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The transfer of loads in structures takes place predominantly due to bending, shear and axial load. In beams, the load transfer is carried mainly by flexure and shear. Of these, shear failure is considered to be critical than that of flexural failure. Shear failure is a brittle failure, meaning that failure will be sudden and without much warnings (in the case of members without transverse reinforcement). So, the structures are designed to fail in flexure than in shear, as this will provide ample warnings, in the form of cracks, as compared to latter before impending failure.
From various researches undergone in the past century, the design of flexure formulas incorporated in various codes are similar and provide results which are more or less in conjunction with experimental output. This leads to saving of material and thereby making the design economical, whereas, when the predicted values of shear differ very much from the test results and as a result a high factor of safety has to be associated for shear and so leading to wastage of materials.
Several theories starting from the truss model of Ritter and Morsch were proposed including
MCFT, but most of the code provisions are empirical in nature and formulated using regression methods. (ShilangXu1 et.al.)There has been several shear failures in history , famous of which are failure of beams in Wilkins Air Force Depot in Shelby (Evan C. Bentz et.al), the recent failure of Laval Overpass in Qubec (Michael P. Collins et.al. 2008)etc. The failure was due to the underestimation of shear carrying capacity by the relevant codes. For example in the case of Laval overpass at the time of construction, as per the codes, the factor of safety against shear failure was 1.83 but after revision of codes it was 1.33. Even then due to long term degradation the factor of safety was not enough to avoid the catastrophic failure.
The primary aim of this project will be to look into the shear provisions of EC 2 code both the latest one, published in 2004, as well as the one preceding this, which is (ENV 1992-1-1: 1992). A comparison with the code for design of concrete structures for India, IS456:2000, will also be made to see the difference in the theories adopted and the accuracy of the same in predicting the shear stress for various parameters affecting shear. The Indian code follows the standard angle method while the latest Eurocode follows the variable strut angle method.
Figure 1-1:Collapse of Laval Overpass Quebec (Michael P. Collins, Dennis Mitchell and Evan C. Bentz, 2008)
1.1 AIM OF THE DISSERTATION
To assess the performance of current shear design provisions of various national and international codes and propose possible improvements
1.2 OBJECTIVES OF THE DISSERTATION
Understand various shear theory and shear design equations.
Understand the different shear transfer mechanisms involved in the transfer of shear across reinforced concrete beams members.
Collect results of beam failing in shear from literature and critically analyze them to arrive at specific conclusion regarding the performance of the same with respect to various parameters affecting shear failure such as a/d ratio, percentage of steel , compressive strength of concrete etc.
Evaluate the performance of the various shear design equations from international and national codes, mainly old Eurocode 2, (ENV 1992-1-1: 1992), and the newly published (EN 1992-1-1: 2004) as well as Indian code for concrete design, IS 456:2000.
A shear database including beams with and without vertical shear reinforcement or web reinforcement has been compiled. The results from various experiments conducted around the world on shear failure are available in the database. This database includes deep beams, T- beams as well as rectangular beams. The performance of codes variation in results for T-beams and rectangular beams will be compared. The same will be performed for deep beams with slender beams will be investigated. Next, the influence of shear parameters in the equations provided in different codes will be compared. The results of the same will be critically evaluated to arrive at a conclusion on how different shear parameters have been represented in the empirical equations for shear in different codes.
4 su groups and performance of these database with codes will be examined in deetial
2.1. RESEARCH IN SHEAR
In reinforced concrete before the advent of cracks, it is considered as elastic material with homogeneous properties and shear stress can be calculated as
V= shear force,
b= breadth of section
I= second moment of area and
A = area of section.
y = distance from neutral axis to the point at which stress is calculated.
Once the crack forms the, stress distribution becomes complex and predicting the shear stress is difficult (Park and Paulay).
