3D Coordinate Transformation Calculations. Space Truss Member

Transcription

1 3D oordinate Transformation alculations Transformation of the element stiffness equations for a space frame member from the local to the global coordinate system can be accomplished as the product of three separate transformations. In these slides, we will develop the details for these calculations considering both a space truss member and a space frame member. Space Truss Member Fig.. Typical Space Truss Member onsider the space truss member shown in Fig. with six local coordinate degrees of freedom identified. We desire to transform the element stiffness equations { F'} [k']{ '} () into the global coordinate element stiffness equations { F} [k]{ } () The global coordinate element stiffness equations of () are related to () via T {F} [ ] {F'} { '} [ ]{ } T [k] [ ] [k'][ ] [ ] ] [ ] (3) [ = local-to-global coordinate transformation matrix for a truss 3 member; (see (5.) in MGZ); {F} = <F x F y F z F x F y F z > T ; {F} = <F x F y F z F x F y F z > T ; {} = <u v w u v w > T ; {} = <u v w u v w > T ; and the 6x6 local coordinate element stiffness matrix [k] contains only four nonzero coefficients, i.e., k = k 44 =EA/L, k 4 =k 4 = -EA/L. Only lx' [ ] ly' lz' lx', mx', and nx' mx' my' mz' nx' n y' nz' of [] multiply 4

2 nonzero coefficients in the element stiffness matrix. onsequently, only these rotation components are required to construct the global coordinate stiffness matrix and these three elements are the same as the direction cosines for the member itself and can be found from the coordinates of the member ends as x x l x' x L y y m x' y L z z n x' z L (4) The lengthlcanalsobecalculated from the member end point coordinates as L (x x ) (y y ) (z z (5) ) 5 Substituting (4) into [] and using the third equation of (3) leads to x x y EA x [k] L x x y x y x y y y x y y z x z y z x z x x x y x x x y x y x y y y x y y zx y zx zx (6) Fig.. Rotation Axes for a Space 6 Truss Member For geometric nonlinear analysis and for loads directly applied to the truss members, it is necessary to specify the orientation of the cross section axes y and z. Figure shows the rotational geometry of a typical space truss member. It is desired to determine the relationship between the local coordinate axes and the global coordinate axes. This relationship can be determined using two two-dimensional coordinate transformations for space truss members. The first two-dimensional transformation is about the y-axis and relates the x cos y z sin sin x y cos z 7 8 global axes to the -axes, i.e. { x } [ ]{x} (7) The direction cosines of (7) can be expressed in terms of x and xz as cos x xz sin z xz (8) Equation (7) expresses the rotation of the global x- and z-axes such that global x-axis lies under the projection of the x-axis onto the x-z plane. To complete the desired transformation, the x -axis must be rotated into the x-axis.

3 This second two-dimensional transformation is about the z - or z-axis and can be expressed as x' cos y' sin z' sin cos x y z x x' x y y' xz z' xz y xz x y y xz z x xz () { x'} [ ]{x} (9) The direction cosines of (9) can be expressed in terms of y and xz as cos xz sin y () ombining (7) () leads to { x'} [ ]{x} [ ]{x} [ ][ ]{x} Fig. 3. Rotation Axes for a Vertical Truss Member 9 The transformation matrix given in () is valid for all space truss member orientations with the exception of a vertical truss member as shown in Fig. 3. For the vertical truss member, x = z = xz = and () is not numerically defined. Inspection of Fig. 3 shows that [ ] y y vert () Space Frame Element A space frame member has twelve degrees of freedom as discussed in MGZ (Sections 4.5 and 5.). Our purpose is to extend the space truss transformations () and () to include the principal cross section axes of the member, which is not necessary for the space truss member. The form of the rotation matrix [] depends upon the particular orientation of the member axes. In many instances a space frame 3

4 member will be oriented so that the principal axes of the cross section lie in horizontal and vertical planes (e.g., I-beam with its web in a vertical plane). Under these scenarios, the y- and z-axes can be selected in the exact same fashion as the space truss member. There are other instances in which a space frame member has two axes of symmetry in the cross section and the same moment of inertia about each axis (e.g., circular or square member either tubular or solid). In these cases, the y- and z-axes can 3 again be selected to be the same as the space truss member. This choice is possible because all axes in the cross section are principal axes. However, a space frame member may have its principal y- and zaxes in general directions. Two Fig. 4. Rotation About the x-axis 4 methods are typically used to describe this more general case. The first choice is to specify the angle about the x-axis. This choice is normally reserved for cases in which can easily be visualized, e.g., horizontal or vertical members. Another choice is to specify the coordinates of a point p as shown in Fig. 4 and calculate the angle. {x } [ ]{x } [ ][ ]{x } [ ][ ][ ]{x} [ ]{x} (3) where x' y' z' cos sin x sin y cos z { x'} [ ]{x } (4) In either case, the transformation process for a space frame member can be expressed in terms of three two-dimensional transformations as shown schematically in the Example 5.6 figure. This three-step process 5 can be expressed as x x y cos sin [ ] xz x y sin cos xz y xz cos xz sin z y cos x sin xz y sin x cos xz (5) 6 4

