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## 2017IABSESYMPOSIUM_ConferencePaper (2017-09-16 13:33:13)
 分类： 论文专利

Ultimate load of cylindrically curved panels under uniform compression at straight edge and the influence of curvature

Hui Guo1, 2, Zhibin Zeng1, 2, Xiaoguang Liu1, 2, Xiaochen Ju1, 2, Xinxin Zhao1, 2

1Railway Engineering Research Institute, China Academy of Railway Sciences, Beijing, China

2State Key Laboratory for Track Technology of High-speed Railway, Beijing, China

Contact: superhugo@163.com

# Abstract

Curved panels are widely used in modern novel steel bridges and buildings. Stability shall be considered for these structures under compressive loads. While it still has many challenges on how to consider stability since there are no guidelines in specifications. In this paper, ultimate load of the cylindrically curved panels were studied. First, accuracy of numerical model was verified by theory and model test. Then the influence of curvature on ultimate load at the straight edge of curved panel was studied. Results showed that the deformation of curved panel was obvious in the direction perpendicular to the panel. The position of maximum deformation was located near the central region and perpendicular to the curved panel. Ultimate load of curved panel is smaller than the straight one. And the ultimate load decreases as the curvature increases with a linear relationship between the reduction factor of local stability and the curvature.

Keywords: curved panel; stability; steel pylon of cable-stayed bridge; elasto-plastic buckling; full-scale model test; ultimate load; curvature; reduction factor; empirical formula.

# 1        Introduction

Curved panels have been used widely in steel bridges and buildings, etc. Stability design of those structures often encounters difficulty by a lack of specifications. Research on stability of curved panels mainly deals with the axial or external compression of cylindrically curved panels or shells, as shown in Figure 1a and 1b. Tran et al. carried out series of numerical simulations to study the characteristics of curved panels’ elasto-plastic behavior and derived the semi-empirical formulae for predicting the elastic buckling and ultimate strength1. It appeared that the behavior of cylindrically curved panels usually depended on its curvature, its slenderness and its imperfection1. Martins et al. also studied the ultimate load of simply supported cylindrically curved panels subjected to pure compression and in-plane bending by numerical method. Results showed that the ultimate behavior of cylindrically curved panels was similar to flat plates (similar curves) 2. Xu et al. derived approximate solutions by shallow shell theory and Galerkin method for buckling stress of curved web plate of a steel bridge under wheel compression. Solutions of straight plate as the special case were then proved to be reasonable compared to the calculated value from Japan’s structural stability handbook3. Research group from Institute of Mechanics, China Academy of Sciences had carried out theoretical study and test on buckling of stiffened cylindrical shells under external pressure. Model tests for stiffened cylindrical shells subjected to external pressure by gas cell were conducted and it revealed that the ultimate load of test was accord with the calculated value from membrane theory under the boundary condition of simply support at the straight edge with longitudinal restrictions and sliding simply support at curved edge. Test results also influenced by additional bending moment at the loading end, initial imperfections and movement of boundary at the straight edge4, 5.

(a) Cylindrically curved panel under uniform axial compression

(b) Cylindrically curved panel under uniform external compression

(c) Cylindrically curved panel under uniform compression at straight edge

Figure 1. Cylindrically curved panel under uniform compression in different directions

Different from typical buckling cases of curved panels as shown in Figure 1a and 1b, buckling problems of Figure 1c was another circumstance encountered in actual structural design such as curved pylon, tower or arch rib. Such a curved panel has the characteristic that the direction of load changes with the deformation of the panel. Hence theoretical solutions of ultimate load would be impractical for such a case. And relevant research was also limited and scarce.

In this paper, full-scale model test and non-linear numerical method were adopted to deal with the buckling problems of curved panel with compression at the straight edge (Figure 1c). Influence of curvature on ultimate load was also studied to obtain the relationship between curvature and reduction factor of local stability.

