1、Cracking Load and Ultimate Moment 开裂荷载和极限弯矩,教学目标,了解预应力梁在开裂荷载下的反应 了解有（无）粘结力的预应力构件的极限弯矩能力 熟悉预应力构件中的专业词汇 熟悉科技类文献中的常用句型,It has been shown that a variation in the external load acting on a prestressed beam results in a change in the location of the pressure line（压力线）for beams in the elastic range. This i
2、s a fundamental principle（规律）of prestressed construction. In a normal prestressed beam, this shift（移动）in the location of the pressure line continues at a relatively uniform rate（速度）, as（随着）the external load is increased, to the point where cracks develop in the tension fiber. After the cracking load
3、 has been exceeded, the rate of movement in the pressure line decreases as additional load is applied, and a significant（显著）increase in the stress in the prestressing tendon（预应力钢筋束）and the resultant concrete force begins to take place.,Action Under Overload-Cracking Load 超载（-开裂荷载）下的反应,已经显示作用在一根预应力梁上
4、的外部荷载的变化会导致梁在弹性阶段时其压力线位置的变化。这是预应力结构的基本规律。在一根普通的预应力梁中，随着外力的增加，其压力线位置以一个相对均匀的速度不断移动直至受拉纤维形成开裂的位置。在已经超过开裂荷载后，随着附加荷载的施加，压力线移动的速度便会降低，且使预应力钢筋束中的应力和合成的混凝土力开始发生显著的增加。,This change in the action of the internal moment continues until all movement of the pressure line ceases（停止）. The moment caused by loads th
5、at are applied thereafter（其后）is offset（抵消）entirely by a corresponding and proportional change in the internal forces, just as in reinforced-concrete construction. This fact, that the load in the elastic range and the plastic range is carried by actions that are fundamentally different, is very signi
6、ficant（重要的） and renders（使变得）strength computations essential for all designs in order to ensure that adequate safety factors exist. This is true even though the stresses in the elastic range may conform to（符合）a recognized（公认的）elastic design criterion.,内部弯矩的作用不断变化直到所有压力线的移动都停止。其后施加的荷载引起的弯矩完全通过相应的、且按比例
7、的内力变化来抵消，正如在钢筋混凝土的结构中。这个事实，即弹性阶段和塑性阶段的力由基本不同的作用来传送是非常重要的，且使强度计算变得对所有的设计都是必要的，以确保存在足够的安全系数。这是正确的，即使弹性阶段的应力可能符合公认的弹性设计准则。,It should be noted that the load deflection curve is close to（接近于）a straight line up to the cracking load and that the curve becomes progressively（逐渐）more curved as the load is inc
8、reased above the cracking load. The curvature（弯曲）of the load-deflection curve for loads over the cracking load is due to（由于所造成的）the change in the basic internal resisting moment action that counteracts（抵消）the applied loads, as described above, as well as to plastic strains that begin to take place i
9、n the steel and the concrete when stressed to high levels（受到很大压力）.应该注意到荷载挠度曲线在开裂荷载以内是接近直线的，且在超过开裂荷载后，该曲线随着荷载的增加逐渐变得更弯曲。正如上面所描述的，当超过开裂荷载时，其荷载-挠度曲线的弯曲是由于抵消施加荷载的基本内部抵抗弯矩作用的变化以及当受到很大压力时钢筋和混凝土中开始发生的塑性应变所造成的。,In some structures it may be essential that the flexural members remain crack free（保持构件不开裂）even u
10、nder significant（明显的）overloads. This may be due to the structures being exposed to exceptionally corrosive（特别腐蚀）atmospheres during their useful life（有效寿命）. In designing prestressed members to be used in special structures of this type, it may be necessary to compute the load that causes cracking of
11、the tensile flange（受拉翼缘）, in order to ensure that adequate safety against cracking is provided by the design. The computation of the moment that will cause cracking is also necessary to ensure compliance with（符合）some design criteria（criterion的复数形式，标准）.,在一些结构中，保持受弯构件不开裂可能是必要的，甚至在明显的超载下。这可能是由于结构在它们的有效
12、寿命期间被暴露在特别腐蚀的空气中的原因。当设计预应力构件用于这种类型的特殊结构时，可能有必要计算引起受拉翼缘开裂的荷载，以确保该设计提供足够抵抗开裂的安全性。也有必要计算会导致开裂的弯矩，以确保符合一些设计标准。,Many tests have demonstrated（证明）that the load-deflection curves of prestressed beams are approximately linear（近似直线的）up to and slightly in excess of（稍微超过）the load that causes the first cracks in
13、 the tensile flange. (The linearity（直线性）is a function of the rate at which the load is applied.) For this reason, normal elastic-design relationships（关系式）can be used in computing the cracking load by simply determining the load that results in a net（净）tensile stress in the tensile flange (prestress
