Chapter 50: Large Rotation Analysis of a Riveted Lap Joint

50 Large Rotation Analysis of a Riveted Lap Joint

Summary 1038

Introduction 1039

Modeling Details 1040

Solution Procedure 1045

Results 1046

Modeling Tips 1049

Input File(s) 1049

Video 1049

MD Demonstration Problems

CHAPTER 501038

Summary

Title Chapter 50: Large Rotation Analysis of a Riveted Lap Joint

Features • Use an empirical formula to characterize the rivet stiffness• Compare point-to-point and patch-to-patch connection types• Compare CFAST and CWELD patch-to-patch connection types• Demonstrate large rotation capability of the connector elements• Hookean, isotropic material

Geometry

Material properties ,

Analysis characteristics Quasi static analysis using geometric nonlinearity due to large displacements and large rotations

Boundary conditions • Left end of the lower plate clamped• Symmetry conditions on the strip edges

Applied loads • Apply a total tensile load of 2400 N to the right end of the upper plate, which is normal to this edge and parallel to the plate.

• Prior to this loading make a rigid body rotation of 45° about the global y-axis

Element types CQUAD4, CWELD, CFAST, or CBUSH

FE results • Equivalent stress in lap joint model with patch-to-patch CWELD/PWELD

• Load transfer through the rivets• Deformed shape of the lap joint

1 2 3

rivet diameter = 4Units: mm

rivet pitch = 20

plate length = 160plate overlap = 60plate thickness = 1.2

E 60GPa= 0.3=

1039CHAPTER 50

Large Rotation Analysis of a Riveted Lap Joint

IntroductionThis example demonstrates the modeling and analysis of a lap joint. Two plates are joined using a riveted connection. Three methods of modeling the rivets are considered, resulting in three different analysis models. In the first two, the rivets are modeled with bushing elements since their flexibility is given by an empirical expression. They are connected to the plates using a point-to-point or a patch-to-patch connection. The third method models the rivets with beam elements and connects them to the plates using patch-to-patch connections.

The first method uses a point-to-point connection and requires the bushing elements to be defined explicitly as CBUSH elements, together with its grids. The grids of the bushing elements need to coincide with grids of the plate elements, so this imposes a limitation on how the plates can be meshed, since plate grids must be present at locations where a connection is desired. Furthermore this method leads to a strongly localized load transfer, especially when the plate mesh is relatively fine.

The second method uses a patch-to-patch connection, which is modeled using CFAST. This method generates the bushing elements internally and does not require their grids to be coincident with plate grids. In addition to the bushing element, a set of constraints is generated internally to connect the bushing grids to the plate elements on each side of the connection. This eliminates the need of nearly congruent meshes on both sides with grids at the location of the connection.

The third method uses a patch-to-patch connection, which is modeled using CWELD. This method internally generates beam elements instead of bushing elements, but the way of connecting the beam grids to the plates is the same as for CFAST. In this case, the stiffness of the rivets is given by the standard beam stiffness formulations for a beam with circular cross-section having linear elastic material behavior.

The lap joint has three rows of rivets in the loading direction. For this analysis only, a strip (one rivet pitch of 20 mm wide) of the lap joint is modeled with proper symmetry boundary conditions along the edges of the strip that are parallel to the xz-plane.

The shear flexibility (see Vlieger, H., Broek, D., “Residual Strength of Cracked Stiffened Panels, Built-up Sheet Structure”, Fracture Mechanics of Aircraft Structure, AGARD-AG-176, NATO, London, 1974) is calculated as follows:

the axial rivet stiffness is calculated using a simple formula:

These values are entered as the translational stiffness values of the bushing elements. Their rotational stiffness values are assumed to be zero, but a small torsional stiffness is added to avoid singularities. Beam elements have bending and torsional stiffness given through their formulation, so there is no risk of singularities

Cs1

Ervd----------- 5 0.8

Ervd

Epl tpl--------------

Ervd

Eputpu----------------+

+ 4.3x10 5– mmN

---------= =

KaEA

L 2.4mm= --------------------------------- 314160

Nmm---------= =

MD Demonstration Problems

CHAPTER 501040

Modeling DetailsA numerical solution has been obtained with MD Nastran's solution sequence 400 performing a nonlinear static analysis. The details of the finite element model, the material, load, and boundary conditions and the solution procedure are discussed below.

