GFRP (glass-fiber-reinforced polymer), as a composite material, possesses many favorable properties including high strength and low weight and is amenable to unique processing methods; therefore, it is a potential free-form surface material. However, the complex design theory owing to anisotropy limits its application. Thus, a simplified material solution becomes significant. In this study, the strength and stiffness of orthotropic symmetrical GFRP laminates are derived theoretically, and a simplified material solution is proposed to simplify the anisotropy as isotropy. Then, using the numerical simulation of an actual orthotropic symmetrical GFRP laminate free-form facade structure, the effectiveness of the simplified material solution is analyzed and evaluated. This solution can provide guidance for similar GFRP facades and further promote the application of GFRP in engineering.
Recent times have seen the construction of a number of special-shaped buildings around the world. However, the realization of these complex architectural shapes poses great challenges for engineers. Traditional facade materials like glass, metal, and concrete cannot be used to construct free-form facades without significant construction time and costs considering the current technical conditions. GFRP (glass-fiber-reinforced polymer), as composite material, possesses many favorable properties including high strength and low weight and is amenable to unique processing methods. It seems to be a potential solution for free-form facades.
GFRP represents a series of composite materials that employs a polymer as the matrix and glass fiber as the reinforcing material. Although it is a relatively new material compared to traditional materials such as steel and concrete, it has been applied in civil engineering ever since its invention. However, its applications were limited until recent years. Of late, GFRP has been used to fabricate bridge decks, structural reinforcements, bars, and decoration materials.
As reinforcement material, Kurt [
FRP was first employed as a building facade material in the 1950s, when composite materials were just beginning to find civilian applications. Although several creative FRP house prototypes were developed, only a few of them were applied successfully. In 1956, the Maison en Plastique, which was regarded as the first ever structure with load-bearing FRP members, was set up in Paris [
The inherent anisotropy of FRP is not very difficult for researchers. However, it is too complicated for applications in actual engineering. It results in complicated design theories that, in turn, require extensive calculations, thus directly limiting its applications. In recent years, some simplified calculation methods were proposed for composite materials. In 2014, a simplified reliability-based determination procedure on the strengthening ratio of a deteriorated concrete girder with CFRP strips was proposed [
As facade material or load-bearing single shell members, the utilization value of orthotropic symmetrical GFRP laminates is the highest in terms of cost, process, and performance. The orthotropic symmetrical GFRP laminates in this paper are reinforced equally in two mutually perpendicular directions, such as
This study involves the theoretical derivation of the strength and stiffness of orthotropic symmetrical GFRP laminates and obtains the off-axis strength and stiffness. Bending experiments and corresponding finite element method analysis are carried out, and their results are compared to verify the consistency. Numerical simulation of an actual GFRP free-form facade is performed, in which the GFRP material and simplified material properties are employed, respectively, to examine the accuracy and effectiveness of the simplified solution. The simplified material solution is ultimately applied to the design of the actual GFRP free-form facade, and the construction of the free-form facade was completed during early December 2017.
Theoretical research on the anisotropic properties of composite materials has always been the focus for composite material research. Although some theories are in good agreement with the actual performance of composites materials under specific conditions, their value in practical application is limited and experiments are always required to further examine FRP products. Generally, engineers are cautious when assuming composite material property values. Most current FRP-related codes and standards in civil engineering mainly deal with structural reinforcement, and some of them involve FRP pultruded profiles [
Unidirectional reinforced GFRP can only bear load in the longitudinal direction as its properties in the transverse direction are too weak, and when applied to unidirectional members, it is almost identical to traditional isotropic materials. For multidirectional load-bearing members, orthotropic symmetrical GFRP laminates are mostly applied due to their relatively low cost, ease of processing, and relatively stable properties. The anisotropy of orthotropic symmetrical GFRP laminate is significant, which distinguishes it from conventional building materials. Generally, the stiffness of FRP is calculated based on the classical laminate theory, and failure criteria include the maximum stress criterion, maximum strain criterion, Tsai–Hill criterion, and Tsai–Wu criterion. The Tsai–Hill and Tsai–Wu criteria [
In addition, even if designers are familiar with GFRP, it is still difficult to analyze orthotropic symmetrical GFRP laminate structures directly using composite material properties, especially when the calculation involves a large number of panels of various shapes as in a GFRP free-form facade. As a multilayer material, a single GFRP panel can be considered as a composite member, so that the calculation requirement is equivalent to that of a small structure. Therefore, the calculations pertaining to an entire structure is equivalent to several structures at least. For buildings or structures, macroscopic indexes like the maximum deformation are the most relevant factors, and the working environments of buildings are complicated so that a certain error in the calculation is always acceptable. Therefore, the practical significance of laminate-level calculation is relatively limited, and it is necessary to propose a simplified material solution.
