Investigation of the probability of losses in the external insulation layer of residential engineering projects using the weather test

Abstract

Increasing attention has been drawn to the delamination of the external thermal insulation layer, which severely impacts the residents’ living quality and insurers’ claim risks. Therefore, the probability distribution of losses caused by external insulation layer delamination is investigated in this study through weather tests, considering that such delamination occurs due to a bond strength of zero, in order to provide scientific support for the inherent defect insurance (IDI). Using a thin-coat external thermal insulation system with expanded polystyrene (EPS) boards (500*500mm) as the test subject, weather tests were designed to simulate the monsoon climate in China. Bond strength testing was then performed to quantify the degradation level of the insulation layer. The results indicate that bond strength degradation and the initial bond strength under non-load conditions follows logistic distribution and Weibull distribution respectively, with 25% of the test blocks exhibiting Grade I damage. Additionally, the delamination failure probability is counted out 0.164 based on the structural reliability theory. Further analysis indicated that IDI claim frequencies can be approximated through a Poisson distribution using parameter λ, thereby establishing a quantitative model for insurers’ premium pricing and risk management. Moreover, the probabilistic patterns of insulation layer delamination were clarified by integrating experimental and theoretical methods, thereby not only addressing a critical knowledge gap in the IDI’s loss assessment for external insulation systems, but also directly enhancing the accuracy and reliability of construction quality insurance practices and IDI risk evaluation.

1. Introduction

Recently, there has been increased public concern on the potential defects in the construction quality owing to various developments in the construction industry. Higher demands have been placed on the housing quality as the living standards improve. This is evident from data from the People’s Insurance Company of China (PICC Property and Casualty Co., Ltd.), a leading Chinese property insurer, which handled 1,243 quality-related insurance claims in 2021, ranging from foundation works, structural frames, insulation, waterproofing, electrical systems, and interior finishes, significantly impacting the residents’ quality of life and consumer rights. ¹

IDI refers to the coverage purchased by construction entities that allows insurers to assume liability for losses caused by latent defects during specified coverage periods. ¹ IDI originated in France but has been widely adopted in Europe owing to its evolution into a diversified insurance service that is essential in the construction sector and is currently being implemented globally in mature quality assurance mechanisms. ² China initiated IDI pilots in major cities such as Shanghai, which reported 1,364 insured residential projects by 2021 with an insured amount of 502.6 billion yuan and coverage spanning 140 million square meters. However, 118,000 potential risks were identified in over 11,470 inspection reports. ³ Similarly, Guangzhou’s housing authority reported 118 insured projects since late 2020 that provide quality guarantees for 300,000 households with 800 million yuan in premiums. Currently, multiple provincial governments have introduced IDI-related policies. ³

The detachment and damage of the external insulation layers in residential buildings, which are similar to leakage issues, are among the most frequent inherent defects that cause substantial losses. According to statistics from the Wuxi Construction Engineering Quality and Safety Supervision Station, 92.7% of the 55 projects that were surveyed exhibited wall cracking, detachment, or hollowing defects. ⁴ Additionally, the analysis of the utilization of the maintenance fund across 165 residential communities in Chengdu revealed that wall detachment repairs accounted for 19.4% of the maintenance cases, which ranked it among the top maintenance issues. ⁵ These findings demonstrate the widespread occurrence and significant losses associated with failures in the external insulation layer, making it a primary claim concern for insurers. This necessitates systematic research into the prevalence and severity of such defects in Chinese residential projects, with a focus on the loss patterns and a risk assessment of how external insulation failures can impact the IDI from an insurer’s perspective.

However, the current research on the damage of the external insulation layer of residential projects is not extensive. Some scholars have investigated the connection performance of external walls. Liu conducted a full-scale test on cold-formed steel (CFS) load-bearing walls with external insulation under fire conditions and analyzed the insulation layer’s connection behavior during fires. ⁶ Similarly, Chen conducted experimental studies on the thermal behavior of non-load-bearing CFS walls with external insulation under various fire conditions and proposed criteria for predicting insulation detachment. ⁷ However, fire-related studies on external wall connections are classified as force majeure in insurance practice because they do not provide practical guidance or evidence for promoting IDI. Therefore, Liao proposed a vertical pull-off test methodology to evaluate the connection performance by converting it into a bond strength analysis, where a bond strength of zero indicates connection failure. ⁸ Additionally, Zhao investigated the bearing capacity and temperature effects of exterior formwork systems with rock wool composite insulation to address persistent detachment and fire risks in conventional external insulation systems. ⁹ However, such systems have not been widely implemented because of insufficient experimental data to establish reliable detachment patterns that can be used to determine their implications for IDI. Krentowski comprehensively analyzed the durability of external insulation connections by constructing computational models and verifying the stress state of structural components. ¹⁰ Consequently, structural mechanics approaches were proposed to investigate the insulation durability. However, the main focus of this study was theoretical modeling without practical engineering-based detachment pattern summaries. In order to better explore the shedding law of the external insulation layer of the residential project, which can provide guidance for the operation of IDI, this study introduces the weather test to investigate the shedding law of the external insulation layer of the residential project.