The case control section of the input contains the following options for a nonlinear analysis:

TITLE = MD Nastran job with connectorsSUBTITLE = lap joint with 3 rivets modeled by CWELDLABEL = riveted lap jointSET 1 = 337,338,339SET 2 = 354,365,376,387,398,409,420SET 3 = 1,12,23,34,45,56,67DISPLACEMENT(SORT1,REAL)=ALLSPCFORCES(SORT1,REAL)=3OLOAD(SORT1,REAL)=2STRESS(SORT1,REAL,VONMISES,BILIN)=ALLFORCE(SORT1,REAL,VONMISES,BILIN)=1SUBCASE 1 TITLE=SC1$ Tensile load in (1,0,0) direction STEP 1 ANALYSIS = NLSTATIC NLPARM = 2 SPC = 2 LOAD = 2SUBCASE 2 TITLE=SC2$ Rigid body rotation over -45 degrees about y-axis STEP 1 ANALYSIS = NLSTATIC NLPARM = 1 SPC = 2 LOAD = 50$ Tensile load in (1,0,1) direction STEP 2 ANALYSIS = NLSTATIC NLPARM = 2 SPC = 2 LOAD = 20

The analysis contains two subcases essentially analyzing the same type of loading but in different spatial positions. The first subcase performs one step by applying the tensile load in x-direction. The second subcase performs two steps: the first step rigidly rotates the lap joint through 45° about the model y-axis and the second step applies the tensile load in this rotated position. It is clear that the CBUSH or CBEAM forces in the connector elements as well as the stress state in the plates at the end of each subcase must be the same, thus illustrating the large displacement capability of these connections. Each step defines a nonlinear static analysis via ANALYSIS, has a definition of convergence control via NLPARM, fixed displacements (or single point constraints) via SPC, forced displacements (in this case a rotation) and applied loads via LOAD. The displacement and stress results and other output requests for the .f06 (output) file apply to both subcases. Some output requests are limited to sets via the use of SET.

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Large Rotation Analysis of a Riveted Lap Joint

The mesh of the lap joint is shown in Figure 50-1 where each plate is meshed by 28 x 6 CQUAD4 elements with 18x6 elements in the overlap region. Figure 50-1 also displays a zoomed in view of one of the rivets in a patch-to-patch connection and a top view of the overlap region displaying the locations of the auxiliary grids in the connection.

Figure 50-1 Finite Element Mesh of the Lap Joint and Locations of the Rivets

Large displacement effects are included in the nonlinear analysis using the option:

PARAM LGDISP 1

This parameter is needed to account for all geometrically nonlinear effects and is essential even if no large rigid body rotation is applied prior to loading of the joint.

Plate Element ModelingThe standard options to define the element connectivity, the grid locations and the element properties are used in the bulk data section of the input:

$ Elements and Element Properties for region : lower_platePSHELL 1 1 1.2 1 1$ Elements in: "lower_plate"CQUAD4 1 1 1 2 13 12CQUAD4 2 1 2 3 14 13...$ Elements and Element Properties for region : upper_platePSHELL 2 1 1.2 1 1$ Elements in: "upper_plate"CQUAD4 169 2 211 212 231 230CQUAD4 170 2 212 213 232 231

MD Demonstration Problems

CHAPTER 501042

...$ Nodes of the Entire ModelGRID 1 0. 0. 0.GRID 2 10. 0. 0....

Modeling the ConnectionsThe input for the three different methods of the connection is summarized below:

Connection method 1: Explicitly define the CBUSH elements and their properties to make point-to-point connections between the plates.

$ Connector elements and properties, point-to-pointPBUSH 3 K 3.1416E52.3226E42.3226E4 100.0CBUSH 337 3 138 271 1.0 0.0 0.0CBUSH 338 3 144 277 1.0 0.0 0.0CBUSH 339 3 150 283 1.0 0.0 0.0

Each CBUSH element has two grids entered in fields 4 and 5 and references its properties through the property ID in field 3. In this application, the local x-direction of the CBUSH element is from the first grid to the second. Fields 6,7, and 8 define an orientation vector lying in the x-y plane (similar to the CBEAM element). The properties are entered through PBUSH and only stiffness values are entered in this model. The K in field 3 indicates that the next values are stiffness values. The first three are the translational stiffness values: first the axial stiffness followed by two shear stiffness values. The next three are the rotational stiffness values of which only the first (.e., the torsion value) has been entered so the bending values are zero.

Connection method 2: Define patch-to-patch connections between the plates using bushing elements generated through CFAST and their properties through PFAST.