The simplified material solution should satisfy three requirements: The calculated results pertaining to members or structures that apply simplified material properties are in good agreement with the original properties The simplified material properties can be obtained conveniently through specific experiments The simplified material solution should guarantee the safety of the structure, and so it should be a relatively conservative solution
Therefore, the target is to simplify orthotropic symmetrical laminates into the isotropic material, thus avoiding anisotropic properties, specifying appropriate stiffness and strength parameters for design reference, reducing design difficulty, and further promoting the application of GFRP in civil engineering.
According to the classical laminate theory, the load-strain matrix of the laminate is shown in the following equation:
An orthotropic symmetrical laminate is a special laminate having coupling stiffness matrix
The thickness of a panel is far less than its length and width; therefore, the analysis of the panels could be assumed as plane stress problems:
Material axis.
The axes directions and off-axis directions are as follows:
The off-axis stiffness is lower than that in the reinforcement direction, and the load may be transmitted along the off-axis direction for some irregular shell members. Therefore, the off-axis properties actually determine the lower limit of material properties.
In Figure
Therefore, the off-axis stiffness properties can be expressed as follows:
For orthotropic symmetrical laminates,
Therefore, when
The stiffness properties of orthotropic symmetrical GFRP laminates applied in the actual structure described in chapter 4 are shown in Table
Stiffness properties of
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16 GPa | 16 GPa | 3 GPa | 0.14 |
Substituting the properties in Table
Off-axis elastic modulus variation curves with
Off-axis shear modulus variation curves with
Off-axis Poisson’s ratio variation curves with
When
Here, the Tsai–Hill criterion is applied as the failure criterion of GFRP laminate. The general form of the formula is given in following equation:
In the case of plane stress problems,
So, equation (
When the laminates are subjected to unidirectional load,
Then, equation (
The derivation of
Therefore,
Strength properties of orthotropic symmetrical GFRP laminates applied in the actual structure described in chapter 4 are shown in Table
Strength properties of
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|
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300 MPa | 250 MPa | 60 MPa | 120 MPa | 70 MPa |
where
Substitute the properties of Table
Off-axis tension strength variation curves with
Off-axis compression strength variation curves with
Appropriate failure criteria and strength parameters must be selected for the simplified isotropic material. The maximum stress criterion assumes that a material fails when the maximum principal stress exceeds the uniaxial tensile strength, and it is commonly applied among brittle materials like concrete. GFRP is essentially a brittle material and therefore, the maximum stress criterion might be appropriate for the simplified isotropic material. The stress-strain curves of brittle materials in uniaxial load experiments are always oblique lines terminated at the ultimate strain values, as shown in Figure
Stress-strain curves. (a) Brittle materials (including FRP) in uniaxial load experiment. (b) FRP in bending test.
