# Research on Seismic Test and Calculation Model for Improving Factor of Composite Board with Heat Preservation Function

Authors: Shaochun Ma; Hongyuan Fang
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

Since the composition and stress state of thermal-insulation composite plates are extremely complex, it is difficult to conduct calculation using conventional methods. Further calculation on its composition structure is even more difficult. To settle this difficulty, this paper conducted experimental investigations on thermal-insulation composite plate and conventional composite plates, and a simplified calculation method with improved factors was deduced based on the experimental results. In this study, the influence of nonstructural thermal insulation system on the composite plate was specially considered. The calculated expressions are deduced, while the stiffness and damping (properties) are determined. Case analysis was carried out to validate the reliability and accuracy of the simplified calculation method of improvement factor. The research results indicate that the modeling difficulty and computational effort are greatly reduced, with relatively small errors. Hence, it is beneficial to the calculation and analysis of composite plates and global structures.


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## Body

## 1. Introduction

Thermal-insulation composite plates consisted of conventional composite plates and thermal insulation systems [1]. Conventional composite plates are formed by filling concrete in the pores of plaster plates [2]. Polyphenyl insulation plate and outside plasterboard shields construct the thermal insulation system of composite plates. The thermal insulation system makes the composite plate have the ability of heat insulation, thus, achieving the purpose of building energy conservation [3, 4]. During the normal operation of composite plates, the thermal insulation system can not only protect composite plates from erosion of severe environment, but also extend the service life of composite plates [5]. Plaster plates have good flatness and can be directly used as the leveling course of building outside finish, as well as avoiding the out leakage of thermal insulation plates. Plaster plates protect and isolate thermal insulation plates at the same time. Hence, they can block the spread of fire during the breaking out of fire [6, 7]. Hence, applications of green energy saving plaster and polyphenyl materials are wide, which can promote the innovation of new wall types around the world [8].In practical engineering, the structural load is mainly born by conventional composite plates. The thermal insulation system, as the envelope enclosure, is the secondarily stressed nonbearing unit [9]. The load applying on the outsider thermal insulation system is relatively small, or even no load is applied. Hence, during analyzing composite plates, the influence of thermal insulation system is usually ignored. Also, reliability design of nonstructural units still draws few attentions from designers [10]. However, scholars around the world have conducted massive experiments on similar composite plates [11–14]. The results indicated that the bearing and deformation capacities of composite plates with thermal insulation systems are slightly higher than those of conventional composite plates [15]. Hence, during analyzing this type of composite plates, it is unscientific to ignore the influence of the thermal insulation system.A calculated model for the engineered cementitious composite reinforced walls was proposed by Ref. [16], and it can be used to calculate and estimate the maximum lateral strength. Literature [17] presents the seismic assessment of the Cathedral of Matera in southern Italy, and modal identification tests were performed in order to identify and characterize the main dynamic properties of the structure. Literature [18] concludes with a statistical analysis of the seismic damage conducted with the tool of Damage Probability Matrixes (DPM) and fragility curves. The results of these assessments can be used to establish criteria for the restoration and reinforcement of buildings.In this paper, experimental investigations were conducted on thermal-insulation composite plate and conventional composite plates, and a simplified calculation method of improvement factors was deduced based on experimental results. Thus, the difficulty in the calculation of complex composite plates is settled, and the influence of thermal insulation system is also considered. The research results are of great importance for the further calculation of thermal-insulation composite components and more complex structures and can provide important references for the study of similar plates and structures.


## 2. Experimental Setup

### 2.1. Component Design

Figure1 is a schematic diagram of a composite board. Thermal-insulation composite plates consist of bidirectional porous plasterboard, hidden beams, embedded columns, and thermal insulation systems. Figure 2 is the planar view. The diameter of rebars in composite plates and load beams is all 14 mm. Rebars in the fixed beam of the component bottom are mainly load-bearing rebars (diameter of 20 mm) and stirrup rebars (diameter of 8 mm). Figure 3 is the front reinforcement diagram. The thickness of thermal-insulation plates is 120 mm, and the thickness of plaster plates is 13 mm. The material test results are listed in Table 1.Figure 1
Schematic diagram of composite board. Notes: 1, vertical gypsum cavity; 2, insulation board; 3, horizontal gypsum cavity; 4, externally bonded single gypsum board; 5, gypsum board.Figure 2
Planar view.Figure 3
The front reinforcement diagram.Table 1
Material mechanical properties.
Material propertiesPlasterConcreteThermal insulation plateRebarC8C14C20Compressive strengthN/mm25.5224.630.206———Tensile strengthN/mm20.3681.6420.22654.00669.45676.92Elastic modulusN/mm24350272002.3200000Densitykg/m314002500337850Figure4 shows the test apparatus for composite plates. The loading system contains one horizontal loader and two vertical loaders. The measurement system mainly consists of a horizontal displacement meter, a load sensor, a dial indicator, and the strain gauges for different materials. The specimen fixation system mainly consists of bottom compression beam, top beam, and pull rods. The load control system contains a loading control system (earlier stages) and a displacement control system (later stages). Tests are conducted by strictly following regulations described in Ref. [19].Figure 4
Experimental apparatus. (a) Positive. (b) Back. (c) Side.
(a)(b)(c)The material test results of polystyrene insulation boards are listed in Table1. Specimen of polystyrene board is of diameter 39.1 mm, with 80 mm high cylinder of polystyrene. The results of the triaxial shear test of polystyrene insulation board are shown in Figures 5 and 6.Figure 5
On-site experimental loading instrument.Figure 6
Stress-strain curve.
(a)(b)(c)


