Material-Dependent Shear Capacity of Threaded Rods

: Currently, the determination of the shear capacity V 0 Rk , s of post installed adhesive anchors is already included in the current standardization and approval documents. Considering that EAD 330499-00-0601 allows for determining the shear resistance of a fastener only based on the characteristic steel ultimate tensile strength f uk determined from the material tensile tests, and without considering the actual ductility of the material used, this leads to a severe underestimation of the actual steel shear resistance. In order to e ﬃ ciently determine the shear strength by calculation based on material characteristics, tensile and shear tests were carried out on metallic threaded rods and rebars to show their correlation with the material properties. A new correlation between tensile and shear strength is presented, which is based on the plasticity module E pl and allows a good prognosis of the shear strength.


Introduction
The approval of a new post installed injection anchor is time consuming and expensive. The approval as such is based almost exclusively on the results of approval tests. The design of the approval procedure is based on several years of research, the results of which can be found in approval documents such as Technical Reports (TRs) or European Assessment Documents (EADs). A review of the last few years alone shows the major progress in transforming European Technical Approval Guidelines (ETAGs) into EADs or TRs. At this stage, several changes have already been made to the test procedures, and explanations have been added. Nevertheless, there are still some open questions regarding the determination of steel shear resistance of injection anchors. Determination of shear resistance in accordance with the current standard [1,2] is recorded in the test description of the test series V1 which offers two alternative options: 1.
performing at least five shear tests with the fastening elements in concrete; 2.
The load-bearing behavior of the fastening elements in concrete has been researched since the beginning of the 20th century. In the 1930s, B.F. Friberg [3,4] applied the approach of beams on elastic foundations to describe the load-bearing behavior of fasteners. Later, the fundamental failure mechanisms were described by Paulay [5]. Following further experimental and theoretical 0.55 0.60 0.60 -EAD 330499-00-0601 [2] --0.50 -At this point, it must also be noted that the comparison with the theoretical value α v = 1/ √ 3 according to Hubert-Mises does not take place here, because this theory does not allow large plastic shear distortions: "With plastic deformations, stress always remains at the elastic limit" [23] (p. 588).
In summary, the maximum shear strength τ u of threaded rods must be dependent on at least one other material parameter in addition to tensile strength. Based on the shear and tensile tests, we found a new correlation to estimate the shear strength of threaded and reinforcing steels based on their material properties.

Materials and Methods
To compare the steel properties of threaded bars and reinforcing bars, five different steel grades were used: 4.8, 5.6, 8.8, A4/70 and B500 B. To demonstrate the influence of the shape factor, the investigations were carried out on several bar diameters (see Table 2).

Method-Plasticity Module
The modulus of elasticity E el is known to be an indispensable measurement parameter in civil engineering, and the basis for deformation calculations and the determination of the resulting stress states in the elastic or elastic-plastic stress state of the material. Shear strength of individual anchors in fastening technology is determined based on plastic material failure by either determining the shear stress τ u as a result of practical shear tests in concrete, or theoretically according to Equation (1).
According to Equation (1), the shear stress τ u of a threaded rod depends on tensile stress σ u . This material dependence is defined by the reduction factor α v . It is assumed here that the value of the reduction factor α v depends on at least one additional steel parameter. Observing the mechanisms of plastic deformations allows for the identification of such a parameter.
Microscopic examination reveals a crystalline structure of most metals. In the case of elastic deformation, the crystal must change its atomic structure. The crystal structure is retained during plastic deformation, while the outer shape is simultaneously changed [24]. The change of the atomic structure is illustrated in Figure 1.

