Figure
1.
Layout of the Longtanhe Great Bridge.
| Citation: | Li Li, Yuancheng Peng, Xiaohong Long, Ping Liao. Nonlinear Stability Analysis of Long-Span Continuous Rigid Frame Bridge with Thin-Wall High Piers. Journal of Earth Science, 2005, 16(1): 72-78.
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By utilizing the current finite element program ANSYS, two types of finite element models (FEM), the beam model (BM) and shell model (SM), are established for the nonlinear stability analysis of a practical rigid frame bridge—Longtanhe Great Bridge. In these analyses, geometrical and material nonlinearities are simultaneously taken into account. For geometrical nonlinearity, updated Lagrangian formulations are adopted to derive the tangent stiffness matrix. In order to simulate the nonlinear behavior of the plastic hinge of the piers, the multi lines spring element COMBIN39 is used in the SM while the bilinear rotational spring element COMBIN40 is employed in the BM. Numerical calculations show that satisfying results can be obtained in the stability analysis of the bridge when the double coupling nonlinearity effects are considered. In addition, the conclusion is significant for practical engineering.
With the current development of the west of China, new speedways are being built. The mountainous areas of the west have slowed these projects, because of the high, deep and steep landform features. A special bridge shape, the long-span continuous rigid frame bridge with thin-wall high piers, is being widely adopted in these projects. In the past, a linear buckling analysis on the continuous rigid frame bridge has often been carried out (Li, 1996), but this method does not consider the effects of the geometrically large deformation induced by the initial deflection and the material nonlinearity at some local locations. Some scholars tried to solve the ultimate load using the divided layer beam element (Pan et al., 2000). Although this method can obtain a satisfying result, there exist some localizations to an extent because the code must be programmed manually. A nonlinear stability analysis is carried out using the excellent current FEM software ANSYS, taking Longtanhe Great Bridge in Hubei Province as an example. The geometrical nonlinearity effect is considered by choosing elements with a large deformation function, and the corresponding items are set, while the material nonlinearity effect is taken into account by using the nonlinear spring element at the bottom of high piers.
Longtanhe Great Bridge, built on the Hu-Rong speedway (connecting Chengdu with Shanghai) in the west of Hubei, is a long-span continuous rigid frame bridge. The plane layout of the bridge is shown in Fig. 1, and the double-limb thin-wall hollow piers are adopted from pier ⑤ to pier ⑧, the corresponding heights are 70 m, 179 m, 170 m, and 116 m, respectively. Pier ⑥ is 179 m, which is the highest of recently completed, similar bridges in China and overseas. The main bridge spans are 106+3×200+106 m, using a cantilever construction technique consisting of C50 pre-stressed reinforced concrete (RC) box girder sections. The section height changes according to the 1.8 cubic parabola, it is 4 m mid-span while the end is 12 m. C40 RC is adopted for the lower piers. In addition, pier sections are amplified according to the scale 1∶100 in the longitudinal direction of the bridge and the scales of the transverse direction are 1∶100, 1∶60 and 1∶40, respectively. There is a short connected beam every 60 m along the length of the piers ⑥, ⑦, ⑧. Because the bridge is provided with thin-wall and high features, the stability analysis will play an important role in the design and calculation.
A beam element is often adopted in the structural analysis because the transmitting force path is explicit and establishing a model is simple, while the supporting force capability and buckling modes of the thin-wall structures can be perfectly simulated for the shell element. In order to investigate the buckling capability of the bridge the 3-D finite element analysis models are established by utilizing the beam and shell elements, respectively.
In the beam model (BM), two nodes of the 3-D beam element BEAM188, based on the Timoshenko beam theory, are employed for the simulations of the variational section girder and piers. Element BEAM188 has six or seven degrees of freedom (DOFs) at each node and the sheared deformation effects and warping degree of freedom are included. The model boundary conditions are dealt with as follows: all double-limb piers are fixed at the base and a rigid connection is formed between the piers and the main girder. Additionally, only the main bridge part (from pier ④ to pier ⑧) is considered in the structure buckling analysis. The main girder ends are assumed to be hinge links in the vertical and transverse directions of the bridge. Finally, the finite element analysis (FEA) model contains 2 498 BEAM188 elements and the total DOFs are 27 492.
