Non-uniform thermal behavior of single-layer spherical reticulated shell structures considering time-variant environmental factors: analysis and design

Wucheng XU, Xiaoqing ZHENG, Xuanhe ZHANG, Zhejie LAI, Yanbin SHEN

Research Article

Non-uniform thermal behavior of single-layer spherical reticulated shell structures considering time-variant environmental factors: analysis and design

1College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China2Center for Balance Architecture, Zhejiang University, Hangzhou 310028, China3Innovation Center of Yangtze River Delta, Zhejiang University, Jiaxing 314100, China4Zhejiang Provincial Key Laboratory of Space Structures, Zhejiang University, Hangzhou 310058, China

Contrary to conventional design methods that assume uniform and slow temperature changes tied to atmospheric conditions, single-layer spherical reticulated shells undergo significant non-uniform and time-variant temperature variations due to dynamic environmental coupling. These differences can affect structural performance and pose safety risks. Here, a systematic numerical method was developed and applied to simulate long-term temperature variations in such a structure under real environmental conditions, revealing its non-uniform distribution characteristics and time-variant regularity. A simplified design method for non-uniform thermal loads, accounting for time-variant environmental factors, was theoretically derived and validated through experiments and simulations. The maximum deviation and mean error rate between calculated and tested results were 6.1 ℃ and 3.7%, respectively. Calculated temperature fields aligned with simulated ones, with deviations under 6.0 ℃. Using the design method, non-uniform thermal effects of the structure are analyzed. Maximum member stress and nodal displacement under non-uniform thermal loads reached 119.3 MPa and 19.7 mm, representing increases of 167.5% and 169.9%, respectively, compared to uniform thermal loads. The impacts of healing construction time on non-uniform thermal effects were evaluated, resulting in construction recommendations. The methodologies and conclusions presented here can serve as valuable references for the thermal design, construction, and control of single-layer spherical reticulated shells or similar structures.

Non-uniform temperature field; Non-uniform thermal load; Non-uniform thermal effect; Single-layer spherical reticulated shell; Time-variant environmental factor

1 Introduction

Because of its robust mechanical properties and efficient material strength utilization, the single-layer spherical reticulated shell is extensively used in large infrastructures. The temperature of such a structure varies significantly non-uniformly throughout its life-cycle, including the construction and service periods, due to the dynamic interplay of various environmental factors like solar radiation, wind, and atmospheric temperature (AT). Previous experiments have revealed that under intense solar radiation, the temperature of steel components can significantly exceed ATs, with the disparity even surpassing 30 ℃ (Liu and Chen, 2016). Traditional design methodologies often assume that temperature changes in a building structure occur uniformly and progressively (MOHURD, 2010, 2012). However, this simplification can result in discrepancies between the design and practical conditions, and neglecting the non-uniformity and time-variability of temperature changes may pose significant safety hazards. Therefore, it is vital to study the changing patterns and distribution characteristics of non-uniform temperature variations. We see that it is necessary to devise an efficient and systematic method for designing non-uniform thermal loads that would account for multiple environmental factors, and apply it in structural thermal analysis.

Previous structural engineering studies on non-uniform thermal behavior have focused on bridges (Zhou and Sun, 2019), components (Liu et al., 2012a), and radio telescopes (Greve and Kaercher, 2009). Many elements such as radiation, AT, and convective heat transfer, complicate the temperature field within bridge structures (Xia et al., 2013). Significant thermal effects, potentially causing hidden safety issues, can be instigated by the high time-variance and non-uniformity of a bridge's temperature field (Kim et al., 2015). Ascertaining the distribution and trend of temperature changes in bridges plays a pivotal role in guaranteeing their safety. In steel components, the temperature is uniform longitudinally and across thickness, but displays clear non-uniformity cross-sectionally (Liu et al., 2012b). Experiments have shown that under specific conditions, the maximum thermal stresses of steel components can reach 67 and 58 MPa under rigid and hinged constraints, respectively (Chen et al., 2017b). The uneven temperature distribution frequently results in pronounced thermal effects on steel components, which is a fact corroborated by extensive research. Non-uniform thermal deformation, caused by factors such as solar radiation and wind, can alter a telescope's geometric shape during operation from its original design (Chen et al., 2017a). This discrepancy can negatively impact the accuracy of reflector surface precision and pointing accuracy, thus leading to observational errors, as demonstrated in the research involving the Five-hundred-meter Aperture Spherical Telescope (Fan et al., 2009), Shanghai 65 m Radio Telescope (Qian et al., 2016), and the IRAM-30 m Telescope (Greve and Bremer, 2010). The thermal analysis and control frequently constitute a critical aspect of telescope design.