Method-Plasticity Module
The modulus of elasticity is known to be an indispensable measurement parameter in civil engineering, and the basis for deformation calculations and the determination of the resulting stress states in the elastic or elastic-plastic stress state of the material. Shear strength of individual anchors in fastening technology is determined based on plastic material failure by either determining the shear stress as a result of practical shear tests in concrete, or theoretically according to Equation (1).
According to Equation (1), the shear stress of a threaded rod depends on tensile stress . This material dependence is defined by the reduction factor . It is assumed here that the value of the reduction factor depends on at least one additional steel parameter. Observing the mechanisms of plastic deformations allows for the identification of such a parameter.
Microscopic examination reveals a crystalline structure of most metals. In the case of elastic deformation, the crystal must change its atomic structure. The crystal structure is retained during plastic deformation, while the outer shape is simultaneously changed [24]. The change of the atomic structure is illustrated in Figure 1. The dislocation motion between the atoms causes the formation of slip systems (slip plane or slip lines) in the atomic structure. The beginning of plastic flow resulting from tensile and shear stresses also characterizes the beginning of the dislocation movement. The Schmidian stress law according to Equation (2) states that when tensile stress is applied, the resulting shear stress in the slip system decides the dislocation movement and not the applied tensile stress [25]    The dislocation motion between the atoms causes the formation of slip systems (slip plane or slip lines) in the atomic structure. The beginning of plastic flow resulting from tensile and shear stresses also characterizes the beginning of the dislocation movement. The Schmidian stress law according to Equation (2) states that when tensile stress is applied, the resulting shear stress in the slip system decides the dislocation movement and not the applied tensile stress [25] (p. 228). The geometrical relations of Equation (2) are shown in Figure 2. The magnitude of the resulting shear stress describes the shear stress in the slip system which is critical for the origin of the dislocation movement. An extensive description of the physical processes can be found in the literature [24][25][26].
Based on this knowledge, an ideal failure free model can be imagined in which a shear failure is key. With regard to dislocation movements in slip systems, the maximum shear stress can be determined directly in a shear test. Thus, Equation (2) can be idealized and simplified as follows: The position and orientation of the slip plane in the considered material remain unknown at first. However, they can be described by a positional change function of the atoms during shear loading both at microscopic and macroscopic levels. The position change function is the change in distance of the atoms relative to the original position ( , ), which can be approximately measured as strains during a tensile test. It is assumed here that the maximum tensile stress thus describes the maximum shear stress dependent on the material strains . The real level of tensile stresses is due to the many different strengthening mechanisms. The dislocation movement mechanisms make a significant contribution to the increase in strength [27].
The determination of the modulus of elasticity is based on considerations to determine the elastic deformability of the material based on its tensile stresses and strains . A new parameter, the modulus of plasticity , is introduced to describe the plastic deformability of the material. Based on the assumption that the determination of can be characterized as a transition point from an elastic to a plastic state, the same procedure is used to determine the modulus. However, the maximum stress determined in a tensile test and the corresponding strain at the maximum load characterizes the transition point of the plastic uniform strain along the tensile test to the local material necking-softening. The ratio of tensile stress to longitudinal strain at this inflection point is named the modulus of plasticity according to Equation (4) and is shown schematically in The magnitude of the resulting shear stress τ R describes the shear stress in the slip system which is critical for the origin of the dislocation movement. An extensive description of the physical processes can be found in the literature [24][25][26].
Based on this knowledge, an ideal failure free model can be imagined in which a shear failure is key. With regard to dislocation movements in slip systems, the maximum shear stress τ u can be determined directly in a shear test. Thus, Equation (2) can be idealized and simplified as follows: The position and orientation of the slip plane in the considered material remain unknown at first. However, they can be described by a positional change function f (ε) of the atoms during shear loading both at microscopic and macroscopic levels. The position change function f (ε) is the change in distance of the atoms relative to the original position (ε x , ε y ), which can be approximately measured as strains during a tensile test. It is assumed here that the maximum tensile stress σ u thus describes the maximum shear stress τ u dependent on the material strains ε. The real level of tensile stresses σ u is due to the many different strengthening mechanisms. The dislocation movement mechanisms make a significant contribution to the increase in strength [27].
The determination of the modulus of elasticity E el is based on considerations to determine the elastic deformability of the material based on its tensile stresses σ u and strains ε. A new parameter, the modulus of plasticity E pl , is introduced to describe the plastic deformability of the material. Based on the assumption that the determination of E el can be characterized as a transition point from an elastic to a plastic state, the same procedure is used to determine the E pl modulus. However, the maximum stress R m determined in a tensile test and the corresponding strain at the maximum load A gt characterizes the transition point of the plastic uniform strain along the tensile test to the local material necking-softening. The ratio of tensile stress to longitudinal strain at this inflection point is named the modulus of plasticity E pl according to Equation (4) and is shown schematically in Figure 3.
The introduced quantity is intended to represent a direct dependency of the results from tensile and shear tests in order to determine the shear strength of threaded rods based on the parameters obtained in the tensile test. It is important to note that the total extensions and not only the plastic extensions are considered. While determining the shear strength, the rod undergoes both elastic and plastic strains which are retained up to the failure point. The metallurgical outlook is intended to provide a general understanding of the experimental results presented in Section 3. The correlations between shear strength and tensile strength are empirical. The theoretical background under the inclusion of the plasticity theory is complex and is promised for further investigations. The theory of plasticity is not discussed in this article.

