Notes on Beam Theory

NOTES ON BEAM THEORY J.E. Meyer Revision of Apr-96 Beam Geometnr and Notation: Consider a beam with the beam axis in the zdirection and with a constant cross sectional area denoted by A. Positions in A are denoted by (x, y) coordinates; the origin (x, y) = (0,O) defines the beam axis. . If the beam axis goes through the geometric centroid of A and if the beam is constructed of a single elastic material, certain simplifications occur. However, if multiple materials are used to consttuct the beam, and/or if creeplplasticity effects are present, then the simplifications may not be available. Any non-centroidal effects are incorporated directly in the equations used here. The x,y,z coordinates denote positions fixed in the unstressed beam at the reference temperature (zero st), I Figure 1: Coordinate System and Beam Geometry The displacements of a point on the beam axis relative to the reference state are denoted by (u, v, w) in the (x, y, z) directions, respectively. Thus (u, v ,w) are functions of z. The displacements of points on the beam axis are defined by giving the following quantities as a function of z: I u the xdirection displacement (m); 8, the slope (radians) of the beam axis (a small rotation that can be represented by a vector parallel to the y-axis); and k, the curvature (llm) of the beam axis (the reciprocal of the radius of curvature) in the plane normal to the y-axis. Figure 2: Slope of Beam Axis A positive sign for curvature (16) and for displacement (u) are indicated in Figure 3. The equality of the reciprocal of k, and the radius of cuwature is also shown schematically. Symbols for forces and moments are also defined, as follows: f, the external load in the positive xdirection per unit axial l e n m (Nlm); V, the shear force (N) on a positive face that has a normal in the zdirection (with a positive sign illustrated in Figure 4); I Beam Theory Y Page 2 z My a moment (N-m) about an axis in the y-direction that passes through the beam axis on a positive face (a positive sign for the moment is illustrated in Figures 3 and 4); and Fz an axial force (N) in the z-direction (acting on a positive face). Figure 3: Sign Conventions (a positive sign is illustrated) Force-Deformation (force-defo): Consider a case in which the predominant stresses are normal stresses that lie in the z-direction. Then the z-direction tensile strain is: where effects of a, and a, are neglected; and where ,E is the strain in the z direction that would exist if, locally, a, went to zero. Contributors to Ezo include thermal strain (ST), mechanical strain ( t . ) , from plasticity and creep, and strains (E~), from other deformation mechanisms (such as radiation swelling and radiation growth): I Figure 4: Forces and MomentsI The terms ElET, (E,,,),, and (E& in Eqs 1 and 2 are evaluated for the local material, for the local temperature, and for the local creep/plastidirradiationhistory at the particular (x, y, z) position of interest. Deformation-Displacement (defo-disol): Beam displacements' are defined by giving u and w as a function of z. The additional deformation and displacement measures (slope (O,), curvature (ky),and axial strain at the beam axis (a,))are obtained as follows, on a small slope and small ' If the beam axis is also displaced in the y-direction, then additional equations must be solved. For the present development, it is assumed that v = 0. Page 3 Beam Theory curvature basis (these equations also require the assumption that the beam is slender; if thick beams are encountered, then shear deformations may also be important): ky -- dey ; and dz A next step is to invoke the fundamental assumption of beam theory (Popov, Ref. P-1, page 122), that "each plane section that is normal to the beam axis in the original reference state is also plane and normal to the beam axis in later deformed states." This assumption is satisfied if (in treating strains at a given axial position in the case of curvature represented by a vector parallel to the y axis): Eauilibrium (earn): Force measures in the beam are the moment My, the shear force V, and the axial force F,. Equations of statics are (see Figure 4): The force measures are related to the stresses in the z-direction as follows: Discussion: Consider that the quantities f, F, and,E are known. Two boundary conditions must also be supplied at each end of the beam (to give information about u or 8, or M, or V, at each location. One statement about axial displacements is also needed (e.g., w = 0 at z = 0). Note that other combinations of input may also provide well posed problems. The solution may be thought of in the following way: integrate Eq (7) to obtain V;, integrate Eq (8) to obtain My; combine Eqs (A), (6), (9) and (10) to obtain a relation between M, and k,; integrate the k, relation using Eq (4) to obtain 8,; and Page 4 Beam Theory integrate Eq (3) to obtain u. In performing these steps. the boundary conditions and the strain relations of Eqs (1) and (6) must be satisfied. The equations may be simplified by noting that: A if the beam axis is centroidal with respect to the cross-sectional area and the y-axis; and A may be used to define the symbol ly(the area moment of inertia about the y-axis). Reference: P-1 E.P. Popov. "Mechanics of Materials." 2nd ed.. SI version. Prentice- Hall. Englewood Cliffs, NJ, 1978. CORE SUPPORT DETAILS Page 1 IN-CORE INSTRUMENTATION I ( BOSTON IDISON n?t-.: - r.-.?, - I PLATE I.., I Figure Reactor Vertical Arranqement BOSTON EDISON Pilgrim Station Figure 4.1-2' REACTOR CORE CROSS SECTIOI - -. I ayc 3 TOP VIEW BOTTOM VIEW I( BOSTON EDlSON Pilgrim Station \ I I FUEL ASSEMBLY \;GRID 1- .. . . j , i,.SPRING GRID PERIMETER j5TRIP STRIP =---.- BOSTON EDISON D:I--:m <*"bi.... /. CEA - , - d - . _. , --- .--.- --- .- ... - . - - Tc -*': GUIDE TUBE LOCATION' - . .:;. 8 FUEL ROD 1 ,'- - L A RIEL SPACER GRID Figure CORE SUPFOHT DETAILS