Chapter 23: Bolted Plates

23 Bolted Plates

Summary 426

Introduction 427

Solution Requirements 427

FEM Solutions 429

Modeling Tips 436

Input File(s) 436

Video 437

MD Demonstration Problems

CHAPTER 23426

SummaryTitle Chapter 23: Bolted Plates

Contact features • Deformable-deformable contact• No friction

Geometry

Material properties , , , , Linear

elastic material

Analysis type Quasi-static analysis

Boundary conditions Small plate is supported at one side. Normal contact conditions applied between the two plates and between the large plate and the bolt, glued contact between the small plate and the nut. Rigid rotation and translation of the plates is suppressed

Applied loads Load step 1: Bolt is fastened by pre-tension force .Load steps 2-4: Cyclic loading of plates. Two different cases:

• uniform pressure • thermal load, temperature increase

Element type 3-D solid 8-node linear elements

FE results 1. Deformed shape and von Mises stress distribution2. Plot of bolt forces

X

Y

Z

1

X Y

Z

4

XXXXXXXXXXXXXXXXXXXXXXXX YYYYYYYYY

ZZZZZZZZZZ

44

Units: mmLarge plate 60x20x6Small plate 20x20x2Bolt hole radius = 5Bolt shaft radius = 4Bolt head radius = 6Bolt head thickness = 2Nut thickness = 2Nut outer radius = 6

Eplates 210kN mm2= Ebolt 21kN mm2= plates bolt 0.3= = plates 105– C

1–=

F 200N=

P 0.125MPa=

T 50C=

427CHAPTER 23

Bolted Plates

IntroductionA small and a large steel plate are bolted together. Initially, the smaller plate is in full contact on one side with the larger plate. The opposite side of the smaller plate is supported. Furthermore, the bolt head is touching the larger plate and the nut is glued to the smaller plate. It is assumed that the material behavior for both the plates and the bolt is linear elastic.

In the first load step, the bolt is fastened by applying a pre-tension force ( ) to the bolt in the basic Z-direction. In three subsequent load steps, the bolt is locked (that is, further shortening of the bolt is suppressed) and the plates are subjected to cyclic loads. Two types of loads will be presented: a mechanical load that consists of a uniform pressure equal to applied to the larger plate and a thermal load in which temperature of the plates is

increased by .

Solution RequirementsTwo solutions, one involving a uniform pressure equal to applied to the larger plate and one involving

a temperature increase by of the two plates, are:

• Bolt shortening during fastening in the first load step• Bolt forces during the loading cycle• Bolt stresses

These solutions demonstrate:

• Bolt modelling• That the bolt force is largely unaffected by the applied pressure to the larger plate• That the bolt force increases with increasing temperature of the plates, due to thermal expansion

The analysis results are presented with linear elements.

Bolt ModelingIn various engineering applications, it is necessary to define a pre-stress in, for example, bolts or rivets before applying any other structural loading. A convenient way do this is via multi-point constraints. The idea is to split the element mesh of the bolt across the shaft in two disjoint parts, such that duplicate grid points appear at the cut, and to connect the duplicate nodes again by multi-point constraints (see Figure 23-1). The constraints are chosen such that an overlap or a gap can be created between the two parts in a controllable way. If the motion of the parts is somehow constrained in the direction in which the gap or overlap is created, then an overlap (a “shortening” of the bolt) will introduce a tensile (pre-)stress in each of the parts and a gap (an “enlongation” of the bolt) will result in a compressive stress.

The multi-point constraints have one slave and two master grid points. The slaves are the grid points at the cut from the bottom part of the bolt (see Figure 23-1). The first master grids are the corresponding grid points from the top part of the bolt on the other side of the cut. The second master in the constraints is a unique third grid point, called the control grid point of the bolt. This is often a free grid point (that is, not part of the element mesh) and is shared by all multi-point constraints on the cut.

F 200N=

P 0.125MPa=

T 50C=

P 0.125MPa=

T 50C=

MD Demonstration Problems

CHAPTER 23428

Figure 23-1 Pre-stressing a Structure by Creating an Overlap Between the Top and the Bottom Part Using Multi-Point Constraints.

The multi-point constraints impose the following constraint equations on the model:

.

in which , and are the displacement degrees of freedom of a grid point from the bottom part, its

corresponding grid from the top part and the control grid point, respectively. It immediately follows from this equation that is the displacement difference of the bottom and top grids and is equal to the size of the overlap or gap

between the parts. Hence, by enforcing the displacements of the control grid point, an overlap or gap of a particular size can be created between the two parts.

