tailieunhanh - Fundamentals of Finite Element Analysis phần 2

Các định lý đầu tiên của Castigliano cũng áp dụng đối với chuyển vị quay. Trong trường hợp quay, đạo hàm một phần năng lượng căng thẳng đối với một chuyển luân phiên tương đương với thời điểm / mô-men xoắn được áp dụng tại các điểm quan tâm trong ý nghĩa của tự quay. Ví dụ sau đây minh họa ứng dụng trong các điều khoản của một thành viên xoắn đơn giản. | I Text Hutton Fundamentals of Finite Element Analysis 2. Stiffness Matrices Spring and Bar Elements The McGraw-Hill Companies 2004 Strain Energy Castigliano s First Theorem 41 The first theorem of Castigliano is also applicable to rotational displacements. In the case of rotation the partial derivative of strain energy with respect to a rotational displacement is equal to the moment torque applied at the point of concern in the sense of the rotation. The following example illustrates the application in terms of a simple torsional member. EXAMPLE A solid circular shaft of radius R and length L is subjected to constant torque T. The shaft is fixed at one end as shown in Figure . Formulate the elastic strain energy in terms of the angle of twist 0 at x L and show that Castigliano s first theorem gives the correct expression for the applied torque. Solution From strength of materials theory the shear stress at any cross section along the length of the member is given by Tr T T J where r is radial distance from the axis of the member and J is polar moment of inertia of the cross section. For elastic behavior we have T Tr y G JG where G is the shear modulus of the material and the strain energy is then L r If If Í Tr Tr U ydv y JXJ d Ỳ T 2 2 J 2 G ỉ ỉ r2 d A dx T2L 2 JG 0 A where we have used the definition of the polar moment of inertia r 2d A Figure Example Circular cylinder subjected to torsion. I Text Hutton Fundamentals of Finite Element Analysis 2. Stiffness Matrices Spring and Bar Elements The McGraw-Hill Companies 2004 42 CHAPTER 2 Stiffness Matrices Spring and Bar Elements Again invoking the strength of materials results the angle of twist at the end of the member is known to be so the strain energy can be written as Ue 1 L 1 y ỈG 02 2 JG L 2L Per Castangliano s first theorem dUe _ JGQ IT L which is exactly the relation shown by strength of materials theory. The reader may think that we used circular reasoning in this example since we utilized