The weather test was used to evaluate the material durability and stability under various environmental conditions, such as natural factors (sunlight, rainwater, thermal cycling, and weathering) and artificial factors (extreme temperatures and humidity). ¹¹ The primary objective was to determine the performance evolution over time and the environmental resistance capabilities. Weather testing is typically conducted on construction materials, coatings, plastics, and electronic devices to assess how these materials withstand environmental stressors like UV radiation, moisture, and thermal fluctuations. ¹² For external wall insulation systems in residential projects, rainwater infiltration and temperature variations emerge as dominant environmental factors that degrade the bond strength of insulation layers, ultimately compromising the construction quality owing to the inherent defect risks.

2. Methodology

2.1. Research settings

Based on IDI coverage provisions, IDI covers damage claims for cracking and detachment of external insulation layers.^1,2 The insurance claims data of PICC in Shanghai in 2021are shown in Table 1.

Table 1.

Claims details of each engineering division and subdivision in Shanghai in 2021.

Scope of claims	Claims number	Proportion	Claim amount (yuan)	Proportion	Average claim amount (yuan)
10-Year Term Foundation and Main Structure	10	0.80%	134444.00	1.25%	13444.40
5-Year Term Waterproofing and Insulation	817	65.73%	7219099.04	67.30%	8836.11
5-Year Term Window and Door	1	0.08%	6150.00	0.06%	6150.00
5-Year Term Wall and Ceiling Plastering Layer	2	0.16%	8366.00	0.08%	4183.00
2-Year Term Electrical, Water Supply and Drainage	145	11.67%	1842486.87	17.18%	12706.81
2-Year Term Window and Door	83	6.68%	241773.00	2.25%	2912.93
2-Year Term Wall and Ceiling Plastering Layer	44	3.54%	282834.99	2.64%	6428.07
2-Year Term Renovation	141	11.34%	990910.30	9.24%	7027.73
Total	1243	100.00%	10726064.20	100.00	8629.17

From the insurance claims data of PICC, it can be seen that cracking and detachment is the claim item of insurance claims, which is also the focus of insurance companies. Other failure mechanisms affecting ETICS performance are not discussed in the study because they are not included in the claims.

Therefore, the specific focus in this study is the cracking and detachment phenomena in external insulation systems, excluding other insulation-related defects such as thermal conductivity alterations owing to material aging or the fire safety issues of insulation materials. Among them, the attenuation of bond strength is directly related to cracking and shedding, and it is also the most intuitive reflection. Therefore, it is selected as a random variable in the research process, and the performance of cracking and detachment is reflected by its value. The primary external impacts on the residential building insulation are precipitation and thermal stresses induced by seasonal temperature variations and potential structural damage from water infiltration. Accordingly, the thermal stress effects were investigated in this study by employing weather tests to simulate environmental variations. The testing protocol incorporates high-temperature water spray cycling tests (to simulate precipitation-induced thermal stress) and thermal-freeze cycling tests (to simulate winter-cold conditions). The water spray procedure also facilitates the investigation of precipitation-induced leakage mechanisms. ¹³ Insulation layer damage typically manifests as surface blistering or cracking, observable through post-test specimen examination. Detachment indicates complete bond failure between the insulation and substrate walls. Therefore, the bond strength between insulation layers and concrete substrates is consequently adopted as the critical evaluation parameter for both the experimental measurements and finite element simulations.

Currently, the most widely used exterior wall insulation system in China is the expanded polystyrene (EPS) panel thin-layer plaster exterior wall insulation system, which accounts for 53% of the domestic building insulation material market share. ¹⁴ This system is lightweight and has a high strength, ensuring the structural safety and stability of the insulation system. ¹⁵ In addition to the EPS plaster system, integrated insulation-structure systems occupy a significant market share and represent the future development direction of the construction industry. However, statistical data from quality inspection departments indicate that the bonding strength of these integrated systems significantly exceeds that of residential EPS insulation systems, ¹⁶ with correspondingly fewer inherent defects and minimal relevance to insurance practices. Therefore, such systems are unsuitable for IDI loss distribution analysis. Consequently, the EPS panel thin-layer plaster system for designing experimental components is adopted in this study, and its structural configuration is shown in Figure 1.

Figure 1.

The cross section of the specimen of EPS board thin plastering external thermal insulation system.

