Bolt-joint modeling with finite-element analysis uses fastener stiffness to develop optimal load distribution through a part.

**Edited by Martha K. Raymond**

**Alexander Rutman**

**Lead Stress Engineer**

**Joseph Bales-Kogan**

**Stress Analyst**

**Boeing Commercial Airplane Group**

**Wichita, Kans.**

In most designs, load-distribution calculations can literally make or break the part. Distributing loads through a structure typically depends not only on the mechanical properties and dimensions of materials but also on the stiffness of the fasteners connecting components. Traditionally, engineers have used analysis based on single spring rate calculations to model fastened joints. The methods determine joint flexibility for a particular combination of fastener and plate properties. This approach is adequate for one or two dimensional joint models, but can’t describe 3D joint models when fasteners and joints don’t share the same reference plane.

However, a finite-element model provides more realistic load distribution between components of the model. The new approach considers the effect of independent components on joint flexibility and represent assemblies with a complete platefastener system of the joint.

In 3D joint representation, structural components are modeled by plates with membrane, transverse shear, and bending stiffness. The plates are positioned at the components respective midplanes without offset. Fasteners are modeled with the actual properties. A coordinate system with one of the axes parallel to fastener axial direction must exist to orient modeling elements. Using this coordinate system, fastener- plate interaction is modeled by spring elements representing bearing stiffness of the joint. Each plate is connected rigidly to the fastener in its axial and both bending directions.

When beginning a fastener joint-stiffness analysis, there are three basic stiffness components to consider: plate-bearing stiffness, fastener-bearing stiffness, and fastener shear and bending stiffnesses. Plate *i* bearing flexibility is calculated as:

where *E _{cpi}* is the plate i material compression modulus and

*t*is the plate i thickness. The fastener bearing flexibility at plate

_{pi}*i*is also included and calculated by:

where *Ecb* is the fastener material compression modulus.

The combined fastener and plate bearing flexibility at plate *i* is *C _{bi}* =

*Cb*+

_{pi}*C*and the combined bearing stiffness at plate i is:

_{bbi}

The fastener shear flexibility between mid planes of plates *i* and *i* + 1 is determined by:

where *G _{b}* is the fastener material shear modulus;

*t*,

_{pi}*t*is the thickness of plates

_{p(i + 1)}*i*and

*i*+ 1 in the joint and

*A*is the fastener cross-sectional area described by:

_{b}

where *D _{b}*is the fastener diameter.

The fastener shear stiffness between mid planes of plates *i* and *i* + 1 is calculated as

**FASTENER-JOINT MODELING**

The new approach to fastener joint modeling improves fastener behavior in a joint and generates easily readable output that doesn’t require a lot of postprocessing.

A plate-fastener system takes into account three parameters including: elastic bearing deformations of the plate and fastener at the contact surface; bending and shear deformations of fastener shanks; and joint-displacement compatibility between fastener and connected plates.

The modeling process includes two basic steps. The first one models fasteners separately and then models the interaction between the fastener and plates.

In MSC/Nastran by MacNeal- Schwendler, a finite value of fastener shear rigidity *A _{b}G_{b}* is created by entering the appropriate value of

*G*for the fastener material, calculation of the actual cross-sectional properties, and giving area factors for shear value of

_{b}*K*= 1.

A fastener is modeled by beam elements with cross-sectional properties calculated as:

where *D _{b}* is the fastener diameter and

*I*and

_{1}*I*are moments of inertia of the fastener.

_{2} Then the required fastener shear flexibility *C _{s (i,i + 1)}* is determined by:

To define a fastener shear plane and axial direction correctly, you must have a coordinate system with one of its axes parallel to the fastener axis. Points defining connectivity between plates and fasteners with the same orientation reference this coordinate system.

Verify compatibility of fastener-plate system displacements using the following rules of thumb: A plate can’t slide along a fastener, and a plate and fastener have the same slope under bending. Then, the fastener is connected to each plate of the joint by spring elements with stiffness:

The software program called FEM joint analyzes fastener joint stiffness and generates grid points and elements for modeling fasteners and by automatically determining interface conditions between fasteners and plates, for integration into MSC/Nastran FEM.

The approach can be applied to different types of models, by changing values that affect joint flexibility. However, it doesn’t account for fastener pretension and fit.