What is the nominal shear stress

Shear design of beams and slabs made of reinforced concrete, reinforced concrete with reinforcements and prestressed concrete

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1 Shear measurement of beams and slabs made of reinforced concrete, reinforced concrete with tensioning supports and prestressed concrete Author (s): Object type: Bachmann, Hugo / Thürlimann, Bruno Article Journal: Schweizerische Bauzeitung Volume (year): 84 (1966) Issue 33 PDF created on: Persistent Link : Conditions of use The ETH-Bibliothek is the provider of the digitized journals. She has no copyrights to the contents of the magazines. The publisher usually owns the rights. The documents published on the e-periodica platform are freely available for non-commercial purposes in teaching and research as well as for private use. Individual files or printouts from this offer can be passed on together with these terms of use and the correct designations of origin. The publication of images in print and online publications is only permitted with the prior consent of the rights holder. The systematic storage of parts of the electronic offer on other servers also requires the written consent of the rights holder. Exclusion of liability All information is provided without guarantee for completeness or correctness. No liability is assumed for damage caused by the use of information from this online offer or the lack of information. This also applies to third party content that is accessible via this offer. A service of the ETH Library ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland,

2> 84th year issue 33 SCHWEZERSCHE BAUZETUNG August 18, 1966 ORGAN OF THE SCHWEZERSCHEN NGENEUR- AND ARCHTEKTEN-VERENS S..A. AND THE SOCIETY FORMER STUDERENDER OF THE EDGENÖSSSCHEN TECHNSCHEN UNIVERSITY G.E.P Shear measurement of beams and slabs made of reinforced concrete, reinforced concrete with reinforcements and prestressed concrete By Hugo Bachmann, dipl. ng., and Prof. Dr. Bruno Thürlimann, Institute for Structural Analysis, ETH, Zurich DK: Introduction The shear design of reinforced concrete girders according to S..A.- Norm No. 162 (1956) [1] is essentially based on that of E. Morsch in the years 1903 to 1929 performed attempts back. As a model for the mode of action of reinforced concrete beams in the shear area, W. Ritter and E. Morsch introduce a parallel-belt framework, the struts of which are inclined at 45 ° to the beam axis. The vertical or inclined tensile forces that occur in it must be fully absorbed by steel inserts. The "covering" of oblique main tensile stresses leads to the same result, as can be calculated in the usual way in the neutral axis of a beam set through by bending cracks. However, reinforced concrete has undergone a great deal of development since the experiments mentioned. Thanks to modern concrete technology, the quality of the concrete has increased significantly. The strength values ​​and composite properties of the reinforcing steels have been significantly improved. After the Second World War, intensive push research began, especially in the USA, to adapt to these changed conditions. Tests on more than 2000 girders gave a deeper insight into the extremely complicated shear load-bearing behavior. Systematic series of tests have also been carried out in Europe in recent years. They essentially confirm the results of the American investigations. For prestressed concrete, according to S..A.-Norm No. 162 (1956) [1], the shear design is carried out on the basis of the inclined main tensile stresses in the state of use. The shear research in prestressed concrete has not yet reached the relatively high level as in reinforced concrete. In particular, there is a lack of systematic shear tests on girders with concentrated tension cables and adequate shear reinforcement. Nevertheless, it can be stated that the shear problem in prestressed concrete is much more similar to that in reinforced concrete than was previously assumed. In summary, the current situation can be characterized as follows: 1. Up to now, no all-round satisfactory theory for the calculation of the shear fracture torque (shear fracture theory), which covers all types of fracture, has prevailed. 2. On the basis of the knowledge gained in recent years, on the other hand, rules for shear design can be specified which contain significant improvements compared to the theory of Ritter-Mörsch or the method with the inclined main tensile stresses in prestressed concrete. On behalf of the commission for the revision of the S..A.