# 2        Design case of curved pylon

Beijing Yongding River Bridge is a municipal bridge which is now under construction. It is the key project of Chang’an Avenue West extension line over Yongding River linking Gucheng Avenue and Sanshi Road. The main bridge is a combined cable-stayed-rigid-frame system bridge with two curved pylons. Towards the direction of Tiananmen Square, arrangement of the left span of the main bridge is (50+133+280+120+56) m and the right span is (50+158.1+280+94.9+56) m. Two individual steel box beams with maximum height of 10m are connected by transverse girder. The two pylons have different heights which the higher pylon with 120m is fixed with pier and beam and the lower pylon with 73m is fixed with the beam and separated with the pier. And a bearing moveable in single direction is arranged under the pier at the lower pylon. Aesthetic effects are considered during the design of two pylons. The pylon is a composite structure which the lower part is reinforced concrete structure and the upper part is steel structure. And it has a special arch with spatial surface and inclined opposite to the river course. The inclined angle of left part is 60 degree and 78 degree for higher and lower pylon respectively and the right part is 80 degree and 63 degree. Such a novel design scheme has a special meaning, named‘gate of force’, as shown in Figure 2.

Structural stability was a key challenge during the bridge design. So series of study were carried out including overall stability of the whole bridge, web plate stability of steel box girder, overall stability and local stability of curve pylon, etc.

Figure 2. Rendering of Beijing Yongding River Bridge

Detailed design of the steel curved pylon (the higher one) was shown in Figure 3. It only showed the local inner layout of steel structure near the pylon top which had the largest curvature. And the local layout showed that a number of stiffeners were arranged in the longitudinal and transverse direction of the curve pylon to satisfy the requirements of strength, stability and structural details. Noticeably, the curved panel surrounded by long stiffeners and diaphragms was under the compression at the straight edge like Figure 1c and its stability shall be considered carefully due to lack of specifications and experience. In Figure3, the distance between two diaphragms connecting the outmost elliptical-shape panels was 1300mm and 1350mm respectively.  And the transverse width of plate element between two longitudinal stiffeners was 700mm. Clearly, the axial compression in the vertical direction of the pylon accounted for a major part of the total inner force. So it’s necessary to carry out detailed study to have a better understanding on the local stability and ultimate load of the curved panels. Since the curvature changed along the curved pylon, it’s also important to obtain the relationship between the ultimate load or the reduction factor of local stability and curvature. In this paper, several methods were adopted to get accurate results including theoretical study, model test and numerical simulation. First, the accuracy of numerical model was verified by model test and theoretical analysis for a straight plate as a special case. Then ultimate load of curved panels was studied and compared to the straight plate by model test and numerical simulation.

Figure 3. Local structural design of curved plate at the top of curved pylon (higher pylon)

# 3        Buckling of cylindrically curved panels under uniform compression at straight edge

3.1        Validation of numerical model

Validation of numerical model was proposed to check the accuracy of elastic buckling and elasto-plastic buckling. The straight plate (S1) with length L=1350mm, width B=700mm and thickness t=32mm was first chosen to be compared to the curved pylon’s plate element. Element SOLID185 in ANSYS was adopted in modelling. Boundary conditions of the solid element model were set to be simply supported at the mid-line of four edges of the plate. It was a widely accepted hypothesis for panel stiffened by diaphragms and stiffeners. The results of elastic buckling were shown in Figure 4 and Table 1. Under the uniformly compressive load at the side of width and the condition of simply support, the number of half wave of first five modes was in sequence of 2, 3, 1, 4, and 5. Such results were in consistent with the theory. From Table 1, results showed that the critical stress of elastic buckling obtained by numerical model was accurate enough compared to the theoretical value. The maximum relative error was 3.6% for first five modes.

