Nonlinear Analysis of Torsion in Reinforced Concrete Members after Developing Initial Crack

Background/Objectives: Nonlinear analysis of torsion in reinforced concrete members after developing initial crack is a significant effect on piece life time. Methods/Statistical Analysis: As a result of integration and cohesion of members in the reinforced concrete structures Torsion typically occurs. Reduced torsional stiffness problem was presented after semibrittle break out in concrete due to presence of initial cracks in outer surfaces of concrete element in determining torsional stiffness. Findings: The idea was that reduced torsional stiffness is approximately one-tenth of its value before cracking and this is very effective on the distribution of anchors in the structure. After these studies, different torsion theories were presented which measured the reduced stiffness by using different methods. In this paper, by using these theories, a code was written for torsional analysis of reinforced concrete after cracking called “nlarctm” and 23 rectangular cross-sections were controlled and compared by this program. Comparisons of anchor changes was based on the torsion angle and results indicated that cross-section number (18) has higher torsional strength than similar cross-sections. Applications/ Improvements: The results of this research shows that by comparing three effects the resistance of reinforcements, compressive strength and the distance of collars, it’s concluded that all of them are very effective in increase or decrease.


Introduction
Torsion usually occurs due to integration and cohesion of members in the reinforced concrete structures.For this reason, this problem has less interested the designers of the armed concrete structures.In fact, torsion was considered as a secondary issue and as a result, its effect in designing was only seen as relying on large reliability coefficient which were used in the computational methods.During recent years, the effect of torsion and design for it has been emphasized and its certain criteria form a significant part of regulations.There is two basic reason for this change.First, design methods are based on more realistic concept of loads and the behavior of the structure than older methods and as a result, smaller reliability coefficients govern the design, and second, number and diversity of the structures in which torsion is the axis of the structure behavior-not a secondary effectis developing.Therefore, widespread researches has been conducted about the designing concrete structures under the effect of torsion in the few last years and now we can claim that the results of these researchers take the form of regulation criteria which can lead to secure and economic designs.In the structural systems, torsion is usually divided into two groups: Determined static torsion which sometimes is called balance torsion and undetermined static torsion which is called compatibility torsion.In the first type, i.e., anchor torsion, torsional anchor is not possible only by using the balance of forces or in other words, the static of the problem, but using torsion compatibility of connected members like circumference beam, slab or column is necessary.It is clear that for this case, value of the torsional anchor along with the beam which is equal with the bending anchor in the slab, Nonlinear Analysis of Torsion in Reinforced Concrete Members after Developing Initial Crack is the function of slab rotation in the connected end with the beam or in other words, it is a function of slab and beam stiffness.Generally, in stable undetermined structures, torsional anchor can be measured only when the torsional stiffness of reinforced concrete members is known.In a stable determined structure, like balcony cantilever slab, torsional anchor can be obtained from (static) balance equations and regulations' criteria can be used without problem; but the calculation of torsional anchor in a stable undetermined structure like a beam, needs using compatibility conditions, in addition to balance equations.In other words, torsional stiffness of beam should be considered.
Torsional stiffness of reinforced concrete crosssection is calculated with acceptable accuracy by Saint-Venan using elastic theory.But despite researches about torsion or a combination of torsion, bending and shear, our knowledge of torsional stiffness after cracking is limited and the need to this knowledge is highlighted by this fact that torsional members often crack under service loads and regulations for designing concrete structure remained silent about the torsional stiffness of cracked members.Besides, torsional stiffness after cracking is only a minor fraction of its value before cracking (approximately %10).In a fixed undetermined structure, this effective reduction in torsional stiffness after cracking will influence the distribution of anchors.In this paper, first we study the torsion theory.For this purpose, first the behavior of prismatic members under torsion will be evaluated by the inverse and semi-inverse methods; then, membrane analogy analysis for solving torsion problems was will be introduced and by using it, torsion problem and calculation of maximum shear stress caused by rectangular cross-sections will be analyzed.In following, plastic torsion will be calculated for rectangular crosssections using plastic theory and sand pile method.
Then, torsion theories for concrete members will be studied and compared for concrete members like simple concrete, reinforced concrete, pre-stressed concrete and different loading conditions like torsion, combination of torsion and bending and general condition of combination and bending and shear.In the torsion of reinforced concrete member, the basic problem is finding the thickness of shear flow zone or (t d ) which is similar to the depth of compressive zone or (c) in bending condition.Rausch classic theory calculates the torsional strength of reinforced concrete members as unrealistically higher than real value.Error in this theory is caused by error in determining the position of shear flow zone which directly depends upon the thickness of this zone i.e., (t d ).In this paper, a simple method is presented to calculate shear flow zone (t d ).This method is based on the truss model theory and balance equations and compatibility conditions.Nonlinear analysis of reinforced concrete members under torsion is conducted by using truss model theory.Based on this assumption, 16 basic equations have been obtained for non-prestressed concrete members based on balance equations, compatibility conditions and aggregates' stress-strain dependencies (steel and concrete).Using these 16 basic equations, a method is presented to draw T -θ or τ -γ diagram for reinforced concrete members under torsion after cracking.
In following, 23 rectangular cross-section of the reinforced concrete with different thicknesses were analyzed by help pf nlarctm program.This program is written based on the nonlinear analysis of reinforced concrete members which is presented in the appendix.By referring to the results obtained from nonlinear analysis of 23 rectangular cross-sections, first the thickness of shear flow zone and its changes based on the total percent of longitudinal and transversal steel reinforcements will be discussed.This study leads to equations that by help of them, we can calculate the thickness value of shear zone with good accuracy.In addition, after comparing the parameter for shear flow zone thickness by using Whitney equivalent stress block height in bending, a relation will be presented in order to calculate this parameter which has high accuracy and significantly simplifies the calculation of shear flow zone thickness parameter.Then, cracks 'angle, torsional strength and cross-section torsional stiffness will be evaluated based on the changes in different quantities.

