Concepts of Structural Analysis
Structures can be classified in a variety of ways. The casual observer might first consider classifying structures according to their respective functions: buildings, bridges, ships, aircraft, towers, and so on.
This basis for structural classification is in fact fundamental; all structures have some functional reasons for existence. It is the need to fulfill some function that prompts the designer to give life to a structure. Furthermore, 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.
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). The form and material of a structure in turn dictate its behavior, which in turn dictates the character of the analytical model. Fig. 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. At 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. A 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, and (2) geometric nonlinearitis that arise when the configuration of the structure under load is markedly changed from the unloaded configuration. 2）(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. Elastic 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. A 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.
All these behavioral aspects of the structure will have a significant influence on the nature of the analysis used in studying the structure. In 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. (the physical properties of homogeneous materials are the same at each point; those of nonhomogeneous material are not. The physical properties of isotropic materials are the same in all directions at a point; those 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. These include brittleness, ductility, flammability, texture, color, hardness, and machinability.
Finally, the nature of the loading, which is dependent on the function of the structure, will also influence the analysis. The only truly static loading on a structure is the dead, or gravity, loading. However, if other loadings are applied gradually enough, they are called quasi-static loadings and may be considered static for analysis purposes. Whether 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. Loads usually need to be treated as dynamic only if they are periodic in nature or if they are applied very suddenly. Even then, sometimes an “impact factor” 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. Thermal loading is an example of an internal initial distortion (or initial strain) loading.
Unfortunately, the picture of structural behavior is generally not so clear as that just painted. That is, materials are not either “linear” or “nonlinear” and “elastic” or “plastic”; instead, their behavior depends on circumstances such as environment and rate of loading. The 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. For 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.
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. Thus we will consider only the displacement and stress response due to static loading of linear, elastic, conservative structures. We will further restrict our attention to discrete-membered structures (rigid-and pin-jointed frameworks) as opposed to continuous structures. However, 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. Furthermore, 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. Finally, 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.
Types of Ground Movement and Causes of Settlement
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.
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.
It is useful to start a study of settlement by considering briefly a number of ground movement mechanisms which are potential causes of settlement.
Compaction
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. An 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. In earthquake zones, seismic shock waves may have a similar effect. The most susceptible soils are loosely-packed sands or gravel-sands and fill material, particularly that which has been placed without adequate rolling or tamping.
Consolidation
In saturated cohesive soils the effect of increasing the load is to squeeze out some of the porewater, this process is called consolidation. A 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. A 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.
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.
Elastic Volumetric Settlement
In overconsolidated clays increases in effective stress which do not exceed the yield point cause elastic (approximately) compression. As the stress increases beyond the yield point, non-linear (consolidation) settlement occurs. In 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. The elastic behavior of clays is probably attributable to the flexing of thin and flaky clay particles.
Immediate or Undrained Settlement
Immediate or undrained settlement is that amount that takes place during the application of loading, but before any significant volume change has occurred. Although it theoretically occurs in all loading situations, with slowly applied loading, it is masked by consolidation settlement as volume changes occur. The calculation of amounts of immediate settlement are therefore normally related to quickly applied loading e.g., beneath building structure. The undrained stiffness (Eu) can be assumed as an elastic constant for a given depth and so estimates can be obtained using elastic theory.
Moisture Movement
Some types of clay show a marked increase or decrease in volume as the water content is respectively increased or decreased. Clays exhibiting these characteristics are alternatively called shrinkable clays or expansive clays and are found in certain areas of the southern and eastern countries.
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 .
