Chapter 63: Beam Reinforced Shell Structure using Offsets

63 Beam Reinforced Shell Structure using Offsets

Summary 1187

Introduction 1188

Modeling Details 1188

Solution Procedure 1191

Results 1192

Modeling Tips 1197

Input File(s) 1197

1187CHAPTER 63

Beam Reinforced Shell Structure using Offsets

SummaryTitle Chapter 63: Beam Reinforced Shell Structure using Offsets

Features • Case 1 – Reinforced shell structure with beam shell offsets• Case 2 – Reinforced shell structure with RBE2 elements

Geometry

• Plate Structure: • Plate1: Length 4000mm x Breadth 4000mm x Thick 70 mm• Plate2: Length 2000mm x Breadth 4000mm x Thick 35 mm• Beam1: Length 4000mm x Diameter 100mm x Thick 25 mm• Beam2: Length 4000mm x Diameter 125mm x Thick 40 mm

Material properties Elastic-perfectly plastic material for beams and shells

E = 2.1e4 N/mm2, , with yield stress σ y = 40 N/mm2 (for both case-1 and case-2)

Analysis characteristics Case-1: Nonlinear analysis of pressure loaded reinforced structure of shell with beams using in-built shell offset and beam offset

Case-2: Nonlinear analysis of pressure loaded reinforced structure of shell with beams using RBE2 constraints

Boundary conditions For case-1 and Case-2, fixed conditions on one side

Applied loads For case-1 and Case-2, pressure on the top surface of the shell

Element type • 4 node thick shell element• 2 node beam

FE results • Deformed shape• Stress plot

0.3=

MD Demonstration Problems

CHAPTER 631188

IntroductionThis release of MD Nastran incorporates enhanced capabilities of beams and shells with offsets. The offset option had been included in earlier versions of MD Nastran, which allows elements being offset from the external grid points to element connection points, with some limitations. This version eliminates the limitations of some characteristics not being accounted for in case of offset, viz. differential stiffness (for buckling analysis); effects of thermal, pressure and gravity loads; mass matrix computation; etc. The aim of this chapter is to demonstrate the various features available in MD Nastran regarding in-built beam/shell offsets, which can be employed to analyze beam/shell structures.

The problem presented here is a stiffened plate with reinforcements, fixed at one end and subjected to uniform pressure load. In analyzing such cases, it is common to model the beams and shells at a geometric location that is different from the actual physical location. Such cases are common when shells or beams of varying thicknesses are adjacent to each other and the top/bottom shell surfaces or beam flanges are to be aligned with each other. In such cases, it is convenient to model all the shell nodes at the mid-surface of one of the shells or the beam nodes at the neutral axis of one of the beams. The alignment of the top shell surfaces or beam flanges is then achieved by providing a suitable shell or beam offset to the elements. Another common instance is when beams are used as stiffeners for shells. It is most convenient to model the beam elements at the mid-surface of the shell and sharing the shell nodal connectivity. The fact that the beam is actually offset by sum of half the plate thickness and half the height of the beam section is achieved by providing a suitable beam offset.

This demonstration problem is analyzed using two methods - one using the offset option and the other using the conventional RBE2 approach.

The RBE2 approach is only used to compare the accuracy of the solution obtained using in-built beam/shell offsets and the emphasis in this chapter is placed on describing the setup and solution using the actual in-built beam/shell offset capabilities of MD Nastran.

Modeling DetailsAn overhanging flat plate structure that is reinforced by beams is subjected to a top face load. The plate structure has a variable thickness along the length and the top surfaces of the thick and thin sections are aligned at the same level. The top portion of the reinforcement beam cross-sections are welded to the bottom surface of the thicker plate. In the geometric model (corresponding to Case-1), all the elements are modeled at the mid-surface of the thicker plate. Suitable beam/shell offsets need to be provided to account for the difference between the geometric model and the physical model.

