AppendixA - Appendix A Frame Analysis Program A1. How to...

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Unformatted text preview: Appendix A Frame Analysis Program A1. How to Run the Program Frame.Cmp 1. 2. 3. 4. 5. Run CMAP Open the program Frame.Cmp found in CMAP-directory Execute the program. Select Data-menu to start the process of data input /editing. Select Plot-Moments-menu to plot the bending moment diagram. A2. Modeling and Data Input for an Example System 1. Fundamental of modeling 1. A frame system is an assembly of nodes and members. A member is defined by a line connecting two nodes. Thus, the nodes serve to connect the members together as well as to connect the members to the supports. 2. In general, a node may translate in x- and y-directions as well as to rotate. Each free displacement will generate a nodal equilibrium equation expressing the balance between external forces and resisting member forces acting on the node. 3. A member-end that is fixed to the node will provide moment resistance to the node (unless its moment of inertia is negligible). Thus, a moment connection between members is modeled by fixing the ends of the members to the common node. 4. For modeling pinned and mixed connections, a member-end may be pinned to the node, and thus it will not provide any moment resistance to the node. 2. Example The above principles are used in modeling the example frame (Fig.A1) as shown in Fig.A2. Followings are some explanations of the model. o Appendix A Since members 2 and 3 are fix-fix type (see Note 1 below), the pinned connection at c must be modeled by a pair of coupled nodes (2 & 3). These nodes share the same x- and y-translations, but they have A.1 independent rotations (i.e. the nodes are coupled in translations but uncoupled in rotations as required by the pinned connection). This couplednode model has the advantage that the computed result will include the rotations of both nodes. Note that alternate models for pinnedconnections exist as shown in Fig.A3(b). 40 kN I = 554×106 mm4 a 10 kN/m 20 kN.m c b Girdir abcd: • negligible axial deformations • I = 554×106 mm4 Supporting members fb and fd: • Pinned at both ends • A = 10800 mm2 e d 8m A = 10800 mm2 f 12m 10 kN.m E = 200 GPa all members 6m 6m 12m Figure A1: Example System y Nodes coupled in translations Node numbers Member axes ij 6 8 7 x 1 1 2 2 3 4 5 4 6 6 5 7 Restrained nodal translations 3 Member numbers Restrained nodal rotation Figure A2: A Model for Analysis of the Example System o o Members 5 and 6 are pin-connected to the nodes at both ends. They are pin-pin members (type 3). o Appendix A Members 1, 2, 3 and 4 have their ends rigidly fixed to the nodes. They are fix-fix members (type 0. See Note 1 below). The restrained translations at nodes 1, 6 and 7 are as required by the support conditions. A.2 o The rotation of node 7 is artificially restrained because the pin-pin members 5 & 6 provide NO rotational resistance to the node 7. This rotational restraint is necessary in order to stabilize node 7. Data input for this model and are described below. Any consistent set of units (e.g. meters and kN) may be used. This Sample Data can be invoked within the Frame.Cmp program by the menu SampleData. Menu items General Sub-menu items • Title • Elastic modulus • No. of elements • No. of nodes • No. of restrained nodes Data input Example frame in Fig.A1 200e6 6 7 Remarks Max 255 characters Units: kPa An Element is a member Nodes 1, 6 and 7 where nodal displacements are to be specified (e.g. zero). For book-keeping only 3 • No. of loading cases 1 • No. of pairs of coupled nodes x y Element-Connectivity ElCon (How members are connected together) MemberProperties SupportRestraints Loading Appendix A • Area • Inertia Geometry 1 0, 12, 18, 18, 24, 36, 18 0, 0, 0, 0, 0, 0, -8 1, 2, 0 2, 3, 0 4, 5, 0 5, 6, 0 7, 2, 3 7, 5, 3 0, 0, 0, 0, 10800e-6, 10800e-6 554e-6, 554e-6, 554e-6, 554e-6, 0, 0 • Support conditions 1, 6, 7, R, R, F R, R, F R, R, R • Coupled nodes • Load case No. • No. of loaded nodes(See Note 3) 3, 1 4, 2 1, 1, 0 1 pair (Nodes 2 and 3 are coupled in x- & y-translations). x-coordinates (m) of nodes 1, 2, .... 7 y-coordinates of nodes One row per member: Col.1: Node i Col. 2: Node j Col. 3. End fixity to nodes (See Note 1 below). • Pin-pin (type 3) Cross-sectional areas in m2. (Zero is used if axial deformation in a member is neglected. This zero will be automatically replaced by a suitably large number) Moments of inertia in m4. Inertias for pin-pin members are not used in the computation. See Note 2(a) below. One row per node. See Note 2(b). One row per pair. For book-keeping only Two nodes (3 and 6) have external forces. A.3 • No. of element loads (see Note 4). • Nodal load data (See Note 3) • Element load data (see Note 4). One element load on member 4 . 1 3, 6, 4, 0, 0, 2, -40, 0, -10, -20 10 0, • Forces on node 3 • Forces on node 6 Uniform load on member 4. 