In the shell model (SM), the element SHELL63 has both bending and membrane capabilities. Both in-plane and normal loads are permitted. The element has six DOFs at each node: translations in the nodal x-, y- and z directions and rotations about the nodal x, y, and z-axes. Stress stiffening and large deflection capabilities are included. A consistent tangent stiffness matrix option is available for use in large deflection (finite rotation) analyses. The appointment of the boundary conditions is the same as the above beam model. After the structure is meshed the analysis model contains 48 201 SHELL63 elements and 289 206 DOFs in all.
Structure dynamic characteristic analysis is a method to check the FEA model, thus the modal comparison analyses between the BM and SM are conducted, and two cases are analyzed: the maximum cantilever construction stage and completed bridge stage. The structural fundamental frequencies of the two models for the construction stage are 0.158 4 Hz (BM) and 0.151 3 Hz (SM), and 0.175 1 Hz (BM) and 0.166 8 Hz (SM) for the completed stage. Furthermore, the first modes of the two models under the two different cases are all the transverse bending of the high piers. From the above modal analyses it can be shown that the dynamic characteristics of the two models are basically identical and are in accord with practical engineering design values.
The updated Lagrangian (UL) method is employed, the equilibrium equation is expressed as equation (1) (He and Lin, 1994), the left subscript t in the matrixes indicates that all the structure configurations in this format are appointed to the reference configuration at t. Moreover, in this paper the geometrical and material nonlinearities are simultaneously included in the structure double coupling nonlinearity analysis.
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(1) |
where [K]0 is the elastic stiffness matrix induced by small deformation [K]0=∫v[B0]T[D]ep[B0]dv; [Kσ]d{δ}=∫v[Bn]{σ}dv; [K]σ is the initial stress stiffness matrix (geometry stiffness matrix); [K]L is the elasto-plastic stiffness matrix, [K]L=∫v[B0]T[D]ep[Bn]+[Bn]T[D]ep[B0]+[Bn]T[D]ep[Bn]dv.
If the non-zero solution exists in equation (1), the following buckling distinguishing rule must be met, i.e., [K]T=0. In this paper the structure deformation and supporting force capability are investigated by increasing the load increment step by step. As the load increases, the bottom sections of the high piers will firstly begin to yield under the large deformation, so the structure material nonlinearity can be considered by only setting the plastic hinge (plastic zone) at the bottom sections of the high piers. The structure stiffness matrix continually changes under continued loading until the structure tangent stiffness matrix is close to the singularity because of the compressive stress induced by the external load. At that time the structure supporting force will reach its limits and lose its state of stability equilibrium. Finally, the load-deformation curves can be obtained after the whole process of stability analysis.
In the BM, the structure material nonlinearity is considered by setting plastic hinges at the bottom of the high piers, while the other parts are assumed to be elastic. The plastic hinge is simulated by the bilinear rotational spring element COMBIN40 (Long and Li, 2004).
According to the demand for the bilinear hysteretic loop model due to related parameters (see Fig. 2), the moment-curvature relationship must be defined for the pier bottom sections, i.e., the pre-yielding stiffness (K1+K2), the post-yielding stiffness K2 and yielding moment My.
When bottom sections of a pier yield, the section compressive height xy and yielding moment My are obtained by establishing the force and moment equilibrium equations according to the constitutive relationship of the reinforcing steel bar and concrete material, and the formulae (zhang, 2003) are defined as follows.
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(2) |
When xy > 0.5h0, then
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(3) |
where fcmk, fykand f'yk are concrete compressive strength standard value and longitudinal reinforcing steel bars yielding strength standard value; b and h are the section width and height; As and A's are the longitudinal reinforcing steel bar total areas of the pier section, and as and a's are the distances from the single side reinforcing steel bar to the close side of the pier section, respectively. The sum of h0 and as (a's) is section height h.