As structures with high static indeterminacy, spatial structures exhibit considerable sensitivity to temperature variations (Chen et al., 2021; Xu et al., 2021). In the past, numerous scientists have conducted studies on various actual spatial structure projects, and investigated changes in their temperature during construction and service periods (Xu et al., 2020). These projects include the Tien Rice Cube, China (Liu et al., 2015), Yujiabu Railway Station, China (Zhao et al., 2017), Xi'an Silk Road International Convention and Exhibition Center, China (Chen et al., 2020), and Beijing Daxing International Airport, China (Zhou et al., 2020). The research findings indicate that during construction, spatial structures directly exposed to solar radiation exhibit noticeably non-uniform temperature field distributions due to the coupling effects of various dynamic environmental factors, and that during service, as a result of factors such as transmission properties of roofing materials, thermal conduction between the roof and structure, and temperature box effect, spatial structures may experience even more complex temperature changes and distribution. Thus, the design of spatial structures should take into account the non-uniformity of temperature changes. However, previous research usually analyzed the non-uniform temperature change characteristics of one specific engineering and its impact on structural performance through monitoring and simulation data. As of yet, no published paper has touched on the design of non-uniform thermal loads in spatial structures. The work in this paper is intended to fill this research gap, and specifically, to provide a simplified design methodology for the non-uniform thermal loads of single-layer spherical reticulated shells, which is a common form of spatial structure.

This study employs a systematic numerical method to model the temperature fields of a single-layer spherical reticulated shell, taking into account dynamic environmental factors and shadow changes. The non-uniform distribution and time-variant patterns are revealed through simulation. A theoretically-derived, and experimentally- and numerically-validated design method for non-uniform thermal loads is proposed. The structure's non-uniform thermal effects and the impact of healing construction time on these effects are analyzed, resulting in specific construction recommendations.

2 Analysis of non-uniform temperature field

2.1 Numerical simulation method

In this study, a thermal analysis strategy for single-layer spherical reticulated shells was applied. It represents the single-layer spherical reticulated shell using a spherical steel skin of the same size and material, equating the steel component's temperature with that of the steel skin at the same position. Similar approaches have been used in previous studies (Liu et al., 2014, 2015). We used a single-layer spherical reticulated shell of Q345 steel with a span of 30 m and rise of 12 m, built in Beijing, China, as the study object. ANSYS was applied for thermal analysis, and Shell57, Link31, and Surf152 elements were used to simulate the heat conduction, convection, and radiation. Fig. 1 presents the simulation process of the temperature field. National Aeronautics and Space Administration (NASA)-provided data on AT, solar radiation intensity (SRI), and wind speed (WS) were adopted as the environmental boundary conditions for the process (GMAO, 2015).

2.2 Shadow analysis algorithm

For precise temperature field simulation under solar radiation, it is crucial to identify shaded regions on the steel skin. In this study, a shadow analysis algorithm was proposed to address this, which can be described as follows:

Step 1: A global coordinate system,, is created using geographic and geometric data. Theplane is parallel to the ground and the,, andaxes point to the west, south, and sky.

Step 2: Solar altitude and azimuth angles are calculated using Eqs. (1) and (2); a local coordinate system,, is defined (Fig. 2), with Eq. (3) outlining its conversion from.

Fig. 1 Simulation process of the temperature field

Step 3: Triangular skin elements are projected onto theplane (utilizing the sunlight direction) with vertex and barycenter coordinates calculated for each element.

Step 5: Take one element as the study object. In theplane, if the element's barycenter is located within other elements, these are selected for further analysis; otherwise, it is a sunlit element.

Step 6: Under thesystem, the study object's relationship to the selected elements is evaluated in terms of their-values. If the study object's-value is the highest, it is a sunlit element; otherwise, it is shaded.

Fig. 2 Conversion relationship between OXYZ and oxyz

Step 7: Steps 4?–6 are repeated for all elements to decide whether they are sunlit or shaded, thus identifying the sunlit and shaded regions.