Tensile Tests
The steel properties were determined based on tensile tests carried out at the Technical University of Kaiserslautern on prepared tensile specimens. Form A and B tensile tests according to DIN 50125:2016-12 [28] were carried out on both threaded and reinforced bars. Table 3 shows the geometries of the tensile specimens in   The tensile tests with the tensile specimen form A are considered as a preliminary test series. These results are evaluated in conjunction with the results of the main tests on tensile specimen form The introduced quantity E pl is intended to represent a direct dependency of the results from tensile and shear tests in order to determine the shear strength of threaded rods based on the parameters obtained in the tensile test. It is important to note that the total extensions A gt and not only the plastic extensions A g are considered. While determining the shear strength, the rod undergoes both elastic ε el and plastic ε pl strains which are retained up to the failure point.
The metallurgical outlook is intended to provide a general understanding of the experimental results presented in Section 3. The correlations between shear strength and tensile strength are empirical. The theoretical background under the inclusion of the plasticity theory is complex and is promised for further investigations. The theory of plasticity is not discussed in this article.

Tensile Tests
The steel properties were determined based on tensile tests carried out at the Technical University of Kaiserslautern on prepared tensile specimens. Form A and B tensile tests according to DIN 50125:2016-12 [28] were carried out on both threaded and reinforced bars. Table 3 shows the geometries of the tensile specimens in Figure 4. Table 3. Geometries of the tensile specimens.
The introduced quantity is intended to represent a direct dependency of the results from tensile and shear tests in order to determine the shear strength of threaded rods based on the parameters obtained in the tensile test. It is important to note that the total extensions and not only the plastic extensions are considered. While determining the shear strength, the rod undergoes both elastic and plastic strains which are retained up to the failure point. The metallurgical outlook is intended to provide a general understanding of the experimental results presented in Section 3. The correlations between shear strength and tensile strength are empirical. The theoretical background under the inclusion of the plasticity theory is complex and is promised for further investigations. The theory of plasticity is not discussed in this article.

Tensile Tests
The steel properties were determined based on tensile tests carried out at the Technical University of Kaiserslautern on prepared tensile specimens. Form A and B tensile tests according to DIN 50125:2016-12 [28] were carried out on both threaded and reinforced bars. Table 3 shows the geometries of the tensile specimens in     The tensile tests with the tensile specimen form A are considered as a preliminary test series. These results are evaluated in conjunction with the results of the main tests on tensile specimen form B.
Both strain-controlled and displacement-controlled tensile tests were carried out. In both cases, the test speed was chosen by [28]. In the elastic range, . e was 0.00007 s −1 and in the plastic range until steel failure, the strain rate was constant . e 0.00025 s −1 . The test setup is shown in Figure 5a. Using the RTSS | Video extensometer of the company LIMESS [29] both the elastic and the plastic steel strain up to steel failure were measured without contact, shown in Figure 5b. The individual material properties determined with tensile tests according to DIN EN ISO 6892-1 [30] can be found in Appendix A. They are summarized in Tables A1 and A2. CivilEng 2020, 1, FOR PEER REVIEW 6 B. Both strain-controlled and displacement-controlled tensile tests were carried out. In both cases, the test speed was chosen by [28]. In the elastic range, was 0.00007 s −1 and in the plastic range until steel failure, the strain rate was constant 0.00025 s −1 . The test setup is shown in Figure 5a. Using the RTSS | Video extensometer of the company LIMESS [29] both the elastic and the plastic steel strain up to steel failure were measured without contact, shown in Figure 5b. The individual material properties determined with tensile tests according to DIN EN ISO 6892-1 [30] can be found in Appendix A. They are summarized in Tables A1 and A2.