inverse and Semi-Inverse Methods
Often, elasticity general equations with border conditions cannot be solved by direct method.In order to solve these problems, inverse and semi-inverse methods are useful and efficient methods.In inverse method, every function which held in relevant differential equations, will be considered as a solution for it.In an indirect method which was first introduced 1 , simplifying assumptions about the stress components or displacements were considered which converts high-order differential equations to equations that their mathematical solution is not complex as previous equations.In prismatic members torsion, it is assumed that displacement functions u,v,w have determined form.By imposing this assumption, governing equations convert to a differential equation.Consider prismatic bar with longitude L which is pinned in one end in plate and id under torsional moment around axis z (Figure 1).Rotation of member in the pinned end is prevented.The amount of rotation in each cross-section depends on the distance of cross-section to its pinned end.Because deformation are slight, it is reasonable that torsion angle α in each cross-section is corresponding to the distance from cross-section to pinned end.In other words: z θ α = whereθ is torsion angle to length unit of member.It is assumed that torsion angle α is small.Consider a crosssection with distance z from pinned end of member.After deformation, P' with coordinates (x, y, z) will displace to P' 1 with coordinate (x+u, y+v, z+w).Figure 2 shows this condition clearly 1 .By imposing small displacements (α~0), we have: Where in Equation ( 2), is torsion function.In following, by assuming that prismatic member is under centralized torsion without any external loading, stress equations can be expressed as Equation ( 3 By considering infinitely small element on the outer surface of the beam and regarding that stresses are zero on outer surfaces and considering beam stress tensor and cosines of angles for S-shape cross-section from Figure 2 with coordinate axis, we will have:  5) and shear stress Equations ( 6), we will have.
This equation shows that Prandtl stress function (torsion) should have fixed value along with surface border.In following, we can show that torsional anchor M t that its half is caused by shear stress and the other half by is equal with: Nonlinear Analysis of Torsion in Reinforced Concrete Members after Developing Initial Crack