14、minus the effects of the applied loads) that is equal to the tensile strength of the concrete. It is customary to assume that the flexural tensile strength of the concrete is equal to the modulus of rupture（断裂模量）of the concrete when computing the cracking load.,很多试验证明，在引起受拉翼缘最初开裂的荷载以内或稍微超过时，预应力梁的荷载-
15、挠度曲线是近似直线的。（直线性是荷载施加速度的函数。）因此，通过简单地确定导致受拉翼缘中产生一个净的受拉应力的荷载（预应力减去施加荷载的效应 ），普通的弹性设计关系式能用来计算开裂荷载，其值等于混凝土的抗拉强度。当计算开裂荷载时，习惯上假定混凝土的受弯抗拉强度等于混凝土的断裂模量。,In should be recognized that the performance（性能）of bonded（有粘结的）prestressed member is actually a function of the transformed section（换算截面）rather than the gross
16、（毛）concrete section. If it is desirable（想要）to make a precise estimate of the cracking load, such as is required in some research work, this effect（影响）should be considered.应该承认有粘结的预应力构件的性能实际上是一个换算截面的函数，不是混凝土毛截面的函数。如果想要对开裂荷载作一精确估计，例如在一些研究工作中需要的，该影响应该被考虑。,Principles of Ultimate Moments Capacity for Bon
17、ded members 有粘结的构件极限弯矩能力的规则,When prestressed flexural members that are stronger in shear and bond than in bending are loaded to failure, they fail in one of the following modes（方式）:当剪切和粘结比弯曲强的预应力受弯构件受荷失效时，它们会以下列方式中的一种失效：,（1） Failure at cracking load In very lightly prestressed members, the cracking
18、moment may be greater than the moment the member can withstand in the cracked condition and, hence, the cracking moment is the ultimate moment. This condition is rare（很少的）and is most likely to occur in members that are prestressed concentrically with small amounts of steel. It can also occur in holl
19、ow or solid（实心）prestressed concrete members that have relatively low levels of reinforcing. Determination of the possibility of this type of failure is accomplished by comparing the estimated moment that would cause cracking to the estimated ultimate moment, computed as described below. When the est
20、imated cracking load is larger than the computed ultimate load, this type of failure would take place if the member were subjected to the required loads. Because this type of failure is brittle failure, it occurs without warning designs that would yield（产生）this mode of failure should be avoided.,（1）
21、开裂荷载下的失效 在施加非常少量预应力的构件中，其开裂弯矩可能大于构件在开裂状态下能承受的弯矩，因此，开裂弯矩为极限弯矩。这种情况是很少的，且在构件中最可能发生用少量钢筋同心地施加预应力。在空心或实心的预应力混凝土构件中也会出现相对低等级的钢筋。 通过比较引起开裂的弯矩估计值与按下面描述（的方法）计算的极限弯矩估计值来确定这种失效的可能性。当估计的开裂荷载大于计算的极限荷载，则如果构件承受要求的荷载，这种失效将会发生。由于这种失效是脆性失效，因此其发生时没有警告-产生这种失效方式的设计应该被避免。,（2）Failure due to rupture of steel In lightly re
22、inforced（少量加筋）members subjected to ultimate load, the ultimate strength of the steel may be attained before the concrete has reached a highly plastic state（高度塑性状态）. This type of failure is occasionally encountered in the design of structures with very large compression flanges in comparison to（与相比）t
23、he amount of prestressing steel, such as a composite（复合）bridge stringer（纵梁）. Computation of the ultimate moment of a member subject to this type of failure can be done with a high precision. The method of computation, as well as the determination of which members are subject to this mode of failure,
24、 is described below.,（2）钢筋断裂引起的失效 在少量加筋并承受极限荷载的构件中，钢筋的极限强度可能在混凝土达到高度塑性状态之前就达到。这种失效在与预应力筋的数量相比有很大受压翼缘的结构设计中会偶然遇到，如一个复合的桥梁纵梁。可以高精度地计算易遭受这种失效的构件的极限弯矩。这种计算方法以及确定哪根杆件易遭受这种失效方式将在下面描述。,（3）Failure due to strain The usual underreinforced（配筋不足）, prestressed structure that are encountered in practice are of su
25、ch proportions（尺寸）that, if loaded to ultimate, the steel would be stressed well into（早已进入）the plastic range and the member would evidence（显示）large deflection. Failure of the member will occur when the concrete attains the maximum strain that it is capable of withstanding. It is important to understa
