This study proposed a prediction formula for the torsional strength enabling to reflect the tensile strength of ultra high performance concrete (UHPC) beams based upon the thin-walled tube theory. The remarkable ductile behavior of UHPC can also be attributed to the steel fiber reinforcement. This feature must be considered to provide rational explanation of the torsional behavior of UHPC structures. In this study, the proposed torsional design adopts a modified thin-walled tube theory so as to consider the tensile behavior of UHPC. And torsion test was conducted on thin-walled UHPC box beams to validate the proposed formula through comparison of the predicted torsional strength with the experimental results. The comparison of the predicted values of the cracking torque and torsional moment resistance with those observed in the torsional test of UHPC verified the validity of the design method. The contribution of the steel fibers to the torsional strength and cracking load was larger than that of the stirrups, but the stirrups appeared to contribute additionally to the torsional ductility. Accordingly, it is recommended that design should exploit effectively the contribution of the steel fiber rather than arrange a larger number of stirrups in UHPC structures subjected to torsion.
Ultra high performance concrete (UHPC) is an advanced cementitious composite exhibiting low permeability in which the matrix with high density and strength is reinforced by steel fibers [
Studies reported that the design of torsional reinforcement focused essentially on stirrups by disregarding the tensile behavior of concrete in the case of NSC structures [
Many researchers led investigation on the improvement of the tensile strength and ductility of NSC by steel fiber reinforcement and found out that the adoption of steel fiber achieved improved flexural ductility and ten- sile strength [
The remarkable ductile behavior of UHPC can also be attributed to the steel fiber reinforcement. This feature must be considered to provide rational explanation of the torsional behavior of UHPC structures. The current design specification of ACI 318-11 describes the torsional design of structures based upon the thin-walled tube theory [
In this study, the proposed torsional design adopts a modified thin-walled tube theory so as to consider the tensile behavior of UHPC and the validity of the method is verified through comparison with experimental data. Most of the recent studies on UHPC focused only on the difference in the constitutive materials as compared with normal concrete. Despite of the insufficiency of experimental research utilizable for the analysis of the tor- sional behavior of UHPC beams, Yang, Joh, Lee and Kim conducted the experimental analysis of the pure tor- sional behavior of UHPC beams with rectangular cross-section [
Accordingly, this experimental study intends to propose a torsional design method based on the thin-walled tube theory and considering the tensile strength of UHPC. UHPC beams with thin-walled box cross-section were fabricated and subjected to pure torsion test for comparison with the theoretical data and to verify the validity of the proposed design method.
In order to obtain the ultimate strength of the UHPC beam considering the tensile behavior of UHPC, the beam undergoing cracking due to the torsional moment T is idealized as a thin-walled tube (
The total shear forces
and, the total shear forces
where
The number of stirrups including the inclined crack surface at the leftward face of the UHPC beam (
where
Accordingly, assuming that all the stirrups yield at ultimatetorsion limit state and that UHPC reaches its ten- sile strength
where
Moreover, Equation (5) expresses the tensile force of UHPC acting vertically to the inclined crack surface of
where
Therefore, the vertical component in Equation (5) that is the contribution
Consequently, the total shear force
In addition, the corresponding torsional moment
By distinguishing the contributions of the steel reinforcement and UHPC, Equation (8) becomes
The cracking torque
where
Moreover, if the member transverse compressive stress
If this principal stress
Hence, the cracking torque
In addition, the inclination angle
Ordinary Portland cement is used in the mix of steel fiber reinforced UHPC, and reactive powder and silica fume are also introduced. A water-to-binder of 0.2% is applied. Sand with grain size of 0.5 mm is used as fine aggregate, and coarse aggregate is not used. The filler is made of materials with average grain size of 10 μm, SiO2 content larger than 98% and density of 7500 kg/m3. Straight steel fiber is adopted with density of 7500 kg/m3, yield strength of 2500 MPa and diameter of 0.2 mm. Two fiber lengths of 16.5 mm and 19.5 mm are used concurrently at volume fractions of 1% and 1.5%.