Membrane Analogy Analysis
Membrane analogy analysis method was first used by L. Prandtl in solving torsion problems 1 .Consider a homogenous film which has support on its edges and its circumference is similar to the cross-section of twisted member.In addition, assumed film is placed under monotonic tension in its edges and monotonic lateral pressure is imposed to it (Figure 3).If the pressure on the surface of membrane is q and tension on circumference is s, result of tension forces imposed to ad and dc sides from small component abcd by assuming minor deformations is: Similarly, the result of tension forces imposed on two other sides of element is: By imposing forces' balance equation for element abcd from above membrane, we have: By comparing two Equations ( 6) and ( 10) and considering the border condition for film deformation (Z=0), we can find the similarity between these two relations.It is enough that in Equation (6), we use q/s instead of and by determining Prandtl stress function, we express peak of film as function of x and y.For more description, we considered a rectangular membrane sample with dimensions 2a*2b using membrane analogy analysis and determined displacement, torsion anchor, shear tensions and maximum shear tension functions.
Regarding that Z deformation function has symmetry in relation with axis of membrane, border conditions x = ± a and necessitates that Z = 0; therefore, deformation function is expressed as unlimited series: In this Equation, b n is fixed coefficient and Y n are functions of y.Now, we will express the right side of Equation (10) as Fourier series in interval -a<x<a, and we have: By substituting Equations ( 11) and ( 12) in differential Equation (10) and solving it and by imposing border conditions and substituting 2 G θ for q/s, Prandtl stress function is: By assuming b>a, we can conclude that the maximum shear stress occurs in the middle of larger side; therefore, maximum shear stress in points is equal with: Equation ( 4) gives maximum shear stress value for different b/a ratios.Table 1 demonstrates the maximum shear stress for different b a ratios.Based on this, maximum shear stress is considered as a coefficient of maximum shear stress for b a → ∞ and we have: Now, by having Prandtl stress function of Equation ( 13) and placing it in Equation ( 7), torsional anchor is:  16) indicates the value of torsional anchor to different b/a ratios.Table 2 demonstrates torsional anchor for different b a ratios.Based on this, we will consider torsional anchor value as a coefficient of torsional anchor for b a → ∞ and we have:  16)

Plastic Theory and Sand Pile Analogy Method
By assuming an elastic-plastic curve for stress-strain curve, we can obtain the stress values in those parts of cross-section which have reached to yield limit by help of plasticity theory.If in the torsion of prismatic members, shear stress reaches to yield point, shear stress components i.e., τ yz , τ xz should held in following formula: On the other hand, regarding stress balance equations and plastic function F(x,y), we have: By substituting Eequation (19) in yield surface equation, we have: Since none of the stress components are vertical on the surface, we have: τ yz , τ xz By using above Equation and Equation (19), we can show that dif (F) = 0; Therefore, plastic stress function has fixed value along with the mass border.This value if usually considered zero because its value has no effect on the values of stresses.Since the slope of assumed stress function is fixed, the volume surrounded by F curve is similar to sand pile which is accumulated on the vertical plate with similar surface to the cross-section of membrane under torsion and for this reason, it is called sand pile analogy method.If all cross-section reaches to plastic state, we can reach following conclusions using sand pile analogy method: • Shear stress is a fixed value throughout the cross-section.
• Total torsional anchor imposed to the membrane is corresponding with sand pile volume.• Stress function value is independent of torsion angle.
• A tall and thin member has lower resistance compared to a member with square cross-section with same cross-section area.• Plastic torsional strength of all tall cross-section with C, L, T, U shape is approximately same with plastic torsional strength of rectangular cross-section with same thickness and a length equal with total length of previous cross-section.By using sand pile analogy method, we can obtain the plastic torsion value for members with rectangular crosssection as below: Where, T u : Plastic torsional anchor.τ max : Maximum shear stress of aggregates.d: Depth of cross-section.b: Width of cross-section.
If only some parts of a cross-section reaches to plastic limit and other parts remain in elastic state, stress function and torsional anchor value can be obtained such that first we draw the sand pile surface, then elastic membrane extended inside it.Membrane contact point with sand pile cross-section shows the border between elastic and plastic states.