Therefore, the pre-yielding stiffness and post-yielding stiffness in the bilinear model are expressed as
Pre-yielding stiffness
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(4) |
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(5) |
In the SM, multi-lines spring element COMBIN39 is adopted to simulate the plastic zone of the pier bottom section, that is, the structure material nonlinearity is considered by using the multi-spring model (Li and Kubo, 1998; Li and Otani, 1993), but the other parts stay elastic. Furthermore, this model has been well validated for simulating the RC load experiment. The axial springs are set in the end sections in order to simulate the reinforcing steel bar and concrete stiffness in the multi-spring model. That is, each reinforcing steel bar represents one steel springKs, while enough concrete elements are meshed in the section concrete zone and each element is simplified as one concrete spring Kc. The corresponding material constitutive relationships of the reinforcing steel bar and concrete can be considered as follows, respectively.
(1) For the reinforcing steel bar, the bilinear isotropic strengthening stress-strain relationship is given as equation (6)
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(6) |
where εsis the yielding strain and setting εs=0.001 7.In equation (6), ET is the tangent elastic modulus after yielding and the value is 0.02E.
(2) For the concrete, the multi-lines model isemployed.The corresponding stress-strain constitutive relationship is shown in Fig. 2 and equation (7), respectively.
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(7) |
C40 concrete is adopted in the high piers. The related parameters are defined as follows: the peak value compressive strain εc= 0.002, ultimate compressive strain εu= 0.003 3, compressive strength σc=fc=23 N/mm2, σu= 0.2, fc= 4.6 N/mm2. And in the tensional aspect σc=ft=2.15 N/mm2, εt=-6.5×10-5, εt0=-1.2×10-4.
Finally, structure nonlinear finite element analysis comes down to the solution of a group of nonlinear equations. The methods of the solution are so many that we must make a rational choice according to some factors of the analysis problem, i.e., the nonlinear degree and convergence characteristics, and so on. In this paper, the arc-length method is employed in the BM and SM. This method can make the NR equilibrium iterative converge along an arc segment. Even though the slope of the structure tangent stiffness matrix is zero or a negative value the convergence condition can also be satisfied.
In the second kind of stability analysis on the Longtanhe Great Bridge, the initial deflection is only considered as the deformation induced by unilateral sunlight action. Because the fundamental buckling mode in the first stability analysis (eigenvalue buckling) is a transverse bending of the high piers, the initial deflection is very disadvantageous for the nonlinear stability analysis of the bridge. Furthermore, the deflection values at the construction stage and completed bridge stage are solved through the thermal-structure coupling field analysis.
During the current nonlinear stability analysis, the following two kinds of estimation rules are adopted to confirm the ultimate load. Rule 1:the sudden reduction point of the variation rate on the load-deformation curve is defined as the ultimate load (zhang, 2003). Rule 2:when edge fibre stresses of the structure component reach the yielding strength the corresponding load is defined as the ultimate load (tang, 1989).
Because rule 2 is fit for the linear elastic material and the local material nonlinear behavior of high piers is considered in this paper, rule 1 is adopted to confirm the ultimate load so as to compare the results of the geometrical nonlinearity with that of the double coupling nonlinearity.
According to the concept that the second kind of stability is equal to the loss of the supporting capacity, the stability and the ultimate supporting capacity are consistent during the bridge design by the ultimate state method (Li, 2000), so the stability and strength safety factor should also be current. For the calculation of the structure bearing capacity ultimate state according to the Code for Design of Highway Reinforced Concrete and Prestressed Concrete Bridges and Culverts (JTJ023-85, China), the load factor is 1.2, the safety factor of the concrete and reinforcing steel bar is 1.25 and the structure working condition factor is 0.95 (Ministry of Communications of People's Republic of China, 2000), so the final structure safety factor K should be satisfied as follows
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(8) |
Besides, according to the admitted stress calculation method in the Specification for Design of Steel Structure and Timer Structure Highway Bridge and Culverts (TJT025-86, China), the safety factor K≥1.70. Therefore, the stability safety factor must be greater than both the above in the nonlinear stability analysis.
