J. Cent. South Univ. Technol. (2008) 15(s1): 342-345
DOI: 10.1007/s11771-008-377-0
Elastoplastic analysis of thin-walled structures in reservoir area
DUAN Shao-wei(段绍伟)1, LUO Ying-she(罗迎社)1, ZHU Yu-xiong(朱育雄)2
(1. College of Civil Architectural Engineering and Mechanis, Central South University of Forestry and Technology,
Changsha 410004, China;
2. Transportation Bureau of Xiangtan County, Xiangtan 411200, China)
Abstract: In the structural design of the high pier, in order to analyze the strength and structure stability, the pier was often considered a thin-walled structure. Elastoplastic incremental theory was used to establish the model of elastoplastic stability of high pier. By considering the combined action of pile, soil and pier together, the destabilization bearing capacity was calculated by using 3-D finite element method (3-D FEM) for piers with different pile and section height. Meanwhile, the equivalent stress in different sections of pier was computed and the processor of destabilization was discussed. When the pier is lower, the bearing capacity under mutual effect of pile, soil and pier is less than the situation when mutual effect is not considered; when the pier is higher, their differences are not conspicuous. Along with the increase of the cross-sectional height, the direction of destabilization bearing capacity is varied and the ultimate capacity is buildup. The results of a stability analysis example are almost identical with the practice.
Key words: high pier; thin-walled structure; increment theory; elastoplastic finite element analysis; stability
1 Introduction
A hydropower station locates in the middle and lower reaches of Qing-Shui River of Jinping country, Guizhou province, with the normal pool level being 475 m and installed capacity 1000 MW. The water storage area covers to 19 towns of four counties, namely, Jinping, Liping, Jianhe and Taijiang. Because the original county- level highways were overwhelmed by the impoundment, around, 187 km third-class and forth-class roads should be rebuilt, includeing, 20 super bridges. Zhanjing rebuilding project in this area is a part of the bridge design. There are 12 bridges in all. The bridges situate in the erosion valley, have a sharp gradient and asymmetry terrain at the two sides. Their piers change a lot in height, and range from 34 to 88 m. Because the piers are relatively higher, we should take not only strength and stiffness into account, but also the stability problem in the course of bridge analysis[1-3]. At present, the plane elastic method is often utilized, which is of great difference from the actual situation in the analysis of the stability of high pier. Even if elastoplastic method is adopted, the mutual effects of the pile, soil and pier are not considered[4-5]. In this work, a stability model of high pier was constituted by using elastoplastic increment theory. 3-D finite element method was used by considering the mutual effects of the pile, soil and pier, and geometry size impact on the pier stability was researched.
2 Foundation of piers elastoplastic equation
Assume the pier yield function as follows:
(1)
where σij is a stress state and K is a sclerosis function. when the materials reach the buckling state, the strain increment consists of elastic increment and plastic increment[6-8]. We write them in the form:
(2)
In Eqn.(2), the elastic strain increment and stress increment still obey the Hook law, namely
(3)
Because the plastic deformation is changeable, according to the associated flow rule, the plastic strain should be presented as follows:
(4)
where λ is instantaneous proportion of non-negative factor. Associating Eqn.(3) and Eqn.(4), Eqn.(2) can be rewritten as
(5)
By doing total differential with Eqn.(5),we conclude
(6)
Eqn.(6) can be rewritten as
(7)
where
，
(8)
Using matrix [De] to multiply Eqn.(5), so
(9)
Substitute Eqn.(8)into Eqn.(9), then
(10)
where
(11)
Eqn.(11) is the incremental form of elastoplastic matrix, while the elastoplastic finite elemental equation of incremental form is
(12)
where
(13)
Thus, we can take nonlinear stability analysis on the high pier by Eqn.(12).
3 3-D finite element elastoplastic model
In Zhanjing line there are 12 super bridges, 6 of which are continuous rigid frame bridges. The pier is double thin-walled reinforced concrete structure. The pier height ranges from 34.0 to 88.0 m and the based platform thickness is from 2.0 to 2.5 m. The foundation is man-made hang-dug pile. The pile diameter is 1.5 to 2.0 m and the pile length is 15.0 to 30.0 m.
The pier simplification computation model of Gubenxi No 2 bridge is shown in Fig.1. The height of the pile is 34.0 m, the platform thickness is 2.0 m, the dug pile diameter is 1.5 m and the length of pile is 19.0 m. The west of number 2 pier is riverbed and the east is valley with a slope of about is 45?. For the convenience and accuracy of calculation, we selected the pile calculation scope starts from the centre of the platform, the radius is 25.0 m, the thickness is 25.0 m with the east inclination angle being 40?.
We adopted Duncan-Zhang model to imitate the soil
Fig.1 Simplified computation model of pier
body constructive relationship[9-11]. As to the mutual effect of soil and pile mutual, we imitated their dislocation movement by using contact element with thickness. The entire zone was divided by 8 node hexahedron units and the 3-D infinite model is shown in Fig.2.