$ Connector elements and properties, patch-to-patchPFAST 3 4. -1 3.1416E52.3226E42.3226E4 100.0CFAST 337 3 PROP 1 2 421CFAST 338 3 PROP 1 2 422CFAST 339 3 PROP 1 2 423

Each CFAST fastener element internally generates a CBUSH element and a number of RBE3 elements to connect the CBUSH grids through a number of auxiliary grids to the plates. The approximate location of each fastener is entered as the GS grid in field 7 of the CFAST input. The two plates on each side of the connection are identified through their property IDs in fields 5 and 6 and the connection method PROP is specified in field 4. Each CFAST references its properties through a PFAST property ID entered in field 3. The properties of the fastener are defined in the PFAST input. The fastener diameter is entered in field 3 and is used to compute the locations of the auxiliary grids. Fields 4 and 5 make specifications about the element coordinate system of the internally generated CBUSH element. The default is a local system with its first direction from the first grid to the second of the CBUSH element. Field 6, 7, and 8 specify the translational stiffness values, where the first is the axial stiffness (in the element local x-direction) and the next two are the shear stiffness values. The next three fields specify the rotational stiffness values of which only the first (i.e., the torsion value) has been entered, so the bending values are zero. The GS grids are used to determine the end node locations of the CBUSH elements. The surface on each side of a connection is identified by a PSHELL property ID. The nearest projection point of the GS grid on the shell elements sharing this property ID defines a grid point of the internally generated CBUSH element. Four auxiliary grids are positioned around each projection point

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forming a square auxiliary patch. The connection is established by connecting the CBUSH grids to the auxiliary patches with RBE3 elements and connecting the auxiliary grids to the plate structure with RBE3 elements. Thus each fastener involves one CBUSH and ten RBE3 elements which are being generated internally. Any unspecified CBUSH grids and the auxiliary grids are also generated internally.

Connection method 3: Define patch-to-patch connections between the plates using beam elements generated through CWELD and their properties through PWELD.

$ Connector elements and properties, patch-to-patchPWELD 3 2 4.CWELD 337 3 421 PARTPAT 1 2CWELD 338 3 422 PARTPAT 1 2CWELD 339 3 423 PARTPAT 1 2

Each CWELD weld element internally generates a CBEAM element and a number of RBE3 elements to connect the CBEAM grids through a number of auxiliary grids to the plates. The approximate location of each weld is entered as the GS grid in field 4 of the CWELD input. The two plates on each side of the connection are identified through their property IDs in fields 12 and 13 (i.e. fields 2 and 3 of the second input line) and the connection method PARTPAT is specified in field 5. Each CWELD references its properties through a PWELD property ID entered in field 3. The properties of the weld are defined in the PWELD input. The weld diameter is entered in field 4 and is used to compute the locations of the auxiliary grids and the cross-section properties of the beam. The weld material data are referenced through a material ID in field 3. The GS grids are used to determine the end node locations of the CBEAM elements. The surface on each side of a connection is identified by a PSHELL property ID. The nearest projection point of the GS grid on the shell elements sharing this property ID defines a grid point of the internally generated CBEAM element. Four auxiliary grids are positioned around each projection point forming a square auxiliary patch. The connection is established by connecting the CBEAM grids to the auxiliary patches with RBE3 elements and connecting the auxiliary grids to the plate structure with RBE3 elements. Thus each weld involves one CBEAM and ten RBE3 elements which are being generated internally. Any unspecified CBEAM grids and the auxiliary grids are also generated internally.

The internally generated grids get IDs with high offsets w.r.t. the grids entered in the input. In a similar way, internally generated RBE3s get IDs with high offsets with regard to the elements entered in the input. The internally generated CBUSH or CBEAM elements retain the element ID of the CFAST or CWELD from which they originate.

Material ModelingThe isotropic, Hookean elastic material properties of the plates and rivets are defined using the following MAT1 options:

$ Material Record : plate_materialMAT1 1 60000. .3$ Material Record : rivet_materialMAT1 2 60000. .3

The Young's modulus is taken to be 60000 MPa with a Poisson's ratio of 0.3.