5 Obviously, if the angle is input directly, then (5) is used directly in computing the local to global coordinate transformation of the three-dimensional frame element stiffness equations. However, if the angle is not readily available, then the coordinates of a reference point p must be input in order to calculate (this procedure is typically easier than trying to visualize the angle ). In order to calculate from the coordinates of point p of Fig. 4, you first compute x ps x p x yps yp y zps zp z (6) 7 in which x p, y p,z p = global coordinates of point p; and x, y, z = global coordinates of node. Next, calculate coordinates of p in terms of {x } coordinate system as x p x ps yp [ ] [ ] yps zp zps x p x x ps y yps zps xy y yp x ps xz yps zps xz xz z x p x ps zps xz xz (7) Equations (7) provide the coordinates of point p with respect 8 to the -axes. These coordinates are then used to calculate the direction cosines for the angle of Fig. 4 as sin zp y p z p cos yp y p z p (8) Equations (8) can be substituted into (5) to produce the desired transformation for a space frame member. As in the space truss member, (5) will not work for a vertical space frame element since x = z = xz =. In such cases, the rotation Fig. 5. Rotation of Axes for a Vertical Space Frame Member element can be obtained by inspection from Fig. 5 to give y [ vert ] ycos sin (9) y sin cos Equation (9) is valid for both orientations shown in Fig. 5 and matrix for a vertical space frame 9 5

6 if =, then (9) is the same as () (vertical space truss member). In those cases where it is desired to begin with the coordinates of a point p that is known to lie in a principal plane, it is possible to calculate sin and cos in (9) directly from the coordinates of the point. Figure 6 shows the two cases of Fig. 5 with a point p specified. The sine and cosine of for both cases can be calculated from Fig. 6. Use of Point p for a Vertical Space Frame Member sin zps x ps zps cos x ps y x ps zps where y = for Fig. 6(a) and y = - for Fig. 6(b). () Unsymmetric ross Section Analysis Linear stiffness equations for element e (e.g. 4.34) can be expressed in terms of the principal cross section axes as e e e ef {F } [k ]{u } {F } () Element Transformations and Assembly where k ] = principal coordinate element stiffness matrix given in (4.34) of MGZ; { F e } = principal coordinate element force vector; and { F ef } = principal coordinate element fixed-end force vector. [ e 3 Fig.. Unsymmetric Angle Section and ross Section oordinates Equations () are expressed in terms of the uncoupled principal cross section modes of deformation. That is, () axial, 4 6

7 () torsion, (3) bending about the strong axis (defined as the z-axis), and (4) bending about the weak axis (defined as the y-axis). In order to assemble various element equations, the equations must be expressed in terms of a common set of coordinate axes. For members with doubly symmetric cross sections, the centroid and shear centers coincide and no additional transformations are required. However, when the member cross section is singly symmetric or unsymmetric, e.g., the unequal leg angle shown in Fig., 5 a transformation of displacement and force degrees of freedom is required (e.g., hen and Blandford 989). Such transformations are necessary since (u x, F x ), ( y, M y ), and ( z, M z ) are evaluated in terms of the cross section centroid whereas displacement and force pairs (u y, F y ), (u z, F z ), and ( x, M x ) are evaluated in terms of the cross section shear center. Due to the different evaluation points for these displacement and force pairs, subscripts and S will be introduced to refer to the centroid and shear center, respectively. 6 Transforming the principal and S coordinate stiffness equations is normally performed such that the final equations are expressed in terms of the principal centroidal axes shown in Fig.. Expressing the principal coordinate element displacement vector as { u e } = e e T u u where { u e } i = ux usy usz T Sx y i = element nodal principal coordinate displacement vector with i =, = element node points; then the transformation from the principal coordinates to the centroidal principal coordinates can be expressed as 7 e u } i e [Tyz]{u } i { () [I] [Tyz] [] [t yz] [I] = principal coordinate to centroid transformation matrix; S ys [t yz ] = z ; [I] = 3x3 identity matrix; {u e } = <u x u y u x y > T i = element nodal centroidal displacement vector; and y S, z S = coordinate distances from the cross section centroid to the shear center as shown in Fig.. Using the transformation of (), the element stiffness equations can be expressed in terms of the element centroidal 8 7

8 dof as e e ef e [k ]{u } {F } {F } (3) [k e ] = [T] T e [k ] [T] = element e stiffness matrix; {u } = [T] {u e }; [T yz ] [] [T] [] [T yz ] = element principal coordinate to centroid transformation matrix; {u e } = element displacement vector at the element cross section centroid (i.e., element centroidal dof); ef T ef {F } [T] {F } = element fixed-end force vector measured at the 9 e T e element centroid; and {F } [T] {F } = element force vector measured at the element centroid. Transformation of the element stiffness equations of (3) into the global coordinate system and then assembly into the structure stiffness matrix follows the usual procedures. 3 8