Chapter1 Structural Mechanics

1、Chapter1 Structural Mechanics2.Structural StabilityA fundamental consideration in designing is that of assuring stability under any type of possible loading conditions.设计时的一个基本考虑是确保结构在任何可能荷载条件下的稳定性。All structures undergo some shape changes under load.所有结构在荷载作用下均会产生形状的变化。In a stable structure the def

2、ormations induced by the load are typically small, and internal force generated in the action of the load tend to restore the structure to its original shape after the load has been removed.在一个稳定结构中，由荷载产生的变形通常很小，当卸载后，由荷载作用产生的内力常常致力于使结构恢复原状。In a unstable structure the deformations induced by a load a

3、re typically massive, and often tend to continue increasing as long as the load is applied.在一个不稳定结构中，由荷载产生的变形通常很大，并常常持续增加，在荷载作用下。An unstable structure does not generate internal force s that tend to restore the structure to its original configuration（形状，外形）。一个不稳定结构不会产生使其恢复原状的内力。Un stable structures

4、often quite often collapse completely and instantaneously（瞬间的） as a load is applied to them.当施加荷载时，不稳定结构常常会瞬间全部倒塌。It is the fundamental responsibility of the structural designer to assure that a proposed structure does indeed form a stable configuration.结构设计师的基本责任就是要保证一个建议结构确实形成一个稳定结构。Stability is a

5、 crucial（关键的） issue in the design of structures that are assembling of discrete elements.互不联系的构建组合而成的结构在设计时稳定性是一个关键问题。For example, the post-and-beam structure illustrated in Figure 1.2a is apparently table.例如，图1.2所示的梁柱结构明显是稳定的。Any horizontal force ,however tends to cause deformations of the type ind

6、icated in Figure 1.2b.任何水平力常引起图1.2b所示的变形。Clearly ,the structure has no capacity to resist horizontal load ,nor does it have any mechanism that tends to restore its original shape after the horizontal load is removed.明显的，这种结构没有抵抗水平荷载的能力，也没有任何卸掉水平荷载后使其恢复原状的构造。The large changes in angle that occur betw

7、een members characterize an unstable structure that is beginning to collapse.发生在各杆件之间的大角度变化是一个不稳定结构开始倒塌的特征。This particular structure will collapse instantaneously under load.这种特殊结构在荷载作用下会瞬间倒塌。Consequently ,this particular pattern of members is referred to as a collapse mechanism.因此，这种特殊类型的构件组合被称为一种倒

8、塌结构。 There are really only a few fundamental ways of converting a self-standing structure of the general type shown in Figure 1.2a-c from an unstable to a stable configuration.将图1.2a-c所示的典型独立结构从不稳定变成稳定确实有几种基本方法。These are illustrated in Figure 1.2d. 如图1.2d所示。The first is to add a diagonal member to t

9、he structure.首先是给结构增加一个斜杆支撑。The structure cannot now undergo the parallelogram（平行四边形） indicated in Figure 1.2b without a dramatic release in the length of the diagonal member (this would not occur if the diagonal were adequately sized to take the force involved.) 如果斜杆在长度上没有一个戏剧化的放松，结构就不会经历图1.2b所示的平行

10、四边形（如果斜杆尺寸满足受力要求，这种情况就不会发生）。Another method used to assure stability is though shear walls.另一种用来保证稳定性的方法是通过剪力墙。These are rigid planar surface that inherently resist shape changes of the typically illustrated.这些是刚性的面层，能从根本上抵抗所示的形状变化。A reinforced concrete or masonry（砖石） wall can be used as a shear wall

11、.钢筋混凝土或砖墙均可用作剪力墙。Either a full or partial wall can be used (the required extent of a partial wall depend on the magnitudes（量） of the force involved).全部或部分剪力墙可用（部门剪力墙所需长度取决于受力的大小）。A final method used to achieve stability is though stopping the large angular changes between members that are associated

12、 with collapse by assuring the nature of the connections between members is such that their angular relationship remains a constant value under any loading.最后一种达到稳定性的方法是通过阻止与倒塌有关构件之间较大角度变化，保证构件连接处的基本特征是在任何荷载作用下他们的角度关系是一个定值。This is done by making a rigid joint between members.通过将构件结点做成刚性便可做到。This is

13、a very common form of joint.这是一种非常常见的结点形式。There are ,of course ,variants（变体） on these basic methods of assuring stability.当然还有很多从这些基本保证结构稳定方法中的演变。Still most structures composed of discrete elements rely on one or the other of these basic approaches for stability.由互不联系构件组成的结构大多数仍然依赖于这些基本方法中的这种或那种来保证稳

