Prepared For:

Prepared By: Charles R. McCrearyCRM Engineering Services13774 Highway 322Kilgore, TX 75662

CRM Engineering Services

Stress Analysis Structural Design Heat Transfer Software Development

June 1, 1998

Ken WalkerThe MacNeal-Schwendler Corp.

Stress Linearization in Patran

The MacNeal-Schwendler Corp.Introduction

The evaluation of ASME BPV Code primary-plus-secondary (P+Q) stress limits requires the calculation of membrane and bending stresses. Linearized stresses reduce a complex stress state into the equivalent membrane stress which represents a force divided by the area on which it acts and a bending stress which represents a moment acting on a unit strip of material.

The Patran PCL program documented in this report permits the generation of mem-brane and bending stresses from within Patran. The stress results from which the linearized stresses are derived are independent of the source of the results as long as the results are loaded in the Patran database. The stress classification line is inde-pendent of the mesh, i.e. the stress classification line is not required to be along ele-ment faces. This capability permits the linearization of stresses from arbitrary locations without regard to the mesh details.

Algorithm

Stresses are linearized along a section in an axisymmetric model and along a line through a section in a three-dimensional mesh. This origin of the section is the mid-point of the stress classification line as shown in Figure 1. Thus the section coordi-

nates are from . The axial or in the section coordinate system along

the associated linearized stresses are shown in Figure 2. There are two types of stress linearization offered, cartesian and axisymmetric. The cartesian case is appli-cable to arbitrary three-dimensional finite element models. The axisymmetric case is applicable to axisymmetric finite element models in which the xz and yz components of stress are zero and in which certain assumptions are made about the stress distri-bution.

Cartesian Case

Membrane Stress

The membrane stress for each stress component is computed from

(EQ 1)

where

= membrane stress for component .

t2--- Xs

t2--- yy

mij

1t--- ij xsd

t 2

t 2

=

mij ij6/1/98 2Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp. = thickness of section

= coordinate along section.

The integral is evaluated using the extended trapezoidal rule

(EQ 2)

where

= at point k along section.

= the equal interval size determined by dividing the thickness, , by the number of integration points (input to function).

= the number of intervals which is equal to the number of integration points - 1.

Bending Stress

The bending stress is calculated from

Xs

Ys

FIGURE 1. Section coordinates.

t

xs

mij

1t--- h 12---ij xs

1( ) ij xsk( )k 2=

N 1

12---ij xsN( )+ +

=

ij xsk( ) ijh t

N6/1/98 3Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp.(EQ 3)

The bending stress at the other end of the section is defined as

(EQ 4)

Axisymmetric Case

The algorithm for the axisymmetric case assumes that the y-axis is the centerline of the model and the x-coordinate is the radial coordinate. In an axisymmetric struc-

0. .200 .400 .600 .800 1.00 1.20-8000.

-4000.

0.

4000.

8000.

12000.

16000.

LEGENDFEA Element stress distribution for component YYMembrane stress for component YYShifted bending stress for component YY

FIGURE 2. Axial stress and linearized components.

ijb

Point A6t--- ijxs xsd

t 2

t 2

=

ijb

Point Bij

b

Point A=6/1/98 4Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp.ture, there is more stresses material at a large radius than at a shorter radius. Con-sequently the neutral axis for local bending is shifted outward. This effect is more

pronounced for thick shells, . An axisymmetric structure can have two radii of

curvature, the radius in the x-z plane and the radius of curvature of the neu-tral surface in the x-y plane.

The stress tensor extracted along the section is first transformed so that the local x (r) direction is oriented along the section. The geometry of the axisymmetric case is shown in Figure 3.

Axial Stress

The force over a small sector is defined as

(EQ 5)

Rt--- 10

r ( )r z( )

FIGURE 3. Axisymmetric geometry.

Fy yyR xsdt2---

t2---=

Neutral surface

A

B

R_A

R_C

R_B

x_f

Phi

Rho

RARC

RB

xf6/1/98 5Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp.where

= total force over small sector

= stress in yy direction (meriodional)

= radius

= small angle in hoop direction

= thickness of section.

The area over which the force acts is given by

(EQ 6)

where

=

= Radius to point A

= Radius to point B

The average membrane stress is then

(EQ 7)

The neutral axis is shifted by an amount which is given by

(EQ 8)

The bending moment is given by

(EQ 9)

The bending stress is given by

(EQ 10)

where

Fy

yy

R

t

Ay RCt=

RCRA RB+

2-------------------

RA

RB

yym

FyAy-----

yyR xsdt 2

t 2Rct

---------------------------------= =

xf

xft2 cos12Rc

----------------=

M x xf( ) Fdt 2

t 2 x xf( )yyR xsdt 2

t 2= =

yyb Mc

I--------=6/1/98 6Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp.= the distance from the neutral axis, , to the outside or inside surface

= the moment of inertia

=

The bending stress is thus

(EQ 11)

and

(EQ 12)

Radial Stress

The radial stress should equal the applied pressures, if any, at the free surface. In reality, the mesh density is often not fine enough to capture the applied pressure. The membrane stress is given by

(EQ 13)

The bending stress is defined as

(EQ 14)

and

(EQ 15)

Hoop Stress

The membrane hoop stress is equal to

(EQ 16)

c xf

I

112------Rct3 Rctxf2

yyb

Point A

xA xf

Rctt2

12------ xf2

------------------------------- x xf( )yyR xsd

t 2

t 2=

yyb

Point B

xB xf

Rctt2

12------ xf2

------------------------------- x xf( )yyR xsd

t 2

t 2=

rrm

rr xsdt 2

t 2=

rrb

Point Arr Point A

rrm

=

rrb

Point Brr Point B

rrm

=

zzm Fz

Az-----

zz x+( ) xsdt 2

t 2t----------------------------------------------

1t--- zz 1

x

---+ xsdt 2

t 2= = =

6/1/98 7Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp.when such as for a cylinder, cone, or flat plate (any straight section), the membrane stress reduces to

(EQ 17)

The bending stress is calculated from

(EQ 18)

and

(EQ 19)

Shear Stress

The rz membrane shear stress is calculated from

(EQ 20)

The shear stress is assumed to be parabolic through the section with

(EQ 21)

thus

(EQ 22)

Special Case for

When ( ) at the centerline, and . This occurs for verti-cal stress classification lines on the centerline of structures such as spherical and elliptical pressure heads and flat plates.