### 2.2. Test Results

#### 2.2.1. Bearing Capacity and Deformation

Under cyclic loading, three thermal-insulation composite plates were tested, with the stress states and deformation of specimens obtained. The force-displacement curves (hysteretic and skeleton) are shown in Figure7, while test data is summarized in Table 2. The tests results of conventional composite plates without thermal insulation system refer to those reported in Ref. [2]. Based on the comparison, it is implied that the increasing tendency of the force-displacement curves of two types of composite plates is all nonlinear. Compared to conventional composite plates, the stress magnitudes of key points are increased by 1.19, 1.02, and 1.18 times, for thermal-insulation composite plates. Meanwhile, the deformation magnitudes of key points of thermal-insulation composite plates are improved by 1.14, 1.20, and 1.44 times, respectively. The test results indicate that if the thermal insulation system can cooperatively work with conventional composite plates, the global strength and stiffness of composite plates can be effectively improved. Moreover, the elastic-plastic deformation capacity of composite plates can also be enhanced to a certain degree. That is, the global earthquake resistance of thermal-insulation composite plates has an advantage over that of conventional composite plates. According to the skeleton curves displayed in Figure 8, similar conclusions can be drawn.Figure 7
Hysteretic curves of thermal-insulation composite plates.
(a)(b)(c)Table 2
Force and displacement.
Plate typeSpecimen no.Pcr△crPy△yPu△u(kN)(mm)(kN)(mm)(kN)(mm)Without thermal insulation system160.460.84212.603.68311.8511.90260.110.69206.403.50305.0212.42364.740.61211.503.07327.0711.85Avg.61.770.71210.173.42314.6512.06With thermal insulation system178.700.85209.074.02369.6316.08269.430.75220.254.11377.7117.72371.570.83215.764.16366.4518.13Avg.73.230.81215.034.10371.2617.31Improvement0.190.140.020.200.180.44Notes:Pcr denotes the cracking load; Δcr denotes the cracking displacement; Py denotes the yielding load; Δy denotes the yielding displacement; Pu denotes the ultimate load; Δu denotes the ultimate displacement.Figure 8
Comparison of skeleton curves.Figure9 are the destruction graphs of thermal-insulation composite plates. By comparing and analyzing the destruction diagram of the composite wallboard with thermal insulation function, it can be found that the oblique crack in the direction of 45° is generated on the thermal insulation board, as shown in Figure 9. Or the separation phenomenon has basically occurred between the insulation board and the main structure of the composite board. It shows that the nonstructural insulation system can improve the bearing and deformation capacity of the composite board. During the seismic test of the composite panel, the insulation system was able to bear part of the load and played a beneficial role. Hence, during analyzing this type of composite plates, it is unscientific to ignore the influence of the thermal insulation system.Figure 9
Destruction graph of thermal-insulation composite plates. (a) Back. (b) Side.
(a)(b)


#### 2.2.2. Bearing Capacity and Deformation

The average ductility coefficients of composite wallboard with thermal insulation function and ordinary composite wallboards are shown in Table3, which are 4.22 and 3.55, respectively. Compared with the two composite wallboards, the ductility coefficient is increased by 18.87%. It shows that the ductility of composite wallboard with thermal insulation function is relatively better than that of ordinary composite wallboard. The implication is that composite wall panels with thermal insulation function are better than ordinary composite wall panels in terms of elastoplastic deformation, energy consumption, and seismic performance. The same conclusion can be drawn. In the calculation and analysis of such composite panels, it is unscientific to ignore the influence of the insulation system.Table 3
The ductility coefficient of the sample.
Specimen no.Without thermal insulation systemWith thermal insulation system123123Ductility coefficientμ3.233.553.864.004.314.36Avg.3.554.22


## 2.1. Component Design

Figure1 is a schematic diagram of a composite board. Thermal-insulation composite plates consist of bidirectional porous plasterboard, hidden beams, embedded columns, and thermal insulation systems. Figure 2 is the planar view. The diameter of rebars in composite plates and load beams is all 14 mm. Rebars in the fixed beam of the component bottom are mainly load-bearing rebars (diameter of 20 mm) and stirrup rebars (diameter of 8 mm). Figure 3 is the front reinforcement diagram. The thickness of thermal-insulation plates is 120 mm, and the thickness of plaster plates is 13 mm. The material test results are listed in Table 1.Figure 1
Schematic diagram of composite board. Notes: 1, vertical gypsum cavity; 2, insulation board; 3, horizontal gypsum cavity; 4, externally bonded single gypsum board; 5, gypsum board.Figure 2
Planar view.Figure 3
The front reinforcement diagram.Table 1
Material mechanical properties.
Material propertiesPlasterConcreteThermal insulation plateRebarC8C14C20Compressive strengthN/mm25.5224.630.206———Tensile strengthN/mm20.3681.6420.22654.00669.45676.92Elastic modulusN/mm24350272002.3200000Densitykg/m314002500337850Figure4 shows the test apparatus for composite plates. The loading system contains one horizontal loader and two vertical loaders. The measurement system mainly consists of a horizontal displacement meter, a load sensor, a dial indicator, and the strain gauges for different materials. The specimen fixation system mainly consists of bottom compression beam, top beam, and pull rods. The load control system contains a loading control system (earlier stages) and a displacement control system (later stages). Tests are conducted by strictly following regulations described in Ref. [19].Figure 4
Experimental apparatus. (a) Positive. (b) Back. (c) Side.
(a)(b)(c)The material test results of polystyrene insulation boards are listed in Table1. Specimen of polystyrene board is of diameter 39.1 mm, with 80 mm high cylinder of polystyrene. The results of the triaxial shear test of polystyrene insulation board are shown in Figures 5 and 6.Figure 5
On-site experimental loading instrument.Figure 6
Stress-strain curve.
(a)(b)(c)