Members like facade panels are subjected primarily to bending loads. Bending strength and bending modulus, which are obtained from bending tests, are common parameters for FRP. There are many factors that influence bending properties, including the type of polymer and fiber and the ply-stacking sequence. And thus, bending properties are a reflection of the overall material properties. A typical stress-strain curve of FRP in bending test is shown in Figure
The formulas provided in bending test standards, such as ASTM D790 [
Within the elastic range of the curve in Figure
The compression strength of polymers (the matrix of GFRP) is generally higher than its tensile strength, and so, in bending experiments, the polymers in tensile side cracks and fails first (“crack point” in Figure
The elastic bending ultimate strength (stress on the crack point of Figure
The aim of this study is to simplify and idealize orthotropic symmetrical GFRP laminates into isotropic material. Theoretically, the stiffness and strength of orthotropic symmetrical GFRP laminates are both minimum at
A brief solution is shown in Table
Simplified material solution.
Material properties | Elastic modulus | Poisson’s ratio | Failure criterion | Failure strength |
---|---|---|---|---|
Isotropic material |
|
0.3 | The maximum stress criterion |
|
In 2017, a structure with a free-form facade was built in Fuzhou, China. The architectural schematic is a sculpture-like structure representing three morning glories: a considerably free-form facade with strong visual impact. The facade model is shown in Figure
Facade model.
Night view.
The height of the structure is 9.8 m, and the surface area is approximately 1460 m2. The structure is completely exposed to open air, covering the entire sunken plaza and protecting it from rain. The inner supporting structures are steel tubes and
The bending properties of orthotropic symmetrical GFRP laminates applied in this structure are tested to obtain the bending modulus and elastic bending ultimate strength at
Material ingredients.
Batch | Fiber | Fiber volume fraction (%) | Matrix | Additive | Depth (mm) |
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Material A | E-Glass |
50 | Epoxy resin | None | 5.7 |
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|||||
Material B | E-Glass |
50 | Epoxy resin | 10% ammonium polyphosphate | 4.8 |
Axis properties of the tested material.
Batch |
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|
|
|
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Material A/B | 16 GPa | 3 GPa | 0.14 | 300 MPa |
According to ASTM D790 [
Specimens dimensions model. (a) 3D model. (b) Front view.
Nominal specimen dimensions.
Batch | Length ( |
Width ( |
Depth ( |
Span ( |
---|---|---|---|---|
Material A | 110 mm | 12.7 mm | 5.7 mm | 91.2 mm |
Material B | 92 mm | 12.7 mm | 4.8 mm | 76.8 mm |
The loading process and failed specimens are shown in Figures
Three-point bending test.
Failed specimens. (a) Side view. (b) Front view.
The tensile and compressive stresses and strain on specimen surfaces can be calculated using equations (
After the tensile side cracks, equations (
The bending elastic modulus can be obtained from the test data when
Bending elastic modulus.
Specimens | 1 | 2 | 3 | 4 | 5 | Avg. |
---|---|---|---|---|---|---|
Material A | 10.16 GPa | 10.35 GPa | 9.12 GPa | 10.48 GPa | 9.18 GPa | 9.86 GPa |
Material B | 9.86 GPa | 9.21 GPa | 9.69 GPa | 9.76 GPa | 10.01 GPa | 9.71 GPa |
The average elastic moduli are 9.86 GPa and 9.71 GPa, respectively, which is close to 9.074 GPa calculated theoretically but a little higher. The error is probably induced by two factors: 9.074 GPa is calculated based on the properties in Table The test error of
Yet, the little error does not influence the application of the simplified material solution.
In Figure
Numerical simulation of the bending test is conducted in Abaqus™. The multilayer and laminate material simulation functions of Abaqus™ are relatively mature and widely used in the simulation of composite materials. The laminate material parameters and model dimensions used in Abaqus™ are listed in Tables
The loads at the center of the model are 74.35 N and 61.66 N, respectively, and these are the forces when
Deformation nephogram from the finite element method. (a) Material A. (b) Material B.