MD Demonstration Problems

CHAPTER 501044

Loading and Boundary ConditionsThe clamped condition for the left side of the lower plate constrains all six degrees of freedom of the grids on this side:

$ Displacement Constraints of Load Set : clamped edgesSPC1 1 123456 1 12 23 34 45 56 67

The symmetry condition for both edges of the strip (lower and upper plate) constrains the y-displacement and the x- and z-rotations of all grids on these edges:

$ Displacement Constraints of Load Set : symmetry edgesSPC1 3 246 1 THRU 11SPC1 3 246 67 THRU 77SPC1 3 246 79 THRU 96SPC1 3 246 193 THRU 229SPC1 3 246 325 THRU 343SPC1 3 246 345 THRU 354SPC1 3 246 411 THRU 420

The two conditions are combined in SPCADD, so they can be activated simultaneously in the two subcases:

$ Displacement constraints for both subcasesSPCADD 2 1 3

The loading in subcase 1 at the right side of the upper plate is applied as concentrated forces in (1,0,0) direction to the grids on this side. The corner grids only carry half the force, so the loading represents a uniformly distributed load over the edge on this side. The FORCE definitions are combined in one LOAD definition with SID 2:

$ Nodal Forces of Load Set : tensile_load in Subcase 1, Step 1FORCE 3 365 0 400. 1. 0. 0.FORCE 3 376 0 400. 1. 0. 0.FORCE 3 387 0 400. 1. 0. 0.FORCE 3 398 0 400. 1. 0. 0.FORCE 3 409 0 400. 1. 0. 0.$ Nodal Forces of Load Set : tensile_load_corner in Subcase 1, Step 1FORCE 1 354 0 200. 1. 0. 0.FORCE 1 420 0 200. 1. 0. 0.$ Loads for Subcase 1:LOAD 2 1. 1. 1 1. 3

The loading in step 1 of subcase 2 consists of a forced rigid body rotation of 45° about the y-axis of the model. In step 2, this rotated position must be retained but must be combined with the external forces at the right end of the upper plate. Hence the repetition of the forced rotation with two different SIDs (50 in step 1, and 20 in step 2):

$ Rigid rotation about y-axis in Subcase 2, Step 1SPCD,50, 1,5,-0.7854SPCD,50,12,5,-0.7854SPCD,50,23,5,-0.7854SPCD,50,34,5,-0.7854SPCD,50,45,5,-0.7854SPCD,50,56,5,-0.7854SPCD,50,67,5,-0.7854$ Rigid rotation about y-axis in Subcase 2, Step 2

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Large Rotation Analysis of a Riveted Lap Joint

SPCD,20, 1,5,-0.7854SPCD,20,12,5,-0.7854SPCD,20,23,5,-0.7854SPCD,20,34,5,-0.7854SPCD,20,45,5,-0.7854SPCD,20,56,5,-0.7854SPCD,20,67,5,-0.7854

The loading in step 2 of subcase 2 at the right side of the upper plate is applied as concentrated forces in (1,0,1) direction to the grids on this side. The corner grids only carry half the force, so the loading represents a uniformly distributed load over the edge on this side. The FORCE definitions are combined in one LOAD definition with SID 20:

$ Nodal Forces of Load Set : tensile_load in Subcase 2, Step 2FORCE,30,365,0,400.,0.707107,0.0,0.707107FORCE,30,376,0,400.,0.707107,0.0,0.707107FORCE,30,387,0,400.,0.707107,0.0,0.707107FORCE,30,398,0,400.,0.707107,0.0,0.707107FORCE,30,409,0,400.,0.707107,0.0,0.707107$ Nodal Forces of Load Set : tensile_load_corner in Subcase 2, Step 2FORCE,10,354,0,200.,0.707107,0.0,0.707107FORCE,10,420,0,200.,0.707107,0.0,0.707107$ Loads for Subcase 2, Step 2:LOAD 20 1. 1. 10 1. 30

Solution ProcedureThe nonlinear procedure used is defined through the following NLPARM entry:

NLPARM 1 45 PFNT 25 U NO,1.0E-4,1.0E-4NLPARM 2 10 PFNT 25 UP NO,1.0E-4,1.0E-4

PFNT represents the “Pure” Full Newton Raphson technique where the stiffness is updated every iteration. KSTEP (the field following PFNT) is left blank and in conjunction with PFNT, it indicates that stiffness needs to be updated between the end of a load increment and the start of the next load increment. 25 is the maximum number of allowed recycles for every increment. U indicates that convergence testing will be done based on the displacement error. UP indicates that convergence testing will be done based on the displacement error and the load equilibrium error. NO indicates that result output will be produced at the end of every step. The second line of NLPARM indicates that tolerances of 0.0001 will be used for convergence checking.

