A theoretical analysis of upward deflection and midspan deflection of prestressed bamboo-steel composite beams is presented in this study. The deflection analysis considers the influences of interface slippage and shear deformation. Furthermore, the calculation model for flexural capacity is proposed considering the two stages of loading. The theoretical results are verified with 8 specimens considering different prestressed load levels, load schemes, and prestress schemes. The results indicate that the proposed theoretical analysis provides a feasible prediction of the deflection and bearing capacity of bamboo-steel composite beams. For deflection analysis, the method considering the slippage and shear deformation provides better accuracy. The theoretical method for bearing capacity matches well with the test results, and the relative errors in the serviceability limit state and ultimate limit state are 4.95% and 5.85%, respectively, which meet the accuracy requirements of the engineered application.
Bamboo is the most important plant resource in the world, characterized by a fast-growing rate, renewability, biodegradability, and carbon sequestration ability [
Nevertheless, the flexural members made by bamboo scrimber had higher deflection compared with concrete and steel members [
On the other hand, an I-shaped section is a type of highly efficient beam section, which can economically enhance the flexural stiffness and bearing capacity compared with a solid rectangular section [
Thin-walled cold-formed steel is widely used in modern structures with high strength and stiffness, while the steel members tend to buckle before achieving ultimate strength [
Large-span structure is an important form of the modern structural system [
To analyze the bearing capacity and deflection of prestressed bamboo-steel composite beams, 8 I-shaped composite beams were prepared for the test. The specimens are shown in
Specimens | Prestressed load (kN) | Prestress scheme | Load scheme |
---|---|---|---|
B-20-I-1 | 20 | I | 1 |
B-30-I-1 | 30 | I | 1 |
B-20-I-2 | 20 | I | 2 |
B-30-I-2 | 30 | I | 2 |
B-20-II-1 | 20 | II | 1 |
B-30-II-1 | 30 | II | 1 |
B-20-II-2 | 20 | II | 2 |
B-30-II-2 | 30 | II | 2 |
Note: I and II represent bilinear and trilinear prestress schemes, 1 and 2 represent three-point and four-point bending load schemes.
The bamboo-steel composite beam is made of cold-formed thin-walled steel and bamboo scrimber bonded by adhesive. The adhesive is Henkel EA3162A/B component epoxy adhesive, manufactured by LOCTITE. The prestressed strands are made of seven-strand wire, whose nominal diameter is 15.2 mm. The main mechanical properties of the three materials are shown in
Material | Strength (MPa) | Elastic modulus (MPa) |
---|---|---|
Bamboo scrimber | 93.47 (Compression) | 15673 |
124.04 (Tension) | ||
Thin-walled steel | 284 (Yield tension) | 2.0 × 10^{5} |
378 (Ultimate tension) | ||
Prestressed strands | 1860 (Ultimate tension) | 1.95 × 10^{5} |
The fabrication of the specimens is as follows: first, the surfaces of the bamboo panel and steel channel were polished with polishers and cleaned with alcohol wipes (
As shown in
A universal reaction frame was adopted for the test, and the vertical load was applied through a hydraulic jack and measured by a load sensor. The load control strategy with the step of 5 kN was employed for loading. There are two schemes for vertical loading according to
Previous studies about the flexural behavior of bamboo-steel composite beams have indicated that slight slippage occurred between bamboo and steel [
The calculation method of additional deflection has been proposed and matches well with the test results according to the reference [
where
The flexural behaviors of composite beams will be determined by the combined effects of bending moment and shear force [
where
The application of prestressed reinforcement may cause an upward deflection for the beam. According to the equivalent theory of statics, the effect of prestressed reinforcement can be equivalent to upward forces acting on the deviators. Therefore, the bilinear prestress scheme can be seen as a concentrated upward force acting on the midspan of the beam, and the trilinear prestress scheme can be regarded as two symmetric upward forces acting upon the deviators (
Without considering the influence of additional deflection, the mid-span deflection of composite beams can be expressed according to the mechanics of materials as follows:
where
Therefore, the midspan upward deflection
where
According to the test overview above, the midspan deflection of the prestressed composite beam under the four conditions (i.e., the bilinear prestress and three-point bending scheme, the bilinear prestress and four-point bending load scheme, the trilinear prestress and three-point bending load scheme, the trilinear prestress and four-point bending load scheme) are discussed in this paper.