It can be shown (see, for instance, MSC.Marc 2010 Volume A: Theory and User Information, Chapter 9, Section “Overclosure Tying”), that if the multi-point constraints are set up as outlined above, the force on the control grid point equals the sum of the forces on the grid points from the bottom part as well as minus the sum of the forces on the grid points from the top part:

.

Hence, the force on the control grid point is the total force on the cross-section of the bolt. By applying a (pre-tension) force to that grid point, the total force on the cross-section can be prescribed. Moreover, if the shortening of the bolt is prescribed via an enforced displacement on the control grid point, then the reaction force on that grid point is equal to the total force on the cross-section of the bolt.

Note that both types of boundary conditions on the control grid point can be combined in a single analysis as demonstrated in this example. In the first load step, the pre-tension force will be applied to the control grid point of the bolt. This results in a certain amount of shortening of the bolt. At the end of the first load step, the amount of shortening is recorded and is kept constant in subsequent load steps, via a single point constraint on the control grid point.

control grid

bottom part

MPCs

(slave)

(first master)top grids

top part

mesh split

bottom grids(second master)

undeformed

F1,bot F2,bot

F2,top

Fcontroltop part

bottom part

F1,top

u1,top u2,top

u2,botu1,botucontrol

(overlap) ucontrol

deformed

ubot utop– ucontrol– 0=

ubot utop ucontrol

ucontrol

Fcontrol Fbot Ftop–= =

429CHAPTER 23

Bolted Plates

Figure 23-2 Element Mesh and Multi-Point Constraints applied in Target Solution with MD Nastran

There are two ways to define the multi-point constraints for bolt modeling in the bulk data: each constraint can be defined explicitly via the MPC option or the entire set of constraints can be defined via the BOLT option. The latter has been designed specially for bolt modeling and has several advantages over explicit MPCs:

• Provides a much more concise input than explicit MPCs;

• Generates all the required multi-point constraints on all displacement and rotational degrees of freedom automatically;

• Ensures continuity of the temperature field across the cut in the thermal passes of coupled analyses;

• Requires no special provisions in a contact analysis (see below).

FEM SolutionsA numerical solution has been obtained with MD Nastran’s SOL 400 for the element mesh shown in Figure 23-2 using 3-D solid linear elements. The bolt and the nut are assumed to be rigidly connected and are modeled as a single physical body. To fasten the bolt, the element mesh of the bolt is split into two parts across the shaft and the 41 grid point pairs on both sides of the cut are connected by multi-point constraints of the form discussed in the preceding section. Grid ID 1903 acts as the control grid of the bolt.

Two versions of the input are considered. In the first version, the BOLT option is used to generate the multi-point constraints on the cut. In the second version, the constraints are defined explicitly via the MPC option.

The BOLT option requires a bolt ID (5000), the ID of the control grid of the bolt (1903) and the grids at the cut from the top and bottom parts of the bolt. The latter must be entered pair-wise in the TOP and BOTTOM section of the option: the i-th TOP grid should correspond to the i-th BOTTOM grid.

BOLT 5000 1903

Note: The gap between the top and bottom parts of the bolt in the picture on the right is purely for visualization purposes. In reality, the gap is closed although the duplicate grids remain.

Bolt

Small plate

Nut

Large Plate

Grid 1903

MD Demonstration Problems

CHAPTER 23430

TOP 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 BOTTOM 341 353 365 377 389 401 413 425 437 449 461 473 485 497 1394 1406 1418 1430 1442 1454 1466 1478 1490 1502 1572 1584 1596 1608 1620 1632 1644 1656 1668 1680 1747 1759 1771 1783 1795 1807 1819

The equivalent input using explicit MPCs reads:

MPC 1 341 1 1.0 1862 1 -1.0 1903 1 -1.0MPC 1 341 2 1.0 1862 2 -1.0 1903 2 -1.0MPC 1 341 3 1.0 1862 3 -1.0 1903 3 -1.0MPC 2 353 1 1.0 1863 1 -1.0 1903 1 -1.0MPC 2 353 2 1.0 1863 2 -1.0 1903 2 -1.0MPC 2 353 3 1.0 1863 3 -1.0 1903 3 -1.0...$MPCADD 100 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

ContactThe main problem with the use of explicit MPCs is that in a contact analysis, the constraints may conflict with the multi-point constraints due to contact. Special provisions have to be made in the contact setup to avoid that the slave grids of the MPCs can come in contact with other contact bodies. Furthermore, due to the cut in the mesh, it is difficult for grid points of other contact bodies that touch the bolt surface, to slide across the cut from the bottom part of the bolt to the top part or vice versa. The BOLT option addresses both issues, provided that the two parts of the bolt are in the same contact body. Conflicts with contact constraints are avoided and grid points that touch the surface of the bolt can slide without difficulties across the cut.