14、定性。More than one approach can be used in a structure (e.g., a structure having both rigid joints and a diagonal), but a measure of redundancy（多余，重复） is obviously involved.一个结构中不止一种方法可用（例如一个结构可有钢节点和斜撑），但是很明显其中一种是多余的。1.2 Determinate(确定的) and Indeterminate Structures 1.2.1 Statically Determinate Struct

15、ures(静定结构)Structures are said to be statically determinate when the force and reactions produced by a given loading can be calculated using only the equations of equilibrium（平衡）.当给定荷载产生的力和反作用通过平衡方程即可求出，这种结构称为静定结构。The simply supported beam(简支梁) shown in Figure 1.3 is statically determinate.图1.3所示的简支梁

16、是静定的。We can solve for the three unknown reactions using the equations of equilibrium and then calculate the internal force such as bending moment ,shear force, and axial force at any given location along the length of the beam.应用平衡方程我们可求出三个反作用力进而计算出梁长任意位置处的内力，像弯矩、剪力和轴力。1.2.2 Statically Indeterminate

17、 Structures(超静定结构)The shown in Figure 1.4 is statically indeterminate structure.图1.4所示是超静定结构。There are four unknown reactions , .这里有四个未知反作用力。However ,there are only three independent equilibrium equations,然而，只有三个独立的平衡方程。The number of unknown is larger than the number of equations.未知数的个数大于方程个数。ByAAyA

18、xFMFF,0, 0, 0MFFyx1.2.3 Force Method(力法)The force method (also called flexibility method) is used to calculate internal force and reactions in statically indeterminate structures due to loads and imposed（外加的） deformations.力法（也称为柔度法）是用来计算由于荷载和外加变形超静定结构内产生的内力和反作用力。The steps in the force method are lis

19、ted as follows:(1)Determine the degree of statical indeterminate（超静定次数）of the structure. Parameter（参数） n will be used to denote（表示） the degree of indeterminacy.确定结构的超静定次数，用参数n表示。(2)Transform the structure into a statically determinate system by releasing a number of statical constraints equal to the

20、 degree of statical indeterminacy, n.通过放松与超静定次数n相等的静态约束，把结构变成一个静定结构。 This is accomplished by releasing external （外部的）support conditions or by creating internal hinges.这可通过释放外部的支撑条件或构造内部铰来完成。The system thus formed is called the primary system. Number the released constraints from 1 to n. 由此形成的系统称为基本体

21、系。释放的约束从1到n计数。(3)For a given released constraint j ,introduce an unknown redundant force corresponding to the type and direction of the released constraint.对于一个给定的释放约束j,对应释放约束的类型和方向引入一个未知多余力 。(4)Apply the given loading or imposed deformation to the primary system. Calculate displacement due to the g

22、iven loading at each of the released constraints in the primary system. These displacements are called jXjX02010,n对基本体系施加给定荷载或外加变形，计算基本体系在每个释放约束处所产生的位移。这些位移称为(5)For a given released constraint j,apply a unit load to the primary system. Calculate displacements due to at each of the released constrain

23、ts in the primary system. These displacement are called 02010,n1jX1jXn,21对于给定的释放约束j处，施加一个单位力于基本体系。计算基本体系中在每个释放约束处所产生的位移处所产生的位移。这些位移称为(6)Solve for redundant force throughby imposing the compatibility conditions （协调条件）of the original structure. 通过满足原结构的变形协调条件，求解多余力。1jXn,211XnXThese conditions transfor

24、m the primary system back to the original structure by finding the combination of redundant force that make displacement at each of the released constraints equal to zero.多余力在每个释放约束处位移的叠加为0，这些条件将基本体系转变回原结构。The conditions are expressed mathematically as followsThis is a system of n linear（线性的） equati

25、ons in n unknowns. The unknown force are 这些条件的数学表达式如下这是一个有n个未知数，n个线性方程组成的方程组。未知力是jXjXIt can thus be seen that the name of force method was given to this method because its primary computational task is to calculate unknown force ,i.e. (that is), the redundant force through由此看见，力法名称的由来是因为它的基本工作是求多余出未

26、知力。iXnX(7)Calculate force S at a given location in the structure using the follow combinationWhere quantities have been calculated from the n by n system of equations given in step 6, is the force due to the given load or imposed deformation in the primary system, and is the force due to applied to

27、the primary system. jX0SjS1jX通过以下组合计算结构给定位置处的力S，这里 通过第6步中给出的n乘n的方程组已经求出， 是基本体系在给定荷载或外加变形作用下产生的力， 是由于基本体系施加单位力 而产生的力。Force S can be bending moment ,shear , axial force ,or reaction.力S可以是弯矩、剪力和轴力或者反作用力。jX0SjS1jX1.2.4 The Classical Displacement Method(经典位移法)The force method is a method for calculating

28、the response of statically indeterminate structures by which the unknown are force quantities (the redundant forces )and the equations used to solve for the unknowns are based on geometrical(几何的) conditions (compatibility conditions at the location of each redundant force ) .nXXX,21力法是通过未知数是力的大小（多余

