Beams: Composite Materials And Open Cross Sections

Chapter 33: Beams: Composite Materials and Open Cross Sections

33

Beams: Composite Materials and Open Cross Sections 

Summary - Composite Beam



Introduction



Solution Requirements



FEM Solution

519



Modeling Tips

520



Input File(s)



Summary - VKI and VAM Beam Formulations



Introduction



Solution Requirements



FEM Solution



Input File(s)

517

518 518

521

523

524 525

523

522

CHAPTER 33 517 Beams: Composite Materials and Open Cross Sections

Summary - Composite Beam Title Geometry

Chapter 33: Composite Beam Straight Cantilever Beam with load (Fy or Fz) applied at Free-End

Y, Ye

Fy

Fz X, Xe

Z, Ze

Element coordinate (Xe, Ye, Ze) coincides with Basic Coordinate (X,Y,Z)

Material properties

• Linear elastic orthotropic material using MAT8 • Assumptions: E33 = 0.8E22; 13= 23= 12 • Theta on PCOMP/PCOMPG specifies the angle between X-axis of material coordinate and X-axis of element coordinate.

Analysis type

Linear static analysis

Boundary conditions

Cantilever configuration

Applied loads

Bending

Element type

CBEAM3

FE results

• Converted PBEAM3 from PBMSECT • Stress recovery - screened based on max failure index • bdf file for FE mesh of cross section shown here

Z

X

Y

518 MD Demonstration Problems CHAPTER 33

Introduction Composite materials have found increasing applications in many applications and slender structures like rotor blades or high-aspect-ratio wings may be modeled in one-dimension as a 1-D beam provided the complex cross sectional properties (ultimately represented as a 2-D finite element mesh) can be captured properly. Here, a new way for composite beam analysis is introduced. The Variational Asymptotic Method (VAM) computes the properties of a beam’s arbitrary cross section containing composite materials. VAM, the mathematical basis of VABS, splits a general 3-D nonlinear elasticity problem for a beam-like structure into a two-dimensional (2-D) linear cross-sectional analysis and a 1-D nonlinear beam analysis. For details on VAM, refer to Yu, W., Volovoi, V., Hodges, D. and Hong, X. “Validation of the Variational Asymptotic Beam Sectional Analysis (VABS)”, AIAA Journal, Vol. 40, No. 10, 2002 (available at http://www.ae.gatech.edu/people/dhodges/papers/AIAAJ2002.pdf). VAM’s key benefit lies in the ability to model a beam made of composite material with only 1-D elements, namely CBEAM3.

Solution Requirements In general, the solution requires the layup of composite material and the description of this general or arbitrary cross section. PCOMP entries are used to provide the composite layup and PBMSECT entry is utilized to describe the profile of cross section and the link to the composite layup via PCOMP. An example is shown as follows: $ $ Composite case PBMSECT 32 1 OP 0.015 OUTP=101,C=101,brp=103,c(1)=[201,pt=(15,34)] pcomp 101 -0.1 5000. hill 0.0 501 0.05 0.0 501 0.05 501 0.05 -45.0 501 0.05 501 0.05 0.0 pcomp 201 5000. tsai 0.0 501 0.05 -45.0 501 0.05 501 0.05 0.0 $MAT1 501 3.6 .3 mat8,501,2.0e7,2.0e6,.35,1.0e6,1.0e6,1.0e6,0.0,+ +,0.0,0.0,0.0,2.3e5, 1.95e5, 13000., 32000., 12000.

90.0 45.0 SYM 45.0

The theta field on PCOMP is utilized to specify the angle between the X-axis of the material coordinate and the X-axis of the element coordinate. A cutout of the FEM mesh at the intersect of OUTP=101 and BRP=103 illustrates the ply layup shown in Figure 33-1.

CHAPTER 33 519 Beams: Composite Materials and Open Cross Sections

PCOMP 201 -45, 45, 0, 0, 45, -45 Z

X

Y

P 0 C O 45 M -45 P 90 1 0 0 1

Figure 33-1

P 0 C 45 O -45 M P 90 1 0 0 1

Intersection of Ply Layups 101 and 201

FEM Solution The converted PBEAM3 for PBMSECT,32 is as follows: *** USER INFORMATION MESSAGE 4379 (IFP9B) THE USER SUPPLIED PBMSECT BULK DATA ENTRIES ARE REPLACED BY THE FOLLOWING PBEAM3 ENTRIES. CONVERSION METHOD FOR PBARL/PBEAML - . PBEAM3 32 0 4.7202E+00 8.3059E+01 2.9578E+01 -1.5664E+01 3.2316E+01 0.0000E+00 1.8014E+01 4.2136E+00 1.7100E+01 -2.7858E+00 3.8881E+00 -3.5404E+00 4.7202E+00 2.6994E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 1.2253E+08 -2.1160E+05 8.1193E+04 -2.4761E+06 -3.7193E+06 7.9049E+06 -2.1160E+05 2.1792E+06 -1.7859E+06 1.9780E+07 5,4643E+05 -3.5845E+05 8.1193E+04 -1.7859E+06 2.7228E+07 1.7190E+07 2.9835E+04 2.1407E+06 -2.4761E+06 1.9780E+07 1.7190E+07 2.2332E+08 5.8182E+06 -1.2186E+06 -3.7193E+06 5.4643E+05 2.9835E+04 5.8182E+06 2.1349E+09 -4.0706E+08 8.9040E+06 -3.5845E+05 2.1407E+06 -1.2186E+06 -4.0706E+08 7.5602E+08