-Norm No. 162 (1956), the Institute for Structural Analysis at the ETH has carried out various studies on the shear problem since 1962. The results have been presented in several reports to the standards commission and partly in a publication [2]. As a result of this work, a proposal for the shear design of beams and slabs was made, which will be explained in the following. It applies to reinforced concrete and prestressed concrete as well as to steel structures with prestressed allowances1). The use of reinforced concrete with tension allowances') The term "reinforced concrete with tension allowances" is used for concrete structures in which slack and pre-tensioned steel inserts are arranged in any proportion. Cracks are therefore mostly to be expected under the calculated working load. In the case of prestressed concrete, on the other hand, the tensile stresses calculated on the homogeneous structure should not exceed a certain fraction of the concrete flexural strength. still raises various unresolved questions. Nevertheless, an attempt was made to design the following proposal for the shear design in such a way that all transitions between reinforced concrete and prestressed concrete result continuously. The following design rules are based on the calculated breaking load. A formulation for working load would also be possible. The proposal is not written as a standard provision, but in more detail for better understanding. Nevertheless, a certain formulation based on presentation cannot be avoided, since these are design rules that are not based on a closed theory. The suggestion explicitly applies only to beams and slabs. It is not to be used on bowls and slices. 2. Proposal for the shear design 2.1 Assumptions, definitions 1. The shear reinforcement can consist of the following elements: a) vertical stirrups, b) inclined stirrups, whereby the angle a with the beam axis should not be less than 45, c) a combination of vertical and inclined ironing. Instead of inclined brackets, bent steel inserts may also be used. However, your share of the total shear reinforcement should not be more than half. 2. The longitudinal and shear reinforcement must be properly anchored. The bending radius of bent steel inserts must be at least 10 bar diameters. 3. The nominal shear stress (1) Q b0h is used as a measure of the shear stress on a cross-section. Q is the transverse force under the calculated breaking load, h is the static effective height of the cross-section2) and b0 is the minimum web width. 4. The shear force under the calculated breaking load is: (2) Q s (Q, + Qp) + Qv + Qz where: s General safety factor (normally s 1.8) Qg shear force due to dead weight Qp shear force due to payload Shear force due to inclination of a Tension reinforcement against the beam axis: (3) Qv V0 sin

3 Table 1. Values ​​S for determining the shear resistance of the bending pressure zone and the shear stress limits ßw (kg / cm2) T / j (kg / cm2)> To determine the shear resistance of the bending pressure zone and the shear stress limits, the values ​​rs according to Table 1 are required . Here, ßw means the cube compressive strength of the concrete. 6. The shear resistances in a cross-section consist of the following nominal components: a) Shear resistance of the bending pressure zone: (5) Qc = 1 zoo z. but not greater than (6) Qc 1.5 ^ boh. * i ba Koo is the preload force after shrinkage, creep and relaxation, Zs the sum of the forces in all steel inserts in the flexural tensile zone when their yield point is reached. b) Additional shear resistance as a result of the effect of normal stresses in beam areas where, under the calculated breaking load, the tensile stresses at the edge of the tensile zone do not exceed the value ßw / 20: (7) Qn 0.2 ajsr ba h (in fact the stress in the center of gravity of Cross-section as a result of the prestressing force. C) Resistance of the shear reinforcement: For vertical stirrups: FB (8) Qb Qb T, it must be continued beyond these areas by at least the stirrup spacing tb. 9. The shear reinforcement is to be dimensioned in such a way that the sum of all nominal shear resistances is at least equal to the calculated breaking load: (12) Qc + Qn + Qb + Qd> Q 10. The minimum shear reinforcement content must be chosen so that the condition (13) Qb + Qd> *, b0 hy is fulfilled. 11. In the area of ​​supports and individual tubes, the shear dimensioning is to be carried out in the cross-section that is at a distance of h / 2 from the edge of the support or the load. The shear reinforcement calculated there must also be arranged between this cross-section and the edge of the support or the load. 2.3 Maximum shear stresses, maximum distances 12. The maximum permissible shear stresses depend on the concrete's compressive strength and the division of the shear reinforcement. A distinction is made between 2 cases: a) Normal distances The nominal shear stress x may exceed the value t2: (14) t 7), then the influence of the shear stress is on it Behavior of the wearer mostly low (compare e.g. [3]). Flexural cracks arise first in the zones of greatest bending stress and, with increasing load, propagate more or less perpendicular to the support axis. In the area of ​​the maximum moment, as in the case of pure bending, a bending fracture occurs. With a smaller moment-shear ratio (M / Qh <7), inclined cracks also arise. In places where large bending moments act together with large transverse forces, so-called bending shear cracks often run vertically from the edge to the tensile reinforcement and then take an inclined path under the combined influence of bending and shear. Most of the time they reproduce after the next larger point load or bearing force. 584 Switzerland. Bauzeitung 84th volume, August 1966

