Mathematical Problems in Engineering

Volume 2015, Article ID 485710, 7 pages

http://dx.doi.org/10.1155/2015/485710

## The Failure Criterion of Single-Layer Spherical Lattice Shell Based on Kinetic Energy

Spatial Structures Research Center, Beijing University of Technology, Beijing 100124, China

Received 25 April 2015; Accepted 7 July 2015

Academic Editor: Dapeng P. Du

Copyright © 2015 Panfeng Ba et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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. [1–3] identified that dynamic strength failure occurs when the ratio of plastic bar is greater than 42% and maximum node displacement exceeds span 1/100 in the K8 single-layer spherical lattice shell from an overall perspective of integral structure. Du et al. [4] established a double-control principle based on plasticity dissipated energy and the ultimate displacement, which considers the failure of integral structure caused by damage accumulation and thus could identify the strength fracture and stability fracture. Nevertheless, Du’s principle could not go further than providing evaluation of degree of damage. Zhang and Peil [5] defined the stability concept and proposed a method identifying the stability of integral structure by changes in stability, considering the total potential energy of the rod and the ratio between increment in structural strain energy and potential energy. Nevertheless, Zhang’s method is limited to the elasticity problem.

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.

#### 2. The Dynamic Failure Criterion of Single-Layer Spherical Lattice Shells

In finite element calculation, the kinetic energy equation can be written as follows: where is kinetic energy of whole structure, is the mass matrix of structure, is the displacement of matrix, and is the first derivative. During any arbitrary periods , the increment in the kinetic energy can be expressed as follows: is the increment of kinetic energy. The acceleration at any period can be expressed as as follows: is the second derivative. Substituting (3) into (2), we can get the following:

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: , where is a constant in this equation. Then, anther equation can be written as follows:Substituting (5) into (4), we can getBecause *,* when both sides of (6) divided by the time variable of , we can get constant :

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 as stages of A, B, C, D, and E in Figure 1.