being investigated.\u00a0For example, a simpler analytical model may suffice to obtain static displacement and stress results than that which would be required for vibration or buckling results.<\/p>\n
To clarify this picture for purposes of a rational presentation of matrix analysis of structures, we will make simplifying assumptions as to the nature of the behavior structures.\u00a0Thus we will consider only the displacement and stress response due to static loading of linear, elastic, conservative structures.\u00a0We will further restrict our attention to discrete-membered structures (rigid-and pin-jointed frameworks) as opposed to continuous structures.\u00a0However, it is important to recognize at the outset that the concepts that will be presented can be extended to the solution of many other classes of structural problems, including those involving dynamic response, material and geometric nonlinearitys, inelasticity, instability, and continuous systems.\u00a0Furthermore, the same concepts can be applied to problems from other areas of engineering, such as geotechnics, hydraulics, and heat transfer, as well as to problems outside of engineering altogether.\u00a0Finally, to conserve space and time, most of our studies will deal with planar structures subjected to planar loadings in the plane of the structure. This approach will retain enough generality that the resulting analysis methods can be readily extended to three-dimensional applications.<\/p>\n
Types of Ground Movement and Causes of Settlement<\/p>\n
The relationship between ground movement and the stability of related structures is a complex one. First of all, there are several mechanisms which may produce ground movement, and furthermore there are many types of structure, each with a varying potential to withstand or to be distressed by movement. Some buildings, such as those of brick and masonry construction, are exceedingly brittle and may sustain cracks and even structural damage following very small foundation displacements.Others may be constructed to sustain movements of considerable magnitude without suffering real damage.<\/p>\n
It is important to realize that soil conditions are apt to change ,sometimes considerably ,from before ,to during ,and also after construction .It is the prediction of these changes that presents the most difficult task to the designer .Most building damage that occurs because of foundation movement occurs when unforeseen soil conditions arise;;inadequate site investigations and a lack of understanding of soil behavior are largely the root causes .As will be shown in this chapter,there are methods available by which the amount and rate of foundation settlement due to certain mechanisms can be estimate.These estimates will remain reasonably reliable providing that the soil conditions assumed for the calculation are:(a)a fair representation of the actual conditions ,and (b) likely to persist throughout the life of the building.<\/p>\n
It is useful to start a study of settlement by considering briefly a number of ground movement mechanisms which are potential causes of settlement.<\/p>\n
Compaction<\/p>\n
Compaction is a process whereby the soil particles are forced into a closer state of packing with a corresponding reduction in volume and the expulsion of air.\u00a0An input of mechanical energy is required and this is usually the result of self-weight loading or a surface surcharge.Vibrations due to traffic movement, heavy machinery and certain construction operations, such as pile-driving, have also been known to cause compaction settlement.\u00a0In earthquake zones, seismic shock waves may have a similar effect.\u00a0The most susceptible soils are loosely-packed sands or gravel-sands and fill material, particularly that which has been placed without adequate rolling or tamping.<\/p>\n
Consolidation<\/p>\n
In saturated cohesive soils the effect of increasing the load is to squeeze out some of the porewater, this process is called consolidation.\u00a0A gradual reduction in volume takes place until internal pore pressure equilibrium is reached; a reduction in loading may cause swelling providing that the soil can remain saturated.\u00a0A large part of the remainder of this chapter is devoted to detailed study of the consolidation process and to methods of assessing resulting settlements.It is essential to understand that a change in loading is required to start the process and that it may take several years for the final settlement to be achieved.<\/p>\n
The most susceptible soils are normally-consolidated clays and silts,and certain types of saturated fill .Peat and peaty soils can be highly compressible ,resulting in changes in stratum thickness of as much as 20 per cent under quite modest loading.<\/p>\n
Elastic Volumetric Settlement<\/p>\n
In overconsolidated clays increases in effective stress which do not exceed the yield point cause elastic (approximately) compression.\u00a0As the stress increases beyond the yield point, non-linear (consolidation) settlement occurs.\u00a0In heavily overconsolidated clays, therefore, since the yield point will be very high, settlement calculations can be based on elastic theory, using parameters referred to effective stresses.Alternatively, estimates may be based on the slope of the swelling-recompression curve.\u00a0The elastic behavior of clays is probably attributable to the flexing of thin and flaky clay particles.<\/p>\n
Immediate or Undrained Settlement<\/p>\n
Immediate or undrained settlement is that amount that takes place during the application of loading, but before any significant volume change has occurred.\u00a0Although it theoretically occurs in all loading situations, with slowly applied loading, it is masked by consolidation settlement as volume changes occur.\u00a0The calculation of amounts of immediate settlement are therefore normally related to quickly applied loading e.g., beneath building structure.\u00a0The undrained stiffness (Eu) can be assumed as an elastic constant for a given depth and so estimates can be obtained using elastic theory.<\/p>\n
Moisture Movement<\/p>\n
Some types of clay show a marked increase or decrease in volume as the water content is respectively increased or decreased.\u00a0Clays exhibiting these characteristics are alternatively called shrinkable clays or expansive clays and are found in certain areas of the southern and eastern countries.<\/p>\n
In this country ,it has been found that the effects of seasonal variations in water content can extend down to about 0.8m below the ground surface.Annual surface movements in the south-east of England as high as 50mm may be expected .These clays characteristically possess high liquid limits and plasticity indices .<\/p>\n
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