Torsion Theories of Concrete Members
Torsion theories of concrete members can be divided into three general classes based on the type of loading: • Torsion.
• Combination of torsion and bending.
• Combination of torsion and bending and shear.
In Table 3, different theories have been classified 2 .Approximately, all theories less considered the stiffness factor which can be used in analyzing and calculating the Nonlinear Analysis of Torsion in Reinforced Concrete Members after Developing Initial Crack membrane strength in initial cracking and break out.The only exception in this case is classic elastic theory that can calculate the stiffness of concrete membrane before cracking with reasonable confidence.None of these methods used for estimating and calculating the stiffness of a concrete membrane after cracking.

Torsional Stiffness of Reinforced Concrete after Cracking
In stable undetermined reinforced concrete structures, torsional anchor can be measured when torsional stiffness of reinforced concrete members is known.In this section, by using new concept shear module post-cracking, torsional stiffness post-cracking theory is expressed.
Using this new concept leads to a general theory which is useable for each desired cross-section and simplifies the calculation of torsional stiffness after cracking.In ACI-1971 regulation, new criteria have been presented for reinforced concrete members in torsion.If torsional anchors were known, we can easily use these criteria.In a stable determined structure like balcony cantilever slab, torsional anchor can be obtained from static balance equations and regulations' criteria can be used without any problem.But calculation of torsional anchor in a stable undetermined structure like beam, needs compatibility conditions, in addition to balance equations.In other words, beam torsional stiffness should be considered.
Torsional stiffness of reinforced concrete cross-section before cracking has been calculated with reasonable accuracy by Saint-Venant using elastic theory 1 .But instead of research like 3,4 about torsion or a combination of torsion, bending and shear, our knowledge of torsional stiffness after cracking is not high and need to this knowledge highlights with this reality that torsional members often crack under service loads and design regulations for concrete structures remain silent about torsional stiffness of cracked members.Besides, torsional stiffness after cracking is only a small fraction of its value before cracking (%10).In a stable undetermined structure, this effective reduction in torsional stiffness after cracking will significantly influence the distribution of anchors.
By conducting experiments in Portland Cement Association (PCA), has presented empirical equations for calculating the torsional stiffness after cracking.Results of these experiments indicates that torsional stiffness after cracking is a function of torsional reinforcements' percent 5,6 .
Scientific calculation of torsional stiffness after cracking was first done by Robinson.His suggested equations were reasonable, although these equations were only applicable for circular cross-sections and torsional stiffness should be calculated with trial and error method.For rectangular cross-sections which have higher use in practice, torsional stiffness after cracking was obtained.He did it by obtaining compatibility equation for a thinwall rectangular cross-section.His report explicitly shows that torsional stiffness is a function of percent of steels Mojtaba Hosseini, Hamid Reza Ashrafi, Peyman Beiranvand and Behzad Ghanbari used in the cross-section.In this section, by using new concept shear module after cracking, post-cracking torsional stiffness theory is presented.Using this new concept leads to a general theory which can be used for each desired cross-section including rectangular and circular cross-sections.Using this can significantly simplify the calculation of torsional stiffness after cracking.

