衢州学院本科毕业设计（论文）外文翻译

衢州学院本科毕业设计(论文)外文翻译

译文：

受扭构件的强度及变形

Dr.A.美赫.普拉萨德

8.1引言

钢筋混凝土结构中的扭转现象常常是由于构件之间的连续性而产生的。由于这个原因，扭转问题在本世纪前半叶相对来说就未曾受到足够的重视,而在设计中忽视扭转问题看来好象也没有造成什么严重后果。在最近十至十五年内，研究活动的大幅度增长显著提高了对这个问題的理解。目前世界各地巳经或正在对混凝土中扭转问題的很多方面进行探讨。第一次重要的、组织起来的，把这方面的知识和研究成果汇集起来的转动是由士ACI主办的专题讨论会。专题讨论会文集还对许多有价值的开创性系作进行了回顾.

到目前力止，大多数关于扭转的规范参考文献都从各向同性匀质弹性材料的性能中 借用的概念为依据的。现行的ACI规范第一次包含了关于扭转的设计建议。这些建议是以相当数置的实验资料为基础的，但是在综合了更多的得自现代研究成果的资料之后，这些建议大概还要进一步修改。

扭转现象可能是由于一阶效应或二阶效应的结果而产生的一阶扭转情况是发生在外荷载不能由扭转以外的另一种方式来承受的时候。在这种情况下，为了保持静力平衡 而需要的扭矩单一的确定出来。这种情况也可以称为平衡扭转。它主要是一个强度回题，因为如果扭矩强度得不到满足，机构或其部分就会破坏。如图8.1和8.8所示的沿着跨度通过悬臂承受偏心线荷载的简支梁和偏心受荷的箱形都截面梁是一阶扭转或平衡扭矩的例子。

图8.1 一阶扭转或平衡扭转的实例

在超静定结构扭转现象就可能是由于连续住的要求作为二阶效应而产生的。在设计中忽略这种连续性就会导致过宽的裂缝，但不一定有更严重的后果。设计者常常都直觉地忽略这种二阶扭转效应。支承板或次梁的框架边梁就是这种情况的典型(见图8.2 在具有刚性连接的空间结构中几乎没有可能避免由干变形的协调性所引起的扭转现象。某些堵如受边梁弹性约一束的壳体之类的结构较之其它结构对这类扭转就更为敏感。目的知识水平使我似有可能接近实际地估计在钢筋混凝土超静定结构中的不同加载阶段可能产生的扭转效应。

混凝土结构中的扭转现象很少是在没有其它作用的情况下发生的。通常都同时作用有弯矩、剪力和轴向力。大量较为近期的研究工，作都曾试图确定扭转与结构上的其它作用之间的相互影响规律。由于包含的参数很多，要确切地估量这一综合性能的所有方面就还需要做一些努力。

图8.2 超静定结构中的扭转

8.2承受扭矩的素混凝土

钢筋混凝土的受扭性能在开始开裂以前可以取对素混凝土的研究作为基础.因为钢筋在这个时候所起的作用是可以忽略不计的。

8.2.1·弹性性能介绍的众所周知的方法.圣维南(St, Venant)的古典解可以用于一般的矩形混凝土截面。干是最大的扭转剪应力vt便发生在长边的中点，并可由下式求得:

???11Tx2v

（8.1）

式中: T—作用于截面的扭矩

x,y—矩形截面的外轮廓尺寸，x

vt一图8.3中给出的应力系数，它是y/x的函数。

可能同样重要的是要知道一个特定的受扭构件的荷载一位移关系。这可以从下列熟悉的关系式得出:

d?tT? （8.2） dzGC式中: 0t—扭转角

T—作用的扭矩，它可能是沿跨度方向距离的函数 G——公式7.37所定义的剪切模量，

C一一扭转惯性矩，有时称作扭转常数或等效极惯性矩; Z--沿构件的距离。

对于矩形截面可得： C??1x3y （8.3）

其中β1是一个取决于1截面高宽比y/x的系数(图8,3),它考虑了剪应变在截面中的非线性分布。

图8.3 受扭矩形截面的劲度系数和应力系数

这些关系使一个长度为l的构件的抗扭劲度可以定义为在这个长度上导致单位扭转角时所兽的扭矩值，即：

GC Kt?