4 1 Flexural cracks Diagonal cracks i flexural shear cracks Figure 2.Examples of crack formation: flexural cracks, flexural shear cracks and diagonal cracks If the moment is small compared to the transverse force or if the tensile reinforcement is prestressed, oblique cracks are often first visible in the middle of the web, which spread outwards. They are known as oblique tensile stress cracks or diagonal cracks. When such cracks finally reach the tensile reinforcement, they become flexural shear cracks. n Figure 2 shows typical examples of bending cracks, bending shear cracks and diagonal cracks. A distinction can be made between different cases in the fracture behavior: a) Beams without shear reinforcement Beams without shear reinforcement can suddenly fail with the formation of diagonal cracks. The crack spreads almost instantaneously over the middle zone of the girder, whereupon it moves quickly and with a shallow slope towards the tensile reinforcement and into the bending pressure zone. Either the entire pressure zone of the carrier is torn through or a piece of it is compressed. This type of rupture can be catastrophic as it occurs suddenly and causes a total collapse. It can be described as a diagonal shear fracture. If the oblique cracks are flexural shear cracks, the pressure zone is reduced by the expansion of the cracks. Finally, the weakened pressure zone is destroyed, and what is known as a flexural shear fracture occurs. b) Beam with shear reinforcement The inlay of diagonal or stirrup reinforcement does not prevent the development of shear cracks. The crack load at which the cracks form is also not influenced by this. In contrast, the shear reinforcement noticeably inhibits the opening and spreading of the cracks and thus increases the load-bearing capacity. n Beams with shear reinforcement usually suffer from flexural shear failure [4, 5]. After bending shear cracks have formed, the shear force is absorbed jointly by the shear reinforcement and the concrete pressure zone. If the shear reinforcement suffers plastic deformations, a further increase in the transverse force must be borne primarily by the bending pressure zone. The load can usually be increased a little until the bending pressure zone at the end of the inclined shear crack is destroyed under the combined stress of pressure and shear. Another type of break is the so-called shear pressure break. Failure occurs through compressing or breaking out of the concrete in inclined strips that lie parallel to the crooked cracks (corresponding to the compression struts according to the framework analogy) before the shear reinforcement experiences plastic deformations. This type of fracture only occurs when there is a very high level of shear stress or a very strong shear reinforcement. Inadequate structural training can lead to further failures, for example: 1. If the longitudinal reinforcement has insufficient concrete coating, the oblique shear cracks are bent in the horizontal direction along the longitudinal reinforcement, bursting the concrete coating and causing a bond break. 2. If the longitudinal reinforcement is insufficiently adhered or anchored, it can slide, and a bond failure also occurs along the beam. 3. If the shear reinforcement is not anchored sufficiently, its effect can be greatly reduced by sliding. The shear cracks become Switzerland. Bauzeitung 84th volume issue August 1966 open and expand almost unhindered, whereupon a premature break occurs. 4. If the bending radius of bent steel inserts is too small, the concentrated deflection forces can cause a crack. On the other hand, special structural designs can also noticeably increase the shear resistance: 1. n girders with very heavy flanges, with vertical reinforcement ribs and with strong longitudinal reinforcement, a considerable part of the shear stress can be absorbed by the frame effect. 2. If a load acts very close to the support (M / Qh <2), a large part of it is conducted directly into the support through a kind of vault effect. 3.The longitudinal reinforcement is often stiff enough to transfer part of the shear force through the action of anchors. Although each of the cases dealt with in the last two sections can have a strong influence on the shear resistance of a girder, they must be viewed as special cases. Therefore, they cannot be taken into account in the development of a general design method. Fundamentals of the proposal in the fracture area of ​​a beam under combined stress from bending and shear, basically 3 conditions must be met [6]: 1. Equilibrium: The internal forces must be combined with the external loads be in balance. 