Note that the MID field of above PBEAM3 has value of 0 which is a flag for using the Timoshenko 6 x 6 matrix stored from the seventh line of PBEAM3. Timoshenko 6 x 6 matrix includes cross sectional and material properties. The cross-sectional shape and the FE mesh is shown in Figure 33-2. The coordinate shown in the figure matches with element coordinate.

520 MD Demonstration Problems CHAPTER 33

Z

X

Figure 33-2

Y

Cross-sectional Shape and the Corresponding FE Mesh

Full cross sectional stress recovery can be performed with PARAM,ARBMSS,YES in bulk data and FORCE=setid in case control. The stresses screened based on maximum failure index is shown as follows: 1

VAB ALGORITHM USING CORE OF PBMSECT TRANSVERSE TIP LOAD

MARCH

6, 2007

MD NASTRAN

0

3/ 6/07

PAGE

14

SUBCASE 1

S T R E S S E S

ELEMENT ID

I N

GRID ID

PLY ID

NORMAL-1

302 102 1301

2 2 2

2.468E+01 1.685E+01 1.588E+01

L A Y E R E D

D I R E NORMAL-2

C O M P O S I T E

C T S NORMAL-3

E L E M E N T S

T R E S S E S SHEAR-12 SHEAR-23

( BEAM3 )

FAILURE MAXIMUM SHEAR-13 THEORY FAIL. INDEX

STRENGTH RATIO

FLAG 2

1.601E+01 2.670E+00 1.619E+01 -7.230E-01 1.594E+01 -7.167E-01

2.323E+01 4.991E-01 3.724E+00 1.993E+01 -1.377E-01 -5.572E-01 1.938e+-1 -1.162e-01 -5.280e-01

TSIA-WU TSAI-WU TSAI-WU

7.161E-04 7.258E-04 7.193E-04

4.035E+02 4.470E+02 4.569E+02

Modeling Tips CBEAM3 is considered a straight beam if PID points to PBMSECT ID. The third point is ignored during the formation of element matrices. During data recovery, the stresses for the third point are computed based on the forces recovered which may not be correct. PARAM,ARBMSTYP,TIMOSHEN must be present to access VAM for composite beam.

CHAPTER 33 521 Beams: Composite Materials and Open Cross Sections

Input File(s) File Vabcore1.dat

Description Composite beam with MAT1.

522 MD Demonstration Problems CHAPTER 33

Summary - VKI and VAM Beam Formulations Title Geometry

Chapter 33: VKI and VAM Beam Formulations Straight Cantilever Beam with load (Fy or Fz) applied at Free-End

Y, Ye

Fy

Z

0.04 X

Fz

Y

0.5

1.0

X, Xe

Z, Ze

Element coordinate (Xe, Ye, Ze) coincides with Basic Coordinate (X,Y,Z)

Material properties

Linear elastic isotropic material

Analysis type

Linear static analysis

Boundary conditions

Cantilever configuration

Applied loads

Bending load with forces applied at free end

Element type

CBEAM, CBEAM3

FE results

• Converted PBEAM/PBEAM3 from PBMSECT • bdf file for FE mesh of cross section • Stress recovery - screened based on max failure index

Z X

Results

Isotropic with VKI

Isotropic with VAM

Composite with MAT1 using VAM

Disp at free end

49.987

49.974

49.977

Smax at fixed end

74974

74956

75351

Y

CHAPTER 33 523 Beams: Composite Materials and Open Cross Sections

Introduction In MD Nastran, there are two formulations to compute sectional properties. Both formulations use the finite element method. The first one is named after its third party vender, VKI, which solves a series of equations (see documentation of PBMSECT in Quick Look Guide) to obtain sectional properties. The other formulation is Variational Asymptotic Method (VAM), see attached for details on VAM Theory. While VKI formulation is for isotropic material only, VAM is capable to compute beam sectional properties for isotropic and composite material.