## 2.2. Test Results

### 2.2.1. Bearing Capacity and Deformation

Under cyclic loading, three thermal-insulation composite plates were tested, with the stress states and deformation of specimens obtained. The force-displacement curves (hysteretic and skeleton) are shown in Figure7, while test data is summarized in Table 2. The tests results of conventional composite plates without thermal insulation system refer to those reported in Ref. [2]. Based on the comparison, it is implied that the increasing tendency of the force-displacement curves of two types of composite plates is all nonlinear. Compared to conventional composite plates, the stress magnitudes of key points are increased by 1.19, 1.02, and 1.18 times, for thermal-insulation composite plates. Meanwhile, the deformation magnitudes of key points of thermal-insulation composite plates are improved by 1.14, 1.20, and 1.44 times, respectively. The test results indicate that if the thermal insulation system can cooperatively work with conventional composite plates, the global strength and stiffness of composite plates can be effectively improved. Moreover, the elastic-plastic deformation capacity of composite plates can also be enhanced to a certain degree. That is, the global earthquake resistance of thermal-insulation composite plates has an advantage over that of conventional composite plates. According to the skeleton curves displayed in Figure 8, similar conclusions can be drawn.Figure 7
Hysteretic curves of thermal-insulation composite plates.
(a)(b)(c)Table 2
Force and displacement.
Plate typeSpecimen no.Pcr△crPy△yPu△u(kN)(mm)(kN)(mm)(kN)(mm)Without thermal insulation system160.460.84212.603.68311.8511.90260.110.69206.403.50305.0212.42364.740.61211.503.07327.0711.85Avg.61.770.71210.173.42314.6512.06With thermal insulation system178.700.85209.074.02369.6316.08269.430.75220.254.11377.7117.72371.570.83215.764.16366.4518.13Avg.73.230.81215.034.10371.2617.31Improvement0.190.140.020.200.180.44Notes:Pcr denotes the cracking load; Δcr denotes the cracking displacement; Py denotes the yielding load; Δy denotes the yielding displacement; Pu denotes the ultimate load; Δu denotes the ultimate displacement.Figure 8
Comparison of skeleton curves.Figure9 are the destruction graphs of thermal-insulation composite plates. By comparing and analyzing the destruction diagram of the composite wallboard with thermal insulation function, it can be found that the oblique crack in the direction of 45° is generated on the thermal insulation board, as shown in Figure 9. Or the separation phenomenon has basically occurred between the insulation board and the main structure of the composite board. It shows that the nonstructural insulation system can improve the bearing and deformation capacity of the composite board. During the seismic test of the composite panel, the insulation system was able to bear part of the load and played a beneficial role. Hence, during analyzing this type of composite plates, it is unscientific to ignore the influence of the thermal insulation system.Figure 9
Destruction graph of thermal-insulation composite plates. (a) Back. (b) Side.
(a)(b)


### 2.2.2. Bearing Capacity and Deformation

The average ductility coefficients of composite wallboard with thermal insulation function and ordinary composite wallboards are shown in Table3, which are 4.22 and 3.55, respectively. Compared with the two composite wallboards, the ductility coefficient is increased by 18.87%. It shows that the ductility of composite wallboard with thermal insulation function is relatively better than that of ordinary composite wallboard. The implication is that composite wall panels with thermal insulation function are better than ordinary composite wall panels in terms of elastoplastic deformation, energy consumption, and seismic performance. The same conclusion can be drawn. In the calculation and analysis of such composite panels, it is unscientific to ignore the influence of the insulation system.Table 3
The ductility coefficient of the sample.
Specimen no.Without thermal insulation systemWith thermal insulation system123123Ductility coefficientμ3.233.553.864.004.314.36Avg.3.554.22


## 2.2.1. Bearing Capacity and Deformation

Under cyclic loading, three thermal-insulation composite plates were tested, with the stress states and deformation of specimens obtained. The force-displacement curves (hysteretic and skeleton) are shown in Figure7, while test data is summarized in Table 2. The tests results of conventional composite plates without thermal insulation system refer to those reported in Ref. [2]. Based on the comparison, it is implied that the increasing tendency of the force-displacement curves of two types of composite plates is all nonlinear. Compared to conventional composite plates, the stress magnitudes of key points are increased by 1.19, 1.02, and 1.18 times, for thermal-insulation composite plates. Meanwhile, the deformation magnitudes of key points of thermal-insulation composite plates are improved by 1.14, 1.20, and 1.44 times, respectively. The test results indicate that if the thermal insulation system can cooperatively work with conventional composite plates, the global strength and stiffness of composite plates can be effectively improved. Moreover, the elastic-plastic deformation capacity of composite plates can also be enhanced to a certain degree. That is, the global earthquake resistance of thermal-insulation composite plates has an advantage over that of conventional composite plates. According to the skeleton curves displayed in Figure 8, similar conclusions can be drawn.Figure 7
Hysteretic curves of thermal-insulation composite plates.
(a)(b)(c)Table 2
Force and displacement.
Plate typeSpecimen no.Pcr△crPy△yPu△u(kN)(mm)(kN)(mm)(kN)(mm)Without thermal insulation system160.460.84212.603.68311.8511.90260.110.69206.403.50305.0212.42364.740.61211.503.07327.0711.85Avg.61.770.71210.173.42314.6512.06With thermal insulation system178.700.85209.074.02369.6316.08269.430.75220.254.11377.7117.72371.570.83215.764.16366.4518.13Avg.73.230.81215.034.10371.2617.31Improvement0.190.140.020.200.180.44Notes:Pcr denotes the cracking load; Δcr denotes the cracking displacement; Py denotes the yielding load; Δy denotes the yielding displacement; Pu denotes the ultimate load; Δu denotes the ultimate displacement.Figure 8
Comparison of skeleton curves.Figure9 are the destruction graphs of thermal-insulation composite plates. By comparing and analyzing the destruction diagram of the composite wallboard with thermal insulation function, it can be found that the oblique crack in the direction of 45° is generated on the thermal insulation board, as shown in Figure 9. Or the separation phenomenon has basically occurred between the insulation board and the main structure of the composite board. It shows that the nonstructural insulation system can improve the bearing and deformation capacity of the composite board. During the seismic test of the composite panel, the insulation system was able to bear part of the load and played a beneficial role. Hence, during analyzing this type of composite plates, it is unscientific to ignore the influence of the thermal insulation system.Figure 9
Destruction graph of thermal-insulation composite plates. (a) Back. (b) Side.
(a)(b)