(a) Mode1                                    (b) Mode2

(c) Mode3                                    (d) Mode4

(e) Mode5

Figure 4. First five modes of elastic buckling

Table 1. Comparison of critical stress of elastic buckling obtained by numerical model and theory Accuracy of numerical model for elasto-plastic buckling load was also validated by theoretical analysis and model test. First, approximate theoretical solution for elasto-plastic buckling stress of straight rectangular plate (S1) under simply support at four sides was as follows.

Where k was the elasto-plastic buckling coefficient which could be expressed by  among which  was the tangent modulus, and  was the modulus of elasticity. E equalled to 2.1e5 MPa and  equalled to 0.03 according to the linear hardening constitutive model. So k had the value of 0.693. D was the flexural rigidity of plate’s unit width. And t equalled to 32mm, b had the value of 700mm, and  was the Poisson’s ratio 0.5. Thus the value of  was 333.514MPa. So the elasto-plastic buckling load at the width for plate S1 was 333.514×0.7×0.032=7470.7kN.

Figure 5. Verification of numerical model for elasto-plastic buckling by model test

## 3.2        Model test of curved panels

(a) Curved plate S2 at the outmost panels with smaller radius of curvature of the pylon

(b) Curved plate S3 at the outmost panels with larger radius of curvature of the pylon

Figure 6. Curved panels for ultimate load test

Figure 7. Buckling test equipment for curved panels with simply support at four edges

Static load test was first carried out repeatedly for 6 times with the structural response in the curved panel’s elastic deformation range to guarantee the reliability of the test results. After the static load test, the panel was loaded to its carrying capacity. Before the test, several measuring points for strain and displacement were arranged on the panel. The measuring point for longitudinal displacement was at the middle position of the loading edge and the vertical displacement was at the lower side of the central section. Results of measuring displacement were shown in Figure 8. It showed that the load-displacement curve had undergone a complete process from elastic stage to elasto-plastic stage and finally the ultimate load stage. No obvious descending stage appeared in the displacement curves.

(a) Longitudinal and vertical displacement of curved plate S2

Figure 8. Displacement of curved panels obtained by model test

For curved panel S2, the maximum value of vertical displacement was about 10mm which was twice the value of displacement in the longitudinal direction as shown in Figure 8a. And it was nearly the same of the longitudinal and vertical displacement for curved panel S3 since it had a larger radius of curvature. Compared to the longitudinal displacement of the straight panel S1 shown in Figure 5, it could be concluded that the displacement increased much obviously from 3mm to 9.5mm in the elasto-plastic stage than curved panel S2 and S3. The vertical displacement of S1 at the central section was also measured which was much smaller than curved panels with the maximum value of 0.4mm. The ultimate load of curved panel S2 and S3 was 7090kN and 7330kN respectively which decreased by 15.3% and 12.4% respectively compared to the straight panel S1.

Numerical results of elastic-plastic buckling of curved panels S2 and S3 were shown in Figure 9. From Figure 9, ultimate load of S2 panel was 6260kN which was less than the test value of 7090kN by 11.7%. Ultimate load of S3 panel was 6840kN which was also less than the test value of 7330kN by 6.7%. Compared to the numerical result of straight panel S1 in Figure 5, the ultimate loads of panels S2 and S3 had a decrease by 25.2% and 18.3% respectively. Total displacement (USUM) distribution of curved panels S2 and S3 under the ultimate load was given in Figure 10. From Figure 10, it was evident that the maximum displacement was near the central region which was perpendicular to the plate surface.

Figure 9. Comparison of load-displacement curves between model test and numerical model

(a) Load-displacement curve of S2 (numerical)

(b) Load-displacement curve of S3 (numerical)

Figure 10. Total displacement (USUM) distribution of curved panels under the ultimate load

## 3.3        Influence of curvature on ultimate load

From results of model test and numerical simulation, it was certainly confirmed that the curvature had obvious influence on the ultimate load of curved panel. The scope of the curvature was [0.0018, 0.2353] and [0.0046, 0.1479] respectively for the higher and lower curved pylon of Yongding River Bridge obtained from spatial axis equation. Then parametric study was developed to investigate the influence of curvature on ultimate load. Considering the variation scope of curvature from 0.1 to 0.33 with each step of 0.01, 24 groups of curved panels with the same dimension like S3 except the curvature were selected and analyzed. Results were shown in Figure 11. Obviously, the ultimate load of curved panel decreased with increase of curvature.