Series of 6 specimens for tensile strength test were fabricated at each batch during the manufacture of the beam member using the placed UHPC. Direct tensile test was conducted to evaluate the tensile strength of each specimen. The dogbone type tensile specimens shown in
The test members were fabricated with rectangular box cross-section and dimensions of 350 mm ´ 350 mm ´ 3000 mm. For the tested part of the member, the central part (section B-B) was designed to have a wall thick- ness of 50 mm to induce torsional failure. To that goal, both ends of the member (section A-A) were designed with wall thickness of 120 mm, dense arrangement of D10 stirrups at spacing of 50 mm along a length of 650 mm, and the installation of 3 longitudinal reinforcements in each side of the rectangular cross-section. Moreover, apart from the ends of the member, D10 stirrups were disposed in section B-B along the member at spacing of 170 mm or 340 mm.
For the members with prestress, nineteen 7-wire tendons with nominal diameter of 15.2 mm (SWPC7, yield strength of 1860 MPa) were installed at the center of the empty box of the member.
A total of 14 test members corresponding to 2 specimens for each of 7 types were fabricated considering the test variables that are the volume fraction of steel fiber (1.0%, 1.5%), the amount of stirrups and reinforcing steel, and the introduction of prestress.
Specimen designation | Crack initiation strength | Tensile strength |
---|---|---|
SH-P0-F1.5-L1-S1(D13) | 8.23 | 11.48 |
SH-P2-F1.5-L1-S1 | 7.81 | 11.30 |
SH-P4-F1.5-L1-S1 | 8.25 | 11.44 |
SH-P0-F1-L1-S1 | 6.96 | 8.39 |
SH-P4-F1-L1-S1 | 5.90 | 7.09 |
SH-P0-F1.5-L1-S2 | 7.77 | 11.18 |
SH-P0-F1.5-L1-S1(D10) | 8.23 | 11.48 |
Specimen designation | Steel fiber (%) | Steel reinforcement | Stirrup spacing | ||
---|---|---|---|---|---|
16.5 mm | 19.2 mm | Stirrup | Longitudinal | ||
SH-P0-F1.5-L1-S1(D13) | 0.5 | 1 | D10 | D13@4 | 5@340 |
SH-P2-F1.5-L1-S1 | 0.5 | 1 | D10 | D22@4 | 5@340 |
SH-P4-F1.5-L1-S1 | 0.5 | 1 | D10 | D22@4 | 5@340 |
SH-P0-F1-L1-S1 | 0.5 | 0.5 | D10 | D13@4 | 5@340 |
SH-P4-F1-L1-S1 | 0.5 | 0.5 | D10 | D22@4 | 5@340 |
SH-P0-F1.5-L1-S2 | 0.5 | 1 | D10 | D10@4 | 10@170 |
SH-P0-F1.5-L1-S1(D10) | 0.5 | 1 | D10 | D10@4 | 5@340 |
The torque was applied as shown in
For the prestressed member, a hydraulic jack for the introduction of the prestress force in the tendon and a load cell for the measurement of the prestress force were installed in series at the fixed end of the member as shown in
A loading beam was installed at the rotating support. The load was applied at the position of the loading lo- cated at a distance of 0.9 m from the centerline of the member, which corresponds to the lever arm. Loading was applied through displacement control at speed of 0.03 mm/s. The rotating end was fabricated considering the ro- tation radius with respect to the centerline of the member.
The observation of the torsional angle or angle of twist of the beam was done by measuring the deflection at each loading stage using steel frames and LVDTs installed at sections located at 800 mm and 200 mm far from the center of the span as shown in
The propagation of the cracks was observed at each loading stage until failure, and the relation between the tor- sional moment-angle of twist was measured.