l 在超静定结构的一般弹性分析中可能同时需要用到抗扭劲度和抗弯劲度。一根构件的抗扭劲度公式8.4可以和一很远端嵌固的构件的抗弯劲度公式相对照。抗弯劲度是定义为导致单位转角所需的弯矩，即4EI/L，其中El为截面的抗弯刚度。

T形和L形复合截面的性能就更加复杂。不过，根据巴赫(Bach) 的建议，习惯上是假定把这种截面适当地划分成组成它的若干块矩形面积，这对设计应用来说是一个可以接受的近似办法。因此就假定每一块矩形面积只分担外扭炬中与它的抗扭刚度成比例的那一部分扭矩。如图8,4所示，翼缘的挑出部分应取成不与其它面积重叠。在形成梁翼缘的板中，能起作用的矩形面积的有效长度应取成不大于板厚的三倍。对于纯扭的情况来说，这是一种偏于保守的近似方法。

当菜用巴赫(Bach) 的近似方法时，图8.4a中第2号面积在总扭矩T中所承受的那分部分扭矩即为：

T2?Kt2i?1?3T

而由公式8.1得出的最大扭矩剪应力为：

T2 vt2??t22x2y2

这个近似方法之所以是偏于保守的，就是因为忽略了“连接效应”在能够形成剪力流的复合截面中，例如在箱形截面中，必须用另一种方式来划分截面。图8.4c表示了具休 作法。

图8.4 为抗扭分析面划分复合截面

弹性扭转剪应力在复合截面上的分布可以根据普兰德尔(PrandtI)的薄膜比拟法最恰当地想象出来，它的原理可以在弹性力学的一般著作中找到。在钢筋混凝士构中我们很难遇到上面提出的与线弹性性能有关联的各项假定能够得到满足的情况。 8.22 塑性性能

就延性材料而言，有可能达到一种在一个个别横截面的整个面积上都将产生屈服剪应力的状态。如果在整个截面上发生了屈服，那就可以比较容易地计算出塑性扭矩。 我们现在来考虑示于图8.5中的正方形截面，在那里屈服剪应力Vty巳遍布于它的各个象限。这时在一个象限中作用的总剪力vt便是：

vt?b而由截面承受的总扭矩即为：

b1 22

bb3 T?4V1?vty

33

图8.5 正方形截面的扭转屈服 图8.6 那代的沙滩比拟

用那代(Nadai)的“沙堆比拟法”可以得到同样的结果。按照这个比拟法，在一个给定截面上能够堆置的沙量是与该截而所能承受的塑性扭矩成比例的。矩形截面五的沙堆(或屋顶)(见图8,6),的高度为xvty，其中x为截面的较小尺寸。于是正方形截面上的棱锥体体积(图8.6)即为：

bv1yb3?vty T?b332长方形截面上的沙堆体积(图8.6)即为：

T?x2xvtyxvy?(y?x)xt32

x2x(y?)vty ?23所以: vty??tyTxy2 (8.7)

其中 ?ty??2 (8.7a)

1?x/3y

显然，当x/y=1时，ψty =3。当 x/y=0时，ψty =2。可以看出，公式8,7与考虑弹性性能所获得的表达式公式8. 1是类似的

混凝土的延性，特别是当受拉时，尚不足以使剪应力得到理想塑性分布。因此，素混凝土截面的极限抗扭强度将处于由薄膜比拟法(全弹性)和沙堆比拟法(全塑性）所预测的数值之间。剪应力能引起导致破坏的斜向(主)拉应力。因此，·从前述近似方法和混凝土的抗拉强度具有变异性的角度来看，由a.AC1318 - 71提出的用以确定素混凝士截面中由扭矩引起的名义极限剪应力的简化设计公式：

Vty=vta=3Ta/x2y （8.8） 便是可以接受的，其中x≤y而把Ψt或Ψty值为3，这对弹性理论是最小值，对塑性理论是最大值(见图8.3及公式8.7a),复合截面的极限抗扭强度可以近似地取为组成该截面的各个矩形所提供的强度之和。对于像图8.4中那样的截面，近似的结果即为： vtu?3Ta （8.8a） 2xy?其中对每个矩形都都是xy。公式8.8a也忽略了连接效应所起的作用(在相邻屋顶之

