Master and Slave Nodes (Rigid Link Function)

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Rebecca Lester
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1 Master and Slave Nodes (Rigid Link Function) The rigid link function specified in Model>Boundaries>Rigid Link constrains geometric, relative movements of a structure. Geometric constraints of relative movements are established at a particular node to which one or more nodal degrees of freedom (d.o.f.) are subordinated. The particular reference node is called a Master Node, and the subordinated nodes are called Slave Nodes. The rigid link function includes the following four connections: 1. Rigid Body Connection 2. Rigid Plane Connection 3. Rigid Translation Connection 4. Rigid Rotation Connection Rigid Body Connection constrains the relative movements of the and slave nodes as if they are interconnected by a three dimensional rigid body. In this case, relative nodal displacements are kept constant, and the geometric relationships for the displacements are expressed by the following equations: U Xs = U Xm + R Ym ΔZ - R Zm ΔY U Ys = U Ym + R Zm ΔX - R Xm ΔZ U Zs = U Zm + R Xm ΔY - R Ym ΔX R Xs = R Xm R Ys = R Ym R Zs = R Zm where, ΔX = X m - X s, ΔY = Y m - Y s, ΔZ = Z m - Z s The subscripts, m and s, in the above equations represent a and slave nodes respectively. U X, U Y and U Z are displacements in the Global Coordinate System (GCS) X, Y and Z directions respectively, and R X, R Y and R Z are rotations about the GCS X, Y and Z-axes respectively. X m, Y m and Z m represent the coordinates of the, and X s, Y s and Z s represent the coordinates of a slave node. This feature may be applied to certain members whose stiffnesses are substantially larger than the remaining structural members such that their deformations can be ignored. It can be also used in the case of a stiffened plate to interconnect its plate and stiffener. 111

2 ANALYSIS FOR CIVIL STRUCTURES Rigid Plane Connection constrains the relative movements of the and slave nodes as if a planar rigid body parallel with the X-Y, Y-Z or Z-X plane interconnects them. The distances between the nodes projected on the plane in question remain constant. The geometric relationships for the displacements are expressed by the following equations: Rigid Plane Connection assigned to X-Y plane U Xs = U Xm - R Zm ΔY U Ys = U Ym + R Zm ΔX R Zs = R Zm Rigid Plane Connection assigned to Y-Z plane U Ys = U Ym - R Xm ΔZ U Zs = U Zm + R Xm ΔY R Xs = R Xm Rigid Plane Connection assigned to Z-X plane U Zs = U Zm - R Ym ΔX U Xs = U Xm + R Ym ΔZ R Ys = R Ym This feature is generally used to model floor diaphragms whose relative in-plane displacements are negligible. Rigid Translation Connection constrains relative translational movements of the and slave nodes in the X, Y or Z-axis direction. The geometric relationships for the displacements are expressed by the following equations: Displacement constraint in the X-axis direction U Xs = U Xm Displacement constraint in the Y-axis direction U Ys = U Ym Displacement constraint in the Z-axis direction U Zs = U Zm 112

3 Rigid Rotation Connection constrains the relative rotational movements of the and slave nodes about the X, Y or Z-axis. The geometric relationships for the displacements are expressed by the following equations: Rotational constraint about the X-axis R Xs = R Xm Rotational constraint about the Y-axis R Ys = R Ym Rotational constraint about the Z-axis R Zs = R Zm The following illustrates an application of Rigid Plane Connection to a building floor (or any other structural plate) diaphragm to help the user understand the concept of the rigid link feature. When a building is subjected to a lateral load, the relative horizontal deformation at any point in the floor plane is generally negligible compared to that from other structural members such as columns, walls and bracings. This rigid diaphragm action of the floor slab can be implemented by constraining all the relative inplane displacements to behave as a unit. The movements consist of two in-plane translational displacements and one rotational displacement about the vertical direction. 113

4 ANALYSIS FOR CIVIL STRUCTURES floor diaphrarm lateral load 1 2 Before deformation After deformation 1 2 Figure 1.71 Typical structure with floor diaphragm subjected to a lateral load As illustrated in Figure 1.71, when a structure is subjected to a lateral load and the in-plane stiffness of the floor is significantly greater than the horizontal stiffness of the columns, the in-plane deformations of the floor can be ignored. Accordingly, the values of δ 1 and δ 2 may be considered equal. 114