Shear Tests
Based on current research results, the determination of the shear strength of bonded threaded rod and rebar bars is significantly influenced by the embedment depth and the strength of the adhesive mortar used [16]. To avoid these influences, the bars were sheared in a steel fixture to determine the steel shear strength without external influences. In contrast to the standardized shear test according to [31], the test is performed as a single-shear test. Figure 6 shows a schematic drawing of the testing device. The test specimen is located between two hardened pass cylinders, which are inserted in the shear plate and the base plate. Before the test is carried out, both the threaded bars and the rebars are locked with nuts on both sides to prevent the bars from slipping through. The single-sided load causes additional tensile forces in the vertical direction, which can lead to the shear plates lifting off. Lifting of the shear plate is prevented by additional mounted roller bearings. The friction between the pass cylinders in the shear plate and the base plate is reduced to a minimum by the polytetrafluoroethylene (PTFE) foil used. Following [32,33], a 0.50 mm thick PTFE foil was chosen.

Shear Tests
Based on current research results, the determination of the shear strength of bonded threaded rod and rebar bars is significantly influenced by the embedment depth and the strength of the adhesive mortar used [16]. To avoid these influences, the bars were sheared in a steel fixture to determine the steel shear strength without external influences. In contrast to the standardized shear test according to [31], the test is performed as a single-shear test. Figure 6 shows a schematic drawing of the testing device.
CivilEng 2020, 1, FOR PEER REVIEW 6 B. Both strain-controlled and displacement-controlled tensile tests were carried out. In both cases, the test speed was chosen by [28]. In the elastic range, was 0.00007 s −1 and in the plastic range until steel failure, the strain rate was constant 0.00025 s −1 . The test setup is shown in Figure 5a. Using the RTSS | Video extensometer of the company LIMESS [29] both the elastic and the plastic steel strain up to steel failure were measured without contact, shown in Figure 5b. The individual material properties determined with tensile tests according to DIN EN ISO 6892-1 [30] can be found in Appendix A. They are summarized in Tables A1 and A2.

Shear Tests
Based on current research results, the determination of the shear strength of bonded threaded rod and rebar bars is significantly influenced by the embedment depth and the strength of the adhesive mortar used [16]. To avoid these influences, the bars were sheared in a steel fixture to determine the steel shear strength without external influences. In contrast to the standardized shear test according to [31], the test is performed as a single-shear test. Figure 6 shows a schematic drawing of the testing device. The test specimen is located between two hardened pass cylinders, which are inserted in the shear plate and the base plate. Before the test is carried out, both the threaded bars and the rebars are locked with nuts on both sides to prevent the bars from slipping through. The single-sided load causes additional tensile forces in the vertical direction, which can lead to the shear plates lifting off. Lifting of the shear plate is prevented by additional mounted roller bearings. The friction between the pass cylinders in the shear plate and the base plate is reduced to a minimum by the polytetrafluoroethylene (PTFE) foil used. Following [32,33], a 0.50 mm thick PTFE foil was chosen. The test specimen is located between two hardened pass cylinders, which are inserted in the shear plate and the base plate. Before the test is carried out, both the threaded bars and the rebars are locked with nuts on both sides to prevent the bars from slipping through. The single-sided load causes additional tensile forces in the vertical direction, which can lead to the shear plates lifting off. Lifting of the shear plate is prevented by additional mounted roller bearings. The friction between the pass cylinders in the shear plate and the base plate is reduced to a minimum by the polytetrafluoroethylene (PTFE) foil used. Following [32,33], a 0.50 mm thick PTFE foil was chosen. The described procedure corresponds to performing suitability tests according to Technical Report 049 [33], except in these tests the anchor is not bonded in the concrete, but is screwed hand-tight between two metal plates.
A load cell was positioned on the side of the base plate to measure the tensile forces caused by forced deformation of the rod.
The diameter of the clearance hole d f was chosen so that there was as little space as possible between the hole wall and the test specimen. The geometries of the pass cylinders are summarized in Table 4. According to [2,34], the threaded and rebar bars were brought to shear failure within 1 to 3 min by a linearly increasing force. The individual results are shown in Appendix A, Table A3.

Results
The results of the performed tests were first presented as mean values of the maximum tensile X σu and shear stresses X τu . The resulting reduction factors from Table 5 were compared to the α v values from various standards in Figure 7. by a linearly increasing force. The individual results are shown in Appendix A, Table A3.