间的峡谷中还可以堆积的沙量)。这里的vtu值未能与混凝土的任何一个强度特性发生联系。不过，不少的试验均已表明，如果按公式8.8或公式8.8a计算，它的数值便是4.0fe'psi和7.0fe'psi之间。

主应力(抗拉强度)的概念将暗示破坏裂缝应在梁的各个面上沿着与梁轴成45。的段旋螺旋线产生。但这是不可能的，因为破坏面的边界线必须形成一个封闭环。徐增全曾假定在这里围绕一个大致与梁轴成45。并与矩形截面梁的长边表面相平行的轴有弯曲现象发生。这种弯曲作用将在以45。横交于梁的平面内引起压应力和拉应力。拉应力最终要引发表两裂缝。弯曲受拉裂缝一经发生，截面的抗弯强度就要降低，裂缝也就迅速伸延，随后就发生突然破坏。徐增全借助于高速摄影机观察了这个破坏过程。对于大多数结构来说，这种不配筋混凝土构件的抗扭(抗拉)强度几乎一点儿都不能加以利用。

8.2.3，筒形截面

由于剪应力的有利分布，筒形截面在抗扭方面是最有效的。这类截面在桥梁结构中得到了广泛的应用。图8.7示出了用于桥梁结构大梁的一些基本型式。这些大梁的抗扭性能是由图8.7a到图8.7g逐步提高的。

图8.7 用于桥梁截面的基本模型

当壁厚h与截面的外形尺寸相比相对较小时，就可以假定剪应力vt，系沿壁厚均匀分布的。若考虑如图8.8a所示的作用在筒形截面的各个无限小单元上的剪应力对一个适当的点所施加的力矩，则.截面的抵抗扭矩就可以表示为：T??hvtrds

乘积hvt=v0称为剪力流，它是一个常量。于是：

v0?TT或vt?2Aeh?rds

式中A为简壁中心线所包围的面积（图8.8中画有阴影线的面积)。环绕着薄壁筒

的剪力流的概念在考虑配筋的抗扭作用时是有用的。ACI规范建议，只要壁厚不小于x/10(见图8.8c），则适用于实心截面的公式8.8通过以下修改也可以用于空心截面：

vt?(x3Tu)2 4hxy (8.10)

式中x≤y。

公式8.9b是根据基本原理得出来的，它的好处是能用于弹性和全塑性这两种应力状态。

空心裁面的扭矩一扭转变形关系可以很容易地根据率变能原理推导出来。使作用扭杯所做的功(外功)等于剪应力所做的功(内功)，就可以求得筒形截面的扭片常数Cot: 内功=（1/2）*（剪应力之和*发生在筒形截面上的剪应变*单位长度） = 1??vtvthds?1

2G 外功=（1/2）*作用扭矩*单位长度构件上的扭转角 =1/2 *T *θ/l

故通过令此两式相等并利用公式8.9b即可求得扭矩和扭转角之间的关系为： T?因此，这类构件的抗扭劲度即为：

Kt?GCelGCe? l （8.4a）

式中C为筒形截面的当是极惯性矩，并可表示为：

C0?24A0

?ds/h （8.11）

这里要强调的是，前面对弹性及塑性性能的讨论是针对素混凝土的，而且提出的一些建议只适用于开裂前的低荷载情况。它们一可以用来预测斜裂缝的出现。

图8.8 空心截面

8.2受弯兼受扭的无腹筋梁

梁在承受扭转和弯曲作用时的破坏机理取决于哪一种作用比较突出。极限扭矩与极限弯矩的比值Tm/Mm是一个衡量这两种作用相对大小的较为适宜的参数。抗弯强度主要是取决于抗弯钢筋的数量。而在有弯曲作用的情况下无腹筋棍凝土梁的抗扭性能则比较难于佑计。

弯曲应力在有扭转作用的情况下几乎和它在有剪切作用的渭况下一样也起着引发斜裂缝的效应。在有弯曲作用时这些裂缝在受压区兢受到了抑制。因此，斜向开裂的梁仍有能力承担一定数量的扭矩。通过什么方式来承担这一扭矩目前还是个推测。