Since in this case, Eq. 7 becomes

=

zzm 1

t--- zz xsd

t 2

t 2=

zzb

Point A

xA xz

tt2

12------ xz2

------------------------ x xz( )yy 1

x

---+ xsdt 2

t 2=

zzb

Point B

xB xz

tt2

12------ xz2

------------------------ x xz( )yy 1

x

---+ xsdt 2

t 2=

rzm 1

Rct------- rzR xsd

t 2

t 2=

rz Point Arz Point B

0= =

rzb 0=

90= 90= R 0= Rc 0= cos 0=

R Rc=6/1/98 8Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp., (EQ 23)

Eq. 11 becomes

, (EQ 24)

Eq. 12 becomes

, (EQ 25)

Since as , becomes

(EQ 26)

Eq. 26 only applies if the structure forms a perpendicular intersection with the cen-terline. For other intersections, the stress classification line should be placed a small distance away from the centerline.

Usage

The stress linearization tool appears as shown in Figure 4. The results selection form is shown in Figure 5. When the results selection form is entered, if only one load case and sub-case exist in the database, they will be pre-selected. Only tensor results will be shown in the results list box. Thus the stress tensor must be available in the database.

It is important to note that it is possible to have shell element results along with the continuum element results. In an axisymmetric model, axisymmetric shell elements may be on the boundary of the continuum elements to extract accurate surface stresses. In such a case, the shell elements have results at section points (top and/or bottom of the shell element). Since the continuum element results are unlayered, the stress classification line will extract results from the continuum elements and the shell elements but the shell elements only have results at a section point. Thus the

yym

FyAy-----

yy xsdt 2

t 2t

-----------------------------= =

yyb

Point A

xA xf

tt2

12------ xf2

------------------------ x xf( )yy xsd

t 2

t 2=

yyb

Point B

xB xf

tt2

12------ xf2

------------------------ x xf( )yy xsd

t 2

t 2=cos RC------

pi2--- xf

xf

t2 cos12Rc

---------------- if Rc

t1000------------

t2

12--------- if Rct

1000------------

The MacNeal-Schwendler Corp.-1.0

Linearized Stresses

Along an Arbitrary Curve

Select Results...

Default-Step1,TotalTime=0.

Stress-Components

Axisymmetric

Non-Axisymmetric

No. of Integration Points

50

Curve Ids

Curve 3

Curved section

Radius

-1.0

Apply

Post Results...

Cancel

Selecting this button will allow you to select the results. Results must be selected as there is not a default.

Axisymmetric is the default, Non-axisymmetric represents the Cartesian case.

The number of integration points through the section for use with the extended trapezoi-dal numerical integration. The default of 50 should provide adequate accuracy.

Select the stress classification line. This must be a line, not a surface edge. Only one line should be selected at a time.

The default is Straight section (cylinder). Selecting curved indi-cates that the radius of curva-ture of the neutral surface will be given.

Selecting the apply button will perform the calculations. If suc-cessful, the Post Results button will be enabled so that the lin-earized stress components can be viewed, saved to a file, or plotted along with the FEA stress distribution.

FIGURE 4. Patran form.6/1/98 10Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp.Select ResultsLoadcases1-Default

Subcases1-Step1,TotalTime=0.

Results

1.1-Stress, Components

Layers1-(NON-LAYERED)

Ok Cancel

FIGURE 5. Results selection form.6/1/98 11Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp.stresses along the stress classification line will have a discontinuity where it crosses the shell elements.

The solution is to place the continuum elements in a group containing only contin-uum elements and make that group the current group since the results are extracted only from the elements in the current group.

Selecting Post Results after selecting Apply will produce a spreadsheet containing the linearized stress components as shown in Figure 6. The linearized stress can be written to a file by selecting the Output Report button. The stress distribution along with the membrane stress and the shifted bending stress can be plotted by selecting Create XY plots button. The xy plots can be manipulated from within the XY plot area of Patran. The plots appear as in Figure 2.6/1/98 12Stress-Linearization CRM Engineering Services

The MacNeal-Schwendler Corp.FIGURE 6. Results display form.6/1/98 13Stress-Linearization CRM Engineering Services

Stress Linearization in PatranIntroductionAlgorithmFIGURE 1. Section coordinates.FIGURE 2. Axial stress and linearized components.Cartesian CaseMembrane Stress(EQ 1)(EQ 2)

Bending Stress(EQ 3)(EQ 4)

Axisymmetric CaseFIGURE 3. Axisymmetric geometry.Axial Stress(EQ 5)(EQ 6)(EQ 7)(EQ 8)(EQ 9)(EQ 10)(EQ 11)(EQ 12)

Radial Stress(EQ 13)(EQ 14)(EQ 15)

Hoop Stress(EQ 16)(EQ 17)(EQ 18)(EQ 19)

Shear Stress(EQ 20)(EQ 21)(EQ 22)

Special Case for, (EQ 23), (EQ 24), (EQ 25)(EQ 26)

UsageFIGURE 4. Patran form.FIGURE 5. Results selection form.FIGURE 6. Results display form.