كليدواژه :
اعضاي غيرمنشوري , ماده مدرج تابعي , سري تواني , اصل حداقل انرژي پتانسيل
چكيده فارسي :
در تحقيق پيش رو، يك تكنيك عددي جديد جهت تحليل پايداري ستون غيرمنشوري از جنس مواد مدرج تابعي ارائه گرديده است. خواص مواد در راستاي ضخامت تير ثابت در نظر گرفته شده است، درحالي كه در راستاي طولي به صورت پيوسته و تدريجي و طبق قانون تواني تغيير مي كند. همچنين به علت تفاوت ناچيز ضريب پواسون مصالح سازنده تير، مقدار آن در طول عضو ثابت در نظر گرفته شده است. در مرحله نخست به منظور بررسي رفتار پايداري و ارتعاش آزاد اين اعضا، معادله حركت حاكم بر ارتعاش آزاد تيرهاي غيرمنشوري با مقطع متغير نمايي با استفاده از تركيب بسط مك لورن و روش سري هاي تواني حل مي گردد. درنتيجه، فرم تقريبي تغيير شكل ارتعاشي تير به صورت يك چند جمله اي متناهي به دست مي آيد و با جايگذاري شرايط مرزي حاكم بر تير و با استفاده از روش حل مقادير ويژه مقدار فركانس طبيعي ارتعاشي محاسبه مي گردد. در ادامه به كمك اصل حداقل انرژي پتانسيل و با توجه به تشابه فرم تغيير شكل اعضاي الاستيك تحت تحليل پايداري و ارتعاش آزاد، مقدار بار كمانش بحراني تعيين مي گردد. در پايان جهت اعتبار سنجي، نتايج عددي حاصل از تحليل پايداري و ارتعاش آزاد براي اعضاي ساخته شده از مواد همگن و يا تابعي ارائه شده و تاثير تغييرات سطح مقطع، توان ماده تابعي و نوع تكيه گاه بر بار كمانش بحراني و فركانس طبيعي ارتعاش مورد بررسي قرار گرفته است. در تمامي مسائل، كارآمدي، دقت و صحت روش عددي ارائه شده به اثبات رسيده است.
چكيده لاتين :
Owing to efficiency, rising stability of structures, reduction in structural weight and cost, and the improvements in fabrication process, non-uniform beams are extensively adopted in civil and mechanical structures. Furthermore, the use of functionally graded (FG) materials has been increasing in many mechanical components due to their conspicuous characteristics such as high strength, thermal resistance and optimal distribution of weight. In the present paper, a numerical model combining the power series expansions and the Rayleigh-Ritz method is adopted for stability and free vibration analyses of axially functionally graded (FG) columns with variable cross-section. The main purpose of this paper is calculating the critical buckling loads and natural frequencies concurrently for AFG members with exponentially-varying geometrical properties. For this, a mixed power series expansions and the principle of stationary total potential energy as a first endeavor is presented. In this study, the material properties of the non-prismatic beam including Young’s modulus of elasticity and density of material are assumed to be graded smoothly along the beam axis by a power-law distribution of volume fractions of metal and ceramic. Moreover, the cross-sectional area and moment of inertia vary exponentially over the member’s length. In this regard, the power series approximation is applied to solve the fourth order differential equation of motion, since in the presence of variable cross-section and axially non-homogeneous material, stiffness quantities are not constant. All geometrical and material properties and displacement component are developed based on power series of an identified degree. The natural frequencies of the AFG beam with variable cross-section are derived by imposing the boundary conditions and solving the eigenvalue problem. The explicit expression of vibrational shape function is then derived based on this rigorous numerical method. The vibrational mode shapes of an elastic member are similar to the buckling ones. Therefore, the obtained deflected shapes of the considered non-prismatic beams can be used as deformation shape of member for the linear buckling analysis. The critical buckling load of non-prismatic beam can be then estimated by adopting the principle of stationary total potential energy. According to the steps mentioned above, for measuring the accuracy and competency of the proposed numerical procedure, two numerical examples including axially non-homogeneous and homogeneous column with non-uniform section are represented. Numerical results of the critical buckling loads and natural frequencies for various boundary conditions, different gradient index and cross-section variation are represented. Due to lack of similar research for the stability and free vibration analyses of elastic AFG beams with exponential variation of the cross-sectional area and moment of inertia, outcomes of homogeneous members are compared with the results presented in other available numerical and analytical references and those related to non-prismatic beams with material variation are then reported. The accuracy of the method is then remarked. This method has many positive points consisting of efficiency, accuracy and simplicity contrasted with more complex numerical methods. It has to be noticed that the present numerical method can be applied to determine the critical buckling loads and natural frequencies of axially functionally graded (FG) prismatic beams as well as non-prismatic ones.