2. Compatibility: The deformations must be geometrically compatible. 3. Rupture criterion: The state of stress must have reached such an intensity that no further increase in load is possible and thus rupture occurs. Every mechanical model of the fracture should at least meet the equilibrium conditions. According to the current state of knowledge, an accurate assessment of the other two conditions is extremely difficult. Both the breakage criterion and the compatibility conditions in the case of inelastic deformations hardly allow an exact mathematical formulation. An equilibrium consideration is used in the present proposal. The external shear force under the calculated breaking load must be less than or at most equal to the sum of the following shear resistances: shear resistance of the bending pressure zone in areas where bending shear cracks occur under calculated breaking load, shear resistance of the pressure zones in areas where only diagonal cracks occur under calculated breaking load, shear resistance of the shear reinforcement . It is therefore not a closed shear fracture theory, since the inclusion of an actual compatibility condition is dispensed with. In reality, this is responsible for the actual distribution of the external shear force between the shear reinforcement and pressure zones. The resistance of the pressure zones was set in consideration of test results in such a way that the risk of premature shear fractures is extremely low. The formulated resistance of the shear reinforcement corresponds to the idea from the framework analogy. Therefore, the proposed dimensioning of the shear reinforcement can also be referred to as a "modified framework analogy". 4.2 Nominal shear stress The usual methods for dimensioning the shear reinforcement in reinforced concrete are based either on the calculation of the greatest main tensile stress on or the greatest shear stress r in the cross section. To calculate the greatest main tensile stress, a beam element with vertical bending cracks is generally considered and the following relationship is derived: (16) Q b0 V h However, this approach is arbitrary and even contradictory. Because where the shear stress is essential, the cracks do not run vertically, but at an angle. The result is a completely different internal support system than assumed. In addition, the shear stresses calculated in this way are smaller than the shear stresses determined on the uncracked (homogeneous) cross-section. The difference 585

5> Table 2. Calculated values ​​for the shear measurement of the examples from Fig. 1 Beam Reinforced concrete beam Soannbeton beam Prestressed concrete beam (pre-tensioned with tension cable) (prefabricated in the tension bed) Characteristic values ​​ß = 300 kg / cm2; T, 10 kg / cm2 3 = 400 kg / cm2; T, 2 kg / cm2 (3 = 500 kg / cm2, T, = 14 kg / cm2 Yx / Zs = 0 j SsB = 4.0 t / cm2-, h 92 cm V ^ / Z ^ O.5 SN 60 kg / cm2; SsB = 4.0 t / cm; Vm / Zs> 0.5; 3N = 75 kg / cm2 h 52 cm h = 46 cm h g46 cm SsB = 4.0 t / cm2; h = 110 cm h 52 cm h 46 cm beam sections AA BB cc AA BB CC DD EE AA BB Shear force due to utility loads g, p, P: Qg + Qp t 18.0 26.4 21.3 19.0 21.2 21.2 27.5 21.7 58.7 51.1 i, 8 times the lateral force due to g, p, P: Q = l, 8- (Qg + Qp) t 32.4 47.5 38.6 34.2 38 , 2 38.2 49.6 39.2 105.7 92.0 Shear force due to the inclination of a prestressed reinforcement (4): Qv = Vm- sin at, 096-12,, 110-14,, 110-14,, 082- 10,, 082-10.9 Shear force under calculated breaking load (2): Q = l, 8 (Qg + Qp) + QV t 32.4 47.5 38.6 21.4 23.6 23.6 38.7 28.3 105.7 92.0 Nominal shear stress (1): TQ / b -h kg / cm2 Maximum shear stress (14), (15): T2 = 4T1 or T3 = 5T, kg / cm2 Comparison: No post-catfish one Shear reinforcement (10): Tfbo-h Minimum shear reinforcement (13): t 32 ', 6 T2 =' 3 = 47'500 20S9ai 25.8 T2 = 40 38'400 _ "20.9 T2 = 40 18.4 18, 4 18.4 2l'400 "27.4 T2 = 48 l2-l5 -52-o "3 = 9.4 23 * 600 _ p 30.3 T2 = 48 l2-l5-52-o_3 = 9.4 23'600 _" 34.2 T2 = O "3 =" 8.3 »38 ', 1 T3 = 60 T 1/2 T -b0-h t 9.2 9.2 9.2 4.7 4.7 4.2 4.2 4.2 10.8 10.8 Shear resistance of the bending pressure zone ( 5), (6): Qc = (l + Va / Zs) -T, -b0-h but Qc ^ l, 5-t, -b0-h t Additional shear resistance inf. Normal stresses (7): Area where tensile stress at the edge of the tension ßy / 20: QN = 0.2-Gtfb0-h Shear resistance of the shear reinforcement (12): Qb + Qd-Q-Qc-Qn t ma shear coverage by vertical stirrups alone (8) : Maximum possible distance: tb- h / 2 resp. h / 3, but tb 30 resp. 20 cm: cm Selected distance: cm P - BH "3 18.4 18.4 18.4 14,,,," 3 = l, 5-l2-l5-52-o "3 = l, 5-l2- 15-46-ö "3 = 14.0 14.0 12.4 0.2-60-15-52-O -3 = 9.3-1.9 (mass .: 4.7), 2-60 - 15-52-l03 = 9.3 0.3 (massg .: 4.7) see figure 0.2-60-i5-46-Ö "3 = 8.3 2.9 (massg .: 4.2 ) l, 5-l2-l5-46-ö "3 = 12.4, Ö3 = 12.4," 3 = 32.3 see 0.2-75-l4-O-Ö3 = FB - s7B-- h- cm 0.99 1.58 1.53 0.59 0.59 0.53 1.57 0.95 2.28 2.17 two-cut stirrups: mm e,,, 1 50, '00Q "59.7 T3 = 70 T