2.2. Test design

2.2.1. Materials of the specimen

Five 500 x 500 mm specimens were fabricated, and four of the specimens were used to statistically analyze the common characteristics of external insulation layer damage in buildings and to establish its distribution pattern. The last specimen was designated as the control group and used to investigate the adhesive condition of external insulation layers under ambient environmental conditions, allowing the determination of the initial and design values of the bond strength for residential project insulation systems.

The interface agent for specimens adopted HS-1203 stone-like paint emulsion in accordance with relevant tests, ¹³ and the concrete strength grade was specified as C40. To examine the insulation layer failure modes and adhesive strength, a 150 mm-thick plain concrete layer was designed using the concrete mix proportion shown in Table 2 and implemented to prevent concrete layer fracture. ¹⁷

Table 2.

Concrete mixture ratio.

Water-binder ratio	Cement (kg/m3)	Sand (kg/m3)	Microlith (kg/m3)	Water (kg/m3)	Water reducing admixture (kg/m3)	Defoaming agent (kg/m3)
0.40	438.00	723.00	950.00	175.00	3.50	0.43

The performance index of the main supporting materials of the specimen should meet the requirements in relevant standards and specifications. ¹³

2.2.2. Test equipment and environment setting

According to IDI underwriting regulations, the insurance period for exterior insulation layers in residential projects is 5 years. ² The lifetime of insulation panel also affects its performance and failure over time is not considered in the study because the impact of service life on performance within 5 years is basically negligible, and the IDI project will be re-evaluated after 5 years. To simulate seasonal environmental conditions during building operation, the weather test comprises five heating-freezing cycles. According to the Test Methods for the Weather Resistance of External Thermal Insulation Systems (JG/T 429-2014), 80 times high-temperature water spray cycles were also conducted. ¹⁸ During post-cyclical testing, the surface defects (damage, hollowing, cracking) are documented, and bond strength tests are performed. ¹⁸ The weather test employs a high-low temperature humidity test chamber; its external appearance and internal configuration are illustrated in Figures 2(a) and (b), respectively. This equipment operates within a temperature range of −70°C to 150°C, fulfilling the environmental simulation requirements. Additionally, its workspace dimensions (1300 × 1300 × 1400 mm) ensure comprehensive temperature control during testing.

Figure 2.

Test equipment. (a) Appearance of test equipment; (b) Internal environment of test equipment; (c) Selection of test area; (d) Universal testing machine.

To meet the requirements of the high-temperature spray cycling test, a spray system was installed on the high-low temperature humidity chamber (EW2470W). After completing the weather test, the specimens underwent conditioning followed by a 7-day curing period before bond strength testing (Ministry of Housing and Urban Rural Development of the People’s Republic of China, 2014) to evaluate the adhesion between the insulation layer and concrete wall post-weathering. The specimens were proportionally divided into 25 test zones (40 mm × 40 mm) prior to bond strength testing owing to the large specimen size in the weather test. The test zone selection is illustrated in Figure 2(c).

Subsequently, the notch was cut to the lower surface of the anti-crack mortar composite with a rendering layer made of an alkali-resistant glass fiber mesh. This eliminated the influence of the bond strength between the mortar layer and thermal insulation layer, because the test is tailored to measure the bond strength between the thermal insulation layer and concrete wall. The specimen was then connected to a 40 mm × 40 mm iron block on the tensile testing machine for bond strength testing, and the data was recorded. The electronic universal testing machine used for bond strength testing is shown in Figure 2(d).

2.3. Test process

2.3.1. Specimen construction

Based on the specimen dimensions, a 500 × 500 mm mold was fabricated, as shown in Figure 3(a). Concrete was then poured into the mold based on the mix proportions specified in Table 2 (Figure 3(b)). After 7 days of curing, the mold was removed, and a bonding agent was added to the specimen surface. EPS insulation boards of corresponding dimensions were then promptly adhered (Figure 3(c)), followed by the application of an alkali-resistant fiberglass mesh and a mortar overlay to level the surface after sufficient curing (Figure 3(d)). Once the mortar had cured successfully, an exterior finish layer was applied to complete the specimen preparation. Following this procedure, five 500 × 500 mm test specimens were fabricated: samples 1, 2, 3, and 4, and a control sample. The completed specimens are shown in Figure 3(e)–(h).

Figure 3.

The production process of the specimen for the weather test. (a) Form erecting; (b) Pour concrete; (c) Paste insulation layer EPS board; (d) Spread mortar; (e) Sample 1 and 2; (f) Sample 3 and 4; (g) Samples of the control group; (h) Test specimens.