Results
The results of the performed tests were first presented as mean values of the maximum tensile and shear stresses . The resulting reduction factors from Table 5 were compared to the values from various standards in Figure 7.   The calculated reduction factors α v show a controversial situation at first. Within the same steel grade, a relatively large scatter of the α v values is recorded. The difference between the diameters M8 and M20 of steel grade 4.8, which is approx. 0.1, is particularly high. A comparison of the laboratory results with the standard values from Table 1 does not lead to plausible conclusions either.
Although it could be observed in some cases that the increase in tensile strength σ u within a steel grades increases the shear strength τ u proportionally, this could not be transferred to all materials. Further analysis of the laboratory results showed a direct correlation between the strains A gt determined from the tensile tests and the shear strength τ u . By applying the proposed method for determining the modulus of plasticity E pl , it was possible to determine the correlation between shear and tensile stresses as a function of the reduction factor α v . The calculated values of the modulus of plasticity E pl and the reduction factors α v for used materials are summarized in Figure 8.  Although it could be observed in some cases that the increase in tensile strength within a steel grades increases the shear strength proportionally, this could not be transferred to all materials. Further analysis of the laboratory results showed a direct correlation between the strains determined from the tensile tests and the shear strength . By applying the proposed method for determining the modulus of plasticity , it was possible to determine the correlation between shear and tensile stresses as a function of the reduction factor . The calculated values of the modulus of plasticity and the reduction factors for used materials are summarized in Figure   8. To apply the presented procedure to the existing data sets, the following must be observed: •  To apply the presented procedure to the existing data sets, the following must be observed: • The shear force acting on the bar causes forced deformations in the longitudinal direction of the bar. These deformations manifest themselves by the formation of normal forces in the bar. The level of the resulting tensile force can easily be estimated. In Figure 9a, the secondary bending moment of the threaded nut is used to calculate the tensile and compressive components in the bar. The shear force acting on the bar causes forced deformations in the longitudinal direction of the bar. These deformations manifest themselves by the formation of normal forces in the bar. The level of the resulting tensile force can easily be estimated. In Figure 9a, the secondary bending moment of the threaded nut is used to calculate the tensile and compressive components in the bar.
In the tests performed in Figure 9b, the tensile forces are very low because the selected geometry of the hole wall in Table 4 directly limits the bar in space and does not allow any significant deflection. The development of secondary bending moments at the nut undoubtedly influences the loadbearing behavior of threaded connections. To estimate its effect, the quotients of the measured bar tension stresses , and shear stresses were calculated in Table 6.  In the tests performed in Figure 9b, the tensile forces are very low because the selected geometry of the hole wall in Table 4 directly limits the bar in space and does not allow any significant deflection.
The development of secondary bending moments at the nut undoubtedly influences the load-bearing behavior of threaded connections. To estimate its effect, the quotients of the measured bar tension stresses σ N,u and shear stresses τ u were calculated in Table 6. According to this, the level of the tensile force N u is geometry dependent and shows only minimal deviations for different materials. With increasing material strength, the tensile force is reduced according to the influence of the tensile force.
These effects have already been considered in the factor α v . Further efforts will be made to investigate the influence of the longitudinal strains on the failure mechanism. In this article, this topic will not be pursued further.

Discussion
The results in Figure 8 can be described as a regression curve. This means the reduction factor α v can already be taken directly from the diagram in a practical way, using the determined E pl value as a first approximation. The correlations shown here are only valid for the depicted results. To generalize the scope, further investigations must be carried out with a much wider spectrum of materials and rod diameters. It is also advisable to define sample geometry and test speed so that results from different sources can be compared. Finally, the creation of an extensive database can be seen as a goal towards advancing research.
However, the regression analysis also shows that mixing the results of the reinforcement and threaded bars without taking other factors into account is not useful, because, in contrast to threaded bars, the shear failure of the reinforcing bars was determined in the shaft. Additionally, the low variation of the modulus of the plasticity of different rebars combined with a strong variation of the reduction factors α v 0.70 ÷ 0.80 allows the assumption that bar geometry has to be considered.
The influence of bar geometry on the threaded bars could not be investigated extensively because of the small number of tests. Nevertheless, it can be confirmed that the variation of the test results took place within comparable materials with different diameters.
Our findings confirm that brittle materials with a very high modulus of plasticity E pl have a lower shear strength τ u . In contrast, the high-ductility steels with a very low modulus of plasticity E pl have a higher shear strength τ u concerning the tensile strength.
In the future, it may be possible to develop a new design methodology based on the presented correlation between tensile and shear strength, so that laboratory testing of shear resistance of threaded and reinforcing bars as well as screws can be completely omitted.

Conflicts of Interest:
The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.