Figure 2-1: Shear Stress in homogeneous isotropic beam (Park and Paulay)
2.2 Standard Method or 450 Truss Angle Method
In 1899, Ritter proposed the concept truss model followed by Morsch in 1909. This was the beginning of Strut and Tie modelling. In fact, strut and tie method is a modified version of truss analogy wherein, the stresses are transferred by steel stirrups as well as the longitudinal bars across the cracks. According to Ritter, the reinforced concrete beam can be considered as a truss with compression diagonals inclined at 450 with respect to the tensile steel ties. The theories were hence forth known as 450 truss model. Later on it was found that with 450 truss models, cracking could be controlled very well and though it is not an economical choice. The reason for the crack control is because it was found that the strains from the main bars and shear bars were equal at this particular angle (Thomas T. C. Hsu -1996). Once, truss angle was started using in design equations it came to be over conservative and that is when the contribution of concrete was also taken into consideration. V=Vc + Vs. That is total shear carrying capacity of concrete beam (V)is sum of shear carrying capacity of concrete, (Vc) and that of steel (Vs) calculated by truss method (E.G. Sherwood et al.).
Figure 2-2: Ritter's original truss model (1899) Michael. D. Brown et. al. April 2006
According to Morsch, the angle of compression diagonals, which is assumed to be 450, is a conservative value and it is difficult to find out exact angle. The method of finding the minimum reinforcement based on Morsch's assumption was used by several codes for design of shear and was known as truss equation for shear (FJ Vecchio, MP Collins). The truss model by Ritter completely ignored the contribution of concrete in tension and thereby the shear test values compared with truss models were conservative as per Whithey (1907, 1908) (ASCE-ACI Committee 445R, 1999).
Talbot (1909) (ASCE-ACI Committee 445R, 1999)also arrived at the same conclusion. Talbot also found that shear failure is a function of many variables such as reinforcement percentage, strength of concrete, relative length of beam etc. Also, higher shear carrying capacity is found for beams which are short and deep than slender long ones. His findings were left behind until the failure of beams in the warehouse of Wilkins Air Force Depot in Shelby, Ohio ( Bentz et.al 2008)which led to revival of the ACI shear design equations. The failure mentioned above acted as an impetus to the revival of research in shear of reinforced beams.
Figure 2-3: Flowchart of conventional strut-and-tie design methodology (Liang-Jenq Leu et al. 2006)
2.3. KANI (1964)- BENDING OF TEETH BETWEEN FLEXURAL CRACKS..
Kani, in 1964 produced a new rational theory for the shear capacity. He assumed that the concrete in the tension zone between flexural cracks acts like tooth with their root fixed in the compression zone and they are free to act like cantilevers. This was similar to the tooth's in a comb (Kani 1964).
Later on Taylor (1974) pointed out that the assumption that the tooth acts like free cantilevers is not correct as the main reinforcement bars as well as the shear between the cracks prevent this from free movement. The tooth model incorporating these new findings was worked on by many researchers. In 1991, Reineck did a nonlinear analysis with tooth model and it was concluded that dowel action is not important for ordinarily reinforced beams and only beams with a high reinforcement ratio. He also found that interface shear is the major shear mechanism which was similar to what Taylor had found in 1974 (ASCE-ACI Committee 445R, 1999).
Figure 2-4: Kani's tooth model (Kani 1964)
2.4. VARIABLE STRUT INCLINATION METHOD
450 was a conservative figure and Thurlimann and Lampert (1968) later found that it can deviate from the aforementioned value. This led to the development of variable angle truss method. (F. K. Kong). The new EC2 (EN 1992-1-1: 2004) also follows the same method. When a stress distribution takes place the angle of strut can be reduced from 450 and this will make design economical because, more load carrying capacity at ultimate state is then possible with lesser strut inclination. But this is possible only if loading and detailing is provided properly else the structure will fail in an angle which is more than the angle assumed in design. The figure below shows how load near support can cause the member to fail at steep angles. In this case if the designer has assumed a lesser inclination then the beam might not display the predicted shear strength as per design. (A. W. Beeby, R. S. Narayanan-1995)
Figure 2-5: Shear Failure due to nature of loading (A. W. Beeby, R. S. Narayanan. -1995)
2.5. COMPRESSION FIELD THEORY AND MODIFIED COMPRESSION FIELD THEORY
In the truss model though Morsch argued that angle can be less than 450, he was unable to come up with any valid theory to find out the reduced strut angle. His theory was that there are only three available equations which cannot be used to solve 4 unknowns. The figure below depicts the details of Morsch's dilemma in solving for the variable strut angle.