6 34, t Reinforced concrete beams L lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllttttt1 ... 'c. ^ Jfiü g + p 5t / m'.is l H 1000 ZOO L u V2 03 h / T Shear reinforcement diagram T, -bh l / 2 * t?! Kh jq / z-tf ^ -h T, bh "o" i Qb + QoJJ-1 ^ m 32, t QBtP Prestressed concrete beam (prestressed with tensioning cable) g + p 5 t / m / -, = U moments cj ullpunkt * 4 r BC g 1t / m P Z01-3t u fttf / zo S ^ KA / 20 S0> f3w / 20 So <Ö «/ 20 rrandzug - parabola straight line circle straight line 1 SDQnnuna tension tension member length 49.6 t Shear reinforcement diagram / 2-X-b'h T; b_ hf T.-b L_t g 1/2 i; b0-h fl / 2-t, -b0-h 3 = 2 t / m1 PMOt prestressed concrete beams (prefabricated in the prestressed bed) ft = tfroo L -v »= 48 'u <ßw / Z0 5c> Ö« / 20 n Uug ntere Hand jgspannung DBO T.-tvh Shear reinforcement diagram Switzerland. Bauzeitung 84th volume, August 1966 [, 0 Fig. 1. Examples for the shear measurement of reinforced concrete and prestressed concrete beams, with shear reinforcement diagrams 587

7 is around 30% for a rectangular cross-section. From this it can at least be concluded that the shear stresses calculated according to equation (16) do not actually correspond to values. It therefore seems more appropriate to introduce a nominal shear stress according to equation (1) instead of equation (16): 4.4 Differentiation between the bending shear range and the diagonal shear range Q (1) b0h Even with prestressed concrete girders, after the occurrence of cracks, especially bending shear cracks, a Equation (16) or shear stress calculated according to formula (17) bl no longer has a real meaning. The actual circumstances are also quite complex, so that the use of the nominal size (1) is also suggested for prestressed concrete and reinforced concrete with prestressed reinforcements. 4.3 Shear force under calculated breaking load The shear force under calculated breaking load is to be calculated according to equation (2). The transverse force Qv due to the inclination of the tension reinforcement in relation to the girder axis is interpreted as an external force unchanged by the load. This assumption applies to a large extent in areas where only diagonal cracks are present and the edge zones of the cross-section work without cracks. However, if there is a significant bending moment and all or some of the inclined prestressed steel inserts are in the vicinity of the carrier edge, these steel inserts are stressed beyond the prestressing force as the load increases. This means that the corresponding contribution to the transverse force can be slightly larger than Koo sina. The approach Qv Koo sina is therefore on the safe side, provided that Qv reduces the transverse force Qg + Qp. The error committed will usually be minor, since in those areas where the bending moment is finally reached, the pre-stressed steel inserts are inclined either not at all or only very slightly. The case where prestressing steels are inclined in such a way that Qv increases the shear force Qg + Qp is theoretically conceivable, but hardly of practical importance. Then the calculation of Qv with the initial preload force V0 should suffice. Basically, the question arises whether in the design condition (12) the quantity Qv should be listed on the resistance side instead of on the side of the load. Arguments can be given for both solutions. The arrangement selected here in accordance with equation (2) has inter alia. the advantage that the upper shear stress limit could be determined in exactly the same way as for reinforced concrete. j The general safety factor has the function of compensating for all uncertainties that exist in the results of the design. The most important sources of these uncertainties are: 1. Statistical deviations of the quantities used for the design (load, material properties, geometric quantities, etc.), 2. Unforeseeable errors that arise from human inadequacies and are generally not statistically detectable (arbitrary Assumptions and simplifications, lack of care, gross errors, etc.). In addition to the introduction of a single general safety factor, there are other ways of taking into account the uncertainties of the design. For example, the safety factor on- QK.rcy =: h -z T ~ y = h Diagonal shear Bending shear "Fig. 3. Schematic representation of the support systems diagonal shear (left) and bending shear (right) 588 can be divided into individual safety factors that affect the members of the Design condition (12) must be assigned to (2) (load factors, material factors, capacity factors, etc.). For a more comprehensive assessment of the various options, reference must be made to the relevant literature (e.g. [16]) It is essential whether the shear