5 floor diaphragm torsional moment Ф 1 Ф 2 Ф 3 Ф 4 Ф 5 Figure 1.72 Single story structure with a floor (plate) diaphragm subjected to a torsional moment about the vertical axis When a single-level structure, as illustrated in Figure 1.72, is subjected to a torsional moment about the vertical direction and the in-plane stiffness of the floor is significantly greater than the horizontal stiffness of the columns, the entire floor diaphragm will be rotated by, where, Accordingly, the four degrees of freedom can be reduced to a single degree of freedom. 115

6 ANALYSIS FOR CIVIL STRUCTURES Figure 1.73 shows a process in which a total of 24 (6 4) degrees of freedom are compressed to 15 d.o.f. within the floor plane, considering its diaphragm actions. slave nodes floor diaphram floor diaphram UxUyRz slave node U X : displacement degree of freedom in the X-direction at the corresponding node U Y : displacement degree of freedom in the Y-direction at the corresponding node U Z : displacement degree of freedom in the Z-direction at the corresponding node R X : rotational degree of freedom about the X-axis at the corresponding node R Y : rotational degree of freedom about the Y-axis at the corresponding node R Z : rotational degree of freedom about the Z-axis at the corresponding node Figure 1.73 Reduction of d.o.f for floor diaphragm of significant in-plane stiffness 116

7 displaced floor diaphram R Zm R Zs slave node initial floor diaphram U Xm U Ym R Zm U Xs U Ys R Zs : X-direction displacement of : Y-direction displacement of : rotation about Z-axis at : X-direction displacement of slave node : X-direction displacement of slave node : rotation about Z-axis at slave node Figure 1.74 Displacements of an infinitely stiff floor diaphragm due to horizontal loads As illustrated in Figure 1.74, if translational and rotational displacements take place simultaneously in an infinitely stiff floor diaphragm due to a lateral load, the displacements of a point on the floor plane can be obtained by: U Xs = U Xm - R Zm ΔY U Ys = U Ym + R Zm ΔX R Zs = R Zm Reducing number of degrees of freedom by geometric constraints can significantly reduce the computational time for analysis. For instance, if a building structure is analyzed with the floors modeled as plate or plane stress elements, the number of nodes will increase substantially. Each additional node 117

8 ANALYSIS FOR CIVIL STRUCTURES represents 3 additional degrees of freedom even if one considers d.o.f in lateral directions only. A large number of nodes in an analysis can result in excessive program execution time, or it may even surpass the program capacity. In general, solver time required is proportional to the number of degrees of freedom to the power of 3. It is, therefore, recommended that the number of degrees of freedom be minimized as long as the accuracy of the results is not compromised. Figure 1.75 shows applications of Rigid Body Connection and Rigid Plane Connection. Figure 1.75 (a) illustrates an application of Rigid Link using Rigid Body Connection. Here a rectangular tube is modeled with plate elements in the region where a detail review is required, beyond which a beam element represents the tube. Then, Rigid Body Connection joins the two regions. Figure 1.75 (b) shows an application of Rigid Plane Connection for a column offset in a two-dimensional plane. Whenever Rigid Link is used in a plane, geometric constraints must be assigned to two translational displacement components and one rotational component about the perpendicular axis to the plane. If a structural analysis model includes geometric constraints and is used for a dynamic analysis, the location of the must coincide with the mass center of all the masses pertaining to the slave nodes. This condition also applies to the masses converted from self-weights. rectangular tube modeled with plate elements Rigid Link rectangular tube modeled as a beam elemant : slave nodes (12 nodes) * all 6 degrees of freedom of the slave nodes are linked to the. (a) A tube modeled using a beam element and plate elements, and connected by Rigid Body Connection 118

9 slave node eccentricity eccentricity * all slave node s d.o.f. in the X-Z plane are linked to the master node (translational displacement d.o.f. in the X and Z directions and rotational d.o.f. about the Y-axis. (b) Eccentricity of an offset-column linked by Rigid Plane Connection Figure 1.75 Application examples of geometric constraints 119

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