Torsional Stiffness after Cracking
Experiments show that concrete middle zone has no effect on torsional behavior o cross-section after cracking 3 .In other words, resistance and torsional stiffness of a filled cross-section is similar to resistance and torsional stiffness of a hollow cross-section.Based on these observations, we can obtain stiffness equation after cracking for hollow cross-section and then, we can use it for filled crosssections.Consider prismatic thin-wall member with desired cross-section under torsional anchor T (Figure 4).Cross-section wall is shown by dashed line.By imposing this torsional anchor in the wall of member, shear stress τ develops that its value according to thinwall cross-section theory is: Where, A: Area of enclosed part by center line of member wall.h: Thickness of wall.
Because of this shear stress, other stresses develop in the steel and concrete.According to Rausch theory 4 , in order to calculate these stresses, given cross-section is considered as a spatial truss (Figure 5).In this method, which is known as truss method, three following assumptions are considered: • Reinforced concrete acts as spatial truss after cracking that its diagonal members have angle 45 0 and made from concrete; longitudinal and medial members form reinforcements which connect to each other by joint connection in nodes.• Metal concrete members tolerate only pressure (there is no shear strength for them).• Longitudinal and lateral reinforcements tolerate only axial tension force.Regarding above assumptions and using balance equations in nodes, Rausch showed that in all diagonal members, forces are equal with D. In a similar way, all exiting forces in all tights are Y and in all longitudinal reinforcements are X.Balance equation for node B along with tight gives: where s is the distance between tight and longitudinal reinforcement.Therefore, stress in the concrete member (diagonal) is: By assuming linear stress-strain relation between steel and concrete, we can obtain values of strain in the concrete members and reinforcements from Equations ( 24), ( 25) and (26).These strains create shear deformation in member's wall.This shear deformation can be obtained from compatibility of deformations.Figure 6 shows shear deformation caused by 1. Concrete members, 2. Longitudinal reinforcements and 3. Annular reinforcements.Therefore, total shear deformation is: By substituting available strains, the relationship between stress and shear strain or shear module is: where A l is cross-section area of a longitudinal reinforcement and A h is cross-section in steel tight and n is steel to concrete module ratio.This new concept of shear module after cracking is a very suitable concept and describes the general property of each torsional member after cracking.If torsional coefficient of a cross-section shows torsional coefficient of a thin wall cross-section with same thickness, C cr , we can calculate it according to thin-wall cross-section theory as following: In Equation (29), p is the perimeter of a ring which passes the tight central line.We know that torsional stiffness is equal with shear module times to torsional coefficient.Critical torsional stiffness after cracking the concrete and before full rupture is:

Effect of Wall Thickness on the Torsional Stiffness after Cracking
Applying Equation (30) for filled cross-sections raises this question that what is the effect of wall thickness on this equation?In fact, for a hollow cross-section, we cannot use real thickness of wall because only some part of it is effective.Achieving a theoretical method in order to determine effective thickness of wall is difficult.Robinson tried to achieve this but his attempts made the theory more sophisticated and difficult.His suggested solution led to trial and error method.Besides, validity of his assumptions are doubtful.On the other hand, by ignoring the first expression in the denominator of Equation ( 30) which includes thickness of wall, we can simply avoid the determining of effective thickness of wall.Because effective thickness of wall influences only on one of the expressions in the denominator of equation related to the torsional stiffness, it seems that the study of effects of wall thickness is not desired.On the other hand, avoidance of determining the effective wall thickness by ignoring related expression overestimates the torsional stiffness.As a result, in following, effective wall thickness is calculated empirically by comparing theoretical equations and test results.Because test results are available for rectangular cross-sections, these cross-section are used to calculate effective wall thickness.Laboratory observations indicate that effective wall thickness is a function of smaller dimension of rectangular cross-section (x) and volumetric ratio of steels (ρ l + ρ h ).

Changes of Torsional Anchor based on Torsion Angle after Cracking
Torsional anchor-angle of twist curve after cracking is a straight line in the beginning and then, by downward concavity, it approaches the break-out point.Torsional stiffness after cracking (G cr C cr ) is equal with the slope of linear area of curve after cracking.In Figure 8, an instance of torsional anchor-angle of twist curve is shown.Along with linear zone after cracking, we can observe that this line has not passed the origin of coordinates but it cuts the vertical axis in a point with ηT 0 width that T 0 is the share of concrete from torsional strength that after cracking, it reduces from the torsional strength of crosssection.and is an empirical coefficient.Therefore, the relationship between torsional anchor and angle of twist for linear zone after cracking is: Experiments done by Hsu in Portland Cement Association showed that ηT 0 is a function of real thickness of wall which is shown in Figure 9 and Figure 10.