26、nd that research into（的调查）the ultimate bending strength of reinforced and prestressed concrete has led most investigators to the conclusion that concrete, of the quality（特性）normally（通常）encountered in prestressed work（工程）fails when the limiting strain of 0.003 is attained in the concrete.,（3）应变引起的失效
27、在实践中遇到通常配筋不足的预应力结构具有这样的尺寸，以至于如果加荷至极限，钢筋的应力早已进入塑性范围，而该构件将显示出很大的挠度。当混凝土达到其能承受的最大应变时，该构件将发生失效。明白对预应力钢筋混凝土的极限抗弯强度的调查已经导致多数调查者得到了结论，即当混凝土达到 0.003的极限应变时，具有在预应力工程中常遇性能的混凝土会失效，这点很重要。,Since the ultimate bending capacity is limited by strain rather than stress in the concrete, it is a function of the elastic
28、moduli（modulus的复数形式,模量）of the concrete and steel. The magnitude of the ultimate moment for members of this category can also be predicted（预测）, as a rule, within the normal tolerance（正常的允许误差）expected in structuraal design. The ultimate moment of underreinforced sections cannot be predicted with the s
29、ame precision as the lightly reinforced members described above, since the ultimate moments of underreinforced members are a function of the elastic properties of the steel and the effective stresses in the prestressing steel, whereas the ultimate moment capacities of lightly reinforced members are
30、not.,由于极限抗弯能力受到混凝土中的应变而不是应力的限制，因此，它是混凝土和钢筋弹性模量的函数。作为一个规律，这类构件极限弯矩的大小也能被预测在结构设计所预期的正常的允许误差范围内。 配筋不足截面的极限弯矩不能以与上面描述的少量配筋的构件相同的精度来预测，因为配筋不足的构件的极限弯矩是钢筋弹性性能和预应力钢筋中的有效应力的函数，而少量配筋构件的极限弯矩能力则不是。,（4）Failure due to crushing of the concrete Flexural members that have relatively large amounts of prestressing ste
31、el or relatively small compressive flanges are referred to as being overreinforced（超配筋的）. Overreinforced members, when loaded to destruction, do not attain the large deflections associated with underreinforced members the steel stresses do not exceed the yield point and failure is the result of the
32、concrete being crushed. Computation of the ultimate moments of overreinforced members is done by a trial and error （试算） procedure，involving assumed strain patterns（模式）, as well as by empirical relationships（经验关系式）.,（4）混凝土压碎引起的失效 有着相对大量预应力筋或相对小的受压翼缘的受弯构件称为是超配筋的。当受荷至破坏的超配筋构件没有达到与配筋不足构件有关联的大挠度-钢筋的应力没有超
33、过屈服点，因而失效是混凝土被压碎的结果。通过试算过程以及经验关系式来计算超配筋构件的极限弯矩，包括假定应变模式。,It must be emphasized that there is no clear distinction（明显的区别）between the different classifications（类别）of failure listed above. For convenience of design, certain parameters, which are a function of the percentage of steel, are used by differ
34、ent authorities（权威）to distinguish between the different types of failure that would be anticipated（预测）.必须强调在上面列出的不同的失效类别之间没有明显的区别。为了便于设计，通过不同的权威采用某些参数（这些参数是钢筋百分率的函数）来区别被预测的不同类型的失效。,In order to simplify the explanation of the theory related to the computation of the ultimate moments, a rectangular se
35、ction will be assumed throughout（在整个的过程中）the derivation, in order to eliminate（消除）the variable（变化因素）of flange width which is frequently encountered with I or T sections. In addition, the following assumptions, some of which differ slightly from those contained in ACI-318, are made:为了简化与极限弯矩计算有关的理论的解
36、释，在整个推导的过程中将假定一个矩形截面，以消除翼缘宽度的变化因素，其常常被遇到是I或T形截面。而且，作以下假定，其中一些稍许不同于ACI-318中包括的假定：,Plane sections are assumed to remain plane. The stress-strain properties（特点）of the steel are smooth curves without a definite（确定的）yield point. The limiting strain of the concrete is equal to 0.0034, regardless of the st
37、rength of the concrete. The steel and concrete are completely bonded.,假定平截面依然是平面的； 钢筋的应力-应变特点是光滑的曲线，上面没有确定的屈服点； 混凝土的极限应变等于0.0034，不考虑混凝土的强度； 钢筋与混凝土是完全粘结的；,The stress diagram of the concrete at failure is such that the average concrete stress is 0.80fc and the resultant of the stress in the concrete a