The shadow distribution variations on important solar cycle dates are depicted in Fig. 3. Red and blue elements denote sunlit and shaded elements, respectively. The western, northern, and eastern elements are shaded during morning, noon, and afternoon, respectively. At the summer solstice, with the sun closest to the structure, daylight is the longest and shadows are the smallest; at the winter solstice, when the sun is the farthest, daylight is the shortest and shadows are the largest; during the spring and autumn equinoxes, shadow changes are the same. The analysis accurately mirrors real-world shadow effects, affirming the algorithm's accuracy.

Fig. 3 Shadow distribution variations on important solar cycle dates. References to color refer to the online version of this figure

2.3 Seasonal temperature variation and distribution

Fig. 4 Variation curves of structural and atmospheric temperatures

1. The temperature field distribution is non-uniform, with the temperature differences all exceeding 32.0 ℃.

Table 1 Temperature field information for the top 10 highest

Table 2 Temperature field information for the top 10 lowest

3. Atmospheric temperatures are high, all above 30.0 ℃; solar radiations are intense, with all intensities above 750.0 W/m2; WSs are minimal, averaging only 0.8 m/s.

1. The temperature field distribution is uniform, with no TR higher than 0.3 ℃.

3. Atmospheric temperatures are low, all below -?10.0 ℃; there is no or minimal solar radiation, all under 7.0 W/m2in intensity; WSs are not insignificant, reaching an average of 2.8 m/s.

In general, high structural temperatures occur mainly at noon in summer with high ATs, strong solar radiation, and weak winds, when the temperature field is non-uniform and much higher than the AT; low structural temperatures occur mainly in early morning in winter with low ATs, no or minimal solar radiation, and strong winds, when the temperature field is uniform and slightly lower than the AT.

2.4 Daily temperature variation and distribution

Fig. 5 Simulated temperature fields on Jan. 7

Fig. 6 Simulated temperature fields on July 7

Figs. 7 and 8 show the simulated temperature fields on typical days and also demonstrate the sun's position relative to the structure. The variation rules of temperature field distribution are consistent over these two days, shifting as the sun rises and sets. The temperature field distribution reveals a distinct gradient, with the highest temperatures in areas perpendicular to sunlight and the lowest in shaded regions, which is a pattern attributable to uneven absorption of solar radiation by the structure's surface. While the SRI and WS determine the degree of non-uniformity in temperature field distribution, such as temperature field differences of 12.3 ℃ on Jan. 7 and 37.0 ℃ on July 7 at 12:00, the temperature field distribution pattern relies solely on the sun's position relative to the structure.

Fig. 7 Simulated temperature fields on Jan. 7

Fig. 8 Simulated temperature fields on July 7

3 Design of non-uniform thermal load

3.1 Theoretical design method

Numerical analysis revealed that the temperature field of the single-layer spherical reticulated shell, influenced by environmental factors, is significantly non-uniform and above the AT. This emphasizes the importance of considering these factors in thermal load design. As such, this study proposes a simplified thermal load design method that accounts for environmental factors.

Conceptualizing the spherical steel skin as numerous infinitesimal plane steel plates allows us to determine the structural temperature field by calculating the temperature of the plates. The following section outlines the calculation process for any given plane steel plate.

The general form of the transient heat conduction differential equation for a steel plate exposed to the natural environment is expressed as

Addressing the temperature problem requires solving the heat conduction differential equation via initial and boundary conditions. The initial condition defines the steel plate's initial temperature, while the boundary condition outlines the heat exchange at the plate's boundary, as described by:

Due to steel's high thermal conductivity, the temperature gradient across the plate section is negligible, making the heat flux from the interior to the surface zero. Hence, Eq. (5) is expressed as

Convective heat transfer between the plate surface and flowing air is computed by

Solar radiation absorbed by the plate surface is calculated by

Longwave radiation absorbed by the plate surface is calculated by

The temperature of any steel plate can be calculated using the equations above. Taking several steel plates from the structure as the interpolation points, we adopted biharmonic spline interpolation to determine the temperature field (Deng and Tang, 2011).

3.2 Experimental verification

Experiments were conducted to test the accuracy and reliability of the design method. Test platform was a rotatable tripod installed with a steel plate, and nine platforms (at various angles) to simulate steel plates with different orientations within the structure. Real-time environmental boundary conditions and steel plate temperatures are monitored using test equipment, as shown in Fig. 9 and detailed in Table 3. Test site was on a well-ventilated and sunny rooftop, and the test period covered 15 d from July 16 to July 30, 2021. Five types of steel plates with different coating colors were used, as listed in Table 4 (Liu and Chen, 2016).