结构分析的概念
能用各种方法对结构进行分类。
不认真的观察者首先考虑的是根据其相应功能进行分类，如建筑物、桥梁、飞机、塔楼等等。
事实上这种结构分类的根据是基本的。
所有结构物都因其某些功能而存在。正是由于要使它们完成某些功能要求才促使设计者终生致力于结构设计。此外，也正是对某一功能的安全的、适用的、可行的、和美学上满意的实现决定了一个结构的形式、所用材料和加载方式。
一旦结构的形状和建筑材料确定之后，可将结构再按其形式分类（如拱、桁架或悬挂结构）或按其所用材料分类（如钢结构、混凝土结构或木结构）。结构的形式和建材反过来决定了结构的性能，其性能进而又分析模型的特点。图6.1形象地说明了结构的功能、形式、建筑材料、荷载、结构性能、分析模型储因素之间的关系。至此，我们有必要来讨论一下图6.1所示结构性能的一些方面，并我解释一下它们各自与结构的形式和建筑材料的关系。 如果一结构对其加载的响应，譬如某点的位移与所施加的荷载大小成正比，则此结构就是线性的。如果此比例不存在，则该结构就是非线性的。 结构非线笥分为两类材料非线性，此时材料的应力与应变不呈比例；几何非线性，此时在荷载作用下其形状与未加载前发生了很大变化。(例如结构中索的存在往往会引起几何非线性，因为索的下垂会产生位移，可以证明，这种位移与索中的内力并不成线性关系) 因此，结构所采用的建筑材料可能被分类为弹性、塑性或粘弹性。(位置，状态). 当卸除荷载后，弹性材料能回弹以其初始外形，但塑性材料会有一永久变形粘弹性材料的变形与时间有关，因而与加载历史有关，但弹性和塑性材料的变形却与时间无关。 一个结构体系是非保守的或保守的，取决于经过一次加载和卸载该体系中有无能量损失。如果卸载后体系并未回到其初始形状，通常都有能量损失，这是要么是由材料非线性引起，要么是由结构内部或其构件之间存在摩擦力。
结构的所有这些性能都将对研究结构时的分析方法起到很大的影响。而且，在建立分析模型时，必须考虑结构材料是否均质、是否各向同性，还是正交各向异性。均质材料的物理性能在各点都相同的，但非均质材料并非如此。各向同性材料的物理性能在各个方向都是相同的 但各向异性材料却并非如此。正交各向异性材料是一种特殊的各向异性材料，它在其三个主轴方向的特性不同，但在所有其它方向上的特性则取决于其三个主轴方向的特性。)结构的其它方面，尽管也是设计中要考虑的主要因素，通常将对分析方法影响不大。 这些因素包括脆性、延性、可燃性、质地、颜色、硬度和可加工性。
最后讨论一下加载特点，它取决于结构的功能，也会影响结构的分析。结构上真正的静力荷载是恒载，即重力荷载。然而，如果其它荷载施加的足够缓慢，就将其称为伪静力加载，从而分析时可认为是静力的。加载是否足够缓慢取决于加载持续时间是否大于所分析结构的基本周期。通常只有当荷载是周期性的或当共是突然施加的，才将其作为动力荷载处理。 即使在此情况下，有时在分析中采用一个所谓的“动力系数”来考虑突然施加荷载的效应，分析结果仍以静态加载形式结出。荷载还可分为外力或内部初始变形。热负荷就是内部初始变形（如初始应变）加载的典型例子。
不幸的是，通常对结构性能的描述并不象上述如此清楚。也就是说，材料并不是“线性”或“非线性”；也不是“弹性”或“塑性”，其性能取决于环境因素，如外界情况和加载速率。由于分析中所必须考虑的结构性能类型可能取决于要研究的结构响应的类型，这就使这种描述变得更加含糊不清。例如，比较简单的分析模型可能足以得到静态的位移和应力结果，但需要更复杂的模型以得到振动或曲屈分析结果。
