**David Dearth President Applied Analysis & Technology Inc. Huntington Beach, Calif.**

**Edited by Paul Dvorak**

Finite-element techniques can help investigate these abrupt changes. Here’s one way that uses spring elements.

**Choose from two springs**

To illustrate how you can simulate abrupt structural discontinuities, consider a cantilever beam with a joint or splice simulated with a rotational spring. For this simple example, there are generally two FE elements with useful spring qualities: the degree-of-freedom (DOF) spring and the CBush spring.

The DOF Spring applies a single degree of freedom for each element. The DOF Spring is controlled by the coordinate-system nodes used to define each end of the spring. Users set or release particular DOFs for the spring in a menu. What’s more, the DOFs (T_{x}, T_{y}, T_{z}, R_{x}, R_{y}, R_{z}) must coincide with the coordinate system that defines the nodes. (The six DOF are sometimes written as 1, 2, 3 and 4, 5, 6). There is more on this in downloadable Run Notes, available from a URL in the *Try it yourself box*, along with detailed modeling tips on how to define DOF Springs using Femap.

A CBush spring (a Nastran element name) acts more like a conventional beam element in that its orientation uses a local coordinate system defined by the element’s “i-j” directional vector in space. As with a beam element, users must define an orthogonal node “k” in a CBush spring.

A comprehensive discussion of how to perform detailed simulations of complete structural joints or splices is beyond the scope of this article. But this simplified approach introduces the elements and provides insight to the method. To gain confidence in the solution, there are also hand calculations using equations from appropriate engineering texts that correlate results with finite-element idealizations (one for each spring element) and in this case, using NX Nastran and Femap.

The sample problem is a cantilever beam with a rotational spring joint. The beam is subjected to a single concentrated end moment, M, of 50 in.-lb. The beam is a uniform, 10-in.-long aluminum tube with a 2-in. OD and 0.028-in. wall thickness. It also has a modulus, E = 10 10^{6} psi and cross-sectional inertia. The simulated joint is 0.5 in. from the fixed boundary support.

First, find the cross-sectional inertia for the circular section with:

I= | π | D^{4}-d^{4} | ) |

64 |

= | π | ( | 2 ^{4}-1.944^{4} | ( |

64 |

^{4}

For a uniform cantilever beam subjected to a single concentrated moment, *M*, at the free end, the maximum deflection at the free end is found with:

y _{max} = | M_{2}L^{2} |

2EI |

= | (50in.-lb)(10in.)^{2} | |

2 X10 X 10 ^{6 }lb | lb | X 0.08434 in.^{4} |

in. ^{2} |

For this problem, select a rotational spring that produces 1° of rotation from the concentrated moment, M_{z} = 50 in-lb. The corresponding spring rate, K_{θz}, that produces 1º of rotation at the discontinuity is then:

K_{θz} = 50 in.-lb/deg = 2,864.79 in.-lb/rad

With the rotational spring, estimate individual component deflections using conventional hand calculations found in the engineering literature and sum the total. For instance:

Displacement due to rotational spring, y_{1}

y_{1} = (9.5 in.) tan (1º) = 0.165823 in.

Displacement due to 9.5-in. cantilever portion, y_{2}

y _{2} = | (50in.-lb)(9.5in.)^{2} | |

2 X10 X 10 ^{6 }lb | lb | X 0.08434 in.^{4} |

in. ^{2} |

Displacement from the 0.5-in. cantilever portion, y_{3}

y _{3}= | (50in.-lb)(0.5in.)^{2} | |

2 X10 X 10 ^{6 }lb | lb | X 0.08434 in.^{4} |

in. ^{2} |

The sum of displacements is then:

Σ_{y} = y_{1} + y_{2} + y_{3} = 0.16851 in.

Now check this result with a FE model of the beam and joint. The FE solution using DOF Spring for maximum deflection shows y_{max} = 0.168496 in. at node N21, the end of the beam. Results from FEM and hand calculations show good agreement.

Substituting the CBush spring element for the rotational- spring element produces similar results, y_{max} = 0.168538 in. In both cases, the percent difference from hand calculations is less than 0.01%.

**A few cautions** Beam elements can be tricky to idealize because they rely on a local-coordinate system defined by the orientation of the beam in space. When modeling beam elements, first define the “direction” of the beam centroidal axis by connecting one grid point node to another, node “i” to node “j” in adjacent elements. Next, define an orientation vector for the beam by defining a reference node “k”. This “k-node” is used with “i” and “j” to define the beam element’s local coordinate system. Using the right-hand rule, orient the beam’s cross-sectional properties with respect to this local coordinate system. The definition of these “i”, “j” or “k” nodes need not be aligned in the same directions as the axis that define the global X, Y and Z directions.

By default, all six DOF for each beam end are connected. Some cases may call for disconnecting an element from one or more of these six degrees of freedom at each node. One can choose to release particular degrees of freedom from the ends of particular elements. When releasing specific degrees of freedom, pay close attention to the beam’s local coordinate system.

A** DOF spring** connects two nodal degrees of freedom independent of their orientation to each other. When defining DOF springs, be sure the particular DOF being addressed matches the nodal-definition-coordinate system that defines the nodes connecting each end of the DOF spring. Occasionally, it takes a little debugging to get this to work correctly, making it another reason to experiment with simple FEM with known solutions. It’s how full-time analysts get to know when they have modeled systems correctly or made a mistake. Generally, nodes connecting each end of the DOF spring must have the node definition and output-coordinate frames defined by the user. In the example problem, the cantilever beam is aligned with the global X, Y and Z directions. Consequently, DOF spring values correspond to the global X, Y, and Z as well.

**The CBush spring**, similar to a beam element, connects one grid point to another. When defining a CBush spring, be sure to define a vector orientation similar to the way one defines the orientation for beam elements.

**For further reading** Raymond J. Roark and Warren C. Young, *Formulas for Stress and Strain*, Chapter 7, Beams; Flexure of Straight Bars, Case 3a, page 101, McGraw-Hill Book Co., 1975.

**Try it yourself**

Several files that provide more detail on discontinuities can be downloaded from the hot link in the online version of this article at machinedesign.com. The file SpringJoints_HandCalcs.pdf contains hand calculations with summary arithmetic to estimate peak deflections due to a spring joint. Run notes and keystroke summaries are attached in Run- Notes_SpringJoints.pdf. This file contains details for defining and orientating elements using Femap. In addition, users can run the models on a demo version of Femap available at www.plm.automation.siemens.com/forms/femap_demo.shtml.

Downloads also include FEA models for use with the example problem. Models for Femap V9.3.1, or later, may be found in Cantilever_wDOF_Spring. mod and Cantilever_wCBush_Spring.mod. Models for MD Patran 2006r2 or later may be downloaded as input and run files Cantilever_wDOF_Spring_Input. bdf and Cantilever_wCBush_Spring_Input.bdf.