## 2.2.2. Bearing Capacity and Deformation

The average ductility coefficients of composite wallboard with thermal insulation function and ordinary composite wallboards are shown in Table3, which are 4.22 and 3.55, respectively. Compared with the two composite wallboards, the ductility coefficient is increased by 18.87%. It shows that the ductility of composite wallboard with thermal insulation function is relatively better than that of ordinary composite wallboard. The implication is that composite wall panels with thermal insulation function are better than ordinary composite wall panels in terms of elastoplastic deformation, energy consumption, and seismic performance. The same conclusion can be drawn. In the calculation and analysis of such composite panels, it is unscientific to ignore the influence of the insulation system.Table 3
The ductility coefficient of the sample.
Specimen no.Without thermal insulation systemWith thermal insulation system123123Ductility coefficientμ3.233.553.864.004.314.36Avg.3.554.22


## 3. Simplified Calculation Theory

As widely recognized, to meet the seismic requirement, architectural structures not only have to possess enough bearing capacity and stiffness, but also need to have ductility [20]. In the event of an earthquake, strong damping capacity is required to ensure that architectural structures can effectively dissipate earthquake energy. Hence, stiffness, ductility, and damping are three key parameters in seismic design. The center-of-gravity position of thermal-insulation composite plates bearing upside vertical load coincides with the center position of core load-bearing components. The shape, dimension, and reinforcement assembly of core load-bearing components are in full accord with those of conventional composite plates. In this paper, to consider the influence of the thermal insulation system in composite plates, the calculation method, which considers improvement factors on the basis of conventional composite plates, is adopted, in order to simplify the calculation of this composite plate. During calculation, the simplified calculation method described in Ref. [21] is used for conventional composite plates.


### 3.1. The Improved Factor of Stiffness

The stiffness calculation formula of the composite plate is(1)EIq=E∑Iihi∑hi,where E is the elastic modulus; Iq is the corrected inertia moment; Ii and hi are the inertia moment and effective height of some composite plates.First, the shear elastic modulus of material is determined according to the code for design of concrete structure, i.e.,G = 40% E [22]; then, the simplified calculation formula can be obtained via the weighted average of equivalent stiffness under three kinds of load (inverted triangle distributed load, uniformly distributed load, and top concentrated load):(2)EIeqAvg.=EIq1+9μIq/H2Aq.According to this formula, the stiffness only correlates to the equivalent the inertia momentIeq. The magnitude of inertia moment is decided only based the basic geometrical dimension of composite plates. The elementary geometrical dimensions of core load-bearing components and conventional composite plates are completely the same, indicating that these two have the same inertia moment. That is, the inertia moment plays the role of a constant in the formula. Hence, the magnitudes of different equivalent wall elastic modulus directly decide the magnitude of wall stiffness, with a direct proportion. In other words, between two kinds of composite plates, the magnitude of their stiffness only differs with a proportionality coefficient, which is also validated by the test results. According to the above analysis results, to fully consider the role played by the thermal insulation system in composite plates and achieve the reasonably simplified calculation and investigation of composite plates, this paper proposes the concept and expression of stiffness improvement factor. This conclusion is crucial to the smooth calculation of thermal-insulation composite plates and other complex structures.Via calculating the test results of composite plates, the coordinates (X, Y) of key points on the framework curves (crack point A, yield point B, and ultimate point C) can be obtained, i.e.,(3)X=Fi¯=+Fi+−Fi2,Y=δ¯i=+δi+−δi2.The stiffnessKi at the key point can be obtained as(4)Ki=Fi¯δi¯.Similarly, the related dimensionless coordinate (xi, yi) is(5)xi=Fi¯Fimax¯,yi=δi¯δimax¯,where Fi and δi are the load and displacement of the key point; ＋ and － indicate the positive and negative loading directions; Fi¯ and δ¯i are the average values of corresponding load and displacement; Kcr, Ky, and Ku are the cracking, yielding, and ultimate stiffness values.By connecting the initial origin point, pointA, point B, and point C via straight lines, the dimensionless stiffness-displacement trilinear model can be obtained. Afterwards, the tangent value of the angle between each broken line and the x-axis is defined as the stiffness evaluation factor of the composite plate and expressed as ki. That is, the stiffness evaluation factor of the composite plate before cracking, before yielding, and reaching ultimate value is kcr, ky, and ku, respectively. By defining the ratio of the stiffness evaluation factors of these two types of composite plates as the stiffness improvement factor, there is(6)λi=ki′ki,where ki′ indicates the stiffness evaluation factor of composite plate with the thermal-insulation system.