cracks occur at a point of high shear stress as flexural shear cracks or only as diagonal cracks in the web area (cf. 4.11). For both cases, two very different internal load-bearing systems develop, which must be considered separately. Figure 3 shows the two schematically “Bending shear” and “diagonal shear” load-bearing systems, in which the forces acting on the left-hand part are entered in the crack or cut The relevant girder areas are referred to as “bending shear area” and “diagonal shear area”. Bending shear area In the case of girders without shear reinforcement, the entire transverse force in the area of ​​bending shear cracks is transmitted through the bending pressure zone alone. However, if there is shear reinforcement, the bending pressure zone and the shear reinforcement each take on part of the shear force when the external load increases. A quantitative approach is made in Section 4.5 for the shear resistance of the bending pressure zone when plastic deformations occur in the longitudinal reinforcement. Diagonal shear area If the bending tensile zone remains overpressed by pretensioning forces even under a calculated breaking load, both edge zones of the cross-section are still free of cracks. This can include be the case in the vicinity of the support of simple beams with tensioning reinforcement parallel to the beam axis or in the area of ​​the moment zero point of continuous beams with tensioning cable reinforcement. In addition to the bending pressure zone, the overpressed tensile zone can also transfer part of the transverse force. The shear reinforcement must take over the remaining shear force. This also ensures that the shear deformations in the web remain limited and that the inclined cracks that occur near the gravity axis do not penetrate too much into the edge zones of the cross-section. The corresponding design equation is explained under 4.6. Boundary between the bending shear area and the diagonal shear area n zones subject to shear stress exist practically everywhere where bending cracks are to be expected at a certain load intensity and there is a risk of bending shear cracks. For statically determined supported girders with a precisely defined load arrangement, it would be conceivable to define the boundary between the bending shear area and the diagonal shear area where, under a calculated breaking load, edge tensile stresses of the order of magnitude of the concrete bending tensile strength are present. For continuous beams, on the other hand, a considerable expansion of the bending shear range could be required for the dimensioning of the shear reinforcement in order to take into account the plastic moment redistribution that always occurs. In addition, it must be noted that even with statically determined girders, longitudinal forces that can hardly be calculated can occur due to incorrect bearing constructions, shrinkage or other effects. As a simple mean solution, the limit was determined by the edge tensile stress ßwßO in the proposal. 4.5 Shear resistance of the bending pressure zone 4.51 Reinforced concrete girders The study [2] was carried out on the proportion of the bending pressure zone in the shear resistance of reinforced concrete girders in the case of stirrup reinforcement. The equilibrium conditions and the break criterion were formulated for the flexural shear failure. An approach of the compatibility condition for the deformations was deliberately omitted. Rather, the corresponding terms were determined by evaluating experimental results. The shear resistance of the bending pressure zone can be expressed for rectangular cross-sections as follows: Qc K ß Sa b h Here ß means the prism strength, f 0 is the distance of the zero line from the pressure edge based on the static height according to the usual reinforced concrete theory: Switzerland. Bauzeitung volume August 1966

8 (18) f] / («iif- + 2 n fi n / in is the ratio of the moduli of elasticity of steel and concrete and / i is the longitudinal reinforcement content. For (18) (19) 0 3.5] / ß is a good one Approximation, provided that the ratio n assumes the value 10. The empirically determined function K essentially depends on the following parameters: properties of the cross-section under consideration (dimensions, reinforcement content, material strengths), moment-shear ratio M / Qh, moment ratio M / Mbt, shear reinforcement content The equations derived in [2] result in the following range for K: 0.125 * the shear reinforcement Qc rlf30a comes to the full Deduction if> 1.5 ^. The longitudinal reinforcement required for the production of r 1.5 t fj. is so large for common values ​​M / Qh and yield strength os that equation (20) in by far most cases yields larger values ​​than equation (5a). 