Figure 2-6: Shear Equations generated using free body diagrams (Neil M. Hawkins)
In compression field theory it was assumed that compressive stress and strain in principal directions were equal (compatibility condition) and this assumption was used as the 4th equation needed for solving the strut angle, 'θ'.
Compression field theory did not take into account stresses which are tensile in nature. It only considered compressive stresses. This is the main difference between the compression filed theory and modified compression field theory, which took into account the contribution of tensile stresses of concrete as well into consideration. (FJ Vecchio)
Figure 2-7: Experimental setup used in compression field theories. (Frank J. Vecchio,
FACTORS INFLUENCING SHEAR RESISTANCE
3.1 INFLUENCE OF BOND.
Bond has a high influence on the capacity of concrete. Once the crack forms it is the bond between steel and concrete that will influence load carrying capacity. The model shown below is the one proposed by Kani in his tooth model.
Figure 3-1 :The displacement of teeth in the tooth model proposed by Kani
(G.N.J Kani 1964)
The critical moment for the proposed model is calculated as
Where, ft'= tensile strength of concrete
s= length of cracked section
a = shear span
d = depth of section
= spacing of cracks
b = width of beam
As can be observed from the formula above, when the spacing increases the moment capacity increases, which in turn depends on bond strength. It was experimentally verified in the test that lesser the bond strength higher will be and higher the critical moment. With all other parameters being kept same the beams with deformed bar did not approach the flexural failure capacity whereas the one with less bond attained the same. There was 31%increase in the load carrying capacity when compared to the one with deformed bars. (Kani 1964)
3.2 Shear Span Ratio (M/Vd or a/d ratio)
The distance between load and support where the shear force has a constant value is known as shear span. In the case of a point load at middle it will be half the length of beam (L) and in the case of two point loads at one third span, the distance L/3 will be the shear span. From various tests it has been found that shear span has a remarkable effect in determining the strength of beams in shear.
Arthur P.Clark (1951) conducted various tests with a/d ranging from 1.1 - 2.3. From the results of his experiment he commented that if load is near support the strength was increased as compared to beams with load further away from support, that is, as spear span increased shear carrying capacity decreased. He also proposed an empirical formula for shear strength.
p= longitudinal reinforcement ratio
fc' = strength of concrete
d = depth of c/s
a = shear span
r= web reinforcement ratio
Kani (1964) performed several experiments for shear
Figure 3-2 : Minimum and Transition span/depth ratios. (Kani 1964)
The figure above has been drawn from the equations generated from those experiments. The dashed lines show the capacity line of remaining arch and the bold line indicates the capacity of concrete teeth. From the experiments conducted at University of Toronto, Kani defined two critical a/d ratios, one minimum and another transition. Beyond transition, the beams failed in flexure and before minimum value the beams failed in shear. That is after the transition value the beams could attain its maximum flexural capacity before failure. Therefore, diagonal failures cannot be expected after the transition point.
There are three subdivisions in the graph above, a/d < a/d min implies that arch has more capacity than teeth and if a/d is in between a/d min and a/d TR implies teeth has more strength than arch indicating sudden failure as the strength of concrete arch is less than applied load. And, when a/d greater than a/d TR then, beam attains full flexural strength.
3.3DEPTH OF MEMBER OR SIZE EFFECT
There is a wide variation between the cross sections tested in laboratory and the ones cast in site. This was taken into account and the size similar to ones cast in the site were tested by several researchers.
Kani(1967) performed several researches in members without transverse reinforcement. He found that size is effecting the strength to much extent as far as shear is concerned.
Shioya (1989) experimented on size effects and arrived at conclusion that depth does affect the shear strength in members with no shear reinforcement. He also commented that for sections with more than 39 inch depth the shear stress reduced drastically. Their failure stress was less than 50 percent as calculated from the codes. (Brown et.al 2005)
Figure 3-3: Photograph of the largest beam test by Shioya (d = 118 in.)
( Michael D. Brown et al. 2005)
Zdenek P. Bazant and Mohammad T. Kazemi
The size effect was looked into by BaÅ¾ant and Kazemi (1991) and fracture mechanics models were made to predict the strengths. They also found that size effect is important in shear strengths of members without shear reinforcement. From their experiments, they concluded that size effect is important at ultimate failure load though it might not be having much contribution at the first crack formation. So they recommended that design codes should take in to consideration the size effect.