图8.9 沿着边梁的屈服线

很明显，梁的受压区能够承担一个有限数量的扭矩，水平钢筋一也可以通过销栓作用提供一定的抗扭强度。

现己发现〔例如由马托克(Mattock)〕，如果有一定数量的弯矩存在，则巳开裂截面的抗扭强度就大约是未开裂截面极限抗扭强度的一半。因此，也就是说构件在裂缝形成之后还可以承受开裂扭矩的一半。这时承受的扭矩已经到可该把它对弯曲作用的影响忽略不计的程度。而ACl 318一71是把这一有限的扭矩所对应的名义扭转剪应力偏子保守地假定为开裂剪应力6fe'psi(0.5fe'N/mm2)的40%。

'?v?0.4(6f te) （8.12）

x2y因而开裂后仅由混凝土截面承受的扭矩便可由公式8.8展示为： Te?v1e

3同理，对于复合截面，由公式8.8a可得:

x2yvte （8.13） Te??3其中应按图8.4所示对悬出部分进行限制。

当Tm/Mm>0.5时(即扭转作用较为显著时)，观察到的是脆性破坏.而当弯矩较为显著时(即Tm/Mm<0.5时)，就可望产生较为具有延性的破坏。梁的抗扭强度只有在增加腹筋的情况下才能提高。抗弯钢筋的数量看来对混凝土截面的抗扭能力没有影响。 在T形和L形梁中，翼缘的挑出部分对抗扭强度是起作用的。这已通过独立梁得到证实。当翼缘是楼板的一部分时，它的有效宽度是难以确定的。当由千于版中负弯矩的作用面有可能如图8.9所示沿边梁形成一条屈服线时，翼缘的一大部分看来就不大可能再提供什么抗扭强度了。在这种情况下只依靠矩形截面就比较合理。 8.4无腹筋梁中伯扭转与剪切

显然就叠加的意义来说由扭矩和剪应力所引起的剪应力在矩形梁截面的一边是相加，而在对面一边则是相减。这样接着产生的临界斜拉应力又会受到混凝土中弯曲拉应力的进一步影响，因为不可能只作用剪力而同时却没有弯矩产生。现在还没有听说已经研究出了一个在考虑弯曲作用的情况下对剪切与扭转的相互作用进行分析的十分合理的理论.由子这个原因就必须依靠由试验得到的经验数据。在设置的抗弯钢筋多于需要的条件下，就有可通过实验来研究剪扭联合作用时的破坏判别条件.通常在这类试验中都要在荷载增大直至破坏的过程中使扭矩与剪力的比值维持不变，然而实际上却可能是一种作用首先发生，并在另一种作用显著增大之前就使构件产生了与其作用相应的裂分布图形。因此在分析试验结果方而权且偏于保守是合理的。

图8.10绘出了在典型的扭一剪共同作用试脸中获得的试验点子的散布倩况，它还表明只要选用了足够低的剪切.斜向开裂和扭转斜向开裂应力值，一条圆弧形的相互作用曲线(对这一组特定试验进行了公称化处理)是可以用于设计的。对于这些不设腹筋的梁来说，由公式7.5和公式8.8计算出来的、对途中所绘出的那些试验点子形成了近似下限的剪应力和扭转剪应力值分别为

ve2.68fe'psi(0.22fe'N/mm2) vte?4.80fe'psi(0.41fe'N/mm2)

这个圆弧形相互作用关系曲线是现行ACI规范条文的基础。为了方便起见，可以把已开裂截面在极限荷载时所承受的相互作用的剪力值和扭矩值用名义应力表示为：

(vte2.4fe')2?(ve2fe')2?1 （8.14）

式中:vtm—在极限情况下引起的由混凝土承受的名义扭转应力，由公式8.8给出；Vm-在极限情况下引起的由棍凝土承受的剪应力，由公式7.5给出。

图8.10 扭转与剪力的相互作用

原文:

Strength and Deformationof Members with Torsion

Dr. A. Meher Prasad

8.1 INTRODUCTION

Torsion in reinforced concrete structures often arises from continuity between members. For this reason torsion received; relatively scant attention during the first half of this century, and the omission from design considerations apparently had no serious consequences. During ;the last 10 to 15 years, a great increase in research activity has advanced the understanding of the problem significantly. Numerous aspects of torsion in concrete have been,and currently are being, examined in various parts of the world. The first significant organized pooling of knowledge and research effort in this field was a symposium sponsored by the American Concrete Institute. The symposium volume also reviews much of the valuable pioneering work.