### 3.2. The Improved Factor of Damping

During analyzing the energy dissipation of composite the plate, when the deformation degree is relatively large, the deformation gradually goes over the elastic range and enters the stage of elastoplastic deformation. That is, after unloading, there is still part of deformation that cannot recover completely, indicating that some residual deformation occurs during loading, and some energy is dissipated. This is the so-called elastoplastic-plastic dissipated energy in earthquake proofing construction. Earthquake proofing tests of composite plates usually follow the quasistatic seismic test method described in the regulation for building seismic testing method [23]. During testing, the velocity and acceleration are both considered as zero, i.e.,(7)mu¨=cu˙=0,where u˙ and u¨ are the velocity and acceleration, respectively; m is the mass of the composite plate system; c is the viscous damping coefficient.By recording the applied load and plate-end deformation, a hysteresis loop energy dissipation representative diagram reflecting the load-deformation relationship can be obtained, as shown in Figure10.Figure 10
Hysteresis loop energy dissipation representative diagram.
(a)(b)F and δ are the load and displacement of the top point; Fm and δm are the averaged peak values of the top point load and displacement within one hysteresis loop; ζ stands for the equivalent viscous damping ratio.As widely recognized, within one motion period of dynamic structural response, the dissipated energy is equal to the area surrounded by the hysteresis loop. Under the influence of equivalent viscous damping, the energy dissipated by the composite plate within one period isWd, i.e.,(8)Wdamping=4πζSΔOMN.As shown in Figure5(a), the area of the triangle OMNSΔOMN is(9)SΔOMN=12Fmδm=12kδm2,where k stands for the average stiffness during elastoplastic deformation, i.e.,(10)k=Fmδm,Hence, it can be obtained that(11)Wdamping=2πζkδm2.Discussing from the aspect of energy dissipation, the damping dissipated energy can be completely considered as being equal to the elastoplastic dissipated energy. As shown in Figure7, there is(12)Wdamping=W,where(13)SΔOMN=12SΔBEO+SΔDFO,Fm=12FB+FD,δm=12δB+δD.Hence,(14)ζ=12πWkδm2.Afterζ is determined, under the seismic influence, the motion equation of the composite plate system can be directly constructed, i.e., u¨+2ζωu˙+ω2u=−u¨g where(15)ω=km.Thus, the nonlinear calculation of the composite plate system (elastoplastic energy) can be transferred into the equivalent linear calculation. That is, the so-called equivalent linearization method can also obtain similar results. By substitutingk and SΔOMN into the equation, we can acquire that(16)ζ=12πSΔABC+SΔCDASΔBEO+SΔDFO=he,he can be decided according to the test results. The existence of the thermal insulation system leads to the difference of he values of two composite plates. According to the hysteresis curves obtained via tests, he values of two composite plates corresponding to 1△y, 2△y, and 3△y can be obtained, respectively. The ratio is defined as the damping evaluation coefficient ξi, i.e.,(17)ξi=hei′hei,where i is the magnification; hei′ is the damping coefficient of the thermal-insulation composite plate. The average value of ξi is defined as the damping improvement factor ψ, where n stands for the total number of the damping evaluation coefficients, i.e.,(18)ψ=∑ξin.


## 3.1. The Improved Factor of Stiffness

The stiffness calculation formula of the composite plate is(1)EIq=E∑Iihi∑hi,where E is the elastic modulus; Iq is the corrected inertia moment; Ii and hi are the inertia moment and effective height of some composite plates.First, the shear elastic modulus of material is determined according to the code for design of concrete structure, i.e.,G = 40% E [22]; then, the simplified calculation formula can be obtained via the weighted average of equivalent stiffness under three kinds of load (inverted triangle distributed load, uniformly distributed load, and top concentrated load):(2)EIeqAvg.=EIq1+9μIq/H2Aq.According to this formula, the stiffness only correlates to the equivalent the inertia momentIeq. The magnitude of inertia moment is decided only based the basic geometrical dimension of composite plates. The elementary geometrical dimensions of core load-bearing components and conventional composite plates are completely the same, indicating that these two have the same inertia moment. That is, the inertia moment plays the role of a constant in the formula. Hence, the magnitudes of different equivalent wall elastic modulus directly decide the magnitude of wall stiffness, with a direct proportion. In other words, between two kinds of composite plates, the magnitude of their stiffness only differs with a proportionality coefficient, which is also validated by the test results. According to the above analysis results, to fully consider the role played by the thermal insulation system in composite plates and achieve the reasonably simplified calculation and investigation of composite plates, this paper proposes the concept and expression of stiffness improvement factor. This conclusion is crucial to the smooth calculation of thermal-insulation composite plates and other complex structures.Via calculating the test results of composite plates, the coordinates (X, Y) of key points on the framework curves (crack point A, yield point B, and ultimate point C) can be obtained, i.e.,(3)X=Fi¯=+Fi+−Fi2,Y=δ¯i=+δi+−δi2.The stiffnessKi at the key point can be obtained as(4)Ki=Fi¯δi¯.Similarly, the related dimensionless coordinate (xi, yi) is(5)xi=Fi¯Fimax¯,yi=δi¯δimax¯,where Fi and δi are the load and displacement of the key point; ＋ and － indicate the positive and negative loading directions; Fi¯ and δ¯i are the average values of corresponding load and displacement; Kcr, Ky, and Ku are the cracking, yielding, and ultimate stiffness values.By connecting the initial origin point, pointA, point B, and point C via straight lines, the dimensionless stiffness-displacement trilinear model can be obtained. Afterwards, the tangent value of the angle between each broken line and the x-axis is defined as the stiffness evaluation factor of the composite plate and expressed as ki. That is, the stiffness evaluation factor of the composite plate before cracking, before yielding, and reaching ultimate value is kcr, ky, and ku, respectively. By defining the ratio of the stiffness evaluation factors of these two types of composite plates as the stiffness improvement factor, there is(6)λi=ki′ki,where ki′ indicates the stiffness evaluation factor of composite plate with the thermal-insulation system.