The number of increments is provided in the 3rd field of the NLPARM option and since no adaptive load stepping has been activated all increments will be of equal size. The NLPARM with ID = 1 is used to control the rigid body rotation of 45° in step 1 of subcase 2. Thus, each increment makes a rotation of 1°. Since the motion is a rigid body motion, the lap joint remains stress free and there are no loads acting on the joint. Therefore, only displacement convergence testing is done during this phase, no load convergence testing. The NLPARM with ID = 2 is used to control the loading of the lap joint in step 1 of subcase 1 and step 2 of subcase 2. Thus, the total load is applied in ten equal load increments. During this phase, the lap joint no longer remains stress free and both displacement and load convergence testing are activated.

MD Demonstration Problems

CHAPTER 501046

ResultsFigure 50-2, Figure 50-3, and Figure 50-4 show the equivalent stress distribution for the three connection methods. Only the overlap region is shown here because the stresses near the rivets are of primary interest. It can be observed that the maximum equivalent stress in the point-to-point connection displayed in Figure 50-2 is higher than the maximum equivalent stress in the patch-to-patch connection shown in Figure 50-3. This is as expected since the patch-to-patch connection provides a less localized load transfer in the connection. A difference can also be observed between the CFAST and CWELD connections. With the CFAST connection, there is direct control over the stiffness values in the different deformation modes of the element (axial, shear, bending and torsion deformations); whereas with the CWELD, these stiffness values are determined by the underlying beam formulations. There is clearly an advantage for the CFAST when the stiffness values are known from empirical expressions based on a detailed investigation of the connections in question. The stress state shown is at the end of step 1 in subcase 1. It can easily be verified that the stress state at the end of step 2 in subcase 2 is the same, illustrating the proper handling of the large rotation.

Figure 50-2 Equivalent Stress in Lap Joint Model with Point-to-point CBUSH/PBUSH

1047CHAPTER 50

Large Rotation Analysis of a Riveted Lap Joint

Figure 50-3 Equivalent Stress in Lap Joint Model with Patch-to-patch CFAST/PFAST

Figure 50-4 Equivalent Stress in Lap Joint Model with Patch-to-patch CWELD/PWELD

MD Demonstration Problems

CHAPTER 501048

Figure 50-5 Deformed Configuration of the Overlap Region in the Patch-to-patch Connection with CFAST/PFAST

Table 50-1 lists the shear force in the three rivets for the three connection methods. These results are taken from the output at the end of step 1 of subcase 1. It can easily be verified that these results at the end of step 2 in subcase 2 are the same.

Figure 50-5 shows the deformed configuration of the overlap region in the patch-to-patch connection with CFAST at the end of step 1 in subcase 1. Clearly an effect of geometrical nonlinearity can be observed as the joint shows the tendency to align the lower and upper plates in the direction of the external load. The plot shows the deformations in true scale. It can easily be verified that the other two models display a similar behavior.

Table 50-1 Rivet Shear Forces

FRivet-1 (N) FRivet-2 (N) FRivet-3 (N)

point-to-point: CBUSH/PBUSH

825 748 825

patch-to-patch: CFAST/PFAST

843 713 843

patch-to-patch: CWELD/PWELD

919 561 919

1049CHAPTER 50

Large Rotation Analysis of a Riveted Lap Joint

Modeling TipsFor geometrically complicated structures, modeling riveted joints (or similar types of spot connections) with point-to-point connections using CBUSH elements (or other line type elements like CBEAM) can be a labor intensive task since it requires meshes with hard points at the rivet locations. Making such congruent or near congruent meshes may prove to be very difficult. Moreover this type of connection creates stress singularities at the point of connection, because of the highly localized load transfer. CFAST and CWELD connections can eliminate these drawbacks, since more grids near the point of connection are involved in the load transfer. The patch-to-patch type connection methods involving auxiliary patches are preferred when the area of the connector element is large with respect to the size of the element faces to which the connection is made. In general this improves the accuracy of the load transfer between the connected surfaces.

CFAST has more flexibility to define the mechanical properties, because the stiffness values for the different deformation modes of the element (i.e. axial, shear, bending, and torsion deformation) can be specified independently. With CWELD, the stiffness values follow from the underlying beam formulations.

Input File(s)

VideoClick on the image or caption below to view a streaming video of this problem; it lasts approximately 30 minutes and explains how the steps are performed.

Figure 50-6 Video of the Above Steps

File Description

nug_50a.dat Input for the point-to-point connection with CBUSH/PBUSH

nug_50b.dat Input for the point-to-point connection with CFAST/PFAST

nug_50c.dat Input for the patch-to-patch connection with CWELD/PWELD

1 2 3

rivet diameter = 4Units: mm

rivet pitch = 20

plate length = 160plate overlap = 60plate thickness = 1.2