Most code references to torsion to date have relied on ideas borrowed from the behavior of homogeneous isotropic elastic materials. The current ACI code8.2 incorporates for the first time detailed design recommendations for torsion. These recommendations are based on a considerable volume of experimental evidence, but they are likely to be further modified as additional information from current research efforts is consolidated.

Torsion may arise as a result of primary or secondary actions. The case of primary torsion occurs when the external load has no alternative to being resisted but by torsion. In such situations the torsion, required to maintain static equilibrium, can be uniquely determined. This case may also be refer-red to as equilibrium torsion. It is primarily a strength problem because the structure, or its component, will collapse if the torsional resistance cannot be supplied. A simple beam, receiving eccentric line loadings along its span,cantilevers and eccentrically loaded box girders, as illustrated in Figs. 8.1and 8.8, are examples of primary or equilibrium torsion.

In statically indeterminate structures, torsion cart also arise as a secondary action from the requirements of continuity. Disregard for such continuity in the design may lead to excessive crack widths but need not have more serious consequences. Often designers intuitively neglect such secondary torsional effects. The edge beams of frames, supporting slabs or secondary-beams, are typical of this situation (see Fig. 8.2). In a rigid jointed space structure it is hardly possible to avoid torsion arising from the compatibility of deformations. Certain structures, such as shells elastically restrained by edge beams,\

of torsion than are other.

The present state of knowledge allows a realistic assessment. of the torsion that may arise in statically indeterminate reinforced concrete structures at various stages of the loading.

Torsion in concrete structures rarely occurs. without other actions.

Usually flexure, shear, and axial forces are also present. A great many of the more recent studies have attempted to establish the laws of interactions that may exist between torsion and other structural actions. Because of the large number of parameters involved, some effort is still required to assess reliably all aspects of this complex behavior.

8.2PLAIN CONCRETE SUBJECT TO TORSION

The behavior of reinforced concrete in torsion, before the onset of cracking,can be based ors the study of plain concrete because the contribution of rein-force ment at this stage is negligible.

8.2.1 Elastic Behavior

For the assessment of torsional effects in plain concrete, we can use the well-known approach presented inmost texts on structural mechanics. The classical solution of St.Venant can be applied to the common rectangular concrete section. Accordingly, the maximum torsional shearing stress vt is generated at the middle of the long side and can be obtained from

where T=torsional moment at the section

y,x =overall dimensions of the rectangular section, x

It may be equally as important to know the load-displacement relationship for the member. This can be derived from the familiar relationship.

where θt,= the angle of twist

T = the applied torque, which may be a function of the distance along the span G = the modulus in shear as defined in Eq. 7.37

C = the torsional moment of inertia, sometimes referred to as torsion constant or equivalent polar moments of inertia z = distance along member For rectangular sections, we have

in which βt, a coefficient dependent on the aspect ratio y/x of the section (Fig.8.3), allows for the nonlinear distribution of shear strains across the section.

These terms enable the torsional stiffness of a member of length section. l to be defined as the magnitude of the torque required to cause unit angle of twist over this length as

In the general elastic analysis of a statically indeterminate structure, both the torsional stiffness and the flexural stiffness of members may be required.Equation 8.4 for the torsional stiffness of a member may be compared with the equation for the flexural stiffness of a member with far end restrained,defined as the moment required to cause unit rotation, 4EI/1, where EI =flexural rigidity of a section.

The behavior of compound sections, T and L shapes, is more complex.However, following Bach's suggestion, it is customary to assume that a suitable subdivision of the section into its constituent rectangles is an accept-able approximation for design purposes. Accordingly it is assumed that each ,rectangle resists a portion of the external torque in proportion to its torsional rigidity. As Fig. 8.4a shows, the overhanging parts of the flanges should be taken without overlapping. In slabs forming the flanges of beams, the effective length of the contributing rectangle should not be taken as more than three times the slab thickness. For the case of pure torsion, this is a conservative approximation.

Using Bach's approximation,8.5 the portion of the total torque T resisted by element 2 in Fig.

8.4a is

and the resulting maximum torsional shear stress is from Eq. 8.1

The approximation is conservative because the \ Compound sections in which shear must be subdivided in a different way.The elastic torsional shear stress flow can occur, as in box sections,Figure 8.4c illustrates the procedure.distribution over compound cross sections may be best visualized by Prandtl's membrane analogy, the principles of which may be found in standard works concrete structures, we seldom encounter the on elasticity.\In reinforced foregoing assumptions associated with linear conditions under which the elastic behavior are satisfied.