For the present model, the two methods are compared. To avoid problems in the MPC version between the explicit MPCs and the contact constraints, the radius of the bolt shaft is slightly smaller than the radius of the holes in the plates, such that contact between the shaft and plates will not occur.

The three physical components of the model (the two plates and the bolt with the nut) have been selected as contact bodies. The contact bodies are identified as the set of elements in the respective components:

$ contact body: bolt and nutBCBODY 1 3D DEFORM 1BSURF 1 167 168 169 170 171 172 173...$ contact body: small plateBCBODY 2 3D DEFORM 2

431CHAPTER 23

Bolted Plates

BSURF 2 139 140 141 142 143 144 145...$ contact body: large plateBCBODY 3 3D DEFORM 3BSURF 3 1 2 3 4 5 6 7...

The two parts of the bolt are in same contact body (ID=1).

The BCTABLE entries shown below identify the admissible contact combinations, select the slave and master body for each combination, and set associated parameters. It is important to note that:

• The first contact body (bolt and nut) must be selected as the slave (or contacting) body. Since the contact algorithm detects contact between the grid points at the surface of the slave (or contacting) body and the faces of the elements at the surface of the master (or contacted) body, the body with the finer element mesh in the contact region generally should be selected as the slave body and the body with the coarser mesh as the master, as this results in “more points in contact” and thus a better description of the contact conditions than with the opposite definition. The ISEARCH entry is set to 1 to force search order from the slave body to the master.

• The bolt can touch the plates and the plates can touch each other.

• The IGLUE entry is set to 1 for contact between the nut and the smaller plate to activate glued contact conditions (that is, no sliding and no separation) between these two contact bodies.

BCTABLE 0 3 SLAVE 1 0. 0. 0. 1 1 0 0 MASTERS 2 SLAVE 1 0. 0. 0. 0 1 0 0 MASTERS 3 SLAVE 2 0. 0. 0. 0 1 0 0 MASTERS 3BCTABLE 1 3 SLAVE 1 0. 0. 0. 1 1 0 0 MASTERS 2 SLAVE 1 0. 0. 0. 0 1 0 0 MASTERS 3 SLAVE 2 0. 0. 0. 0 1 0 0 MASTERS 3

Materials and PropertiesThe 3-D solid elements with large strain capability available on MD Nastran SOL 400 are chosen by the PSOLID and PSLDN1 entries on the CHEXA option as shown below.

$ platesPSOLID* 1 1PSLDN1* 1 1$$ bolt and nutPSOLID* 2 2PSLDN1* 2 2

MD Demonstration Problems

CHAPTER 23432

The large strain capability and assumed strain formulation (for improved bending behavior) for these elements are activated via the NLMOPTS option.

NLMOPTS ASSM ASSUMED LRGSTRN 1

The two materials are isotropic and elastic with Young’s modulus, Poisson’s ratio and thermal expansion defined as:

$ platesMAT1* 1 2.100000E+05 3.000000E-01* 1.000000E+00 1.000000E-05$ bolt and nutMAT1* 2 2.100000E+04 3.000000E-01

Loads, Boundary Conditions and Load StepsThe loading sequence consists of four load steps. In the first load step. The pre-tension force in the basic Z direction is applied to the control grid point of the bolt via a FORCE option, as follows:

$ bolt-forceFORCE 1 1903 0 200. 0. 0. 1.

At the end of the load step, the shortening of the bolt due to the applied pre-tension force is recorded and kept constant in subsequent load steps by a single-point constraint on the displacement of the control grid in the basic Z direction:

$ bolt-lockSPC1 5 3 1903

Throughout the analysis, the displacements of the control grid in the basic X and Y directions are suppressed by a single-point constraint:

$ bolt-xySPC1 4 12 1903

In all four load steps, the full load is applied in a single increment. The nonlinear procedure used in the load steps is:

NLPARM 1 1 PFNT 1 50 UP NO+ .01 .01+ 0

Here, the PFNT option is selected to activate the pure Newton-Raphson iteration strategy. Convergence of the non-linear iteration process is checked on both displacements and forces, using tolerances equal to 0.01.