Figs. 10?–?13 presents the comparison between the measured and calculated temperatures of some steel plates on July 19, when the highest sample temperature was observed. The measured and calculated results followed consistent trends. At nighttime, measured and calculated temperatures were nearly identical, with a discrepancy of less than 1.0 ℃, while during the daytime, measured temperatures lagged slightly behind calculated ones, with calculated temperatures being marginally higher and error rising with temperature. For the four steel plates, the maximum temperature differences between measurements and calculations were 4.4, 6.1, 4.2, and 4.6 ℃, with peak error rates of 8.8%, 9.7%, 8.8%, and 8.7%, respectively. The error mainly stemmed from calculating temperatures at thermal equilibrium, while tests measured transient temperatures. Based on the comparison of measured and calculated results, Fig. 14 displays the maximum and root mean square (RMS) of temperature differences and error rates for all tested steel plates on July 19. At 13:00, the maximum and RMS of temperature differences reached peaks of 6.1 and 4.4 ℃, and those of error rates achieved peaks of 9.7% and 8.3%, respectively. The mean values of temperature differences and error rates throughout the day for these steel plates were only 1.5 ℃ and 3.7%. In summary, the calculated results were in agreement with measured results, which verified the accuracy and reliability of the design method.

Fig. 12 Temperatures of steel plate 7 on July 19

Fig. 13 Temperatures of steel plate 9 on July 19

Fig. 14 Temperature differences and error rates between measured and calculated results on July 19. Max: maximum

Fig. 9 Test equipment and experimental platforms

Table 3 List of test equipment

Table 4 Test information for the five types of steel plates

RAL is a globally recognized color matching system, originally developed in Germany, which provides a standardized palette of colors for use in various industries such as manufacturing, architecture, and design

Fig. 10 Temperatures of steel plate 3 on July 19

Fig. 11 Temperatures of steel plate 5 on July 19

3.3 Numerical verification

The calculated temperature fields on the typical days are shown in Figs. 15 and 16. Figs. 17 and 18 presents the variation curves of the maximum and minimum of the simulated and calculated temperature fields on the typical days. The simulated and calculated temperature fields shared distribution patterns, and the calculated results accurately echoed the impact of solar motion. For the six moments, the calculated maximums exceeded simulated ones by 1.1?–?5.9 ℃, and minimums were 0.8?–?1.8 ℃ higher, with greater calculation error observed in the summer temperature field compared to the winter one. The calculated results have the same pattern of variation as the simulations, with slightly higher values for the former; the difference between the two is relatively small at nighttime and more significant in the daytime. On Jan. 7, the simulated maximum and minimum were higher than the calculated ones by an average of 0.6 and 0.8 ℃ at nighttime, and by 2.1 and 0.8 ℃ in the daytime, respectively; the differences between simulated and calculated maximums, and simulated and calculated minimums peaked at 2.1 and 2.3 ℃, respectively, at 10:00. On July 7, the calculated maximums and minimums were, on average, higher than the simulated ones by 0.8 and 1.2 ℃ at nighttime, and by 3.1 and 1.7 ℃ in the daytime, respectively; the largest differences between simulated and calculated maximums, and simulated and calculated minimums were 5.9 and 3.3 ℃ at 12:00 and 10:00, respectively. In general, the consistency between simulated and calculated results confirmed the precision and reliability of the design method.

Fig. 17 Simulated and calculated results of temperature fields on Jan. 7. Min: minimum

Fig. 18 Simulated and calculated results of temperature fields on July 7

Fig. 15 Calculated temperature fields on Jan. 7

Fig. 16 Calculated temperature fields on July 7

3.4 Application instruction

Temperature load design entails computing the peak and trough temperature fields, which correspond to the maximum and minimum temperatures experienced by the structure in service. Additionally, the healing temperature field, representing the structure's temperature during its creation, must be determined. Based on the analysis in Section 2.4, we set the environmental boundary conditions of the peak temperature field to the maximum AT, the strongest solar radiation, and zero wind; the trough temperature field was uniformly set to 1 ℃ below the minimum AT. Temperature-rise and temperature-drop loads can be designed by