2.3.2. Test procedure

The tests were conducted in a weather test chamber according to the Test Methods for Weather Resistance of External Thermal Insulation Systems (JG/T 429-2014) guideline. The time and temperature controls for each cyclic test are as follows:

High-temperature water spray cycle (6 h per cycle):

• (1) Use approximately 1 h to increase the specimen surface temperature to 70°C;

• (2) Maintain a constant temperature of $\left(70\pm 5\right)$ °C for 2 hours;

• (3) Spray water onto the specimen surface for 1 h, and maintain the water temperature at $\left(15\pm 5\right)$ °C and the flow rate at $1L/\left(m^2·\hspace{0.17em}\min \right)$ ;

• (4) Leave the specimen to stand for 2 h. ¹⁸

Heating-freezing cycle (24 h per cycle):

• (1) Use approximately 1 h to raise the specimen surface temperature to 50°C;

• (2) Maintain a constant temperature of $\left(50\pm 5\right)$ °C for 7 h;

• (3) Use approximately 2 h to lower the surface temperature to −20°C;

• (4) Maintain a constant temperature of $\left(-20\pm 5\right)$ °C for 14 h, and adjust the humidity to simulate frost and condensation in autumn/winter and nighttime conditions;

• (5) Leave the specimen to stand for 2 h. ¹⁸

The flowcharts for the High-temperature water spray cycle and the Heating-freezing cycle are shown as Figures 4(a) and (b) respectively.

Figure 4.

Flow chart of experiment. (a) The High-temperature water spray cycle; (b) the Heating-freezing cycle.

A 24 h conditioning process was added between the high-temperature water spray and heating-freezing cycles. Starting from the second heating-freezing cycle, specimens were inspected for cracks, hollowing, detachment, and other defects after each heating-freezing cycle, and detailed records were taken. ¹³

After specimen fabrication, weather tests were immediately conducted in a temperature–humidity test chamber (EW2470W). Four specimen samples were evenly placed horizontally in the chamber. The tests followed the designed high-temperature water spray and heating–freezing cycles, and the specimen arrangement and heating conditions are shown in Figure 5(a), and the water spraying is depicted in Figure 5(b). Surface cracks, damage, and detachment were recorded during post-cyclic testing. To ensure accuracy during bonding strength testing, the specimens were transferred for standard curing immediately after the weather tests were concluded.

Figure 5.

The process of weather test. (a) The process of sample placement and heating up; (b) The process of the sample being drenched.

3. Results and analysis

3.1. Test phenomena

After completing high-temperature water spray and heating–freezing cycle testing, the specimens were examined for cracks, hollowing, or detachment, and the observations were documented. The overall condition of the specimens following weather testing is shown in Figure 6(a), and the details of the appearance of Specimen 1 are presented in Figure 6(b).

Figure 6.

The situation of weather test. (a) The overall situation of the specimen after the weather test; (b) The situation of specimen 1 after weather test.

An observation of the exterior surface of Specimen 1 indicates no visible cracks, hollowing, or detachment during and after the 80 high-temperature watering cycles and 5 heating–freezing cycles. Therefore, the analysis of the water leakage or condensation within the external insulation layer is unnecessary.

3.2. Test result analysis

3.2.1. Results of the bond strength tests

Based on the Test Method for Weather Resistance of External Thermal Insulation Composite Systems (JG/T 429-2014) guidelines, all specimens were cured for 7 days, after which bond strength tests were performed on the four sample specimens and control specimen. ¹⁸ A 40 mm × 40 mm test block was then selected from each of the 25 measurement zones at the center of the designated area. Subsequently, the bond strength was measured using the equipment illustrated in Figure 5.

The bond strength is determined by the force applied to the test specimen; thus, the specimen was secured to the testing device using fixtures to ensure no additional stress or deformation was incurred during testing. When the device was activated and loading was applied according to predefined parameters, the tensile force increased, and gradual separation occurred at the bond interface. The force values displayed by the device were then recorded.

For consistent process visualization and comparison, specimens from the second row and second column of each test unit were uniformly selected. During testing, a tangential force, which peaks when the bond strength reaches its limit state (zero value) before gradually decreasing and stabilizing, was applied to the specimen. The test curve shows the time on the horizontal axis and the magnitude of the tangential force on the vertical axis, where the peak value represents the bond strength. However, the morphology and trend in the curve are unrelated to the study. The test results are shown in Figure 7.

Figure 7.

The result of bond strength test. (a) The bond strength curve of sample 1; (b) The bond strength curve of sample 2; (c) The bond strength curve of sample 3; (d) The bond strength curve of sample 4; (e) The bond strength curve of the control group.

Based on the relationship between the force and pressure $P=F/S$ , the tangential force was converted into the bond strength for statistical analysis. The average bond strength of the specimens subjected to the 80 high-temperature water spray cycles and 5 heating–freezing cycles is 0.135 MPa. Comparatively, the control group specimens that did not undergo these cyclic tests maintained an average bond strength of 0.163 MPa.