The BM and SM in the maximum cantilever construction state are established for the given 179 m high pier and the initial deflections induced by the unilateral sunlight thermal effect are applied. According to the design information of this bridge the most disadvantageous load distribution of this construction state is shown in Fig. 3 and the loading steps (Chen and Shou, 1994) are listed as follows. (1) Supposing that the main girder deadweight is not uniform, one end is 1.04 times of the main girder weight while the other is 0.96. (2) The dynamic factor of the hanging basket, casting block and construction machine at one end is 0.8 while the other end is 1.2, and the weight of the hanging basket W=1 050 N. (3) The last cantilever casting RC girder segment is not simultaneously finished, i.e., at one end, loads are zero but the other end is under construction. (4) Some construction tools and material must be placed on the girder for construction convenience, so the uniform load 8.5 kN/m and concentrated load 200 kN are applied at one cantilever end while the other is not considered. (5) The static transverse wind load in one cantilever box girder length range is considered according to the China Bridge Code JTJ021-89. (6) The static longitudinal wind load acting on the pier body is also taken from the China Bridge Code JTJ021-89.
The proportional loading gene is defined as the internal forces under the above load action and the initial deflection induced by unilateral sunlight is simultaneously applied to the piers. The force-displacement curves of the two models are thus obtained (Fig. 4), and the 'eigenvalue solution' represents the results of the linear buckling analysis.
The ultimate load and stability safety factor at construction stage are shown in Table 1.
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The corresponding conclusions can be drawn through the above analysis at the construction stage. For the BM the ultimate load of the geometrical nonlinear analysis is 3.82×108 N and the double coupling nonlinearity is 3.73×108 N, which is a reduction of about 51.8 % and 52.9 % respectively, compared with the eigenvalue load (7.92×108 N). Furthermore, the double coupling nonlinear result is 2.4 % lower than geometrical nonlinearity. For the SM the ultimate load of the geometrical nonlinear analysis is 3.79×108 N and the double coupling nonlinearity is 3.41×108 N, which is a reduction of about 53.3 % and 58.0 % respectively, compared with the eigenvalue load (8.11×108 N). The double coupling nonlinearity result is 10 % less than geometrical nonlinearity.
In the geometrical nonlinear stability analysis the SM calculation result (3.79×108 N) is 1 % less than the BM result (3.82×108 N), whereas the result of the double coupling nonlinear stability analysis is 9 % less. Figure 5 indicates that the force-displacement curves present a remarkable falling trend for the BM model but the SM does not when considering the material nonlinearity. Simultaneously, the difference between the double coupling and geometrical nonlinearity is within 10 %, so the effect of considering the material nonlinearity at the construction stage is not very evident.
The most disadvantageous load disposal is also considered for the nonlinear stability analysis at the completed bridge stage. In total, five load cases must be discussed but only one case is investigated in this paper because of length limitations. The case chosen is engendered maximum eccentricity and moment (unilateral full span load disposal) of the pier sections in the direction of the transverse bridge. The proportional loading gene is defined as internal forces under the above load action and the initial deflection induced by unilateral sunlight. The pier top force-displacement curves of the two models are shown in Fig. 5.
The ultimate load and stability safety factor in the completed bridge stage are shown in Table 2.
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The corresponding conclusions can be drawn through the above nonlinear stability analysis at the completed bridge stage. For the SM, the ultimate load of the geometrical nonlinear analysis is 5.30×108 N and the double coupling nonlinearity is 5.13×108 N, which is a reduction of about 41.8 % and 43.7 % respectively, compared with the eigenvalue load (9.11×108 N). Furthermore, the double coupling nonlinearity result is 2.4 % less than geometrical nonlinearity. But for the BM, the ultimate load of the geometrical nonlinear analysis is 6.06×108 N and the double coupling nonlinearity is 5.16×108 N, which is a reduction of about 38.2 % and 47.3 % respectively, compared with the eigenvalue load (9.80×108 N).The double coupling nonlinearity result is 10 % less than geometrical nonlinearity.