Figure 11. Influence of curvature on load-longitudinal displacement curve and ultimate load

Reduction factor of local stability 𝜌 of straight plate or sub-panel under axially compressive load was given in specifications to consider the influence of local buckling on ultimate load. It had been proved that such a factor was related to width-to-thickness ratio6, 7, 8 Since the curvature had influence on local buckling, it’s necessary to study the relationship between the factor 𝜌 and the curvature k for curved panels. Result was obtained and given in Figure 12. Conclusions could be made that reduction factor 𝜌 decreased as the curvature increased. It conformed to a linear correlation. Correlation between reduction factor ρ and curvature k of the sub-panel of the curved pylon of Yongding River Bridge could be expressed as follows.

Where  R denoted the coefficient of correlation. According to the variation of curvature, the scope of reduction coefficient of curved panel for the higher and lower curved pylon was [0.785, 1] and [0.865, 0.998] respectively.

Figure 12. Relationship between the reduction factor of local stability 𝜌 and curvature k

# 4        Conclusions

Model test and numerical study were adopted to investigate the stability of cylindrically curved panels under the compression at the straight edge. Conclusions were thus obtained as follows.

(1) Ultimate load of curved panel was smaller than the straight panel with the same dimension. And the load-displacement curves obtained from model test and numerical simulation were in good accordance.

(2) The position of maximum displacement was related to the magnitude of curvature. For the straight panel, the maximum value was at the loading edge in the longitudinal direction. While the position would change to the central region of plate and the direction of maximum displacement was perpendicular to the curved surface of the plate with the increase of curvature.

(3) Ultimate load of curved panels decreased steadily as curvature increased. And the reduction factor had a linear correlation with the curvature.

# 5        Acknowledgements

The authors would like to express warmly thanks to the support of designers from Beijing General Municipal Engineering Design & Research Institute Co., Ltd.

# 6        References

        Tran K.L., Davaine L., Douthe C., et al. Stability of curved panels under uniform axial compression. Journal of Constructional Steel Research.2012; 69(1):30-38.

         Martins J. P., Silva L. S. D., Reis A. Ultimate load of cylindrically curved panels under in-plane compression and bending—Extension of rules from EN 1993-1-5. Thin-Walled Structures. 2014; 77(4): 36-47.

         Xu C.Huang J. Y. Elastic buckling of curved web plate of steel girder under wheel compression. Proceedings of the third national conference on urban bridges of the Municipal Engineering Society, China Civil Engineering Society; Shanghai, China, 1991.p. 472-477.

         Plate and shell research group, Institute of Mechanics, CAS. Influence of deformation before buckling on the stability of stiffened cylindrically curved plate under external compression (linear consistency theory). Mechanical Bulletin. 1976; (3): 30-37.

         Plate and shell research group, Institute of Mechanics, CAS. Model test of stability of circularly-stiffened cylindrically curved plate under external compression. Mechanical Bulletin. 1976; (3): 37-42.

         Ministry of Transport of the People’s Republic of China. Specifications for design of highway steel bridge (JTG D64-2015). Beijing:  China Communications Press, 2015; p. 14-28.

         Zhao Q., Gao P. Comparison of calculation method of stiffened compression plate’s stability bearing capacity. Journal of Fuzhou University (Natural Science Edition). 2014; 42(1): 123-127.

         Chen J. Stability of steel structures: Theory and design. Beijing: Science Press, 2006; p. 395-511.

EI检索号：20182905575279。

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