3.2.2. Analysis of the damage level distribution of external wall thermal insulation

After bond strength testing, the recorded data was organized and compiled. The criteria for evaluating the classification of insulation engineering systems were established in accordance with the requirements of the Standard for the Maintenance of the Exterior Insulation Systems of Building Walls (JGJ 376-2015), as shown in Table 3. ¹⁹

Table 3.

Standard for appraisal appraising and classification classifying of thermal insulation engineering.

Damage level	Classification standard	Processing demands
Level one	The defective area is equal to or exceeds 15% of the unit wall, the bond strength of the external wall insulation system is below 70% of the original design value, or evident hollowing or detachment has occurred in the insulation system.	The overall replacement method should be adopted for repair.
Level two	The defective area is less than 15% of the unit wall, or the bond strength of the external wall insulation system is not less than 70% of the original design value.	Local repair should be performed.
Level three	Failure to comply with current national energy efficiency standards and codes, but conforms to original standards, codes, and design requirements, with no evident hollowing or detachment in the insulation system.	No treatment measures can be taken
Level four	In line with the requirements of the current national energy-saving standards and specifications.	No need to take measures

In IDI practice, the claims coverage only applies to the requirements for repair. Damages classified as Grades III and IV inherently fall outside the IDI claim scope and cannot be classified as external insulation layer losses. Consequently, only Grade I and II failures were retained for the classification of the weather test specimens.

To determine the original design value of the external insulation layer bond strength, the control group specimens (that did not undergo weather testing) were used to approximate the bond strength under zero external load conditions. The average bond strength of the 25 test units in the control group is 0.163 MPa and was taken as the original design value.

Following Table 3’s classification criteria, bond strength grading was performed on 100 test units across the four samples. With no detectable voids or detachment observed, 75 test units (75%) retained a bond strength exceeding 70% of the design value (0.163 MPa), indicating a 0.25 probability for Grade I damage and 0.75 for Grade II in the insulation system. Owing to the precision in the bond strength measurement (0.001 MPa), the 100-unit dataset was partitioned at 0.005 MPa intervals to better characterize the probability distribution patterns.

4. Probability model of the detachment loss in the external insulation layers of residential projects

4.1. Construction of a probability model of the shedding loss in the external insulation layers of residential projects

The external insulation layer of residential engineering projects is regarded as a structure on its own; thus, the principle of the reliability design of engineering structures was introduced. The required bonding strength of the external insulation layer to prevent detachment was taken as the resistance function in the structural reliability design, and the degradation of the bonding strength of the external insulation layer, caused by external loads, was taken as the load function. ²⁰ The distribution of the resistance and load functions was determined by taking the bonding strength of the external insulation layer of the residential project as a random variable, which allowed the determination of the probability distribution of the functional function. ²¹

Within the measurable space is a non-empty set composed of some subsets of $\mathrm{\Omega }$ called the sample space, whose elements are called sample points. ²² The measurable space can be obtained by introducing the real-valued function to define its influence on the measurable space. ²³ When $\left(\mathrm{\Omega },F,P\right)$ is taken as the probability space and used as the probability measure, the mathematical description and analysis of random phenomena can be conducted. On this basis, the probability distribution of the resistance, load, and functional functions of the external insulation layer in residential engineering projects can be investigated. ²⁴

For the probability space, assuming the structural reliability design theory is employed, the external insulation layer is regarded as a structural unit. By introducing the concept of functional function in structural reliability theory, the part less than the performance function is marked as structural failure, that is, failure domain. The area of the failure domain is the failure probability, which means the probability of the problem of the external insulation layer. In particular, the functional function, $Z=R-S$ , is a random variable, representing the bonding strength of the external insulation layer of the residential project, with the unit MPa. In the formula of functional function, the probability distribution of the resistance function of the external insulation layer of the residential project is $R$ , and the probability distribution of the load function is $f_S\left(y\right)$ , the functional function of the detachment of the external insulation layer of the residential project is $S$ .This concludes the determination of the functional function of the external insulation layer detachment in residential projects. The failure domain of the functional function, that is $P_f={\int }_{-\infty }^{0}Z\left(y\right)dy$ , can be solved by the micro-element method, which allows the calculation of the failure probability and reliability index of the external insulation layer detachment in residential projects. ²⁵

Given a sample space $\mathrm{\Omega }$ , where the samples are independent and identically distributed sequences of random variables, the corresponding observations are denoted as, each sample value is a measurable function defined in the sample space, and the observation value is the specific implementation of the sample value.