8.2.2 Plastic Behavior

In ductile materials it is possible to attain a state at which yield in shear occur over the whole area of a particular cross section. If yielding occurs over the whole section, the plastic torque can be computed with relative ease.

Consider the square section appearing in Fig. 8.5, where yield in shear Vty has set in the quadrants. The total shear force V acting over one quadrant is

The same results may be obtained using Nadai's ‘sand heap analogy.’ According to this analogy the volume of sand placed over the given cross section is proportional to the plastic torque sustained by this section.the heap (or roof) over the rectangular section (see Fig. 8.6) has a height xv.

where x = small dimension of the cross section.mid over the square section (Fig. 8.5) is

The volume of the heap over the oblong section (Fig. 8.6) is

It is

evident that Ψty=3 when x/y= I and O,y =2when x/y=0

It may be seen that Eq. 8.7 is similar to the expression obtained for elastic behavior, Eq. 8.1.

Concrete is not ductile enough, particularly in tension, to permit a perfect plastic distribution of shear stresses. Therefore the ultimate torsional strength of a plain concrete section will be between the values predicted by the membrane (fully elastic) and sand heap (fully plastic) analogies. Shear stresses cause diagonal (principal) tensile stresses, which initiate, the failure. In the light of the foregoing approximations and the variability of the tensile strength of concrete, the simplified design equation for the determination of the nominal ultimate sections, proposed by shear stress induced by torsion in plain concrete ACI 318-71, is acceptable:

where x ≤y.

The value of 3 for t is or ty,3, is a minimum for the elastic theory and a maxi-mum for the plastic theory (see Fig. 8.3 and Eq. 8.7a).

The ultimate torsional resistance of compound sections can be mated by the summation of the contribution of the constituent sections such as those in Fig. 8.4, the approximation is

where x ≤ y for each rectangle.

The principal stress (tensile strength) concept would suggest that failure cracks should develop at each face of the beam along a spiral running at 450 to the beam axis. However, this is not possible because the boundary of the failure surface must form a closed loop. Hsu has suggested that bending occurs about an axis parallel to the planes that is at approximately 450 to the beam axis and of the long faces of a rectangular beam. This bending causes compression beam. The latter tension cracking eventually and tensile stresses in the 450 plane across the initiates a surface crack. As soon as flexural occurs the flexural strength of the section is reduced, the crack rapidly propagates, and sudden failure follows. Hsu observed this sequence of failure with the aid of high-speed motion pictures. For most structures little use

can be made of the torsional (tensile) strength of unreinforced concrete members.

8.2.3 Tubular Sections

Because of the advantageous efficient in resisting distribution of shear stresses, tubular sections are most resisting torsion. They are widely used in bridge construction .Figure8.7 illustrates the basic forms used for bridge girders. The torsional properties of the girders improve in progressing from Figs. 8.7a to 8.7g.

When the wall thickness h is small relative to the overall dimensions of the section, uniform shear stress across the thickness can be assumed. By considering the moments exerted about a suitable point by the shear stresses,acting over infinitesimal elements of the tube section, as in Fig. 8.8a, the torque of resistance can be expressed.as

The product hvt = vo is termed the shear flow,.and this is constant; thus

where Ao = the area enclosed by the center Jine of the tube wall (shaded area in Fig. 8.8). The concept of shear flow around the thin wall tube is useful when the role of reinforcement in torsion is considered.

The ACI code 8.2 suggests that the equation relevant to solid sections. 8.8, be used also for hollow sections, with the following modification when the wall thickness is not less than x/l0 (see Fig. 8.8c):

where x ≤y.

Equation 8.9b follows from first principles and has the advantage of being applicable to both the elastic and fully plastic state of stress.

The torque-twist relationship for hollow sections may be readily derived from strain energy considerations. By equating the work done by the applied torque (external work) to that of the shear stresses (internal work), the torsion constant CO for tubular sections can be found thus:

Hence by equating the two expressions and using Eq. 8.9b, the relationship between torque and angle of twist is found to be

and the torsional stiffness of such member is therefore

where C0 is the equivalent polar moment of inertia of the tubular section and is given by

where s is measured around the wall centerline. The same expression for the more common form of box section (Fig. 8.8b) becomes

For uniform wall thickness Eq. 8.11 reduces further to

where p is the perimeter measured along the tube centerline.

