Elastic analysis of large urban sculptures in extreme environments

Large-scale urban sculptures are subject to varying degrees of impact in extreme environments, and failure to meet requirements can lead to serious consequences. Next, we will analyze the resilience of large-scale urban sculptures in some extreme environments.

Elasticity analysis under frequent earthquakes

Response spectrum analysis, modal decomposition, and response spectrum method are based on the mode shapes obtained from modal analysis. A typical CQC method is used for mode combination, where the directional combination is SRSS1). Internal force analysis analyzes the internal forces of the structure under seismic loading, mainly examining the ratio A of the inter-story shear force V of vertical members under the standard value of horizontal seismic loading to the representative value ∑G of gravity load, and the base reaction force under each response spectrum condition.

A represents the seismic shear coefficient, which concisely and clearly reflects the relationship between seismic internal forces and static internal forces of members. Based on the above basic parameters, the seismic influence coefficient curve for response spectrum analysis is determined, and then the seismic analysis is carried out using the modal decomposition response spectrum method, and the internal forces and displacements of members are verified respectively.

(1) Base reaction analysis According to the "Code for Seismic Design of Buildings", the total shear force of a small earthquake is a prerequisite for subsequent seismic calculation of the structure. First, by calculating the stress of the structure under dead load and various response spectrum conditions, the base reaction force is obtained. Under the seismic action in the X direction, the base shear force coefficient is 153/6433 = 0.024, and under the seismic action in the Y direction, the base shear force coefficient is 1030643 = 0.016. Both are not less than the lower limit of 0.016 specified by the code for the 2t gravity representative value "0", which can ensure the rationality of the model. However, the base shear force in the two directions differs by 20Q story shear force and 48.5% story shear force. The stiffness in the Y direction is weaker than that in the X direction. When optimizing the model, the structure needs to be adjusted and the stiffness in the Y direction should be appropriately increased.

(2) The following conclusions can be drawn from Figure 3.5 regarding the distribution of inter-story shear force: ① The seismic inter-story shear force calculated by the modal decomposition method for the 8th and 10th stories does not meet the requirements of the code and internal force adjustment is required to ensure the safety of the structure under seismic action. Generally speaking, the internal force adjustment is based on achieving the small seismic shear force coefficient. ② The inter-story shear force tends to decrease continuously with the increase of the number of stories, but it is not as obvious as that of regular structures. This is related to the complex spatial distribution of inter-story shear force (kN) in irregular structures.

2) Deformation analysis: Based on the regulations of various countries, earthquake damage experience, experimental research results and engineering case analysis, it is reasonable to use the inter-story drift angle as an indicator to measure the structural deformation capacity and thus determine whether it meets the building function requirements.

This paper summarizes the trends of story elastic displacement and inter-story drift angle as a function of story height, resulting from the standard values ​​of seismic action in two directions during the elastic stage.

in conclusion

① The seismic response spectrum effects in both principal axis directions are similar with the distribution of the structural floors. The lateral displacement at the top is relatively large. The ratio of the largest displacement value of 14.5 mm in the X direction to the total structural height of 50.2 m is 1/3453, and the ratio of the largest displacement value of 48.2 mm in the Y direction to the total structural height of 50.2 m is 1/1041. Both are less than the limit of 1/500, indicating that the lateral stiffness of the structure is good.

② The horizontal displacement of the structure is mainly bending deformation. The horizontal displacement increases faster with the increase of floors, indicating that the structure is rigid at the bottom and flexible at the top. In particular, the top is prone to large displacement and failure.

③ The larger inter-story drift angles all occurred at the top of the main frame, indicating that the stiffness at the top of the structure is too small. Whiplash effect failure should be prevented.

④ The inter-story drift angles in both the X and Y directions are less than the elastic inter-story drift angle limit of 1/250 for multi-story steel structures, indicating that the structure has not yet entered the plastic stage under frequent earthquakes and the overall stiffness of the structure is good.

Different time history analysis methods are used for different structures.

While these methods are reflected in the seismic codes of various countries, the base shear method and the modal response spectrum method remain the basic methods. For particularly irregular structures, time history analysis should be used as a supplementary analysis method.

The so-called supplementation here refers to comparing the base shear force, inter-story shear force, and inter-story displacement calculated by the basic method. When the time history analysis method yields a result greater than the modal response spectrum, the internal forces of the relevant components should be adjusted accordingly. The code stipulates that when three sets of acceleration time history curves are input, the calculation result should preferably be the larger value of the envelope value from the time history method and the value from the modal response spectrum method; when seven or more sets of time history curves are input, the calculation result can be the larger value of the average value from the time history method and the value from the modal response spectrum method.

1) Seismic Wave Selection: Based on the selection principles, the representative 1940 EL Centro wave, 1952 Taft wave, and Lanzhou artificial wave were selected as input seismic waves. Characteristic parameters of each seismic wave were analyzed. Analysis of the seismic wave curves revealed that the acceleration was relatively small in the latter half. To reduce the computational workload, this paper extracts the acceleration curves of the EL Centro and Taft waves from the first 20 seconds.

2) For internal force analysis, the acceleration time history curves were input bidirectionally, and X:Y = 1:0.85 was defined as time history condition 1, and XY = 0.8511 as time history condition 2. The peak acceleration under frequent earthquakes in the 7-degree intensity region was adjusted to 35 a1. The three seismic waves—EL Centro wave, Taft wave, and Lanzhou artificial wave—were all amplitude-modulated according to this specification, and the damping ratio was set to 0.03, the same as in the response spectrum analysis. The structural base shear force under each time history condition under bidirectional seismic waves was analyzed, and the envelope value was compared with the corresponding condition results in the response spectrum.

in conclusion

① The time history analysis shows that the base shear force in the X direction is larger in working condition 1 and the base shear force in the Y direction is larger in working condition 2, which is consistent with the actual situation, indicating that the selection and input of the time history wave are basically correct.

② The base shear force obtained from the horizontal two-way time history analysis is greater than the result of the response spectrum method under the corresponding working conditions. This indicates that for this actual project, when using the modal decomposition response spectrum method for design, it is necessary to first perform an amplitude-modulated analysis of the seismic internal forces of the overall structure and the inter-story shear force under each working condition of the two-way seismic wave, and compare it with the response spectrum method.

2. Elastic analysis under frequent earthquakes: In case 1, the inter-story shear force in the X direction is greater than that in case 2; in case 2, the inter-story shear force in the Y direction is greater than that in case 1. Therefore, the following analysis extracts the inter-story shear force data in the X direction of case 1 and the inter-story shear force data in the Y direction of case 2.