12 Note 1. Member Fixity code: array ElCon Each member will have its own local member axes defined by the nodes i and j (see Fig. A4) of the two end. When two members share the same node, they are connected via this common node. If the member-ends are (fully) fixed to the common node (Fig. A3a), they also share the node's translations and rotation (i.e. a moment or rigid connection). To model a pinned connection (Fig.A3b), we must uncouple the rotation of the pinned-end from that of the other members (e.g. by pinning the member-end to the node or by using coupled node as shown in the example problem). This feature leads to different member types. Member ends fixed to node Member ends fixed to node (a) Moment connection A B Member end pinned to node (b) Member AB pinned to node A Figure A3 The last column of the array ElCon (ElementConnectivity) contains the Fixity code which specifies the type of member-end fixty conditions as follows: Fixity Fixity Fixity Fixity = = = = 0: 1: 2: 3: Fix-fix member type (both ends are fixed to the nodes) Pin-fix member type (end i is pinned to the node) Fix-pin member type (end j is pinned to the node) Pin-pin member type (both ends pinned to the nodes) Note 2: Support Conditions and Coupled Nodes a) Support Conditions: array aSNode Appendix A A.4 Every node in the system has three free nodal directions unless support condition is given to restrain the individual directions. This support restraint data is defined via the array aSNode (array of SupportedNodes) as follows: Column 1: Enter the node-number where one or more directions are to be restrained. Column 2: If x-direction is restrained, otherwise enter F for free translation. Column 3: If y-direction is restrained, enter R, otherwise enter F for free translation. Column 4: If rotation is restrained, enter R, otherwise enter F for free rotation. b) Coupled nodes: array aCNode When two nodes j and k have equal translation(s) and/or rotation, the nodes are said to be a coupled pair. The data for a coupled pair are specified in the following sequence: Node j Node k 1 if coupled in x-displ, zero otherwise 1 if coupled in y-displ, zero otherwise 1 if coupled in rotation, zero otherwise Note 3: Nodal Loads: array aNLoad All forces acting on a node are specified in one row of array aNLoad: Node No. Force in x-direction Force in y-direction Moment Note 4: Member (Element) Loading: array aELoad Data for each member (Element) load are specified in one row of array aELoad. If a member is subject to 3 different member loads, this will count as 3 Element Loads. Element No. Load type 1 or 2 or 3 (Fig. A4) Fx (Type 1) or w (Type 2) or Tt (Type 3) Fy (Type 1) or a (Type 2) or Tb (Type 3) a (Type 1) or b (Type 2) or D (Type 3) Three types of member load are available in the program: Type 1: concentrated force as specified by: Fx, Fy and a (Fig.A4a). Type 2: uniform load as specified by: w, a and b (Fig.A4b). Type 3: temperature as specified by: Tt, Tb and D (Fig.A4c). Note that external member forces are positive in the positive directions of the member's local axis-system. The member's x-axis goes from node i to node Appendix A A.5 j, and the member's y-axis is 90o counterclockwise from the x-axis. Distances a and b are measured from the end i of the member. Fy y x i Fx Tt j j w y y a i (a) Type 1 x a x b (b) Type 2 Tb D j i (c) Type 3: Temperatures Figure A4: Sign conventions for positive member loading A3. Analysis and Results Once the data have been input/edited and verified, computation is initiated by selecting one of the following item in the main menu panel: • ANALYSE to do only the analysis and the results will be placed in the output document • PLOT DEFORMED SHAPE: To analyse as well as to plot the deformed shape. • PLOT MOMENT DIAGRAM: To analyse as well as to plot the moment diagram. The deformed shape may also be plotted here: (i) First, set a different color by pressing key 'C' and (ii) Press key 'D'. Errors in data input can often be detected by inspection of the plotted deformed shape and moment diagram. Data can be revised/edited by selecting the DATA menu again. Interpretation of the output for member forces Six member-end forces are displayed in the output document. They are given both in local member axes and global (system) axes. The order are always as follows: 1: 2: 3: 4: 5: Appendix A x-Force on end i y-Force on end i Moment on end i x-Force on end j y-Force on end j (local member x-axis or global x-axis) (local member y-axis or global y-axis) (local member x-axis or global x-axis) (local member y-axis or global y-axis) A.6 6: Moment on end j Print or Save the Output Document First you must quit the Frame.Cmp program, and then use the main CMAP menu. Additional Features in Frame.Cmp There are many other features in the Frame.Cmp program. These are easily discovered merely by trying them using the displayed menus. A4. Additional Example Frames Additional sample data files (Frame1.dat, Frame2.dat, Frame3a.dat, Frame3b.dat, Frame4.dat) are available in the compressed file SampleData.zip. The geometry and loading for the frames are as shown in Fig.A.5. 2.4 m 1.2 m 30 kN/m 50 kN 10 kN/m 60 kN a a 1 b 12 kN.m 2 2m d c b c 3m 6m 3 I = 52.9×106mm 4 f e d 3@2m=6m 4.8 m 3m 4.8 m 4.8 m Frame 1 Frame 2 10kN/m 40 kN a 1 b 10 kN/m 2 3 c 6 5 d 4 f 12m 6m 6m 12m Frames 3a, 3b a e 1.5m 8m E = 200 GPa Cable A = 314mm2 f e 2m d c b 2m 2m Beam and posts: I = 10.91×106mm 4 Frame 4 Figure A5: Geometry and loading of sample frames Appendix A A.7 ...
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