Journal of Mechanical Science and Technology 26 (6) (2012) 1811~1816 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-012-0433-4 Limit load analysis of shallow arches made of functionally bi-directional graded materials under mechanical loading † Ali Asghar Atai 1,* , Mohammad Hassan Naei 2 and Shabnam Rahrovan 1 1 Department of Mechanical Engineering, Islamic Azad University, Karaj Branch, Iran 2 Department of Mechanical Engineering, University of Tehran, Tehran, Iran (Manuscript Received April 17, 2011; Revised September 9, 2011; Accepted October 1, 2011) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract Thin shallow arches may become unstable under transverse loading if the built-up internal compressive forces reach a limiting value beyond which the structure undergoes a sudden large displacement towards a new stable configuration. This phenomenon could be both desirable (in toggle switches) and disastrous (collapse of a dome or truss). Hence, the so-called snap- or limit-load analysis becomes im- portant as to which factors influence it to give guidelines in designing structures to behave favorably. By the introduction of functionally graded materials (FGMs) in recent years, and incorporating them into this phenomenon, interesting results can be obtained which can give structures with favorable instability properties. In this work, a thin shallow arch with a modulus that can be varied along the thick- ness or the arch length or both is considered. Based on the governing equations of the deflected arch, the snap load is obtained in a mixed analytical-numerical approach and a parameter study of the critical load is carried out. Several verifying and interesting examples are presented. Keywords: Shallow arch; Bi-directional functionally graded material; Limit load; Mechanical loading ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction Shallow arches are curved one-dimensional structures which can be used as supporting structures for roofs and domes, or stiffeners for car and plane bodies, or as mechanical elements in toggle thermal or pressure switches. The major concern in their behavior under lateral loading is their instabil- ity at a critical load, which can make the structure to collapse or the switch to displace to another configuration and close the circuit. Several researches and investigations have been car- ried out on the subject. Timoshenko [1] in 1935 and Biezno [2] in 1938 were among the first who worked on the subject and presented solutions for the cases of distributed and con- centrated forces respectively. Marguerre [3] in 1938 discussed some cases based on the theory of buckling. Fung and Kaplan [4] considered various types of arches and lateral loads. Since then, several investigations have been carried out on the sub- ject considering variable arch stiffness or variable support stiffness or other deviations from the case of simple arches. With the introduction of the revolutionary functionally graded materials (FGMs) in recent years, some researches have in- corporated these materials into the subject [5-7]. All of these investigations have considered simple unidirectional variation of the material properties. The aim of this paper is to establish the basis for investigating the phenomenon under the condi- tion of biaxial variation of properties by presenting the gov- erning equations and solving them in order to do a parameter study of the phenomenon. Section 2 presents problem formu- lation for the case of thin shallow arch and bi-directional variation of material properties. In section 3, an analytical- numerical procedure for obtaining the critical snap load for the case of property variation along the arch thickness is presented and the procedure for the general case of bi-directional prop- erty variation is briefly discussed. In section 4, several nu- merical examples considering uni- or bi-directional property variation along with a parameter study of the critical load are presented. Wherever possible, results from other numerical and analytical procedures are presented for verification. 2. Problem formulation 2.1 Subsection Fig. 1 shows a pin-ended shallow arch under transverse uni- form loading. It’s made of an FGM material with a Young’s modulus that could vary both in vertical and horizontal direc- tion, i.e. E = E(x,z), in which x represents the horizontal coordi nate of the arch centerline and z is the coordinate of a point on * Corresponding author. Tel.: +60 3 79675210, Fax.: +60 3 79675317 E-mail address: atai@kiau.ac.ir † Recommended by Associate Editor Moon Ki Kim © KSME & Springer 2012