The dynamic failure criterion of single-layer spherical lattice shells has been an important research subject. The paper examines dynamic failures of single-layer spherical lattice shells and proposes the structure dynamic failure criterion based on the kinetic energy. The failure criterion was demonstrated through the dynamic failure test on a single-layer spherical lattice shell. Then, simulation analysis was carried out through two cases with material damage taken into account. The proposed failure criterion can accurately identify failure moments caused either by strength fracture or by stability fracture.

Single-layer spherical lattice shell is highly favored in practice for its light weight and graceful appearance by experts and scholars. The recent high frequency of earthquakes has challenged the extensive application of this light-weighted shell. Thus, the dynamic failure of the single-layer spherical lattice shell has been considered as an important research subject. The intensity fracture and stability fracture were extensively identified by experts as the major dynamic failures of single-layer spherical lattice shell. Shen et al. [

In theory, failure mechanism of single-layer spherical lattice shell and determining the structure of the ultimate load are important subject with maximum displacement and the degree of plastic as a major focus. Despite being highly valued in engineering application, the effective preestimating method of the failure of single-layer for the spherical lattice shell which could be applied to improve structure failure-resisting capacity by strengthening vulnerable spots is yet to be further explored. The authors resort to detecting a macroscopic quantity as a preestimating failure criterion which is easy to calculate and also reflects the characteristics of the overall structural failure. The authors note that there must be vibration and dramatic changes in kinetic energy in dynamic failures. According to the findings stated above, the paper endeavors to propose the dynamic failure criterion based on the kinetic energy and explores whether there exists a failure criterion coefficient applicable to identify dynamic failure and failure moment under strong shock. Moreover, the paper also sets out to verify the validity of the proposed failure criterion coefficient by referring to the collapse test data and the case which stimulates the process of the collapse of single-layer spherical lattice shells.

In finite element calculation, the kinetic energy equation can be written as follows:

The single-layer spherical reticulated shell is a symmetric structure. The mass matrix of single-layer spherical lattice shell is symmetric matrices. An equation could be obtained:

To be clear in discussion, several sign conventions could be stipulated as follows. The upward movement deviating from balance shaft is considered as positive and downward movement away from balance shaft as negative. Signs of node displacement, velocity, and acceleration velocity are in accordance with the above sign convention. The structure vibrates along the balance shaft. Vibration of situations could be vividly demonstrated in trajectory function

The relationship between displacement and coefficient.

At Stage A1, the structure vibrates downward the balance shaft and the trajectory function

When the movement was between Stages A1 and A2 and

When the movement reaches Stage A2 and

At Stage B and when

By such analogy, the changing rule of coefficient

At Stage E, the global vibration of the structure begins continuous downward accelerated movement from zero. During the process, the accelerated speed direction changed,

The above discussion expresses that the coefficient

It can be noted in Figure

From what stated above, conclusions could be made that when the coefficient

Combined with the shell tests conducted by scholars at home and abroad [

The size of test model.

The real test model.

The test was conducted in the Structural Testing Center of Beijing University of Technology. In the test, we used two sets of noncontact displacement acquisition test which are produced by the company of IMETRUM in England. The test applied a tracking shot to record the displacement of all nodes. The process of collapse under the strong earthquake was shown in Figure

The earthquake collapse process of test model.

Particles’ absolute displacement time history was calculated, with particles’ displacement time history curve and the displacement input of bearings taken into consideration. Furthermore, the particle movement velocity history was obtained based on the derivation of discrete points and all particles’ kinetic energy history. Then, the failure criterion coefficient ^{5}, 1.437 × 10^{6}, and 2.007 × 10^{6} and demonstrate a trend of increasing order of magnitude. The maximum ^{4} among all values of

Figure

The test model discriminant coefficient curve of failure.

Figure

The center point displacement curve of test model.

The span of single-layer spherical lattice shell is 40 m in paper [^{2}. The roofing gravity load will be concentrated onto the nodes. Rayleigh damping is used, with its damping factor as 0.02. Finite element method first establishes unit stiffness matrices by dividing the bar into several units and then the whole stiffness matrix. The paper uses finite element software ABAQUS. The model of Figure

The shell model of paper [

Literature [

The displacement history curve of shell model.

The history curve of failure coefficient ^{7}. The failure coefficient ^{9} and −4.48 × 10^{8} at two other subsequent moments. These values, however, had yet not met the failure criterion and thus could be ignored.

Fluctuation developed according to the recorded history curve of coefficient ^{7}. Then, the value of the coefficient remained positive and increased rapidly during 154 consecutive ^{7}, 1.82 × 10^{9}, and 2.6 × 10^{10}, showing a sharply increasing trend. According to the failure criterion, 4.634 s was defined as the failure point of the overall structure through calculation by the linear difference method. As it can be seen from Figure

The history curve of failure coefficient.

The failure moment 4.634 s is slightly ahead of the failure moment identified in literature [

Figure ^{2}, and all ground holds hinged. The software of 3D3S is used for reasonable design and calculation. All bars are hot-rolled steel pipe, of which the main ribbed bar section is ^{5} MPa with Poisson’s ratio as 0.3.

The diagram of overall structure.

The UMAT subroutine is introduced into the model structure shown in Figure

The calculation result shows that when ^{9}, which is comparatively large. When ^{8} and −1.82 × 10^{9}. According to the structure failure criterion, this fluctuation of coefficient ^{8}, 4.18 × 10^{9}, and 1.19 × 10^{10}. And eventually the obtained maximum coefficient is 1.51 × 10^{10}, which is much greater than the maximum value recorded before

The history curve of coefficient (PGA = 1300 Gal).

Figure

The displacement history curve of different point.

The above analysis demonstrates that the vibration of the second calculation model becomes failure when

In this study, the failure identification equation of single-layer spherical lattice shells is deduced based on kinetic energy. The dynamic failure criterion is verified by shaking table test data and two cases. Thus, several conclusions could be drawn as follows:

The structure failure moment under dynamic loading can be identified based on the shift of failure criterion coefficient

The dynamic failure criterion based on kinetic energy is simple, practical, and thus easy to be programmed.

The proposed failure criterion can accurately identify failure moment of single-layer spherical lattice shells damaged either by strength fracture or by stability fracture.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is partially supported by Natural Science Foundation of China under Grant nos. 51178009 and 91315301 and Beijing Lab of Earthquake Engineering and Structural Retrofit.