29、力 ）且求解方程是依据几何条件（每个多余力处的协调条件）来计算超静定结构反应的一种方法。It is possible to consider an analogous（类似的） method for calculating the response of statically indeterminate structures in which the unknown are displacement quantities and the equations used to solve for the unknown are based on statical conditions (equil

30、ibrium conditions).nXXX,21考虑一种类似的方法来计算超静定结构的反应，这种方法的未知数是位移大小且用来求解未知数的方程是基于静态条件（平衡条件）是有可能的。This method is referred to as the classical displacement method.这种方法称为典型位移法。It is helpful to outline the major elements of the method. It is assumed that the joints of the structure do not translational displac

31、ement(线位移).对这种方法的基本原理做一个概述是有用的。假定结构的结点不会发生线位移。(1)For a given structure and loading ,consider the joints to be fully fixed against rotation(旋转).对于一个给定的结构和荷载，考虑结点全面固定，不发生旋转。(2)Calculte the moment in each member of the structure due to the given loads ,assuming full fixity at the joints. These moments

32、are called fixed-end moments (固端弯矩).计算出给定荷载作用下，结构中每一个杆件的弯矩。假定结点充分固定，这些弯矩称为固端弯矩。(3)Caculate moments at the end of each member due to unit displacements of the joints.计算出由于结点单位位移在每个杆端产生的弯矩。(4)Express the total moments at each end of a given member as the sum of the fixed-end moment and the product of

33、unknown joint displacement times the moments produced by unit joint displacements, calculated in step 3. 未知结点位移乘以第3步计算出的单位位移产生的弯矩与固端弯矩之和作为给定杆件杆端总弯矩。(5)Generate an equation of moment equilibrium at each joint.每个结点处构造一个弯矩平衡方程。(6)Solve the system of equations for the unknown joint displacements.求解结点未知位

34、移的方程组。(7)Calculate the member end moments using the expression derived（获得） in Step 4 and the value joint displacements calculated in Step 6.用第4步得到的表达式和第6步的结点位移值计算出杆端弯矩。(8)Calculate all remaining force in the structure (shear force and axial force) .计算出结构剩余内力，剪力和轴力。Theoretically speaking , the can so

35、lve all statically indeterminate structures. The computational complexity of such an approach ,however ,is generally prohibitive for structures with more than three unknown forces. 理论上讲，力法可以求解所有的超静定结构。这种方法计算复杂，通常多于三个未知力便受限。In this regard ,the classical displacement method offers the following advant

36、ages over the force method:在这一点上，经典位移法较力法有以下优点：(1)It allows a solution based on a member-by-member procedure ,rather than one that requires consideration of the structure as a whole.它允许一种杆件和杆件的程序，而不需要把结构看做一个整体。(2)Because it is based on the per-solution of standard cases of intermediate(中间) load and

37、displacement ,it can often reduce the number of unknown in a given solution.因为它是基于先解决中间荷载和位移的标准形式，因此常可减少给定方法下未知数的个数。1.2.5 Moment Distribution Method（力矩分配法）The moment distribution method is used for statically indeterminate beams and frames by simple hand calculations. This is basically an iterative（

38、迭代的） process.力矩分配法用在简单手算法求解超静定梁和框架。它基本是一个迭代过程。The procedure involves artificially（人为） restraining temporarily（暂时） all the joints against rotation and writing down the fixed-end moments for all the members.这个过程需要人为地暂时约束所有节点不旋转并写下所有杆件的固端弯矩。The joints then released one by one in succession（连续）. At each

39、 released joint the unbalanced moments are distribute to all the ends of the members meeting at that joint.然后节点一个接一个的连续释放。在每一个释放的结点处，不平衡弯矩分配给所有连接在该点杆件的端部。 A certain fraction of these distribute moments are carried over to the far ends of members. The released joint is again restrained temporarily be

40、fore proceeding to the next joint.这些分配的弯矩的一部分又传到了杆件的远端。放松的结点再次被暂时约束直到进行下一个结点之前。The same set of operations are carried out at each joint till all the joints are completed . This completes one cycle of operations . The process is repeated a number of times or cycle till all the values obtained are wit

41、hin the desired accuracy.每个结点都要进行同样的饿操作，直至所有结点全部完成。这就完成了分配的一个循环。这个过程要重复很多次或多个循环直到得到的值在目标精度内。 Figure 1.5 indicates a basic problem of moment distribution . The question is ,given a unit moment applied to joint A, what moment are produced in each of the members.图1.5指出了弯矩分配中的一个主要问题。问题是，给定一个单位弯矩施加到结点A,在