Initial cracking occurred in the form of inclined cracks with inclination angles ranging between 46.7˚ and 51.3˚ with reference to the centerline of the beam in the specimens without prestress, which gives an average angle of 49.7˚. For the prestressed members, the crack angles diminished gradually with larger longitudinal prestress to range between 14˚ and 15˚ in the case of the peak prestress of 50 MPa as shown in
After the initiation of the torsional cracks, the cracks exhibited spiral shape along the 4 sides of the members according to the increase of the torsional moment together with the occurrence of numerous torsional cracks. These additional cracks seem to enhance the load bearing capacity of the member owing to the redistribution of stress. Just prior to the peak torsional moment, some specific cracks among the inclined cracks showed increase of their crack width and developed into principal inclined cracks. After the peak load, the torsional moment re- duced with the acceleration of the pullout of the steel fibers and failure occurred as shown in
Specimen designation | Crack inclination (degree) | Prestress (MPa) | |
---|---|---|---|
Test | Theory | ||
SH-P0-F1.5-L1-S1(D13) | 46.7 | 45.0 | 0 |
50.4 | 45.0 | 0 | |
SH-P2-F1.5-L1-S1 | 19.8 | 26.4 | 25.0 |
23.4 | 26.4 | 25.0 | |
SH-P4-F1.5-L1-S1 | 30.9 | 32.1 | 12.5 |
27.3 | 26.4 | 25.0 | |
SH-P0-F1-L1-S1 | 48.5 | 45.0 | 0 |
50.3 | 45.0 | 0 | |
SH-P4-F1-L1-S1 | 14.0 | 18.1 | 50.0 |
15.1 | 18.1 | 50.0 | |
SH-P0-F1.5-L1-S2 | 51.3 | 45.0 | 0 |
50.8 | 45.0 | 0 | |
SH-P0-F1.5-L1-S1(D10) | 49.2 | 45.0 | 0 |
50.6 | 45.0 | 0 |
the bridging effect of the steel fibers at the principal inclined cracks. Similar crack propagation and failure pattern were observed in previous experimental studies on rectangular plain UHPC beams [
In the case where the longitudinal reinforcement is increased from 4 D-10 bars to 4 D-13 bars, the peak tor- sional moment is seen to vary from 104.6 kN-m and 87.2 kN-m for member SH-P0-f1.5-L1-S1(D10) to 96.6 kN-m and 100.7 kN-m for member SH-P0-f1.5-L1-S1(D13). This means that the increase of the longitudinal reinforcement has no effect on the increase of the torsional resistance.
For the transverse reinforcement, when the number of stirrups is doubled, the peak torsional moment is seen to change from 104.6 kN-m and 87.2 kN-m for member SH-P0-f1.5-L1-S1(D10) to 97.5 kN-m and 104.7 kN-m for member SH-P0-f1.5-L1-S2(D10). This indicates that, for the members considered in this study, the increment in the torsional resistance resulting from the increase of the stirrups is smaller than that resulting from the increase of the steel fiber content. Besides, the increase of the stirrups is seen to augment the ductility in the torsional mo- ment-angle of twist relation owing to the generation of a larger number of inclined cracks favoring the redistribu- tion of stress.
The change in the peak torsional moment according to the increase of the amount of steel fiber, increase of transverse reinforcement, and increase of longitudinal reinforcement is similar to that observed by Yang, Joh, Lee and Kim in their experimental study on UHPC beams with plain rectangular cross-section [
Moreover, the peak torsional moment of member SH-P4-f1.0-L1-S1-2 with prestress of 50.0 MPa is 212.8 kN-m. This indicates the larger increase of the peak torsional moment even with a small steel fiber volume frac- tion of 1.0% compared to the case with small prestress and steel fiber volume fraction of 1.5%. For example, the identical member SH-P4-f1.0-L1-S1-1, which underwent reduction of its prestress force due to the dysfunction of the hydraulic jack during the jacking of the prestressing tendon, resisted only up to a peak torsional moment of 167.5 kN-m.