2. The proposed design condition (12) was checked with the results of almost 100 beam tests in which rectangular cross-sections and vertical stirrup reinforcements as well as T-beam and T-beam cross-sections as well as inclined stirrups and bends occurred. The value Qc was set according to equation (5a), Tj being interpolated according to the concrete strength according to Table 1. The numerous recent push attempts by Leonhardt-Walther [3], [7], as well as the attempts by Clark [8] and Moretto [9] were used for this purpose. Beams whose failure was designated as a bending or anchoring break were not taken into account. Figure 4 shows the relationships between the shear force determined in the test at fracture Qbt and the maximum value (21) Qr ßc + Qb + Qd calculated according to the design condition (12) as a function of the compressive strength of the cube. With the exception of three points, all values ​​are above 1.0. In the test, which corresponds to the value 0.78, Qbt

9 2VZS »Fig. 6. Comparison of the deformation behavior of differently reinforced beams with the same bending moment joosssää ^ 5 /.^-zz^-tyz^^ r 8s Jjpi Evco 0.5- ^ 5 ^ ** ^ t ^ - '/" 1 <, Riss i ^ sjf SS / wm WM iü / ^ unier recnn. "Usage load (sl, fp Eb gib - / * f <, ef5 0.85, which corresponds to an overstress of almost 60% and more, the elongations st and the compressive forces in the bending pressure zone of the two then show The deformation behavior in the area under consideration is likely to have become largely similar. If one assumes that the shear reinforcement generally contributes to the transfer of the shear force to the same extent as the expansion e & increases, this comparison could lead to a conclusion That the shear resistance of the bending pressure zone would be roughly the same for both girders, but this would mean that prestressed concrete beams are bending The shear area should be provided with exactly the same shear reinforcement as the corresponding reinforced concrete beams, provided that the shear fracture must not occur before the bending fracture. n Figure 7 shows the distribution of the external shear force on the shear reinforcement and the bending pressure zone according to various standards and suggestions. The entire range of a transverse force is shown, which increases from zero to the point at which the bending moment or the yield point in the longitudinal reinforcement (ßCMW) is reached. The straight line 1 corresponds to the 45-truss analogy with ßc 0 and Qb Q. The straight line 2 shows the division for steel concrete girders according to the present proposal. 3 corresponds roughly to the curve for stress-relieving concrete beams that emerged from the comparison in Figure 6. Here, ßüiSS was set at 0.6ß (Mbt), which roughly corresponds to the behavior of conventional prestressed concrete girders. You can see here the disproportionate increase in the load on the shear reinforcement with increasing shear force and the corresponding decrease in the share of shear force transmitted through the bending pressure zone. The straight lines 4 and 5 correspond to the A.C .. standards or the proposal and are discussed below. b) Standards A.C Compared to the conclusion from the previous comparison, the American standards A.C [10] contain quite a different regulation. There it is essentially stated that when determining the shear reinforcement, the transverse force present when the first crack occurs (bending crack or diagonal crack) may be subtracted from the transverse force under the calculated breaking load. With the designations introduced here, it follows: (22) Qb + Qd> Q ßsiss It is therefore assumed that after the occurrence of a crack, the bending pressure zone is able to continue to transmit the undiminished shear force ßiuss until the bending moment is reached. This assumption ßc ßuiss appears, especially in view of the above considerations in the comparison in Figure 6, to be all too favorable. In fact, an evaluation of shear tests carried out according to (22) on prestressed beams with shear reinforcement [14] results in values ​​(23) Qr Q. ßüiss + Qb + Qd which in some cases are considerably below 1.0. Figure 8 shows the values ​​(23) of 29 beams with tensioning reinforcement arranged parallel to the beam axis, which suffered a shear fracture. As always, the spread is' 590 Switzerland. Bauzeitung 84th volume, August 1966