ResultsThe shortening of the bolt due to the pre-tension force applied in the first load step is listed in Table 23-1. The solution obtained with an equivalent MSC.Marc 2005r3 model is included for reference. This shortening is recorded at the end of the first load step and kept fixed in the subsequent load steps. It is apparent from this table that the MPC version and the BOLT version produce identical results.

433CHAPTER 23

Bolted Plates

Pressure LoadThe pressure load is applied in a cyclic fashion to the large plate in the final three load steps. The plate is loaded in load steps 2 and 4 and unloaded in load step 3. The deformed structure plot (magnification factor 500) as well as the equivalent von Mises stress distribution at the end of the final load step are shown in Figure 23-3. A plot of the bolt force in the basic Z direction is depicted in Figure 23-4. Note that in the first load step, the bolt load is the externally applied pre-tension force; whereas in subsequent load steps, the bolt load is the reaction force on the control grid point.

Figure 23-3 Deformed Structure Plot and von Mises Stress Distribution at Maximum Load Level Due to the Pressure Load (magnification factor = 500)

Table 23-1 Bolt Shortening During Fastening in the First Load Step

MD Nastran (MPC)MD Nastran

(BOLT) MSC.Marc 2005r3

bolt shortening 0.0054 0.0054 0.0054

MD Demonstration Problems

CHAPTER 23434

Figure 23-4 Bolt Forces During Loading Cycle by Pressure Load.

In Figure 23-4, the MD Nastran solution (blue dots) is compared with the solution obtained by MSC.Marc 2005 r3 (the solid line). The good agreement between the two solutions is apparent.

This plot demonstrates the well-known fact that the bolt force is unaffected by the pressure applied to the plate. Due to a slight bending of the larger plate under the pressure load, however, the bolt force is not exactly constant.

0

50

100

150

200

1 2 3 4

Load Step

MSC.Marc 2005 r3MD Nastran

n nn

n

n

Bol

t For

ce [N

]

435CHAPTER 23

Bolted Plates

Thermal LoadThe thermal load is applied in a cyclic fashion to both plates. The plates are heated in load steps 2 and 4 and cooled down in load step 3. The deformed structure plot (magnification factor 100) as well as the equivalent von Mises stress distribution at the end of the final load step are shown in Figure 23-5. A plot of the bolt force in the basic Z direction is shown in Figure 23-6. Again, the MD Nastran solution (blue dots) is compared with the solution obtained by MSC.Marc 2005 r3 (the solid line) and the agreement of the two solutions is apparent.

Figure 23-5 Deformed Structure Plot and von Mises Stress Distribution at Maximum Load Level Due to the Thermal Load (magnification factor = 100)

Figure 23-6 Bolt Forces During Loading Cycle by Thermal Load.

0

50

100

150

200

250

300

1 2 3 4

Bolt

Forc

e [N

]

Load Step

MSC.Marc 2005 r3MD R2 Nastran

n

n

n

n

n

MD Demonstration Problems

CHAPTER 23436

In this load case, the bolt force increases with increasing temperature due to thermal expansion of the plates. It decreases again to the pre-stress force after cooling down.

Modeling TipsMulti-point constraints provide a convenient way to fasten bolts. Either the shortening of the bolt or the total force in the cross-section of the bolt can be controlled via enforced displacements or forces on the control grid point of the bolt. These two types of boundary conditions can be combined in one simulation in which the bolt is first pre-stressed and then loaded by other mechanical or thermal loads.

The BOLT option provides a convenient way to generate the required multi-point constraints. It can be used conveniently in a contact analysis, provided that the two parts of the bolt are in the same contact body.

Input File(s)

File Description

nug_23p_bolt.dat Bolt pre-tension followed by cyclic pressure load (BOLT version)

nug_23p.dat Bolt pre-tension followed by cyclic pressure load (MPC version)

nug_23t_bolt.dat Bolt pre-tension followed by cyclic thermal load (BOLT version)

nug_23t.dat Bolt pre-tension followed by cyclic thermal load (MPC version)

437CHAPTER 23

Bolted Plates

VideoClick on the image or caption below to view a streaming video of this problem; it lasts approximately 58 minutes and explains how the steps are performed.

Figure 23-7 Video of the Above Steps

X

Y

Z

1

X Y

Z

4

XXXXXXXXXXXXXXXXXXXXXXX YYYYYYYYY

ZZZZZZZZ

44

Units: mmLarge plate 60x20x6Small plate 20x20x2Bolt hole radius = 5Bolt shaft radius = 4Bolt head radius = 6Bolt head thickness = 2Nut thickness = 2Nut outer radius = 6