The deformation at the center of materials A and B are 0.650 mm and 0.555 mm, respectively. The bending modulus can be calculated through the following equation:
The bending moduli of materials A and B at
The facade divisions and supporting purlins are shown in Figure
Facade division and supporting purlins.
The fifteen selected panels include extremely twisted shapes as well as several flat ones. Simplified isotropic material properties and original orthotropic symmetrical GFRP laminate properties are given to the panels, respectively, following which the deformations and the stresses are calculated and the values of Tsai–Hill failure criterion of the laminate are evaluated. The specific analysis procedures are shown as follows.
Boundary conditions: panels are relatively thin and the elastic modulus is small; thus, the stiffness is small. Simultaneously, the panels are connected to purlins using structural adhesive. Therefore, the boundaries are assumed to be simply supported on all four sides.
Load: GFRP is light in weight and the panels are mainly subjected to wind loads. The wind load in Fuzhou is considerable, and the wind pressure and wind suction must be considered simultaneously. A load of 4 kN/m2 is applied in the numerical simulation after the combination.
Materials: this analysis focuses on the comparison of panels with different material properties, namely, the simplified isotropic material and original orthotropic symmetrical GFRP laminate. For GFRP laminate material, the panel depth of the structure is 4.8 mm, 0.6 mm for each layer and eight layers in total, and the material properties are the same as the material applied in the actual structure, as shown in Tables
Analysis steps: the finite element method analysis must consider the geometric nonlinearity as the stiffness is typically low; the calculation includes the deformation and stresses of every mesh for all the isotropic materials and GFRP laminate material.
The results of the deformations are listed in Table
Maximum deformation of individual panels adopting different material properties (mm).
Panel | GFRP | Elastic modulus of simplified isotropic material | ||||||
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9 GPa | 10 GPa | 11 GPa | 12 GPa | 13 GPa | 14 GPa | 15 GPa | ||
1 | 2.92 | 2.99 | 2.72 | 2.50 | 2.31 | 2.14 | 2.00 | 1.88 |
2 | 3.36 | 4.06 | 3.71 | 3.42 | 3.17 | 2.95 | 2.75 | 2.58 |
3 | 4.40 | 7.84 | 6.03 | 4.38 | 3.18 | 2.45 | 1.98 | 1.67 |
4 | 6.89 | 8.20 | 7.58 | 7.06 | 6.61 | 6.22 | 5.88 | 5.58 |
5 | 9.81 | 10.71 | 10.05 | 9.46 | 8.95 | 8.49 | 8.06 | 7.67 |
6 | 14.17 | 17.20 | 16.38 | 15.64 | 14.99 | 14.40 | 13.86 | 13.36 |
7 | 17.13 | 20.11 | 19.07 | 18.17 | 17.18 | 16.67 | 16.02 | 15.44 |
8 | 18.11 | 27.15 | 26.26 | 25.47 | 24.82 | 24.25 | 23.72 | 23.23 |
9 | 19.66 | 21.13 | 20.23 | 19.43 | 18.71 | 18.06 | 17.47 | 16.93 |
10 | 20.28 | 22.71 | 22.06 | 21.50 | 20.99 | 20.54 | 20.13 | 19.76 |
11 | 20.79 | 21.41 | 20.40 | 19.49 | 18.64 | 17.83 | 17.05 | 16.33 |
12 | 24.53 | 29.10 | 27.94 | 26.89 | 25.92 | 25.05 | 24.25 | 23.54 |
13 | 25.21 | 27.40 | 26.55 | 25.81 | 25.15 | 24.56 | 24.03 | 23.54 |
14 | 28.49 | 32.39 | 31.41 | 30.55 | 29.76 | 29.06 | 28.41 | 27.81 |
15 | 29.50 | 30.85 | 30.15 | 29.49 | 28.89 | 28.32 | 27.78 | 27.27 |
Maximum deformation curve of all panels (mm).
The lines in Figure
The results of stresses in the panels are listed in Table
Maximum stresses of panels (MPa).