10 ir ^ rj / qr j. "* M 0->» (kg / cm2) ^ ß (kg / cm2] Fig. 8. Relationship between the measured and calculated shear force in the case of shear failure of test beams made of prestressed concrete according to AC-63. Values ​​from [14] taken from Fig. 9. Relation between the measured and the calculated shear force during shear failure of test beams made of prestressed concrete according to the proposal. Values ​​taken from [14] from shear tests, quite considerable. At least 6 values ​​below 1.0 occur, with the The smallest is only 0.74. n Figure 7, the straight line 4 corresponds to the division of the transverse force between the shear reinforcement and the bending pressure zone in the bending shear range according to AC c) Suggestion If the assumption ßc ßiuss according to AC is too favorable, the tests [14] and also [15] it can be concluded that the shear resistance of the biped pressure zone of prestressed concrete girders is greater than that of the corresponding reinforced concrete girders than the prestressing force, the penetration of the cracks into the bending pressure zone is somewhat inhibited and its shear resistance is increased as a result. In the present proposal, the following approaches were therefore chosen: (5) however ßc (l + -Je-) b0 h (6) ßc <1.5 1 ba h The shear resistance of the bending pressure zone of prestressed concrete girders is therefore assumed to be a maximum of 50% higher than that of corresponding reinforced concrete beams. The straight line 5 in Figure 7 corresponds to approach (6), provided that the above preconditions are accepted. It runs between the line according to the comparison in Figure 6 and that according to A.C. Again, the comparison with test results is interesting. It should be noted that (5) or (6) only applies to the state and the corresponding transverse force ß (MBr) where the bending moment or the yield point in the longitudinal reinforcement is reached. For transverse forces that are smaller than β (Mbt), the transverse force transmitted through the bending pressure zone is different from ßc. If line 5 is roughly assumed to be a straight line, the calculated transverse force Qr in the case of premature shear fracture is (24) ßr ßl (Qb + Qd) Q (Mbt) ßr ß (MBr) Qc If ßr is greater than ß (AfBr), so this corresponds to a bending fracture. In this case, the following applies to the evaluation ßr ß (Mbt) If ßc ßiuss were used in formula (24), the value for Qr according to AC n Figure 9 would be the points ßßr / ßr for the same test bars as in Figure 8 . The values ​​corresponding directly to the calculated values ​​ßiuss, ß (MBr) and Qb were taken from the publication [14]. You can see that compared to Figure 8, most of the points have shifted slightly upwards. In particular, there are no longer any values ​​below 0.90. From this it can be concluded that the approach (5) and (6) together with the other design conditions used in the proposal represent a very good basis for determining the shear reinforcement in the bending shear range of prestressed concrete girders. This statement applies under the condition that no shear fracture may occur until the elastic limit has been reached in the longitudinal reinforcement. However, if a beam area subject to deflection and shear is to be able to deform even more in order to obtain a redistribution of moments through plastic deformation (so-called rotation in plastic joints), it is doubtful whether the shear resistance of the bending pressure zone determined with (5) and (6) is maintained remains. As the investigation [13] shows, this is the case for reinforced concrete beams with Qc r1b0h. Based on the comparison in Figure 6, it can be assumed that as the deformation of the longitudinal reinforcement progresses, the effect of an original prestress on the shear resistance of the bending pressure zone disappears. As long as there are no tests on this, it is recommended to use the shear reinforcement of prestressed concrete girders in the area of ​​plastic joints - e.g. B. on intermediate supports of flow girders - measured at least to a length of 2h as that of reinforced concrete girders za. This is especially true for girders that are reinforced with concentrated tension cables in ducts. There it must be expected that in the course of the rotation, under certain circumstances, only a few broad flexural shear cracks occur, which penetrate far into the flexural pressure zone. In conclusion, it can be stated here that systematic shear tests on prestressed concrete girders with cable armouring are still missing. They are necessary in order to be able to check the shear design and the plastic deformability (ability to rotate) Reinforced concrete beams with tensioning allowances The shear resistance of the bending pressure zone of reinforced concrete beams with tensioning allowances lies between the resistances of the corresponding prestressed concrete and reinforced concrete beams. The lower the pre-tensioning force in relation to the total resistance of the tensile reinforcement, the smaller it is. In approach (5) and (6), Kco means the prestressing force still present after shrinkage, creep and relaxation. Zs is the sum of the tensile forces in all steel inserts in the bending tensile zone when their yield strengths are reached. If Fev and asv are the cross-section or the yield point of the prestressing steel, and Fes and o-ss are the corresponding values ​​for the slack reinforcement, then Zs Fev asv ~ \ ~ Fes ass is a constant mathematical transition between fully pre-stressed and given slackly armored straps. Switzerland ends. Bauzeitung 84th volume, August issue