Based on the observed value of a specific sample, the observed values were rearranged to obtain an ordered sample whose empirical distribution function is: ²⁶

$$F_n\left(x\right)=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{I}_{\left\{x_{\left(i\right)}\le x\right\}},\text{where}\left\{x_{\left(i\right)}\right\}\text{is} \text{ordered} \text{sample}$$ (1)

If the cumulative function of the probability measure satisfies $P_n\left(A\right)=\frac{1}{n}{\displaystyle \sum }_{i=1}^n{I}_{\left\{x_i\in A\right\}}$ , the empirical distribution function of the sample is taken as the cumulative function of the probability measure. Based on this, the objective of determining the probability distribution of the loss in the external insulation layer of residential projects is to determine the parametric distribution family according to the probability density function by finding the $\theta \in \mathrm{\Theta }$ that is closest to the true probability measure.

The Akaike information criterion (AIC) is an important means for analyzing the fitting effect of probability distributions.^27–29 By judging the fitting effects of various probability distributions, the probability distribution that is closest to the true probability measure can be found and used as the probability distribution of the loss in the external insulation layer of residential projects, providing a scientific basis for the decision-making of insurance companies.

4.2. Analysis of the probability distribution of the damage to the external insulation layer specimens

Based on the analysis in the previous section, the division point for the first and second damage levels of the external insulation layer of residential projects is 0.1141 MPa. The 25 test blocks, which have a bonding strength lower than 0.1141 MPa, which belong to the first-level damage grade, are classified and statistically analyzed according to different bonding strengths and corresponding degradation degrees. The number of test blocks for each category is shown in Table 4.

Table 4.

Distribution of the first-level damage grade of the test block.

Bonding strength (MPa)	Degree of degradation (%)	Number of measurement blocks (blocks)	Frequency (%)
0.114	30.49	1	4
0.113	31.10	2	8
0.112	31.71	3	12
0.111	32.32	4	16
0.11	32.93	5	2
0.109	33.54	3	12
0.108	34.15	2	8
0.107	34.76	2	8
0.106	35.37	1	4
0.105	35.98	1	4
0.104	36.59	1	4

Based on the descriptive statistics of the control group test blocks, different probability distribution fitting was carried out for the degradation of the first-level damage of the external insulation layer of the residential project. In addition to several common probability distributions introduced in Chapter Two, some probability distributions that may conform in terms of change morphology were also introduced. The fitting results of various probability distribution models are shown in Table 5.

Table 5.

Probability distribution fitting result table of the degradation degree of the first-level damage grade test block.

Probability distribution	AIC	R2
Gaussian distribution	−124.75	0.66
Lognormal distribution	−125.03	0.75
Weber distribution	−121.75	0.36
Logistic distribution	−122.97	0.88
Gamma distribution	−125.98	0.60
Beta distribution	−124.90	0.68
Inverse Gaussian distribution	−124.06	0.73
Fatigue life distribution	−124.06	0.73

Table 5 shows that the AIC value of the logistic distribution is the lowest and the coefficient of determination R² is the highest, which indicates that the change in the degree of bond strength degradation of the first damage grade has the highest fitting degree with the logistic distribution. Therefore, the logistic distribution is selected to fit the probability distribution of the degree of bond strength degradation. The basic form of the logistic distribution is $f\left(x\right)=\frac{e^{-\frac{x-\mu }{\gamma }}}{\gamma ·{\left(1+e^{-\frac{x-\mu }{\gamma }}\right)}^2}$ . The values of each parameter of the logistic distribution fitted by Matlab are $\mu =0.33$ and $\gamma =0.01$ .

From this, it can be determined that under the first-level damage grade, the probability distribution of the degradation degree of the bonding strength of the external insulation layer of the residential project is as equation (2) shown:

$$f\left(x\right)=\frac{e^{-\frac{x-0.33}{0.01}}}{0.01·{\left(1+e^{-\frac{x-0.33}{0.01}}\right)}^2}$$ (2)

From the perspective of practical engineering, the first-level damage of the external insulation layer in residential projects indicates that the degree of degradation of the bonding strength has exceeded 30%. According to the requirements of the “Standard for Repair of External Insulation Systems of Building Exterior Walls” (JGJ 376-2015), the overall replacement method should be adopted for repair. The insulation layer of residential engineering is regarded as a structure. From the perspective of design reliability theory, the distribution of the first-level damage of the external insulation layer of residential engineering can be analyzed as the load function of the functional function. Since the load function reflects the degradation amount of the bonding strength of the external insulation layer of the residential project, the probability distribution of the degradation degree of the bonding strength of the external insulation layer of the residential project must first be converted into the probability distribution of the degradation amount of the bonding strength of the external insulation layer of the residential project. $f_S\left(y\right)$ should be the product of and the original design value; that is:

$$f_s\left(y\right)=\frac{14.91·e^{-\frac{y-0.33}{0.01}}}{{\left(1+e^{-\frac{y-0.33}{0.01}}\right)}^2}$$ (3)

4.3. Analysis of the probability distribution of bond strength of the control group specimens in the weather test

The analysis of the degree of degradation of the bonding strength of the weather test specimens and the damage to the external insulation layer simulates the influence of the external environment on the bonding strength of the external insulation layer and can be regarded as the load function of the external insulation layer in residential projects. Next, the bonding strength of the control group specimens must be analyzed from the inside, and the changes in the bonding strength of the external insulation layer of residential projects must be determined without external loads. The bonding strength of the 25 measurement blocks divided in the control group was classified and statistically analyzed. The number and frequency of test blocks for each category are shown in Table 6.