In the geometrical nonlinear stability analysis theSM calculation result (5.30×108 N) is 14 % less than the BM (6.06×108 N), whereas the result of the double coupling nonlinear stability analysis is 1 % less.
Figure 5 indicates that the force-displacement curves present an evident falling trend for the BM model but the SM does not when considering the material nonlinearity. Simultaneously, the difference between the double coupling and geometrical nonlinearities is within 17 %, so the effect of considering the material nonlinearity at the completed bridge stage stability is very evident.
Through the linear and nonlinear stability analyses of Longtanhe Great Bridge, and the calculations at the construction and completed bridge stages respectively, the following results can be obtained. (1) The nonlinear stability analyses of the two models are all less than the eigenvalue solution. The linear stability analysis is therefore not conservative and the result is also the upper limitation of the ultimate load. (2) The stability safety factor is greater than 1.58 according to the calculation result, and the ultimate load of the BM is greater than that of the SM from the force-displacement curves and tables. The buckling mechanism is summarized as the second rank effect (P-Δ) in the BM while the SM represents compressive debacle buckling. (3) The ultimate load capacity at the construction stage is less than that of the completed bridge stage. The maximum cantilever construction stage is therefore the key to controling bridge stability. (4) The double coupling nonlinear ultimate load is less than that of the geometrical nonlinearity, so it is very necessary to consider the material nonlinear effect in the nonlinear stability analysis.
| Chen, C. Y., Shou, N. C., 1994. Stability Analysis of Sanmenxia Yellow River Highway Bridge under Wind and Construction Load. Bridge Construction, 3: 30-35(in Chinese with English Abstract) |
| He, J. Y., Lin, X. D., 1994. Numerical Analysis about Non-linear Problem in Engineering Structure. National Defence & Industry Publishing House, Beijing (in Chinese) |
| Li, C. Q., 2000. Structure Stability and Stability Internal Force. People's Communications Publishing House, Beijing (in Chinese) |
| Li, G. H., 1996. Stability and Vibrations of Bridge Structures. China Railway Publishing House, Beijing (in Chinese) |
| Li, K. N., Kubo, T., 1998. Reviewing the Multi-spring Model and Fiber Model. Proceedings of the 10th Japan Earthquake Engineering Symposium, 2(2): 369-374 |
| Li, K. N., Otani, S., 1993. Multi-spring Model for 3-Dimentional Analysis of RC Members. International Journal of Structural Engineering and Mechanics, 1(1): 17-30 doi: 10.12989/sem.1993.1.1.017 |
| Long, X. H., Li, L., 2004. Study on Nonlinear Seismic Response and TMD Primary Control of the Cable-Stayed Bridge Section of the Third Macao-Taipa Bridge. Journal of China University of Geosciences, 15(4): 409-415 |
| Ministry of Communications of People's Republic of China, 2000. Specifications for Design of Reinforced Concrete and Prestressed Concrete Highway Bridges and Culverts (JTJ023-85). People's Communications Publishing House, Beijing(in Chinese) |
| Pan, J. Y., Zhang, G. Z., Cheng, Q. G., 2000. Geometrical and Material Nonlinear Analysis for Determining Ultimate Load Capacity of Long-Span Bridges. China Civil Engineering Journal, 33(1): 5-8(in Chinese with English Abstract) http://en.cnki.com.cn/Article_en/ http://search.cnki.net/down/default.aspx?filename=TMGC200001002&dbcode=CJFD&year=2000&dflag=pdfdown |
| Tang, J. X., 1989. Stability Theory of Structure. China Railway Publishing House, Beijing (in Chinese) |
| Zhang, X. P., 2003. Nonlinear Analysis of Earthquake Resisting Reinforce Concrete Structure. Science Press, Beijing (in Chinese) |
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Li Li, Yuancheng Peng, Xiaohong Long, Ping Liao. Nonlinear Stability Analysis of Long-Span Continuous Rigid Frame Bridge with Thin-Wall High Piers. Journal of Earth Science, 2005, 16(1): 72-78.
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