## 3.2. The Improved Factor of Damping

During analyzing the energy dissipation of composite the plate, when the deformation degree is relatively large, the deformation gradually goes over the elastic range and enters the stage of elastoplastic deformation. That is, after unloading, there is still part of deformation that cannot recover completely, indicating that some residual deformation occurs during loading, and some energy is dissipated. This is the so-called elastoplastic-plastic dissipated energy in earthquake proofing construction. Earthquake proofing tests of composite plates usually follow the quasistatic seismic test method described in the regulation for building seismic testing method [23]. During testing, the velocity and acceleration are both considered as zero, i.e.,(7)mu¨=cu˙=0,where u˙ and u¨ are the velocity and acceleration, respectively; m is the mass of the composite plate system; c is the viscous damping coefficient.By recording the applied load and plate-end deformation, a hysteresis loop energy dissipation representative diagram reflecting the load-deformation relationship can be obtained, as shown in Figure10.Figure 10
Hysteresis loop energy dissipation representative diagram.
(a)(b)F and δ are the load and displacement of the top point; Fm and δm are the averaged peak values of the top point load and displacement within one hysteresis loop; ζ stands for the equivalent viscous damping ratio.As widely recognized, within one motion period of dynamic structural response, the dissipated energy is equal to the area surrounded by the hysteresis loop. Under the influence of equivalent viscous damping, the energy dissipated by the composite plate within one period isWd, i.e.,(8)Wdamping=4πζSΔOMN.As shown in Figure5(a), the area of the triangle OMNSΔOMN is(9)SΔOMN=12Fmδm=12kδm2,where k stands for the average stiffness during elastoplastic deformation, i.e.,(10)k=Fmδm,Hence, it can be obtained that(11)Wdamping=2πζkδm2.Discussing from the aspect of energy dissipation, the damping dissipated energy can be completely considered as being equal to the elastoplastic dissipated energy. As shown in Figure7, there is(12)Wdamping=W,where(13)SΔOMN=12SΔBEO+SΔDFO,Fm=12FB+FD,δm=12δB+δD.Hence,(14)ζ=12πWkδm2.Afterζ is determined, under the seismic influence, the motion equation of the composite plate system can be directly constructed, i.e., u¨+2ζωu˙+ω2u=−u¨g where(15)ω=km.Thus, the nonlinear calculation of the composite plate system (elastoplastic energy) can be transferred into the equivalent linear calculation. That is, the so-called equivalent linearization method can also obtain similar results. By substitutingk and SΔOMN into the equation, we can acquire that(16)ζ=12πSΔABC+SΔCDASΔBEO+SΔDFO=he,he can be decided according to the test results. The existence of the thermal insulation system leads to the difference of he values of two composite plates. According to the hysteresis curves obtained via tests, he values of two composite plates corresponding to 1△y, 2△y, and 3△y can be obtained, respectively. The ratio is defined as the damping evaluation coefficient ξi, i.e.,(17)ξi=hei′hei,where i is the magnification; hei′ is the damping coefficient of the thermal-insulation composite plate. The average value of ξi is defined as the damping improvement factor ψ, where n stands for the total number of the damping evaluation coefficients, i.e.,(18)ψ=∑ξin.


## 4. Example Analyses

### 4.1. Material Constants and Improvement Factors

#### 4.1.1. Material Properties

The stress states of core stress components in composite plates considering thermal insulation system are similar to those in conventional composite plates. The material property parameters of core stress components can be acquired according to material (plaster, gypsum, and so on) test results, as well as the equivalent simplified calculation formula of conventional composite plates reported in Ref. [21]. That is, the elastic moduli in the x-direction and y-direction are 22248.41 MPa and 21622.17 MPa, respectively; the xy shear elastic modulus is 8658.20 MPa; the Poisson ratio is 0.196.


#### 4.1.2. The Improved Factor of Stiffness

Average value fitting is conducted on the nondimensional stiffness-displacement triple lines of two types of composite plates, with the results displayed in Figure11. The stiffness evaluation factors of different sections are calculated. As for conventional composite plates, kcr = 3.68, ky = 2.32, and ku = 0.45; for composite plates considering thermal insulation system, kcr’ = 4.20, ky’ = 2.01, and ku’ = 0.55. Finally, the stiffness improvement factors are obtained as λcr = 1.14, λy = 1.00, and λu = 1.23. It is regulated that when individually settling single sections, the stiffness improvement factor should be greater than 1. The advantage of this treating method is that although this section cannot be improved, this section will not decrease, which agrees with the test results.Figure 11
Nondimensional stiffness-displacement relation. (a) Conventional composite plate. (b) Composite plate considering thermal insulation system.
(a)(b)


#### 4.1.3. Calculation Result of Damping Improvement Factor

By conducting data treatment and calculation on the seismic test results of six specimens prepared from the two types of composite plates, relatedHe and ξ can be obtained, with detailed results listed in Table 4. By substituting ξ into the calculation formula of ψ, the damping improvement factor is calculated to be 1.089.Table 4
Damping data sheet.
Composite plate typeWall no.heζψ1△yAvg.2△yAvg.3△yAvg.ζ1ζ2ζ3Without thermal insulation system10.090.070.090.080.100.091.1431.1251.0001.08920.070.080.0930.060.070.07With thermal insulation system10.090.080.080.090.090.0920.090.090.0930.080.090.09Notes:he denotes the equivalent viscous damping coefficient; ζ denotes the damping evaluation coefficient; ψ denotes the damping improvement factor.


### 4.2. Model Establishment and Solution

Figure12 shows the 3D calculation model of the equivalent specimen of the thermal-insulation composite wallboard mainly consisting of three parts: middle equivalent wall (SOLID65 element), and distributive beam and fixed beam with relatively large stiffness in the top and bottom (SOLID45 element). The vertical and horizontal rebars in the wall are meshed using LINK8 bar element [23], with embedded and public nodes. According to the test results, during the loading process of all specimens, the wall fractures, while both the fixed beam and distributive beam are well preserved before and after the test. Hence, to ensure the smooth proceeding of calculation, the material properties are increased to avoid the premature failure of the distributive beam and fixed beam. The simplified stress-strain curve of the wall is shown in Figure 13. The ideal elastoplastic model is utilized for rebar [24]. The boundary condition is set as follows: the bottom and two ends of the fixed beam are fixed. The load applied consists of two parts: a vertical load, which simulates the self-weight of the upper structure and is applied using uniformly distributed area load, and a horizontal load, which simulates the horizontal earthquake action on the wall. In the initial stage of loading, a single cycle can be used to control the displacement load since the wall stressed deformation is small. After yielding, the single cycle is changed into three cycles per stage [25], until wall failure. The displacement control is set by referring to the experimental value of specimen 2.Figure 12
3D model.Figure 13
Stress-strain relationship.