Table 6.

Distribution of the control group test block.

Bonding strength (MPa)	Number of measurement blocks (blocks)	Frequency (%)
0.159	1	4
0.16	2	8
0.161	3	12
0.162	5	2
0.163	3	12
0.164	6	24
0.165	1	4
0.166	3	12
0.167	1	4

Based on the descriptive statistics of the test blocks in the control group, different probability distribution fittings were conducted for the bonding strength of the control group of the external insulation layer of the residential project. In addition to several common probability distributions introduced in Section Two, some probability distributions that might conform in terms of change morphology were also introduced. The fitting results of various probability distribution models are shown in Table 7.

Table 7.

Probability distribution fitting result table of the degradation degree of the control group test block.

Probability distribution	AIC	R2
Gaussian distribution	−234.78	0.73
Lognormal distribution	−234.80	0.73
Weber distribution	−235.04	0.86
Logistic distribution	−233.45	0.58
Gamma distribution	−234.81	0.70
Beta distribution	−234.81	0.70
Inverse Gaussian distribution	−234.82	0.69
Extreme value distribution	−231.84	0.75

Table 7 shows that the AIC value of the Weber distribution is the lowest and the coefficient of determination R² is the highest, which indicates that the change in the degree of bond strength degradation of the first damage grade has the highest fitting degree with the Weber distribution. Therefore, the Weber distribution is selected to fit the probability distribution of the degree of bond strength degradation. The basic form of the Weber distribution is $f\left(x\right)=k·\lambda ·x^{k-1}·e^{-\lambda ·x^k}$ . The values of each parameter of the Weber distribution fitted by Matlab are $k=84.4$ and $\lambda =0.16$ .

From this, it can be determined that the probability distribution of the bonding strength of the control group of the external insulation layer of the residential project is:

$$f\left(x\right)=13.84x^{83.4}·e^{-0.16·x^{84.4}}$$ (4)

The bond strength of the control group specimens of the external insulation layer in residential projects can be regarded as the distribution of bond strength without external force loads, similar to the concept of “resistance” of the structure itself. Therefore, the probability distribution of the bonding strength of the control group specimens of the residential engineering external insulation layer obtained through fitting can be regarded as the resistance function $R$ of the residential engineering external insulation layer. Therefore, to better distinguish different probability densities, the probability distribution of the bonding strength of the control group specimens of the external insulation layer of the residential project is recorded as $f_R\left(y\right)$ .

4.4. Analysis of the probability of loss owing to the detachment of the external insulation layers

Based on the analysis of the degree of degradation of the bonding strength of the weather test specimens and the probability distribution of the bonding strength of the control group in Section 4.3.3, the functional functions of the external insulation layer of residential projects can be constructed in a targeted manner. If the functional function of the external insulation layer of a residential project is denoted as $Z=R-S$ , then. From equations (4) and (5), the resistance function and load function of the functional function are $f_R\left(y\right)$ and $f_S\left(y\right)$ respectively. Therefore, the functional function of the external insulation layer of the residential project is:

$$Z\left(y\right)=f_R\left(y\right)-f_s\left(y\right)=13.84y^{83.4}·e^{-0.16y^{84.4}}-\frac{14.91·e^{-\frac{y-0.33}{0.01}}}{{\left(1+e^{-\frac{y-0.33}{0.01}}\right)}^2}$$ (5)

The functional function reflects the probability strength of the bonding strength of the external insulation layer in residential projects. Therefore, the failure probability of the external insulation layer in residential projects can be defined as the accumulation of the functional function at various intervals using the element method. According to the definition of definite integrals, the failure probability of the external insulation layer in residential projects is:

$$P_f={\int }_{-\infty }^{0}Z\left(y\right)dy=1-{\int }_0^{+\infty }Z\left(y\right)dy\approx 0.164$$ (6)

4.5. Application of the probability model for investigating the detachment in the external insulation layer of residential projects in insurance