For the cases without prestress, the inclined cracks should theoretically form an angle of 45˚ with the centerline of the member. However, slight deviation was reported in the observation of the shear test of beam members. Therefore, the value of the inclination angle
Figures 13 compares the analytic and experimental values of the cracking torque
This study proposed a prediction formula for the torsional strength enabling to reflect the tensile strength of ultra high performance concrete (UHPC) beams based upon the thin-walled tube theory. Torsion test was conducted on thin-walled UHPC box beams to validate the proposed formula through comparison of the predicted torsional
Specimen designation | Tcr (kN-m) | Tn (kN-m) | |||
---|---|---|---|---|---|
Analysis | Test (photograph) | Analysis (q0 accounted) | Analysis (q0 not accounted) | Test (graph) | |
SH-P0-F1.5-L1-S1(D13) | 73.2 | 68.4 | 100.1 | 118.2 | 96.6 |
73.2 | 79.1 | 100.1 | 118.2 | 100.7 | |
SH-P2-F1.5-L1-S1 | 147.7 | 147.6 | 196.0 | 238.5 | 190.6 |
147.7 | 148.5 | 196.0 | 238.5 | 187.9 | |
SH-P4-F1.5-L1-S1 | 116.6 | 115.2 | 157.7 | 188.2 | 145.8 |
147.7 | 157.5 | 196.0 | 238.5 | 192.0 | |
SH-P0-F1-L1-S1 | 62.6 | 66.7 | 76.8 | 90.6 | 76.4 |
62.6 | 68.3 | 76.8 | 90.6 | 84.9 | |
SH-P4-F1-L1-S1 | 165.1 | 165.6 | 190.4 | 245.0 | 167.5 |
165.1 | 144 | 190.4 | 245.0 | 212.8 | |
SH-P0-F1.5-L1-S2 | 73.2 | 65.6 | 112.9 | 133.3 | 97.5 |
73.2 | 70.3 | 112.9 | 133.3 | 104.7 | |
SH-P0-F1.5-L1-S1(D10) | 73.2 | 64.7 | 100.1 | 118.2 | 104.6 |
73.2 | 64.0 | 100.1 | 118.2 | 87.2 |
strength with the experimental results. The following conclusions can be drawn.
1) The thin-walled tube theory was adopted to derive the prediction formula of the torsional strength so as to reflect the contribution of the tensile strength of UHPC provided by the steel fibers to the torsional strength of the beam. The comparison of the predicted values of the cracking torque and torsional moment resistance with those observed in the torsional test of UHPC verified the validity of the design method.
2) The observation of the cracking and failure patterns revealed the occurrence of several additional inclined cracks after the initial cracks according to the increase of the load. The members exhibited post-cracking behavior with continuous increase of the load until the ultimate limit state. This indicated that the torsional behavior of the UHPC beams developed clear ductility owing to the steel fibers even after the initiation of cracks. However, the prestressed members showed brittle failure after the peak load.
3) The contribution of the steel fibers to the torsional strength and cracking load was larger than that of the stirrups, but the stirrups appeared to contribute additionally to the torsional ductility. Accordingly, it is recom- mended that design should exploit effectively the contribution of the steel fiber rather than arrange a larger num- ber of stirrups in UHPC structures subjected to torsion.
4) An average deviation of 4.7˚ from the theoretical angle of 45˚ predicted for the inclined cracks in the case of beam members without prestress. The consideration of this deviation enabled the prediction to approach more accurately the experimental data. The inclination angle of the inclined cracks reduced and the ultimate torsional strength increased with larger amount of prestress. Since the torsional resistance varied more sensitively to the change in the angle of the torsional cracks with larger prestress, attention should be paid on the computation of the crack angle in the design.
This research was supported by a grant (14 AUDP-B069364-02) from Urban Architectural Research Program funded by Ministry of Land, Infrastructure and Transport of Korean Government.