K.H. Tan and H.Y. Lu (1999)
The experiments were on members with shear reinforcement. They made 4 groups of test specimens each having constant width of 140 mm and different heights 1750 mm, 500 mm 1000 mm and 1450 mm. Width was kept constant because of two reasons, one was to keep the fracturing energy same for all the beams tested and also the other reason was that beam width had no effect on ultimate shear strength as per Kani. Apparently, size effect had little effect on diagonal cracking strength..
Bentz 2005 provided various equations for calculation shear stress. Apart from the findings made by researchers before he also added that reinforcement in longitudinal direction as well size of aggregates used also explicitly varies the shear strength results.
3.4 Longitudinal Reinforcement Ratio:
K. G. Moody et. al. (1954) did tests on 42 beams with point loads in which all failed in shear. The reinforcement ratio, concrete strength, shear span and cross section and web reinforcement had effect on the strength at failure. They also found that an increase in span/ depth ratio reduced difference between load at first crack and ultimate load. First cracking determines ultimate strength for beams with large a/d ratio whereas small a/d ratios depend on the strength of compression zone.
Jung-Yo on Lee and Uk-Yeon Kim (2008) in their research, defined a reserve strength which is the ratio of the ultimate shear capacity of beams with reinforcement to those without reinforcement. They found that reserve strength and deflection were increased when ratio of tensile (longitudinal) reinforcement was increased.
Jung Keun Oh and Sung -Woo Shin(2008) did tests on beams with high strength concrete with longitudinal reinforcement.. The longitudinal reinforcement increase was not reflected in ACI equations though the tests showed that there is increase in strength due to longitudinal reinforcement. The reason for increase is because longitudinal reinforcement restricts propagation of flexural cracks from forming shear cracks.
3.5 TRANSVERSE REINFORCEMENT RATIO
Jun-Yo On Lee(2008) found that shear reinforcement has to be increased or decreased depending on decrease or increase of span/depth ratio to maintain reserve strength.
Boris Bresler and A.C. Scordelis (1963) found that web reinforcement helped beams to attain its flexural capacity and indeed the strength was enhanced due to the presence of shear reinforcements.
3.6 CONCRETE STRENGTH
Concrete strength is considered in almost all the code equations for shear.
K.N. Smith and A.S. Vantsiotis (1982) did experiments on deep beams and found that beams with less concrete strength failed prior, to those having more strength and less web reinforcement, as compared to other. They also noted that for beams with less a/d ratios, there was an increase in shear strength as concrete strength increased. But, a increase in a/d ratio led to reduce this effect.
Gregory C. Frantz and Andrew G. Mphonde (1984) conducted tests on beams with low and high strength concrete, with no shear stirrups. From the experiments they found that failure will be explosive in nature and sudden for high strength concrete with small span/depth ratios. That is, concrete strength has a predominant effect on shear failure load as there is reduction in span/depth ratio
DISPARITY BETWEEN RESULTS OBTAINED VIA CODES AND ACTUAL EXPERIMENTS CONDUCTED IN LABS
Figure4-1: Test Capacity VS Predicted Capacity
A database of test results without shear reinforcement was assembled from different tests conducted previously in laboratories across world. The shear capacity of beams without shear reinforcement was predicted using the equation as per Indian code. The shear strength of beams without shear reinforcement is provided by equation
Vc= [SP: 16 Design Aids for Reinforced Concrete]
β= 0.8 fck/6.89 pt
pt = 100 Ast/bwd
To cite an example, a point is chosen from the graph above which is shown in circle. The above graph depicts the results from the tests conducted in labs, compared with the results predicted using the above mentioned formula from the Indian code and the disparity can be observed from the graph above. There is a wide variation in results as can be seen for one of the points marked in the graph. When the test result was 487.1 kN, the prediction as per code is 32.3kN the difference of which is significantly large. If the equations predicting shear capacity were good then almost all the points would have been nearer to the straight line and there would not have been such a scatter of points. It may also be noticed that some points are above the straight line indicating that the predicted results are not safe. No factor of safety has been applied to the values used above, and if that too is taken into account then, the results obtained will be coming below the straight line. But, then for predicted values which are already safe, the difference will be even more. This will make the design uneconomical, as you could have used less material to get a safe solution.