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A Cantilevered Beam Using Beam Element

Table of Contents

  1. Description

  2. Structure Definition

  3. Random Variables

  4. Limit-State Function

  5. Finite Element Parametric Input file

  6. Analysis

  7. Results

  8. Reference

 

Description:

This test problem was designed to evaluate the capability of the integrated software, UNIPASS™-MSC/NASTRAN, by performing finite element-based probabilistic analyses based on the test criteria listed below:

·            Analysis Types: linear elastic static t analysis

·            Response Types: three response types: stress 

·            Element Types: Six MSC/NASTRAN element type, CBAR

·         Problem Types: Component and Series System problems  

·            Random Variable (RV): Both independent and correlated random variables will be solved. 

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 Structure Definition:

This problem demonstrates the component probabilistic static analysis of a cantilevered beam with a static load as shown in the following figure.

Figure 1.1 Problem One Structure Definition

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Random Variables:

In this example, five design parameters are considered to be random variables as shown in Table 1.  The material used for this test problem is steel.  This test problem is solved by considering both independent and correlated random variables.  For correlated case, only the cross sectional height, H and cross sectional base, B are considered correlated with a coefficient of correlation equals to 0.4.  The other random variables are assumed to be independent. The deterministic design parameters include, beam length = 200" and Poisson's ratio = 0.33.

 Table 1 Random Variable Definition

Random Variables

 Name

Distribution

Mean

Stand.  Dev.

Young's Modules of Material, E, psi

 FyoungM

Lognormal

3.0E+07

1.5E+06

Applied Force, P, lb

 Fforce1

Lognormal

8.5E+02

8.5E+01

Cross Sectional Height, H, in

 Fheight

Lognormal

3.0E+00

1.5E-01

Cross Sectional Base, B, in

 Fwidth

Lognormal

4.0E+00

2.0E-01

Yield Strength of Material, Fty, psi

 Fty

Lognormal

9E+03

9E+02

Allowable max. displacement, uZall, in

 Allowd

Lognormal

8

1.35

Allowable max. moment, Mall, lb-in

 Allforce

Lognormal

2.0E+05

2.0E+04

correlation coefficient matrix in original space:

 

Fheight

Fwidth

Fheight

1.000E+00

4.000E-01

Fwidth

4.000E-01

1.000E+00

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Limit State Function:

Table 2 Limit State Function Definition

Response Allowable Value Limit-State function
Deflection, uZmax Allowd, in g = Allowd - uZmax

 Note: Allowd is obtained by calling UNIPASS™ Supported function, NASTRAN.

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Parametric Input File:

init master(s)

PROJ='ARMY98'

ID UNIPASS,BEAM

SOL  101

TIME 20

DIAG  8

cend

TITLE= BEAM FOR STATIC ANALYSIS

SUBTITLE= MODELING AS BEAM ELEMENTS

LABEL= FORCE 100 LB ON FREE END IN Z-DIR

ECHO   = both

DISP=all

force=all

stress=all

strain=all

spcf = all

SPC=1

ANALYSIS = statics

displacement(sort1, punch) = all

stress(sort1, punch) = all

force(sort1, punch,real) = all

strain(sort1, punch) = all

SUBCASE 1

LOAD = 1

BEGIN BULK

GRID,1,,0.

=,*1,=,*10.

=19

CBAR,1,3,1,2,0.,1.,0.

=,*1,=,*1,*1,=,=,=

=18

force*  1               21                              <Fforce1       >*force01

*force01 0.             0.               1.

SPC1,1,123456,1

PBAR*   3               1               <AREA          ><I_1           >*pbar01

*pbar01 <I_2           >                                                *pbar02

*pbar02 2.              0.5

MAT1*   1               <FyoungM       >                0.33            *MAT01

*MAT01  0.1

PARAM,GRDPNT,0

ENDDATA

\EOF

Note: 

AREA = FHEIGHT * FWIDTH 

I_1 = FWIDTH * FHEIGHT ** 3 / 12 

I_2 = FHEIGHT * FWIDTH ** 3 / 12

 

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Analysis: 

Use U-based/X-linearized MPPL method to compute the first-order failure probability and sensitivity.

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Results:

MPP:

variable

names

linearization-point

directional

cosine in u-space

x-space

u-space

Fheight

2.88011E+00

-7.91213E-01

-0.41347

Fwidth

3.75917E+00

-9.83237E-01

-0.51384

FyoungM 2.94301E+07 -3.58869E-01 -0.18752
Fforce1 9.07777E+02 7.09144E-01 0.37048
Allowd 6.45140E+00 -1.20015E+00 -0.62658

 

First-order failure probability:

reliability index ..........................b = 1.9143654E+00 

1st-order failure probability .......... pf1= 2.7786739E-02 

 

Examples of sensitivity measurements:

 

Normalized Factor:

max|(s)d(Pf1)/d(m)|

= 4.9388727E-02

max|(s)d(Pf1)/d(s)|

= 5.3950508E-02

Variable

Name

Normalized

(s)d(Pf1)/d(m)

Normalized

(s)d(Pf1)/d(s)

Fheight

-2.5482340E-01

1.8797030E-01

Fwidth

-7.7112395E-01

8.4006527E-01

FyoungM -2.4748458E-01 9.0607765E-02
Fforce1 4.5104991E-01 2.6583714E-01
Allowd -1.0000000E+00 1.0000000E+00

wpe5.gif (14856 bytes)

Plot of (s)d(Pf1)/d(m)

wpe15.gif (14613 bytes)

Plot of (s)d(Pf1)/d(s)

No. of g-functions called        = 28 

No. of derivatives of g called = 4 

No. of external solvers called = 24

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Reference:

UNIPASS™ NASTRAN.IN Reference Guide,  Version 4.2, PredictionProbe, Inc. Newport Beach, California, 2002

 

Last Updated 02/08/10

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