Taking the structure under study as an example, the roof was designated as either glass or opaque, with a solar radiation transmittance of 0.7 or 0. Healing times were set for 9:00, 12:00, and 15:00 on Mar. 1, 2021, which was the date when the AT would approach its annual midpoint. The application of the design method was illustrated by the six cases. The meteorological data (1922?–?2021) indicated that Beijing's AT ranged from 42.6 to -27.4 ℃. Using the Hottel model (Hottel, 1976), Beijing's strongest solar radiation was identified at 12:00 on June 18, with direct and diffuse intensities of 794.1 and 110.6 W/m2, respectively. We utilized this information to calculate the peak, trough, and healing temperature fields. The non-uniform thermal loads designed for healing times on Mar. 1 are shown in Figs. 19 and 20. For the same healing time, temperature-drop loads were consistent across roof types, but the temperature-rise load was greater for glass roofs compared to opaque ones due to differing peak temperature fields. With varying healing times, thermal load distributions and ranges significantly differed due to diverse healing temperature fields. The glass roof experienced more adverse temperature-rise loads than the opaque roof. In summary, the non-uniform thermal load distribution aptly depicts the temporal variation of the temperature field across the lifecycle, encompassing both construction and service phases.

Fig. 19 Non-uniform thermal loads for glass roof on Mar. 1

Fig. 20 Non-uniform thermal loads for opaque roof on Mar. 1

4 Analysis of non-uniform thermal effect

4.1 Uniform and non-uniform thermal effects

The above analysis reveals that the temperature variation in a single-layer spherical reticulated shell is non-uniform and time-variant, which is an aspect overlooked in traditional design methods that regard it as a uniform and slow process driven by AT. To assess the impact of uniform and non-uniform thermal loads on the structure, we conducted a comparative analysis using both the conventional method for uniform loads and the novel method proposed here for non-uniform loads, as outlined in Table 5. In line with the aforementioned sections, a single-layer spherical reticulated shell, as shown in Fig. 21, constructed in Beijing was designed for structural analysis, and Q345 steel pipes (Φ180 mm×6 mm) were used as principal members, with the structure supported by 16 concrete columns. The structure was assumed to be exposed to natural conditions during healing and covered with an opaque roof in service. ANSYS is used to analyze the structural thermal effects under the given loads, using Beam188 elements to simulate members, and bottom supports are modeled as hinged joints, with the created finite element model displayed in Fig. 21.

Table 5 Definition for four thermal loads

Fig. 21 Finite element model for structural analysis

The uniform and non-uniform thermal effects were analyzed at healing times of 9:00, 12:00, and 15:00, for Mar. 1, 2021, and the results are summarized in Table 6. Under uniform thermal loads, the maximum member stress and nodal displacement were 44.6 MPa and 7.3 mm, reaching 14.4% and 9.7% of the design strength and deformation, respectively; under non-uniform thermal loads, the maximum member stresses reached 53.8, 73.0, and 70.1 MPa, while the maximum nodal displacements were 8.8, 11.1, and 11.3 mm, at healing times of 9:00, 12:00, and 15:00, respectively. Uniform thermal effects were much stronger than non-uniform ones, with a maximum increment of more than 60%. This implies that non-uniform thermal loads enable a more comprehensive analysis of structural thermal effects, preventing potential safety hazards.

4.2 Influence of healing time on non-uniform thermal effect

Table 6 Uniform and non-uniform thermal effects

Fig. 22 Variations of non-uniform thermal effects for healing times on Mar. 1

Fig. 23 Variation curves of non-uniform thermal effects for healing times in 2021

Table 7 Non-uniform thermal effects for healing times in the four seasons

1. Determine healing date: it is recommended to schedule the healing date during winter and avoid it in summer.

2. Determine healing time: for healing dates governed by temperature-rise loads, midday is the optimal time for healing, while for dates controlled by temperature-drop loads, healing is best scheduled at nighttime.

5 Conclusions

In this study, we investigated the non-uniform thermal behavior of a single-layer spherical reticulated shell via numerical analysis, theoretical derivation, and experiments, and we proposed a design method for non-uniform thermal loading. Some conclusions can be drawn as follows:

1. Using a systematic numerical method and considering practical environmental boundary conditions, shadow variations, and the dynamic coupling effect of environmental factors, long-term temperature changes in a single-layer spherical reticulated shell were simulated. The non-uniform distribution characteristics and time-variant regularity of the structure's temperature field were analyzed from the perspective of seasonal and daily variations.