The deformation of composite beams under the external load will cause the stress increment of the prestressed reinforcements, in turn, the stress increment may offset the deformation of beams. Therefore, the final deflection of the prestressed beam is the interaction caused by external load and stress increment of prestressed reinforcements.
The load condition and deformation of beams under the bilinear prestress and three-point bending load scheme are shown in
where
According to the superposition principle and
Similarly, the deformations of beams under the bilinear prestress and four-point bending load scheme are shown in
The deflection can be obtained by solving the equations about
where j = 1 or 2, represents a three-point bending load or four-point bending load scheme, respectively.
The load condition and deformation of beams under the trilinear prestress and three-point bending load scheme are shown in
where
Substituting
The midspan deflection
Similarly, the deformations of beams under the trilinear prestress and four-point bending load scheme are shown in
The deflection can be obtained by solving the equations about
where, j = 1 or 2, represents a three-point bending load scheme or four-point bending load scheme, respectively.
The prestressed bamboo-steel composite beams can be regarded as one-degree of statically indeterminate structures composed of external prestressed reinforcement and bamboo-steel composite beams. The stress increment of prestressed reinforcement can be solved by the force method in structural mechanics. To simplify the theoretical analyses, some basic assumptions are introduced:
(1) The composite beam sections conform to the plane section assumption.
(2) The deformation of the end anchorage device is neglected.
(3) The prestressed reinforcement is considered as an elastic body.
(4) The stress of the prestressed reinforcement is assumed to be equal along the length direction.
The basic diagram of the bilinear prestress and three-point bending load scheme for the force method is shown in
where
where,
The basic diagram of the bilinear prestress and four-point bending load scheme is shown in
The basic diagram of the trilinear prestress and three-point bending load scheme for the force method is shown in
The basic diagram of the trilinear prestress and four-point bending scheme is shown in
Thus, the stress increment
To analyze the flexural load-bearing capacity of the prestressed composite beams, the whole loading process can be divided into two stages. In the first stage, the deflection of beams changes from a precamber state to zero. In the second stage, the deflection of beams changes from zero to the positive ultimate limit state with the combined action of external load and prestressed reinforcement. Thus, the critical state is when the deflection of composite beams reaches zero.
In the first stage, when the bearing capacity reaches the maximum, the deflection of the composite beams is zero, i.e.,
For the bilinear prestress and four-point bending load scheme, the
where
Similarly, the bearing capacity for the other three schemes can be expressed:
For the bilinear prestress and three-point bending load scheme:
For the trilinear prestress and three-point bending load scheme:
For the trilinear prestress and four-point bending load scheme:
where
The ultimate bending moment
where
The ultimate bending moment
For the bilinear prestress and three-point bending load scheme:
For the bilinear prestress and four-point bending load scheme:
For the trilinear prestress and three-point bending load scheme:
For the trilinear prestress and four-point bending load scheme:
For the same specimens, the
The theoretical and experimental results of upward deflection are shown in
Specimens | Δ |
Δ |
|||||
---|---|---|---|---|---|---|---|
B-20-I-1 | 4.09 | 3.732 | 8.75 | 0.065 | 0.132 | 3.929 | 3.94 |
B-30-I-1 | 6.95 | 5.597 | 19.47 | 0.097 | 0.198 | 5.892 | 15.22 |
B-20-I-2 | 3.90 | 3.732 | 4.31 | 0.065 | 0.132 | 3.929 | 0.74 |
B-30-I-2 | 5.88 | 5.597 | 4.81 | 0.097 | 0.198 | 5.892 | 0.20 |
B-20-II-1 | 4.88 | 4.666 | 4.39 | 0.037 | 0.132 | 4.835 | 0.92 |
B-30-II-1 | 7.05 | 6.999 | 0.72 | 0.055 | 0.199 | 7.253 | 2.88 |
B-20-II-2 | 4.77 | 4.666 | 2.18 | 0.037 | 0.132 | 4.835 | 1.36 |
B-30-II-2 | 7.57 | 6.999 | 7.54 | 0.055 | 0.199 | 7.253 | 4.19 |
According to standard [
Due to the step load control strategy adopted in the test, the measuring equipment cannot exactly record the allowable defection and serviceable load. Therefore, the deflection closed to 14 mm, and the corresponding load (
Specimen | Δ |
Δ |
||||||
---|---|---|---|---|---|---|---|---|
B-20-I-1 | 35 | 13.993 | 12.299 | 12.11 | 0.416 | 0.780 | 13.495 | 3.56 |
B-30-I-1 | 40 | 13.613 | 11.910 | 12.51 | 0.479 | 0.891 | 13.280 | 2.45 |
B-20-I-2 | 40 | 14.280 | 12.452 | 12.80 | 0.193 | 0.622 | 13.267 | 7.09 |
B-30-I-2 | 50 | 15.167 | 14.180 | 6.51 | 0.237 | 0.764 | 15.181 | 0.09 |
B-20-II-1 | 40 | 14.773 | 12.621 | 14.57 | 0.479 | 0.891 | 13.991 | 5.29 |
B-30-II-1 | 45 | 14.880 | 12.361 | 16.93 | 0.539 | 1.003 | 13.903 | 6.56 |
B-20-II-2 | 45 | 14.647 | 12.402 | 15.33 | 0.217 | 0.699 | 13.318 | 9.07 |
B-30-II-2 | 55 | 14.981 | 13.707 | 8.51 | 0.264 | 0.852 | 14.823 | 1.05 |
The test results and calculation results of bearing capacity are listed in
Specimen | Serviceability limit state | Ultimate limit state | ||||
---|---|---|---|---|---|---|
B-20-I-1 | 35.01 | 36.27 | 3.59 | 95 | 88.26 | 7.09 |
B-30-I-1 | 40.61 | 39.19 | 3.49 | 88 | 91.75 | 4.26 |
B-20-I-2 | 39.36 | 43.17 | 9.67 | 118 | 124.63 | 5.62 |
B-30-I-2 | 47.55 | 46.38 | 2.46 | 106 | 127.36 | 20.15 |
B-20-II-1 | 38.32 | 37.43 | 2.32 | 100 | 89.92 | 10.08 |
B-30-II-1 | 43.36 | 40.77 | 5.97 | 105 | 93.26 | 11.18 |
B-20-II-2 | 43.54 | 44.94 | 3.22 | 130 | 131.84 | 1.41 |
B-30-II-2 | 52.78 | 51.87 | 1.72 | 134 | 135.68 | 1.25 |
In this study, the theoretical analysis of deflection and bearing capacity of bamboo-steel composite beams is proposed and verified with experiment. The main results and findings are concluded as follows:
(1) The theoretical analysis of upward deflection is built based on the unit load method. The verification result indicates that the theoretical method considering slippage and shear deformation can improve the accuracy; the average errors reduce from 4.67% to 2.03%.
(2) A method is proposed to calculate the midspan deflection of composite beams. The comparisons indicate that the theoretical method provides a feasible prediction of the deflection in the experiment.
(3) The stress increment of prestressed reinforcement was established based on the force method. Then, a practical method is proposed to analyze the bearing capacity of the composite beams. The verification results show that the relative error of the proposed method is 4.05% and 5.85% for the serviceability limit state and ultimate limit state, which meet the accuracy requirement of the engineered application.
This work was supported by the National Natural Science Foundation of China (51978345, 52278264).
The authors declare that they have no conflicts of interest to report regarding the present study.