### 4.3. Results Analysis

Based on the effective area of the hysteresis loop in Figure14, it is implied that the calculation results are slightly greater than the test results. Meanwhile, the global hysteresis curves imply that the calculation results are relatively plump. The framework curve in Figure 15 indicates that the calculation results are greater than test results. The cause of this phenomenon is discussed as follows. During simulation, the slippage between the insulation plate and wall is ignored. The wall material is ideal, without considering errors generated during actual production (insufficient vibration of plaster and concrete), or defects produced during transportation and installation (such as damage of wall corner and wall side).Figure 14
Comparison of hysteresis curves.Figure 15
Comparison of framework curves.Via comparing the load and displacement of key point, it is indicated that the calculation results agree with the test results, as listed in Table5. Based on error analysis, the local error and global error are both controlled within 5%, which meets the requirement of engineering calculation. The research results indicate that it is feasible to convert composite plates, which have complex structures and are difficult to be modeled and calculated, into the equivalent wall with similar properties, and utilize the simplified calculation method of improvement factor to conduct calculation analysis. This method can also be applied in the calculation analysis of more complex global structures.Table 5
Comparison of test and calculation results.
Result typesSpecimen no.Pcr△crPy△yPu△u(kN)(mm)(kN)(mm)(kN)(mm)Test value178.70.85209.074.02369.6316.08269.430.75220.254.11377.7117.72371.570.83215.764.16366.4518.13Avg.73.230.81215.034.1371.2617.31Calculation value272.570.78224.043.98382.4917.66Local error (%)4.5241.723.161.270.34Global error (%)0.913.74.192.853.022.02As we all know, the seismic performance test research is a destructive test. Under the action of simulated seismic force, the composite plate has gone through the elastic stage and the elastoplastic stage. After unloading, there is still part of deformation that cannot recover completely, indicating that some residual deformation occurs during loading, and some energy is dissipated. In the simulation calculation of components, the influence of nonlinearity on the solution must also be considered. If not considered, the specimen will not be damaged. It is difficult to derive and solve the theory of nonlinearity. In order to solve this problem, this article uses the equivalent linearization method. Thus, the nonlinear calculation of the composite plate system (elastoplastic energy) can be transferred into the equivalent linear calculation. By using this method to analyze the composite board by example, the calculation effect is better.The same conclusion can be drawn by comparing the test results with the calculated results. The damage occurred at the foot position of the composite wallboard specimens. The gypsum-concrete composite material at this location was crushed, and the steel bars were bent. The test piece showed good seismic performance. It is suggested that composite wall panels can be popularized and used in actual projects. The composite wallboard proposed in this article has the advantages of energy saving and environmental protection and meets the current requirements of countries around the world to vigorously encourage the development of green buildings.


## 4.1. Material Constants and Improvement Factors

### 4.1.1. Material Properties

The stress states of core stress components in composite plates considering thermal insulation system are similar to those in conventional composite plates. The material property parameters of core stress components can be acquired according to material (plaster, gypsum, and so on) test results, as well as the equivalent simplified calculation formula of conventional composite plates reported in Ref. [21]. That is, the elastic moduli in the x-direction and y-direction are 22248.41 MPa and 21622.17 MPa, respectively; the xy shear elastic modulus is 8658.20 MPa; the Poisson ratio is 0.196.


### 4.1.2. The Improved Factor of Stiffness

Average value fitting is conducted on the nondimensional stiffness-displacement triple lines of two types of composite plates, with the results displayed in Figure11. The stiffness evaluation factors of different sections are calculated. As for conventional composite plates, kcr = 3.68, ky = 2.32, and ku = 0.45; for composite plates considering thermal insulation system, kcr’ = 4.20, ky’ = 2.01, and ku’ = 0.55. Finally, the stiffness improvement factors are obtained as λcr = 1.14, λy = 1.00, and λu = 1.23. It is regulated that when individually settling single sections, the stiffness improvement factor should be greater than 1. The advantage of this treating method is that although this section cannot be improved, this section will not decrease, which agrees with the test results.Figure 11
Nondimensional stiffness-displacement relation. (a) Conventional composite plate. (b) Composite plate considering thermal insulation system.
(a)(b)


### 4.1.3. Calculation Result of Damping Improvement Factor

By conducting data treatment and calculation on the seismic test results of six specimens prepared from the two types of composite plates, relatedHe and ξ can be obtained, with detailed results listed in Table 4. By substituting ξ into the calculation formula of ψ, the damping improvement factor is calculated to be 1.089.Table 4
Damping data sheet.
Composite plate typeWall no.heζψ1△yAvg.2△yAvg.3△yAvg.ζ1ζ2ζ3Without thermal insulation system10.090.070.090.080.100.091.1431.1251.0001.08920.070.080.0930.060.070.07With thermal insulation system10.090.080.080.090.090.0920.090.090.0930.080.090.09Notes:he denotes the equivalent viscous damping coefficient; ζ denotes the damping evaluation coefficient; ψ denotes the damping improvement factor.


## 4.1.1. Material Properties

The stress states of core stress components in composite plates considering thermal insulation system are similar to those in conventional composite plates. The material property parameters of core stress components can be acquired according to material (plaster, gypsum, and so on) test results, as well as the equivalent simplified calculation formula of conventional composite plates reported in Ref. [21]. That is, the elastic moduli in the x-direction and y-direction are 22248.41 MPa and 21622.17 MPa, respectively; the xy shear elastic modulus is 8658.20 MPa; the Poisson ratio is 0.196.


## 4.1.2. The Improved Factor of Stiffness

Average value fitting is conducted on the nondimensional stiffness-displacement triple lines of two types of composite plates, with the results displayed in Figure11. The stiffness evaluation factors of different sections are calculated. As for conventional composite plates, kcr = 3.68, ky = 2.32, and ku = 0.45; for composite plates considering thermal insulation system, kcr’ = 4.20, ky’ = 2.01, and ku’ = 0.55. Finally, the stiffness improvement factors are obtained as λcr = 1.14, λy = 1.00, and λu = 1.23. It is regulated that when individually settling single sections, the stiffness improvement factor should be greater than 1. The advantage of this treating method is that although this section cannot be improved, this section will not decrease, which agrees with the test results.Figure 11
Nondimensional stiffness-displacement relation. (a) Conventional composite plate. (b) Composite plate considering thermal insulation system.
(a)(b)


## 4.1.3. Calculation Result of Damping Improvement Factor

By conducting data treatment and calculation on the seismic test results of six specimens prepared from the two types of composite plates, relatedHe and ξ can be obtained, with detailed results listed in Table 4. By substituting ξ into the calculation formula of ψ, the damping improvement factor is calculated to be 1.089.Table 4
Damping data sheet.
Composite plate typeWall no.heζψ1△yAvg.2△yAvg.3△yAvg.ζ1ζ2ζ3Without thermal insulation system10.090.070.090.080.100.091.1431.1251.0001.08920.070.080.0930.060.070.07With thermal insulation system10.090.080.080.090.090.0920.090.090.0930.080.090.09Notes:he denotes the equivalent viscous damping coefficient; ζ denotes the damping evaluation coefficient; ψ denotes the damping improvement factor.