Fig.2 3-D infinite model of pier
4 Analysis of elastoplastic stability
4.1 Influence of pier height
To analyze the impact of the pier height on the elastoplastic stability, we took the pier section size of Gubenxi No 2 bridge as the pier section size, the width is 5.5 m and the height is 4.1 m, the pier height is in the scope of 34.0 to 88.0 m. To imitate the force situation of the high pier, the load which was exerted to the middle point of the pier top section along the direction of its width was eccentric pressure. Using the models in Fig.2, the limited strength of pier under different height of the pier was calculated by finite element method and the calculation result is shown in Fig.3. From Fig.3 it can be
Fig.3 Influence of pier height on limited strength
seen that, with the increase of the pier height, the limited strength sharply reduces. The highest limited strength of Mantianxing Bridge is only 56% of Gubenxi Bridge. Especially when the pier height reaches 70 m, the limited strength reduces sharply. Thus, in the design of the pier, the disadvantages should be fully considered due to the pier height. What is more, in order to compare the pile influence on the stability of pier, the limited strengths with and without the pile-soil effects were calculated respectively. The results indicate that when the pier height is low, the ultimate bearing capacity under the pile-soil effect is less than that without the pile-soil effect. But there is only a little different when the pier height is relatively higher. This main reason is that the soil is elastoplastic body and it will produce elastoplastic deformation, reducing the resistance deformation capacity of the high pier. When the pier height is higher, the main factor on the limited strength is not the soil but the pier height.
4.2 Influence of section height
Assume the pier height is 34.0 m, the section height is 3.1-to 5.6 m and width of the section is 5.5 m. The limit strength of the pier under every situation was calculated as shown in Fig.4.
From Fig.4 it can be seen that, with the increase of the section height, the limited strength of pier increases gradually. This illustrates that the section size has a great influence on the stability of pier height. When the section height is close to its width, the limited strength approaches the stable value 19.6 MN. According to the theory of stability, when the height of section reaches or exceeds its width, it will change the destabilization direction. So in a certain scope, the increase of the section height does not produce the increment of the limited strength. Therefore, in the design of the pier, in order to enhance the stability of the bridge pier, any excessive increase in section height will not yield better
Fig.4 Influence of section height on limited strength
results; instead it will lead to wasting materials and increasing cost. When the soil effect is neglected, the limited strength of pier is apparently larger. Thus in the calculation of pier, the pier bottom cannot be taken as the fixed but should be fully consider the soil effects.
4.3 Variation of equivalent stress
There are a lot of factors that affect the stability of the pier, and different elements result in different results. So it is complicated to study the mechanism of the pier. Taking the second pier of Gubenxi Bridge for an example, its height is 34.0 m and the width of section is 5.5 m, height is 4.1 m. Using finite element method, the equivalent stress of different sections of pier was gotten, the result is shown in Fig.5. According to Fig.5, for the identical section, from draw area to suppress area, the equivalent stress of each point in the section increases gradually.
Fig.5 Variation of equivalent stress
Divided by the middle of the section, it is 2.2 m away from the edge, the variation of the equivalent stress is little when the width of the sector is below 2.2 m; while it is over 2.2 m, the equivalent stress increases rapidly. The maximal of the equivalent stress appears at 12.0 m away from the bottom of the pier, and then varies along the directions of the bottom and the height of the pier. As a result, the destabilization of the pier begins from 12.0 m away from the bottom of the pier, rather than the middle of the bridge. Once it achieves at elastic plasticity deformation, it will develop to the two ends of the pier.
Since the stability of the high pier is complicated, in this work, only the effect of the geometry dimension on the pier’s stability was studied. Furthermore, different materials like concretes with different strengths have a large effect on the stability. Currently, the stability of the pier is only researched in terms of elasticity and elastoplastic. However, the pier is not a single structure, composes the main body of the pier, the platform and soil rock, etc. Under the effect of the high stress, various materials like soil rock may undergo plastic deformation. What is more, the pier structure may have to bear hydrodynamic pressure and earthquake load, and the stability of the pier under the effect of dynamic loading is a worthwhile problem.
5 Conclusions
1) When the pier is higher, the analysis on the strength and rigidity should not only be considered, but also the buckling problem of the pier in design.
2) The calculated model of the pier stability is a space model. As a result of the material nonlinear, the common functions of the platform, the soil and the pier should be considered. At the same time, finite element model should be used.
3) When the section is identical, with the increase of the pier height, the limited carrying capacity of the pier decreases. Especially when the pier height is over 70.0 m, the limited carrying capacity decreases rapidly.
4) When the pier height is constant, with the increase of the section height, the limited carrying capacity of pier increases gradually. And when the height of the section is close to the width of the section, the carrying capacity tends to a constant.
5) Along with the transition from the draw area to the suppress area, the equivalent stress of different points on the section increases gradually. The maximum of the section’s equivalent stress appears at the place where is at 12.0 m from the bottom and the height of the pier, it varies along the direction of the bottom pier and the height of the pier. As a result, the pier destabilization does not begin at the middle of the pier, but at the bottom of the pier, that is 12.0 m.
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(Edited by YANG Hua)
Foundation item: Project(06JJ5080) supported by the Hunan Natural Science Foundation of China; Project(05026B) supported by the Young Science Foundation of Central South University of Forestry and Technology
Received date: 2008-06-25; Accepted date: 2008-08-05
Corresponding author: DUAN Shao-wei, PhD, Professor; Tel: +86-732-8290637; E-mail: duanshaowei8@sohu.com