2. By integrating numerical analysis results, a simplified design method for non-uniform thermal loads of single-layer spherical reticulated shells was proposed via theoretical derivation. The method effectively evaluates the impacts of time-varying environmental factors like non-uniform solar and longwave radiation absorption, and convective heat transfer between the structure and flowing air, on the structure's temperature field.

3. Verification experiments revealed a maximum temperature difference of 6.1 ℃ and a maximum error rate of 9.7% between the calculated and tested results. The distribution patterns of simulated and calculated temperature fields align, with the deviations not exceeding 6.0 ℃. Through comparative analysis, the design method's accuracy and reliability were validated, and its application was demonstrated in several cases.

4. The design method was employed to examine non-uniform thermal effects on a single-layer spherical reticulated shell. Under non-uniform thermal loads, the maximum member stress and nodal displacement reach 119.3 MPa and 19.7 mm, respectively, with an increase of 167.5% and 169.9% compared to uniform loads. Furthermore, the impacts of healing construction time on non-uniform thermal effects were assessed, leading to construction recommendations.

This work is supported by the National Natural Science Foundation of China (Nos. 51578491 and 52238001).

Wucheng XU designed the research and wrote the first draft of the manuscript. Xiaoqing ZHENG supervised and administrated the project. Xuanhe ZHANG and Zhejie LAI helped to organize the manuscript. Yanbin SHEN revised and edited the final version.

Wucheng XU, Xiaoqing ZHENG, Xuanhe ZHANG, Zhejie LAI, and Yanbin SHEN declare that they have no conflict of interest.

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考慮時變環境因素的單層球面網殼結構的非均勻熱行為：分析與設計

許武成1,3,4，鄭曉清2，張軒赫1,3,4，來哲杰1，沈雁彬1,3,4

1浙江大學，建筑工程學院，中國杭州，310058；2浙江大學，平衡建筑研究中心，中國杭州，310028；3浙江大學，長三角智慧綠洲創新中心，中國嘉興，314100；4浙江大學，浙江省空間結構重點實驗室,中國杭州，310058

由于多環境因素的動態耦合作用，單層球面網殼的溫度變化具有強時變性和顯著的非均勻特征。然而，傳統的設計觀念將結構的溫度變化等效為氣溫的均勻升降，這可能導致潛在的安全隱患。通過數值模擬、實驗研究和理論推導，本文旨在分析單層球面網殼溫度的時變規律和分布模式，研究非均勻熱效應及其影響因素，并針對該類結構提供非均勻熱荷載的設計方法。

1. 根據真實熱邊界條件，模擬單層球面網殼的長期溫度變化并剖析其時變規律和分布模式；2. 基于理論推導、數值和實驗驗證，提出單層球面網殼非均勻熱荷載的設計方法；3. 對比結構均勻與非均勻熱效應，分析影響熱效應的施工因素。

1. 通過仿真計算，運用真實熱邊界條件模擬單層球面網殼的長期溫度變化，分析溫度場的時變規律與分布模式；2. 通過理論推導，構建單層球面網殼非均勻熱荷載的設計方法，并通過現場試驗和數值模擬驗證方法的有效性；3. 通過應用熱荷載設計方法，研究結構的熱效應及其影響因素。

1. 單層球面網殼溫度場具有強時變性和非均勻性特征，不均勻程度取決于熱邊界條件，分布模式取決于太陽與結構之間的位置關系；2. 運用理論推導完成溫度場的簡化計算，實現非均勻熱荷載的設計，并通過實驗和模擬對方法的有效性進行驗證；3. 單層球面網殼的非均勻熱效應顯著強于均勻熱效應，施工合攏時間是決定結構熱效應的關鍵因素，合理安排合攏時間可以有效降低結構的熱應力和變形。

非均勻溫度場；非均勻熱荷載；非均勻熱效應；單層球面網殼；時變環境因素

https://doi.org/10.1631/jzus.A2300143

https://doi.org/10.1631/jzus.A2300143

Yanbin SHEN, https://orcid.org/0000-0002-2148-9138

Received Mar. 18, 2023;

Revision accepted June 23, 2023;

Crosschecked Jan. 18, 2024

? Zhejiang University Press 2024