Frame Element Stiffness Matrices

Transcription

1 Frame Element Stiffness Matrices CEE 421. Matrix Structural Analysis Department of Civil and Environmental Engineering Duke University Henri P. Gavin Fall, 214 Truss elements carry axial forces only. Beam elements carry shear forces and bending moments. Frame elements carry shear forces, bending moments, and axial forces. This document picks up with the previously-derived truss and beam element stiffness matrices in local element coordinates and proceeds through frame element stiffness matrices in global coordinates. 1 Frame Element Stiffness Matrix in ocal Coordinates, k A frame element is a combination of a truss element and a beam element. The forces and displacements in the local axial direction are independent of the shear forces and bending moments. N 1 V 1 M 1 N 2 V 2 M 2 = q 1 q 2 q 3 q 4 q 5 q 6 = 12EI 3 6EI 2 12EI 3 6EI 2 6EI 12EI 2 3 2EI 6EI 2 u 1 u 2 u 3 u 4 u 5 u 6

2 2 CEE 421. Matrix Structural Analysis Duke University Fall 214 H.P. Gavin 2 Relationships between ocal Coordinates and Global Coordinates: T The geometric relationship between local displacements, u, and global displacements, v, is u 1 = v 1 cos θ + v 2 sin θ u 2 = v 1 sin θ + v 2 cos θ u 3 = v 3 or, u = T v. is The equilibrium relationship between local forces, q, and global forces, f, q 1 = f 1 cos θ + f 2 sin θ q 2 = f 1 sin θ + f 2 cos θ q 3 = f 3 or, q = T f, where, in both cases, T = c s s c 1 c s s c 1 c = cos θ = x 2 x 1 s = sin θ = y 2 y 1 The coordinate transformation matrix, T, is orthogonal, T 1 = T T.

3 Frame Element Stiffness Matrices 3 3 Frame Element Stiffness Matrix in Global Coordinates: K Combining the coordinate transformation relationships, q = k u T f = k T v f = T T k T v f = K v which provides the force-deflection relationships in global coordinates. stiffness matrix in global coordinates is K = T T k T The K = c2 cs 3 s 2 12EI 3 s2 c2 cs 6EI 2 s 12EI 3 c 2 6EI 2 c cs 3 s 2 cs 6EI 3 2 s cs s2 cs 12EI 3 c 2 6EI 3 6EI 2 s c2 6EI 2 c 2 c 2EI cs s 2 12EI 6EI 3 2 s s2 3 c 2 6EI 2 c

4 4 CEE 421. Matrix Structural Analysis Duke University Fall 214 H.P. Gavin 4 Frame Element Stiffness Matrices for Elements with End-Releases Some elements in a frame may not be fixed at both ends. For example, an element may be fixed at one end and pinned at the other. Or, the element may be guided on one end so that the element shear forces at that end are zero. Or, the frame element may be pinned at both ends, so that it acts like a truss element. Such modifications to the frame element naturally affect the elements stiffness matrix. Consider a frame element in which a set of end-coordinates r are released, and the goal is to find a stiffness matrix relation for the primary p retained coordinates. The element end forces at the released coordinates, q r are all zero. One can partition the element stiffness matrix equation as follows q p q r k pp k rp k pr k rr u p u r The element displacement coordinates at the released coordinates do not equal the structural displacements at the collocated structural coordinates, since the coordinates r are released. Since the element end forces at the released coordinates are all zero (q r = ), the element end displacements at the released coordinates must be related to the displacements at the primary (retained) coordinates as: u r = k 1 rr k rp u p The element end forces at the primary coordinates The element stiffness matrix equation relating q p and u p is q p = [ k pp k pr k 1 rr k rp ] up The rows and columns of the released element stiffness matrix corresponding to the released coordinates, r, are set to zero. The rows and columns of the released element stiffness matrix corresponding to the retained coordinates, p, are [k pp k pr krr 1 k rp ]. This is the element stiffness matrix that should assemble into the structural coordinates collocated with the primary (retained) coordinates p. The following sections give examples for pinned-fixed and fixed-pinned frame elements. Element stiffness matrices for many other end-release cases can be easily computed.

5 Frame Element Stiffness Matrices Pinned-Fixed Frame Element in ocal Coordinates, k... (r = 3) k = Pinned-Fixed Frame Element in Global Coordinates, K = T T kt K = c2 + 3 s 2 cs c2 3 s 2 s2 cs + c c2 + 3 s 2 cs s 2 + s2 c 2 c 2 3 cs EI s 2 s2 c 2 c 2 3

6 6 CEE 421. Matrix Structural Analysis Duke University Fall 214 H.P. Gavin 4.3 Fixed-Pinned Frame Element in ocal Coordinates, k... (r = 6) k = Fixed-Pinned Frame Element in Global Coordinates, K = T T kt K = c2 + 3 s 2 cs s 2 3 cs s2 + 3 c 2 2 c c2 3 s 2 + cs 2 s c2 + 3 s 2 + cs s2 3 c 2 2 c cs s2 + 3 c 2

7 Frame Element Stiffness Matrices 7 5 Notation u = Element deflection vector in the ocal coordinate system q = Element force vector in the ocal coordinate system k = Element stiffness matrix in the ocal coordinate system... q = k u T = Coordinate Transformation Matrix... T 1 = T T v = Element deflection vector in the Global coordinate system... u = T v f = Element force vector in the Global coordinate system... q = T f K = Element stiffness matrix in the Global coordinate system... K = T T k T d = Structural deflection vector in the Global coordinate system p = Structural load vector in the Global coordinate system K s = Structural stiffness matrix in the Global coordinate system... p = K s d ocal Global Element Deflection u v Element Force q f Element Stiffness k K Structural Deflection - d Structural oads - p Structural Stiffness - K s For frame element stiffness matrices including shear deformations, see: J.S. Przemieniecki, Theory of Matrix Structural Analysis, Dover Press, (... a steal at $12.95)