The finite element mesh of the beam-plate structure is shown in Figure 63-1 and Figure 63-2. The geometric model where the beams are at the shell mid-surface and in-built beam/shell offsets are used is shown in Figure 63-1. The

Case-1: The first method is to use the in-built beam/shell offset capability.

Case-2: The second method is to place the beams and shells at the actual offset position and then tie the nodes of these elements back to the original position through manually defined RBE2 links. While this method is quite accurate, it is quite cumbersome for large models. Furthermore, if the offset elements have to contact other bodies, it is not possible since all degrees of freedom of the offset element nodes are already tied through the RBE2 links.

1189CHAPTER 63

Beam Reinforced Shell Structure using Offsets

physical model with the beams and shells at their actual offset locations is displayed in Figures 63-1 and 63-2. This model can be used with RBE2 links set up between the offset beams and the shell.

Figure 63-1 Reinforced Shell Structure with Beams Modeled at the Midsection of Structure

Figure 63-2 Reinforced Shell Structure with RBE2 Elements with Beams Modeled at Original Location

The plate is of length 6000 mm and width 4000 mm. The plate has a variable thickness along the length (70 mm over the first 4000 mm and 35 mm over the remaining 2000 mm). The top surfaces of the thick and thin shells are aligned at the same level. One reinforcement beam with a cross-sectional radius (mean) of 100 mm and thickness of 25 mm is placed across the plate at the point where the plate thickness transition occurs. Two other reinforcement beams, each with a cross-sectional radius of 125 mm and thickness of 40 mm, are placed along the length on either side of the plate. The top portion of the beam cross-sections are welded to the bottom surface of the plate.

Element ModelingCase-1: The plate of both the cross sections are modeled with lower order shell element (CQUAD4) and for the beam reinforcements, lower-order beam elements (CBEAM) are used. The offset values are specified in the corresponding field of the CQUAD4 entry. The non-linear extensions are activated by using the PSHLN1 property option in conjunction with the regular PSHELL property option in the manner shown below:

MD Demonstration Problems

CHAPTER 631190

.

.CQUAD4 101 2 113 124 125 114 17.5CQUAD4 102 2 114 125 126 115 17.5CQUAD4 103 2 115 126 127 116 17.5..CBEAM 151 3 113 114 1. 0. 0. BBB 0. 0. -135. 0. 0. -135. CBEAM 152 3 114 115 1. 0. 0. BBB 0. 0. -135. 0. 0. -135. ..PSHELL 1 1 70. 1 1PSHLN1 1...PBEAML 3 1 TUBE 112.5 87.5PBEMN1 3 1 N+ C2 BEAM LCC..

Similarly, for the two beam cases, PBEAML property is used with the nonlinear extension for the beam PBEMN1 being activated.

Case-2: Except for the offsets values being zero, all the properties are identical to that of Case-1.

.CQUAD4 131 2 157 168 169 158CQUAD4 132 2 158 169 170 159CQUAD4 133 2 159 170 171 160..CBEAM 101 4 122 124 0. 1. 0.CBEAM 103 4 124 126 0. 1. 0...PSHELL 1 1 70. 1 1PSHLN1 1..PBEAML 3 1 TUBE 112.5 87.5PBEMN1 3 1 N+ C2 BEAM LCC..

1191CHAPTER 63

Beam Reinforced Shell Structure using Offsets

Material ModelingAll the elements of the structure are modeled with isotropic, elastic perfectly-plastic material using the MAT1 and

MATEP options. The yield stress is taken as 40 N/mm2.

MAT1 1 21000. .3 1.MATEP 1 Perfect40. Isotrop Addmean

Loading and Boundary ConditionsFor both the cases – Case-1 and Case-2 – the loading and boundary conditions are identical. One side of the plate structure as shown in Figure 63-3 is constrained for displacement and rotations degrees of freedom using SPC1. A uniform pressure load is applied on the top surface in the downward direction using the LOAD entry.