## 4.2. Model Establishment and Solution

Figure12 shows the 3D calculation model of the equivalent specimen of the thermal-insulation composite wallboard mainly consisting of three parts: middle equivalent wall (SOLID65 element), and distributive beam and fixed beam with relatively large stiffness in the top and bottom (SOLID45 element). The vertical and horizontal rebars in the wall are meshed using LINK8 bar element [23], with embedded and public nodes. According to the test results, during the loading process of all specimens, the wall fractures, while both the fixed beam and distributive beam are well preserved before and after the test. Hence, to ensure the smooth proceeding of calculation, the material properties are increased to avoid the premature failure of the distributive beam and fixed beam. The simplified stress-strain curve of the wall is shown in Figure 13. The ideal elastoplastic model is utilized for rebar [24]. The boundary condition is set as follows: the bottom and two ends of the fixed beam are fixed. The load applied consists of two parts: a vertical load, which simulates the self-weight of the upper structure and is applied using uniformly distributed area load, and a horizontal load, which simulates the horizontal earthquake action on the wall. In the initial stage of loading, a single cycle can be used to control the displacement load since the wall stressed deformation is small. After yielding, the single cycle is changed into three cycles per stage [25], until wall failure. The displacement control is set by referring to the experimental value of specimen 2.Figure 12
3D model.Figure 13
Stress-strain relationship.


## 4.3. Results Analysis

Based on the effective area of the hysteresis loop in Figure14, it is implied that the calculation results are slightly greater than the test results. Meanwhile, the global hysteresis curves imply that the calculation results are relatively plump. The framework curve in Figure 15 indicates that the calculation results are greater than test results. The cause of this phenomenon is discussed as follows. During simulation, the slippage between the insulation plate and wall is ignored. The wall material is ideal, without considering errors generated during actual production (insufficient vibration of plaster and concrete), or defects produced during transportation and installation (such as damage of wall corner and wall side).Figure 14
Comparison of hysteresis curves.Figure 15
Comparison of framework curves.Via comparing the load and displacement of key point, it is indicated that the calculation results agree with the test results, as listed in Table5. Based on error analysis, the local error and global error are both controlled within 5%, which meets the requirement of engineering calculation. The research results indicate that it is feasible to convert composite plates, which have complex structures and are difficult to be modeled and calculated, into the equivalent wall with similar properties, and utilize the simplified calculation method of improvement factor to conduct calculation analysis. This method can also be applied in the calculation analysis of more complex global structures.Table 5
Comparison of test and calculation results.
Result typesSpecimen no.Pcr△crPy△yPu△u(kN)(mm)(kN)(mm)(kN)(mm)Test value178.70.85209.074.02369.6316.08269.430.75220.254.11377.7117.72371.570.83215.764.16366.4518.13Avg.73.230.81215.034.1371.2617.31Calculation value272.570.78224.043.98382.4917.66Local error (%)4.5241.723.161.270.34Global error (%)0.913.74.192.853.022.02As we all know, the seismic performance test research is a destructive test. Under the action of simulated seismic force, the composite plate has gone through the elastic stage and the elastoplastic stage. After unloading, there is still part of deformation that cannot recover completely, indicating that some residual deformation occurs during loading, and some energy is dissipated. In the simulation calculation of components, the influence of nonlinearity on the solution must also be considered. If not considered, the specimen will not be damaged. It is difficult to derive and solve the theory of nonlinearity. In order to solve this problem, this article uses the equivalent linearization method. Thus, the nonlinear calculation of the composite plate system (elastoplastic energy) can be transferred into the equivalent linear calculation. By using this method to analyze the composite board by example, the calculation effect is better.The same conclusion can be drawn by comparing the test results with the calculated results. The damage occurred at the foot position of the composite wallboard specimens. The gypsum-concrete composite material at this location was crushed, and the steel bars were bent. The test piece showed good seismic performance. It is suggested that composite wall panels can be popularized and used in actual projects. The composite wallboard proposed in this article has the advantages of energy saving and environmental protection and meets the current requirements of countries around the world to vigorously encourage the development of green buildings.


## 5. Conclusions

By investigating the testing and simplified calculation methods for thermal-insulation composite plates, the following conclusions can be drawn:(1)
Focusing on the calculation difficulty of complex composite plates, a simplified calculation method, which deduces the improvement factor from test results, is proposed in this paper. The influence of the nonstructural thermal insulation system on the composite plate is considered, and the successive modeling and analysis of the global structure of composite plate are completed. In addition, this method can also be used for the calculation and research of thermal insulation composite wallboard samples, and the calculation process is more convenient than ordinary calculation methods.(2)
By analyzing the test results of two types of composite plates with/without thermal insulation systems, theoretical deduction of the simplified calculation method of improvement factors is carried out. The calculation formulas of stiffness and damping are developed. Through comparative analysis of theoretical derivation and experiment, the correctness has been proved.(3)
Under the influence of cyclic loading, simulation and calculation of composite plates are conducted by using the simplified calculation method of improvement factor. The obtained calculation results agree well with the test results. The errors are all within engineering tolerance ranges. Thus, the accuracy and reliability of this method are validated. Compared to the calculation method described in Ref. [21], the calculation method proposed in this study significantly reduces the calculation difficulty and workload. The research results can provide important references for the study of similar plates and structures.



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*Source: 5546387-2021-04-23.xml*