Solution Requirements PBMSECT bulk data entry is utilized to describe the shape of I section and PARAM,ARBMSTYP is used to control the selection of formulation. Note that default value for PARAM,ARBMSTYP select VKI formulation to compute sectional properties of arbitrary cross section with isotropic material. However, PARAM,ARBMSTYP,TIMISHEN must be present in the bulk data section if PBMSECT entry with Core and/or Layer keywords exists in the file. $ to select VAM PARAM,ARBMSTYPE,TIMOSHEN . $.......2.......3.......4.......5.......6.......7.......8.......9.......10..... $ Section profile $ $ 1 -- 2 -- 3 | | $ 4 -- 5 -- 5 $ point 1 -0.50 0.23 point 2 0.00 0.23 point 3 0.50 0.23 point 4 -0.50 -0.23 point 5 0.00 -0.23 point 6 0.50 -0.23 $ $.......2.......3.......4.......5.......6.......7.......8.......9.......10..... SET1 101 1 2 5 6 SET1 201 2 3 SET1 102 5 4 $ $ Ply properties $.......2.......3.......4.......5.......6.......7.......8.......9.......10..... $MAT8 501 20.59e6 1.42e6 0.42 0.89e6 0.89e6 0.89e6 $MAT1 501 1.+7 .3 $ $ isotropic case using T keyword PBMSECT 31 1 OP + OUTP=101,t=0.04,BRP(1)=201,BRP(3)=102 $ $ isotropic case using C and MAT1 PBMSECT 32 OP + OUTP=101,CORE=301,CORE(1)=[101,PT=(1,2)],CORE(2)=[202,PT=(5,6)],+ BRP(1)=201,CORE(3)=[201,PT=(2,3)], + BRP(3)=102,CORE(3)=[102,PT=(5,4)]

524 MD Demonstration Problems CHAPTER 33

FEM Solution The converted BEAM for PBMSECT,31 from VKI is as follows: *** USER INFORMATION MESSAGE 4379 (IFP9A) THE USER SUPPLIED PBEAML/PBMSECT BULK DATA ENTRIES ARE REPLACED BY THE FOLLOWING PBEAM ENTRIES. CONVERSION METHOD FOR PBARL/PBEAML - FINITE ELEMENT METHOD. PBEAM3 31 1 9.6800E-02 4.4896E-03 6.6689E-03 -8.0299E-19 5.2448E-05 0.0000E+00 2.5000E-01 5.0000E-01 2.5000E-01 -5.0000E-01 -2.5000E-01 -5.0000E-01 -2.5000E-01 5.0000E-01 1.5197E-01 6.9769E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 3.6170E-04 3.6170E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 4.1043E-11 7.5134E-10 4.1043E-11 7.5134E-10

The converted BEAM/PBEAM3 for PBMSECT,31 and 32 from VAM is as follows: *** USER INFORMATION MESSAGE 4379 (IFP9A) THE USER SUPPLIED PBEAML/PBMSECT BULK DATA ENTRIES ARE REPLACED BY THE FOLLOWING PBEAM ENTRIES. CONVERSION METHOD FOR PBARL/PBEAML - FINITE ELEMENT METHOD. PBEAM3 31 1 9.6800E-02 4.4902E-03 6.6696E-03 0.0000E+00 5.5566E-05 0.0000E+00 2.5000E-01 5.0000E-01 2.5000E-01 -5.0000E-01 -2.5000E-01 -5.0000E-01 -2.5000E-01 5.0000E-01 1.5346E-01 7.0201E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 3.5121E-04 3.4121E-04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 *** USER INFORMATION MESSAGE 4379 (IFP9B) THE USER SUPPLIED PBMSECT BULK DATA ENTRIES ARE REPLACED BY THE FOLLOWING PBEAM3 ENTRIES. CONVERSION METHOD FOR PBARL/PBEAML - . PBEAM3 32 0 9.6800E-02 4.4902E-03 6.6696E-03 0.0000E+00 5.5566E-05 0.0000E+00 2.5000E-01 5.0000E-01 2.5000E-01 -5.0000E-01 -2.5000E-01 -5.0000E-01 -2.5000E-01 5.0000E-01 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 9.6800E+05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 2.6041E+05 -5.9944E-04 1.5708E-04 0.0000E+00 0.0000E+00 0.0000E+00 -5.9944E-04 5.6910E+04 -7.1497E-05 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 4.4898E+04 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 6.6693E+04

Note that the MID field of above PBEAM3 has value of 0 which is a flag for using the Timoshenko 6 x 6 matrix stored from the seventh line of PBEAM3. Timoshenko 6 x 6 matrix includes cross-sectional and material properties. The cross-sectional shape and the FE mesh is shown in Figure 33-3.

Z X

Figure 33-3

Y

Cross sectional Shape and the Corresponding FE Mesh

CHAPTER 33 525 Beams: Composite Materials and Open Cross Sections

Regular beam stresses at extreme point from different formulation is shown in following table.

Isotropic with VKI

Isotropic with VAM

Composite with MAT1 using VAM

Disp at free end

49.987

49.974

49.977

Smax at fixed end

74974

74956

75351

Results

Input File(s) File

Description

nug_33a.dat

Isotropic and Composite beam with MAT1 using VAM

nug_33b.dat

Isotropic beam using VKI