It is emphasized that the preceding discussion on elastic and plastic behavior relates to plain concert .and the propositions are applicable only at low load intensities before cracking. They may be used for predicting the one of diagonal cracking.

8.3 BEAMS WITHOUT WEB REINFORCEMENT SUBJECT TO

FLEXURE AND TORSION

The failure mechanism of beams subjected to torsion and bending depends on the predominance of one or the other. The ratio of ultimate torque to moment, TJ/MU,is a suitable parameter to measure the relative magnitude of these actions. The flexural resistance depends primarily on the amount of flexural reinforcement. The -torsional behavior of a concrete beam without web reinforcement is more difficult to assess in the presence of flexure.

Flexural stresses initiate diagonal cracks in the case of torsion, much as they do in the case of shear. In the presence of flexure these cracks are arrested in the compression zone. For this reason a diagonally cracked beam is capable of carrying a certain amount or torsion. The manner in which this torsion is resisted is, at present, a matter, of speculation. Clearly the compression zone of the beam is capable of resisting a limited amount of torsion,.and horizontal reinforcement can also contribute to torsional resistance by means of dowel action.

It has been found (e.g., by Mattock\approximately one-half the ultimate torsional strength of the uncracked section, provided a certain -amount of bending is present.

Thus one half the torque causing cracking can be sustained after the formation of cracks. The torque thus carried is so small that its influence on flexure on can be ignored.

The nominal torsional shear stress, corresponding to this limited torsion is conservatively assumed by ACI 318-71. to be 40 % of a cracking stress of

and the torque supplied by the .concrete section only, after the onset of cracking, is revealed by Eq. 8.8 to be

Similarly, for compound sections, Eq. 8.8a gives

with the limitations on overhanging parts as indicated in Fig. 8.4.

When T./M > 0.5 (i.e., when torsion is significant), brittle failure has been observed. When the bending moment is more pronounced, (i.e., when T/Mu < 0.5), a more ductile failure can be expected. The torsional strength of abeam can be increased only with the addition of web reinforcement. The amount of flexural reinforcement appears to have no influence on the torsional capacity of the concrete section, T .

In T or L beams the overhanging part of the flanges contribute to torsional. strength. This has been verified on isolated beams. The effective width of flanges, when these are part of a floor slab, is difficult to assess.When a yield line can develop along an edge beam because of negative bending moment in the slab, as illustrated in Fig. 8.9 it is unlikely that much of the flange can contribute toward torsional strength.

8.4 TORSION AND SHEAR IN BEAMS WITHOUT WEB REINFORCEMENT It is evident that in superposition ; the shear stresses generated by torsion and shearing force are additive along one side and subtractive along the opposite side of a rectangular beam section .The critical diagonal tensile stresses that ensue are further affected by flexural tensile stresses in the concrete, because it is impossible to apply shearing forces without simultaneously inducing flexure. A fully rational theory for the interaction of shear and torsion in the presence of bending is not known to have yet been developed. For this reason reliance must be placed on empirical information derived from tests. By providing more than adequate flexural reinforcement, it is possible to experimentally study the failure criteria for combined shear and torsion. It is usual in such tests to keep the torsion to, shear ratio constant while the load is being increased to failure. However, in practice one action may occur first,imposing its own crack pattern before the other action becomes significant. For the time being, it is advisable to be conservative in the interpretation of test results.

Figure 8.10 plots the scatter obtained in typical combined torsion-shear tests. It also indicates that a circular interaction relationship (normalized for this particular group of tests) can be useful for design purposes, provided sufficiently low stress values for diagonal cracking by shear and torsion are chosen. For these beams,which contained no web reinforcement, the shear and torsional stresses which formed an approximate lower bound for the plotted experimental points, as computed from Eqs. 7.5 and 8.8, were found to be, respectively,

The circular interaction action relationship is the basis of the current ACI code provisions. 8.2

For convenience, the magnitude of the interaction shear and torsional forces carried by a cracked section at ultimate load can be expressed in terms-of nominal stress as

where

vt = induced nominal torsional stress carried by the concrete at ultimate, given by Eq. 8.8

vu = induced nominal shear stress carried by the concrete at ultimate given by Eq. 7.5

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