SPCADD 2 1LOAD 2 1. 1. 1$ Displacement Constraints of Load Set : apply1SPC1 1 123456 1SPC1 1 123456 4 THRU 13$ Pressure Loads of Load Set : apply2PLOAD4 1 1 -.0075 THRU 150

Figure 63-3 Loading and Boundary Conditions for Cases-1 and -2

Solution ProcedureThe offset formulation is invoked with MDLPRM, OFFDEF, and LROFF in the bulk data section. This ensures that the shell normal directions are used to define the offset direction at each shell grid point and that the effects of offset like differential stiffness, loadings aspects of offset are taken into account.

MDLPRM OFFDEF LROFF

MD Demonstration Problems

CHAPTER 631192

The SOL400 nonlinear procedure is employed for both the cases, the parameters of which are defined through the following NLSTEP entry:

NLSTEP 1 1.00 GENERAL 25 1 10 ADAPT 0.01 0.01 0.05 20 1.2 0 0 0 MECH UPV 0.010 0.010 0.01 PFNT 1

The NLSTEP keyword is followed by the identification number entry and by the total time of the analysis which is 1.0 in this case.

The second line gives the general stepping parameters associated with the analysis. The maximum number of iteration (=25), maximum number of iteration needed for each increment (=1) and the maximum number of bisections allowed in current step (=10).

The keyword ADAPT defines the adaptive time (load) stepping procedure which is followed by the parameters viz. initial time step, minimum time-step as a fraction of total time, maximum time step, number of desired iterations per increment, factor for increasing the time step, output flag, etc.

The keyword MECH stands for a mechanical analysis appended with the parameters such as flags for convergence criteria selection followed by the error tolerance for displacement, load, and work, respectively. UP stands for convergence criteria checking with respect to displacement, load and work. The PFNT character parameter stands for “Pure Full Newton Raphson” which is the method for controlling stiffness updates.

ResultsThe displacement at center of the free end of the shell is compared for Case-1 and Case-2 in the Table 63-1.

The plot of displacement field is shown in Figures 63-4 and 63-5.

Table 63-1 Comparison of Displacement at Center Node of Free End

Quantity Case-1 Case-2 % Difference

Displacement 446.6432 447.3165 0.0015

1193CHAPTER 63

Beam Reinforced Shell Structure using Offsets

Figure 63-4 Deformation Plot for Case-1

Figure 63-5 Deformation Plot for Case-2

The displacement at the center of the free end is shown against time for both the cases in Figures 63-6 and 63-7.

MD Demonstration Problems

CHAPTER 631194

Figure 63-6 Displacement at Center of Free End Against Time for Case-1

Figure 63-7 Displacement at Center of Free End Against Time for Case-2

1195CHAPTER 63

Beam Reinforced Shell Structure using Offsets

The displacement at the nodes of free end is shown against y co-ordinate for both the cases in Figures 63-8 and 63-9. The non-linear stress plots of both the cases are shown in Figures 63-10 and 63-11.

Figure 63-8 Displacement at Free End Against y Coordinate for Case-1

Figure 63-9 Displacement at Free End Against y Coordinate for Case-2

MD Demonstration Problems

CHAPTER 631196

Figure 63-10 Stress Plot for Case-1

Figure 63-11 Stress Plot for Case-2

1197CHAPTER 63

Beam Reinforced Shell Structure using Offsets

Modeling TipsThe key aspect in this analysis is the offsets of beams and shells which can be invoked using the appropriate offset parameters in the CBEAM and CQUAD4 bulk data entries.

It is to be noted that MD Nastran follows numerous element defaults options (please refer to NLMOPTS, SPROPMAP entries in MD Nastran Quick Reference Guide). In particular to this analysis, if PSHLN1 option or PBEMN1 options are not specified in the input, MD Nastran assumes these options as the MATEP option is specified in the material properties for these elements.

Input File(s)

Files Description

nug_63a.dat MD Nastran input for “Reinforced shell structure with beam shell offset” (Case-1)

nug_63b.dat MD Nastran input for “Reinforced shell structure with RBE2 elements” (Case-2)