42、每个杆件中会产生什么弯矩。One way to proceed is to solve the problem for a unit joint rotation and then scale the resulting solution for a unit moment. For a unit rotation the moment in each member at joint A is just its stiffness . The applied moment must then be 一种继续的方法是发生一个单位旋转，然后将结果按比例对应单位弯矩。发生一个单位转动，结点A处每个杆

43、件中的弯矩正是它的刚度 ，施加的弯矩是iiLEIK)/4(iiLEIK)/4(memberiKMmemberiKMWhich is called the joint stiffness. For an applied unit moment this solution scales so that the moment in each member at joint A is 这被称为结点刚度。对于施加的单位弯矩，这个结果按比例分配，因此结点A处每个杆件的弯矩是memberiiKK /memberiiKK /This is called the distribution factor of m

44、ember I at joint A .这称为结点A处i杆件的分配系数。The problem in Figure 1.6 can be solved. First , the center joint is fixed (the rotation is set to zero) which gives the so called fixed-end moment solution for beam on the right and no response in the beam on the left. 这个问题在图1.6中能得到解决。首先，中间结点固定（旋转归零），因此使梁的右端产生固端弯

45、矩解而左端没有反应。This solution is valid（合理的） except that it requires an external moment to be applied to the center joint .这个解是合理的，除非给中间结点施加一个外部的弯矩。The final solution is constructed by releasing or balancing the center joint which is equivalent to applying a clockwise moment of to this joint . Using the id

46、ea of distribution and carried over ,the solution is completed in this figure.最终解的建立通过释放或平衡中心结点，这同在该节点处施加一个顺时针弯矩 是等价的。应用弯矩分配和传递的方法，完成该图的最终解。12/2L12/2LNote that the sign convention （常规）used implies that a counter-clockwise（逆时针） moment on the end of a member is positive（正）. Care must then be exercised

47、 in drawing the final moment diagram（图） which uses a different sign convention.注意常用符号表示一个构件端部逆时针弯矩为正。在绘制最后的弯矩图时必须要小心，它采用不同的常用符号。1.3 Structural dynamics（结构动力学）1.3.1 Equations of Motion for Linear Single-Degree-of-Freedom System线性单自由度体系的运动方程The essential physical properties(物理性质) of a linearly elastic

48、 structural system subjected to external dynamic loading are its mass ,stiffness properties ,and energy absorption capability or damping（阻尼）. 线弹性结构体系受外部的动荷载作用时的基本物理性质是它的质量、刚度性质、能力吸收能力或阻尼。The principle of dynamic analysis may be illustrated by considering a simple single-storey structure as shown in

49、Figure 1.7. The structure is subjected to a time-varying force tfk is the spring constant （弹簧常数）that relates the lateral storey deflection（挠度） x to the storey shear force , and the dash pot （减震器）relates the damping force to the velocity（速度） by a damping coefficient c.K是弹簧常量，把侧面的楼层挠度x同楼层剪力联系在一起。减震器通过

50、阻尼系数c将阻尼力和速度联系在一起。If the mass ,m ,is assumed to concentrate at the beam ,the structure becomes a single-degree-freedom system. The equation of motion of the system may be written as 如果假定质量m集中，结构变成一个单自由度体系。该体系的动力方程可写作：If the structure is subjected to a sinusoidal motion such as a ground acceleration

51、of ,it will oscillate （震荡）and after some time the motion of the structure will reach a steady state. For example ,the equation of motion due to the ground acceleration （地面加速度）W is the resonate natural frequency (固有频率)of the system.twFxfgsin.如果结构受到像地面运动加速度这样一个正玄运动，它会震荡并且经过一段时间后，结构运动将达到一种稳定状态。例如，由于地面运

52、动加速度的运动方程其中w 是该体系的固有频率。The solution to the above equation consists of two parts , the general solution and the particular solution .上述方程的解包括两部分，通解和特解。If the system is damped , oscillation corresponding to the general solution will decay（衰减） with time . After some time , the motion will reach a stead

53、y state and the system will vibrate（颤动）e at a constant amplitude and frequency.如果系统受到阻尼，对应于通解的震荡会随时间衰减。一段时间后，运动会达到一种稳定状态且该体系会以固定的振幅和频率颤动。This motion ,which is called forced vibration（受迫振动） ,is described by the particular solution expressed as 这种运动称为受迫振动，通过特解描述，表达式为Substituting equation(1.3.3) into e

54、quation(1.3.2) ,the displacement amplitude can be shown to be 将方程（1.3.3）带入方程（1.3.2），位移振幅可表示为When the dynamic force is applied at a frequency much lower than the natural frequency of the system , the response is quasi-static. The response is proportional（按比例） to the stiffness of the structure ,and the displacement amplitude is close to the static deflection.当施加的动力频率远低于系统固有频率时，反应时准静态的。反应是和结构刚度成正比的，且位移振幅同静态挠度是接近的。When the force is applied at a frequency much higher th