Bolted Plates

Chapter 23: Bolted Plates

23

Bolted Plates



Summary



Introduction



Solution Requirements



FEM Solutions



Modeling Tips



Input File(s)



Video

437

426 427

429 436 436

427

426 MD Demonstration Problems CHAPTER 23

Summary Title

Chapter 23: Bolted Plates

Contact features

• Deformable-deformable contact • No friction

Geometry

Material properties

Units: mm Large plate 60x20x6 Small plate 20x20x2 Bolt hole radius = 5 Bolt shaft radius = 4 Bolt head radius = 6 Bolt head thickness = 2 Nut thickness = 2 Nut outer radius = 6

Y Z

Z

X X 1

Y 4

–5

E plates = 210kN  mm 2 , E bolt = 21kN  mm 2 ,  plates =  bo lt = 0.3 ,  plates = 10 C

–1

, 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 F = 200N . Load steps 2-4: Cyclic loading of plates. Two different cases: • uniform pressure P = 0.125MPa • thermal load, temperature increase T = 50C

Element type

3-D solid 8-node linear elements

FE results

1. Deformed shape and von Mises stress distribution 2. Plot of bolt forces

CHAPTER 23 427 Bolted Plates

Introduction A 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 ( F = 200N ) 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 P = 0.125MPa applied to the larger plate and a thermal load in which temperature of the plates is increased by T = 50C .

Solution Requirements Two solutions, one involving a uniform pressure equal to P = 0.125MPa applied to the larger plate and one involving a temperature increase by T = 50C 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 Modeling In 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.

428 MD Demonstration Problems CHAPTER 23

top part

top part mesh split

top grids (first master)

MPCs control grid (second master)

bottom grids (slave)

bottom part undeformed Figure 23-1

F1,bot

Fcontrol F2,bot

u1,bot

u2,bot

ucontrol

(overlap) ucontrol u1,top

u2,top

F1,top

F2,top

bottom part deformed

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: u bo t – u t op – u control = 0 .

in which u bo t , u top and u control 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 u control 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: F control =

 F bot

= –  F top .

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.

CHAPTER 23 429 Bolted Plates

Grid 1903 Bolt Large Plate

Small plate Nut

Figure 23-2 Note:

Element Mesh and Multi-Point Constraints applied in Target Solution with MD Nastran 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.

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 Solutions A 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

430 MD Demonstration Problems CHAPTER 23

TOP

1862 1869 1876 1883 1890 1897 341 425 1394 1478 1620 1759

BOTTOM

1863 1870 1877 1884 1891 1898 353 437 1406 1490 1632 1771

1864 1871 1878 1885 1892 1899 365 449 1418 1502 1644 1783

1865 1872 1879 1886 1893 1900 377 461 1430 1572 1656 1795

1866 1873 1880 1887 1894 1901 389 473 1442 1584 1668 1807

1867 1874 1881 1888 1895 1902 401 485 1454 1596 1680 1819

1862

1

-1.0

1862

2

-1.0

1862

3

-1.0

1863

1

-1.0

1863

2

-1.0

1863

3

-1.0

1868 1875 1882 1889 1896 413 497 1466 1608 1747

The equivalent input using explicit MPCs reads: MPC

1

MPC

1

MPC

1

MPC

2

MPC

2

MPC

2

... $ MPCADD

100 8 16 24 32 40

341 1903 341 1903 341 1903 353 1903 353 1903 353 1903 1 9 17 25 33 41

1 1 2 2 3 3 1 1 2 2 3 3 2 10 18 26 34

1.0 -1.0 1.0 -1.0 1.0 -1.0 1.0 -1.0 1.0 -1.0 1.0 -1.0 3 11 19 27 35

4 12 20 28 36

5 13 21 29 37

6 14 22 30 38

7 15 23 31 39

Contact The 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 nut BCBODY 1 3D DEFORM BSURF 1 167 168 ... $ contact body: small plate BCBODY 2 3D DEFORM

1 169 2

170

171

172

173

CHAPTER 23 431 Bolted Plates

BSURF 2 139 140 ... $ contact body: large plate BCBODY 3 3D DEFORM BSURF 3 1 2 ...

141

142

143

144

145

3 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

BCTABLE

0 SLAVE

3

1 1 MASTERS 2 SLAVE 1 1 MASTERS 3 SLAVE 2 1 MASTERS 3 1 SLAVE 1 1 MASTERS 2 SLAVE 1 1 MASTERS 3 SLAVE 2 1 MASTERS 3

0. 0

0

0. 0

0

0. 0

0

0.

0.

1

0.

0.

0

0.

0.

0

0.

0.

1

0.

0.

0

0.

0.

0

3 0. 0

0

0. 0

0

0. 0

0

Materials and Properties The 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. $ plates PSOLID* 1 PSLDN1* 1 $ $ bolt and nut PSOLID* 2 PSLDN1* 2

1 1 2 2

432 MD Demonstration Problems CHAPTER 23

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: $ plates MAT1* 1 * 1.000000E+00 $ bolt and nut MAT1* 2

2.100000E+05 1.000000E-05

3.000000E-01

2.100000E+04

3.000000E-01

Loads, Boundary Conditions and Load Steps The 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-force FORCE 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-lock SPC1 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-xy SPC1 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 .01 0

1 .01

PFNT

1

50

UP

NO

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

Results The 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.

CHAPTER 23 433 Bolted Plates

Table 23-1

Bolt Shortening During Fastening in the First Load Step MD Nastran (MPC)

MD Nastran (BOLT)

MSC.Marc 2005r3

0.0054

0.0054

0.0054

bolt shortening

Pressure Load The 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)

434 MD Demonstration Problems CHAPTER 23

200

n

n

n

n

Bolt Force [N]

150

100

50

MSC.Marc 2005 r3 MD Nastran

0 1

2

3

n

4

Load Step

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.

CHAPTER 23 435 Bolted Plates

Thermal Load The 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) n

n

300

250

Bolt Force [N]

200

n

n

150

100

50 MSC.Marc 2005 r3 MD R2 Nastran

0 1

2

3

n

4

Load Step

Figure 23-6

Bolt Forces During Loading Cycle by Thermal Load.

436 MD Demonstration Problems CHAPTER 23

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 Tips Multi-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)

CHAPTER 23 437 Bolted Plates

Video Click 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. Units: mm Large plate 60x20x6 Small plate 20x20x2 Bolt hole radius = 5 Bolt shaft radius = 4 Bolt head radius = 6 Bolt head thickness = 2 Nut thickness = 2 Nut outer radius = 6

Figure 23-7

Y Z

Z

Video of the Above Steps

X X 1

Y 4