Nonlinear Analysis of Torsion in Reinforced Concrete Members
Nonlinear analysis of reinforced concrete members under torsion is conducted by using truss model theory [7][8][9] .

Nonlinear Analysis of Torsion in Reinforced Concrete Members after Developing Initial Crack
Based on this theory, for non-stresses concrete members, 16 basic equations were obtained based on the balance equations, compatibility conditions and stress-strain dependencies of aggregates (steel and concrete).In this section, by referring to these 16 basic equations, we present a method for drawing T -θ diagram or τ -γ for reinforced concrete members under torsion after cracking.Consider a reinforced concrete member with rectangular crosssection under torsional anchor T (Figure11(a)).Shear flow with q size flows on the external ring with thickness t d which is called shear flow zone.In Figure 11(b), an element is shown along with d-r coordinate axis.Compressive stress σ d along with axis d and tensile stress σ r is imposed along with r axis.The angle between d axis and r axis is shown by α.In Figure 11(c) a square element is shown along with l and t axis that shear stress τ lt equal with d q t influences this element.Finally, an element of member based on the truss model concept located in the shear flow zone, is shown in Figure 11(d).In reference 10,11 , balance equations, compatibility conditions and stressstrain equations are discussed for this member.By referring truss model balance condition, we can write balance equations as following: In these Equations, and, is steel ratios along with axis l and t which are and, respectively.A 0 is enclosed surface by central line of shear flow zone which is a function of thickness of this zone.is central line perimeter of shear flow zone.
Using compatibility condition, seven compatibility equations are obtained as following: where is compressive strain of concrete in concrete diagonal levels.

Stress-Strain Relation of Concrete in Tension and Compression Modes
• In tension mode: Regarding small value of , we can ignore concrete tensile strength in torsion.In other words, σ r is zero.• In compression mode: Stress-strain equation of concrete along with axis d can be written as following: In which, f c ': Compressive strength of cylindrical concrete sample.k l : Ratio of average stress to maximum stress in compressive tension concrete block.η: Softness coefficient equal with.

Simplifying Equations for Achieving Solutions
By looking obtained equations, we find that we have a system with 16 equations and 19 unknowns.By knowing three unknowns, we can solve 16 equations.For a concrete member under torsion, vertical tensions σ t , σ l which are imposed to the element' faces in shear flow zone, are zero.Because compressive strain changes from zero to maximum value, by choosing different ε d values in the allowed range of system, we can solve equation system for 16 remained unknowns.Therefore, by assuming that ε d is Mojtaba Hosseini, Hamid Reza Ashrafi, Peyman Beiranvand and Behzad Ghanbari known, we first calculate t d .Finally, by using cumbersome calculations for chosen ε d and t d , we calculate torsional anchor, shear stress and torsion angle values on length unit.By selecting a new value for ε d , all steps were implemented and or curve can be drawn.

Results of Nonlinear Analysis of Concrete Torsion Members using nlarctm Program
In this section, 23 concrete rectangular cross-sections with different characteristics are analyzed with nlarctm program.This program is written based on the method discussed in previous sections and is introduced in the appendix.In order to prevent the volume, we ignored printing all output files and only drawn related diagrams and analysis of outputs.
In Table 4, characteristics of all cross-sections are shown.Quantities in this table are: b : Width of cross-section (cm).h : Height of cross-section.f ' c : Compressive strength of cylindrical concrete instance (kg/cm 2 ).f ly : Stress for longitudinal reinforcements (kg/cm 2 ).f ty : Stress of closed collars (kg/cm 2 ).s: Distance between closed collars (cm).n: Number of longitudinal reinforcements.:Φ l : Diameter of longitudinal reinforcements (cm).:Φ t : Diameter of closed reinforcement.

Changes of Torsional Anchor based on Angle of Twist on Length Unit ()
Using "nlarctm" program, all above cross-sections are analyzed for torsion loading state.It is necessary to mention that analysis trend presented by this program is nonlinear analysis of cross-section after cracking.Only related diagrams are presented and the study of torsional anchor changes based on the angle of twist on length unit will be discussed later.