For the distribution in insurance claims, the claims in different insurance policies can be regarded as repeated experiments. The claims in one policy cannot affect the claims in other policies. Therefore, the IDI claims process can be seen as a Bernoulli experiment, and the occurrence of claims in a policy can be regarded as a successful Bernoulli experiment. The binomial distribution is a distribution that describes the number of successful Bernoulli trials. The binomial distribution can be utilized to describe the potential defects in the residential engineering projects and the claims situation in insurance. Because the probability of the Bernoulli test is unique, and the failure probabilities of potential defects of different qualities vary, different binomial distributions need to be used to simulate the number of claims. The claims for potential defects in the residential engineering projects are assumed to follow a binomial distribution $B\left(m,p\right)$ . Here, $m$ represents the number of insured policies, and $p$ is the failure probability corresponding to potential defects in the residential engineering projects. From the nature of the binomial distribution, the claim expectation of the potential defect insurance for engineering quality is $E\left(N\right)=mp$ , and the variance is $Var\left(N\right)=mp\left(1-p\right)$ . When the parameter m is sufficiently large, the binomial distribution can be assumed to follow the Poisson distribution, and the parameter of the Poisson distribution is $\lambda =m·p$ .

According to the calculation of the failure probability of the external insulation layer in residential projects in Section 4.4, the failure probability of the external insulation layer falling off in residential projects is approximately 0.164. The claim distribution of the external insulation layer in actual projects can all be simulated and analyzed through this failure probability. Therefore, $p=0.164$ further determines that the claim frequency of the external insulation layer falling off under the potential defect insurance for engineering quality follows a Poisson distribution. Its parameter is $\lambda =0.164m$ . Insurance companies can obtain the expected number of claims based on the number of insured cases and set corresponding premiums in a targeted manner to ensure the normal operation of the insurance company.

5. Conclusion

This study addresses the IDI issue by investigating external insulation layer detachment in residential buildings through an innovative methodology that integrates experimental design and theoretical modeling. The key findings are summarized below.

First, the experimental validation confirms bond strength degradation as the primary risk factor. By conducting accelerated weather tests that simulate a five-year exposure period, this study successfully demonstrated the long-term environmental impacts of the bond strength on the external insulation system performance. The results reveal that although no visible macro-level defects (e.g., cracks or voids) were observed post-testing, the internal bond strength exhibited significant deterioration. The weathered specimens showed an average bond strength of 0.135 MPa, representing a 17.2% reduction compared to the control group’s initial value of 0.163 MPa. These findings establish that insulation layer detachment constitutes a gradual process driven by a degradation in the internal bond performance. This confirms the insufficiency of surface-level inspections for risk assessment and underscores the necessity of monitoring core mechanical properties, specifically bond strength evolution.

Second, a loss probability model was established based on the structural reliability theory. The innovation of this study lies in introducing the engineering structural reliability theory into the risk assessment of the external insulation layer detachment. By defining the system’s initial bond strength as the ‘resistance ( $R$ )’ and the bond strength degradation caused by environmental actions as the ‘load ( $S$ )’ the complex engineering quality issue was successfully transformed into a quantifiable probabilistic model ( $Z=R-S$ ). Through rigorous statistical analyses of the experimental data and selecting models using criteria such as the AIC and coefficient of determination (R²), the Weibull distribution was identified as the optimal fit for the resistance function (control group bond strength), and the logistic distribution best characterized the load function (bond strength degradation under Level 1 damage). This model provides a scientific and rigorous technical framework for the quantitative analysis of the external insulation layer failure.

Subsequently, the failure probability of external insulation layer detachment was quantified. Based on the established probabilistic model and fitted probability density functions, this study calculated the probability of performance function in the failure domain through definite integrals, ultimately determining an approximate 16.4% failure probability for external insulation layer detachment in residential projects. This quantitative value constitutes the core finding, converting the qualitative risk description (“prone to detachment of external insulation layers”) into an actuarially applicable metric. It not only reveals the inherent failure risk level of external insulation systems under standard construction and material conditions during the design lifespan, but also provides critical foundational data for the insurance industry.

Finally, direct application tools for IDI insurance practice were developed. The study integrated the calculated failure probability with insurance claim models, demonstrating that the number of insulation layer detachment claims could approximate a Poisson distribution with parameter $\lambda =m·p$ . This conclusion holds significant practical value, allowing insurers to predict claim frequencies based on their portfolio size, enabling precise risk reserve allocations, optimized premium structuring, and data-driven risk mitigation management (e.g., enhanced technical inspection service during construction). These outcomes ultimately support the sustainable development of IDI business.

In summary, this research not only deepens the understanding of detachment mechanisms in residential external insulation systems but also establishes a comprehensive framework that covers the experimental simulation, probabilistic modeling, and insurance applications. The quantitative findings provide robust scientific support for resolving critical challenges in IDI insurance practice.

ORCID iD

Zeyu Chen https://orcid.org/0000-0002-2210-4770

Data Availability Statement

The data used to support the findings of this paper are included within the article. The data that has been used is confidential.*