38、cts at a distance from the extreme fiber equal to 0.42 of the depth of the compression block. The strain in the top fiber under prestress alone（仅仅）is equal to zero. The section is subject to pure bending. The analysis is for the condition of static loads of short duration（持续时间）.,在混凝土失效时的应力图中，平均的混凝土应
39、力为0.80 fc，而混凝土中的应力合力则作用在离端部纤维的距离为0.42倍的受压区深度处； 仅在预应力作用下顶部纤维的应变等于零； 截面易遭受纯弯； 分析是针对静力荷载在短的持续时间下的情况。,As we stated above, the relationships that were developed are applicable to rectangular sections. These relationships are equally（同样得）accurate for flanged sections（带翼缘的截面）, provided（假如）the neutral axis（
40、中性轴）of the section at ultimate is within the limits of the flange. If the neutral axis falls outside of the flange area（区域）, the same strains distribution applied（采用）as in the case of rectangular sections, but due to the variable width of the section, the distance to the resultant of the compressive
41、 block must be calculated. To facilitate（方便）the calculation of the location of the resultant, the compression block can be assumed to be rectangular rather than curved without introducing significant error（导致重大错误）.,正如我们上面所述，那些被提出的关系式适用于矩形截面。假如在极限状态时截面的中性轴落在翼缘的边界之内，则这些关系式对有翼缘的截面同样得准确。如果中性轴落在翼缘区域的外面，则
42、采用与矩形情况中相同的应变分布，但是由于截面宽度的变化，必须计算（中性轴）至压力区合力的距离。为了方便计算合力的位置，可以假定受压区是矩形的，而不是弯曲的，这不会导致重大的错误。,When small quantities of non-prestressed reinforcement are used in combination with small quantities of prestressed reinforcement, the additional ultimate moment due to the non-prestressed reinforcement can be
43、calculated. For larger amounts of non-prestressed reinforcement or for members with high steel indices（index的复数形式，率）, the moment should be determined by trial and error from the basic strain patterns.当采用少量的非预应力钢筋再加上少量的预应力钢筋，则可以计算由于非预应力钢筋引起的附加的极限弯矩。对较多数量的非预应力钢筋或具有高钢筋率的构件，其弯矩应该根据基本的应变模式通过试算来确定。,Examin
44、ation（研究）will show that small variations（微小变化）in the effective prestress have no significant effect on the ultimate strength of prestressed members. It is important to note that even if errors are made in estimating the losses of prestress, in estimating the stressing（施加应力）friction, or even if the s
45、tressing is not carried out to a high precision in the field due to poor workmanship（低劣的手艺）, the effect on the ultimate moment is generally small for flexural members with bonded tendons.研究显示有效预应力的微小变化对预应力构件的极限强度没有显著的影响。很重要地注意到，即使错误地估计了预应力的损失和施加应力的摩擦，或即使在现场由于低劣的手艺而没有高精度地施加应力，对有粘结钢筋束的受弯构件极限弯矩的影响通常也是小
46、的。,Principles of Ultimate Moment Capacity for Unbonded Members 无粘结的构件极限弯矩能力的规则,Because the prestressing tendons can slip (with respect to the concrete) during loading of an unbonded member, the relationships for ultimate moment capacity do not apply to（适用于）unbonded beams. Because the tendons can sli
47、p with respect to the concrete, other variables（另外的变量）affect the ultimate moment capacity of unbonded prestressed concrete members.由于在对无粘结的构件加荷期间，施加预应力的钢筋束会相对于混凝土滑移，因此极限弯矩能力的关系式不适用于无粘结的梁。由于该钢筋束会相对于混凝土滑移，因此另外的变量影响了无粘结的预应力混凝土构件的极限弯矩能力。,Variables that affect the ultimate moment capacity of an unbonded
48、beam, but which do not affect bonded beams in the same manner or not at all（一点也不）, include the following: Magnitude of the effective stress in the tendons. Span to depth ratio（跨高比）. Characteristics of the materials. Form of loading (shape of the bending moment diagram). Profile（断面）of the prestressin
49、g tendon. Friction coefficient between the prestressing steel and duct（导管）. Amount of bonded non-prestressed reinforcing.,影响无粘结的梁的极限弯矩能力，但不影响或者根本不影响相同方式下的有粘结的梁的变量包括如下： 钢筋束中的有效应力大小； 跨高比； 材料特性； 加荷方式（弯矩图的形状）； 施加预应力的钢筋束的断面； 在施加预应力的钢筋和导管之间的摩擦系数； 有粘结的非预应力钢筋的数量。,A method of computing the ultimate strength of prestressed members (with unbonded tendons) that takes into account（考虑）the variables listed above has been proposed by Pannell. This method is based upon experimental data and is considered slightly conservative.Pannell已经建议用以计算考虑上述列出变量的（具有无粘结的钢筋束的）预应力构件极限强度的方法。该法是基于试验数据，并且考虑时稍有保守。,