Evaluating the Result of Nonlinear Analysis of Reinforced Concrete under Torsion
In this section, first we will discuss about the shear flow zone and its changes based on the total percent of longitudinal and transverse reinforcements 12,13 .Then, we will study the angle of cracks and finally, changes of torsional strength and torsional stiffness of cross-sections will be evaluated based on the changes in the different quantities.All figures were not drawn in Section (6.1) due to their large volume and only 3 first cross-sections and three last cross-section were drawn (Figure 12 to 19).

Thickness of Shear Flow Zone
In Table 5, comparison of Equations ( 40) and (41) with t d values in 14,15 has been done.As seen, ratio of t d in 14,15 to value obtained by Equations ( 40) and (41) for 23 cross-sections, has standard deviation 0.131 and 0.167, respectively.Since in a rectangular cross-section, the highest shear stress caused by torsion occurs in the middle of larger side of cross-section , it is suggested that Equations (40) were used in order to guess the initial value of shear flow zone thickness for certain cross-section.
• Comparing the thickness of shear flow zone with the height of stress block Whitney equivalence in bending About the bending of reinforced concrete members of bending, the height of Whiney equivalent stress block is: In this Equation, A s is the area of longitudinal crosssection located in the tensile zone.Regarding above equation, it is clearly seen that quantity of Whitney stress block height is corresponding with diameter and yield stress of longitudinal reinforcement and has inverse relationship with concrete compressive strength and width of cross-section.By referring to Equations ( 40) and (41), we can find that changes in the thickness of shear flow zone is similar to the changes of Whitney stress block height; in other words, thickness of shear flow zone corresponds with diameter and yield stress of the longitudinal reinforcements and has inverse relationship with compressive strength of concrete.An important point is the effects caused by changes in the diameter and the distance between closed collars on these two quantities.Equation (42) shows that change in the diameter and the distance between closed collars has no effect on the height of Whitney stress block; while values mentioned in the Table 4 and equations ( 40) and (41) indicates that the thickness of shear flow zone has direct relationship with the closed collars and inverse relationship with their distance.Therefore, in order to find a relationship between the height of Whitney stress block and thickness of shear flow zone, we should consider the effect of changes in the diameter and the distance between closed collars.Proposed method indicates the relationship between thicknesses of shear flow zone with the Whitney stress block height, and in order to guess the initial value of shear flow zone thickness, we can simply use it.In Table 6, values obtained for the thickness of shear flow zone from Equation (43) with corresponding values of "nlarctm" program.
As seen, t d ratio to value obtained from Equation (43) for 23 cross-sections in 14,15 has standard deviation 0.082 and mean 0.950 which shows higher accuracy of Equation (43) in order to calculate the thickness of shear flow zone.

Torsional Cracks' Angle
In most previous theories, like 16,17 , truss analogy method has been used in order to analyze reinforced concrete cross-sections under torsion.One of the essential assumptions in this method is that reinforced concrete cross-section after cracking acts like spatial truss that its diagonal members are concrete and have angle 45 0 .In fact, we can claim that by imposing this assumption, the angle of torsional cracks are 45 0 .Results of analyzing 23 given cross-section show that the angle of torsional cracks fluctuates between 34 to 50 0 but if their related values evaluated more accurately, following results are obtained: • By increasing the percent of collars, angle of cracks will increase.• By increasing the percent of longitudinal reinforcements, angle of cracks will decrease.• Changes in the compressive strength of concrete (f ' c 0 and yield stresses of closed collars and longitudinal reinforcements (f ly ,f ty ) has no effect on the angle of cracks.
• By increasing the distance between collars, angle of cracks will decrease.If the tangent on T -θ curve in each point is considered as torsional stiffness of that point, regarding Figures 20 and 21, we can find that in general case, by increase in the percent of longitudinal reinforcements or collars, torsional stiffness of cross-section increases after cracking and final torsional strength of the crosssection will increase 18 .Figure 22 shows that the increase in compressive strength of concrete (f ' c ) increases the final torsional strength and without tangible change in the cracking angle, it increases the torsional stiffness of crosssection after cracking.It seems that in truss theory, the effect of increase in the compressive strength of diagonal members is ignored, while it is clear that by increase in concrete compressive strength (f ' c ), we expect increase in the strength of members.Regarding Figure 23, we can find that by increasing the yield strength of longitudinal and transverse steels, torsional stiffness of cross-section after cracking will not change but the torsional strength will increase.Figure 24 shows that increasing the distance between closed collars will reduce the torsional stiffness of cross-section after cracking and final torsional strength.Therefore, we can conclude that in torsional stiffness of a cross-section after cracking, torsional reinforcement's percent has the highest influence, while change in other physical quantities of cross-section including concrete compressive strength (f ' c ) is also effective.