为了阐明这一问题以讲解清楚结构矩阵分析方法，我们将对结构特性作一些简化假定。因此，我们将讨论线弹性保守结构因静力加载所引起的位移和应力。我们将进一步将注意力集中到离散杆系结构（刚结和铰结框架结构）而非连续结构。然而，重要的是要在开始就认识到我们将要介绍的概念可推广到许多其它结构问题，其中包括动力响应、材料及几何非纯属、非弹性、失稳和连续结构体系。而且，同样的概念也可应用于其它工程领域的问题，如土工学、水力学、热传导以及甚至是工程领域之外的问题。 最后，为了节省时间和篇幅，我们研究的大多数问题将涉及平面内受平面力作用的平面结构。这一方法将保持足够的普遍性，从而使所得到的分析方法能容易地推广到三维空间问题。
地面运动和相关结构稳定性之间的关系比较复杂。首先，导致地面运动的机理有多种，而且，结构形式也多种多样，每一种抵抗地面运动的能力也不同。 一些建筑物，如砖石结构，脆性很大，即使在很小的基础位移发生时就可能导致其开裂，甚至结构破坏。其它结构形式可经受很大的地面运动而不发生真正破坏。
地面移动类型和定居原因
重要的是要意识到,土壤条件是容易改变,有时相当,从之前,期间和之后建设。它是预测这些变化,将最艰巨的任务到设计师。大多数建筑物损害的发生是因为基础运动时发生不可预见的土壤情况的出现不充分的现场调查和缺乏了解的土壤行为在很大程度上是根本原因。我们会看到在这一章,有方法可用的数量和基础沉降速度由于某些机制可以被估计。这些估计仍将相当可靠的提供,土壤条件假设的计算是:(a)公平表示的实际条件,和(b)可能持续整个生命的建筑。
它是有用的开始研究结算通过考虑一个短暂的数量的地面运动机制是解决潜在原因。
弹性体积沉降
夯实是强迫土粒以更密集的形式堆积，从而导致其体积减小和空气被排出的过程， 夯实需要输入机械能，这通常是自重加载或地面附加荷载的结果。而且我们知道，由交通车辆引、重型机械和某些施工工作，如打桩等产生的振动也是夯实沉降的原因。在地震区，地震波也可能具有类似的功效。最易受振动作用的土包括松砂土、松砾砂土和填土，特别是未经充分碾实和夯实时。
瞬时（或不排水）沉降
在饱和粘性土中，增加荷载的效果就是将一些空隙水从土中挤出去，这一过程称为固结。其体积逐渐减小直到空隙水压力达到平衡为止；如果土仍能处于饱和状态，减载可引起土的膨胀。本章剩余的大量篇幅将详细研究土的固结过程及其由此引的起沉降的计算方法。极为重要的就是要明白必须有荷载的变化来产生此过程，且最终沉降可能需要数年才能完成。
最敏感的土壤是正常固结粘土和淤泥,和某些类型的饱和填补。泥炭和泥炭土壤可以是高度可压缩,导致地层厚度的变化多达20%在相当温和的加载。
在超固结土中，有效土压力的增加将导致土的弹性（大约为弹性）的压缩，如果土压力未超过其屈服点。而当其压力增至超过其屈服点后，非线性（即固结）沉降便发生。 因此，对严重超固结粘土，由于其屈服点很高，其沉降计算可根据弹性理论进行，并用其有效应力所对应的诸参数。也可根据其膨胀—再压缩曲线的斜率估计其沉降。粘土的弹性可能是来自其薄的片状颗粒的弯曲。
水分移动
瞬时（或不排水）沉降是加载过程中的沉降，此时土还未发生大的体积变化。尽管理论上讲，在各种加载过程中都发生此类沉降，但它被伴有体积变化的固结沉降所掩盖。因此，瞬时沉降量的计算通常与快速加载有关，例如，建筑物下面的沉降。可将某一给定深度处土的不排水刚度假定为弹性常数，从而用弹性理论估计其沉降量。
当一些粘土的含水量增加或减小时，会伴随着明显的体积增大或减小。具有此特性的粘土也称之为可收缩土或膨胀土，它们常出现在南部和东部的一些地区。
在这个国家,人们已经发现,季节变化的影响在含水量可以延伸至约0.8米以下的地面。年度表面运动在英格兰东南部的高达50毫米可以预期。这些粘土具有高液体限制和特有的可塑性指数。