Panel | Simplified isotropic material | Positive-axis stresses of GFRP | Values of Tsai–Hill failure criterion of GFRP | ||
---|---|---|---|---|---|
Elastic modulus: 13 GPa |
|
|
| ||
1 | 4.682 | 4.912 | 4.487 | 2.338 | 0.03459 |
2 | 8.993 | 9.484 | 7.852 | 2.379 | 0.03867 |
3 | 4.492 | 10.140 | 8.888 | 5.862 | 0.07233 |
4 | 7.695 | 9.560 | 10.020 | 3.100 | 0.05641 |
5 | 10.720 | 9.363 | 9.252 | 4.653 | 0.06730 |
6 | 14.360 | 17.280 | 16.690 | 4.554 | 0.08550 |
7 | 12.190 | 11.160 | 11.450 | 2.830 | 0.05136 |
8 | 26.530 | 19.350 | 17.110 | 8.140 | 0.11800 |
9 | 15.760 | 20.670 | 18.210 | 3.950 | 0.08183 |
10 | 17.270 | 21.780 | 17.540 | 4.013 | 0.08062 |
11 | 17.620 | 17.370 | 20.140 | 8.410 | 0.08820 |
12 | 15.770 | 13.750 | 14.540 | 5.510 | 0.07939 |
13 | 14.770 | 11.940 | 14.380 | 3.630 | 0.06390 |
14 | 16.850 | 16.010 | 15.450 | 4.890 | 0.07846 |
15 | 22.790 | 20.240 | 21.530 | 13.070 | 0.17645 |
where
In Table
In Table
Norms of errors induced by material simplification.
Norm | 9 GPa | 10 GPa | 11 GPa | 12 GPa | 13 GPa | 14 GPa | 15 GPa |
---|---|---|---|---|---|---|---|
1-norm | 37.996 | 26.471 | 18.675 | 17.891 | 19.714 | 23.084 | 28.915 |
2-norm | 12.876 | 10.219 | 8.435 | 7.640 | 7.653 | 8.198 | 9.057 |
The norms of 12 GPa are the least in Table
In Table
In Table
Shape and reinforced direction of panel-8.
The reinforcement direction could not be completely perpendicular to the edges owing to its free-form shape. Rotating the reinforcement direction of panel-8 and recalculating its deformations, the largest deformation rises to 21.2 mm after a
This paper proposed a simplified material solution for orthotropic symmetrical GFRP laminates used in structural facades. Reducing the difficulty of design as well as computational complexity can help promote the application of GFRP in civil engineering. Theoretically, the elastic modulus and strength in
This simplified material solution is convenient to use and recommended for similar structures as reference. The following conclusions can be obtained through this study: The computational complexity and quantity of laminates are relatively large. The stiffness matrix of anisotropic laminates is highly complex, with an 8-layer panel necessitating at least eight times the computational requirement. Coordination conditions between the layers further complicate the calculation, especially for free-form panels considering the recalculation and verification induced by design adjustments. In general, the analysis of integral structures by considering laminate material properties is very difficult, and the accuracy of the result might be doubtful. Even for single panels in this paper, multiple adjustments to element meshing and load steps have to be made to obtain reliable calculation results for some complicated panels when applying GFRP laminate properties. It is almost impossible to calculate GFRP facade structures by applying its original properties; thus, the simplified material solution obtained here becomes significant in promoting GFRP applications. GFRP panels are stiffness-controlled, and the strength is large enough. The applied value of 4 kN/m2 is a relatively large load combination. However, the largest value of Tsai–Hill failure criterion is only 0.1764, while the largest principal tensile stress of simplified isotropic material is only 26.53 MPa, which is far smaller than the The simplified material solution is effective. Deformation of the simplified isotropic material is consistent with that of GFRP laminate material properties and slightly higher considering the safety. Although the
The experimental data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was supported by the National Natural Science Foundation of China (Grant no. 51578404).