Guide for Computer Program
Nonlinear Analysis of Reinforced Torsion Members

Programming Language and Solving Equations in the Program
This program is nonlinear analysis of reinforced concrete torsion members This program is written by MATLAB language and its main steps are considered in Section 5 and Section (2.5) the method to solve equations.

Introducing Input File
Input file is torsion.xlsx in which given quantities in excel file.In order to run a program for a certain cross-section, it is enough that by using program written in common window environment, we select "Import data" option and designed excel file, then we go to Editor and run the program.No particular unit is considered for given quantities but coordination between units is necessary.In general, width and height of cross-section and diameter of longitudinal reinforcements and collars and the distances between collars is in length unit; yield stresses of longitudinal reinforcements and collars and concrete compressive strength and steel elastic module is in length unit and n which shows the number of longitudinal reinforcements has no unit.Concrete compressive strength should be considered with negative sign.

Conclusion
25 percent increase in longitudinal reinforcement strength and closed collars increases torsional strength as 2 percent after cracking which is minor and regarding this, cross-section number (17) is more optimal than cross-section (18).
equations and ignoring body volume forces, torsion function should hold in the differential Equation (4).In fact, torsion function should be harmonic or compatible.
By imposing border condition of Equation (
C along with radial direction necessitates that inward and outward forces along with diameters leading to node were equal; therefore:Nonlinear Analysis of Torsion in Reinforced Concrete Members after Developing Initial Crack

Figure 7 .
Figure 7. Effective wall thickness based on total steel 5 .

Figure 8 .
Figure 8. Anchor changes based on the angle of twist.

Figure 9 .
Figure 9. Anchor changes based on the angle of twist for three cross-sections with different wall thickness 15 .

Figure 10 .
Figure 10.Changes of τ 1 based on the wall thickness 5 .By increasing the thickness of wall, η increases.By drawing η as a function of h x , we can obtain their relationship between these two as following: 0.57 2.86 h x η = + (32)

Figure 11 .
Figure 11.(a) Reinforced concrete torsion member.(b)Main element of member under major stresses.(c)Member element in coordinate system.(d) Truss model for element in shear flow zone 7 .

FiguresFigure 20 .Figure 21 .
Figures 20 to 24 show the diagram of torsional anchor variations based on the torsional angle on length unit for different values of S, f y , f c ', Φ l , Φ t .It should be mentioned that during changing above quantities, all other quantities are fixed.Yield stress of longitudinal and transverse reinforcements are same f y .

Table 1 .
Maximum shear stress for different b/a ratios

Table 2 .
Torsional anchor for different b/a ratios

Table 4 .
Characteristics of all cross-sections.Nonlinear Analysis of Torsion in Reinforced Concrete Members after Developing Initial Crack

•
The study of thickness changes of shear flow zone based on the total percent of steels:

Table 5 .
Comparing the thickness of shear flow thickness obtained by "nlarctm", equation (40) and Equation (41) Nonlinear Analysis of Torsion in Reinforced Concrete Members after Developing Initial Crack

Table 6 .
Comparing the thickness of shear flow zone obtained from "nlarctm" program and Equation (43)