Membrane Bending

(PCALC command). Path items may be differentiated or integrated with respect to any other path item

(PCALC command). Differentiation is based on a central difference method without weighting:

(1913)A

A A

B BS1

2 1

2 1

=

(for first path point)

(1914)A

A A

B BSn

n n

n n

=

+

+

1 1

1 1

(for intermediate path points)

(1915)A

A A

B BSL

L L

L L

=

1

1

(for last path point)

where:

A = values associated with the first labeled path in the operation (LAB1, on the PCALC,DERI

command)

B = values associated with the second labeled path in the operation (LAB2, on the PCALC,DERI

command)

n = 2 to (L-1)

L = number of points on the path

S = scale factor (input as FACT1, on the PCALC,DERI command)

If the denominator is zero for Equation 1913 (p. 1015) through Equation 1915 (p. 1015), then the derivative

is set to zero.

Integration is based on the rectangular rule (see Figure 18.1 (p. 998) for an illustration):

(1916)A1 0 0 = .

(1917)A A A A B B Sn n n n n n

+ = + + 1 1 11

2( )( )

Path items may also be used in vector dot (PDOT command) or cross (PCROSS command) products.

The calculation is the same as the one described in the Vector Dot and Cross Products Topic, above.

The only difference is that the results are not transformed to be in the global Cartesian coordinate

system.

19.4. POST1 - Stress Linearization

An option is available to allow a separation of stresses through a section into constant (membrane) and

linear (bending) stresses. An approach similar to the one used here is reported by Gordon([63] (p. 1102)).

The stress linearization option (accessed using the PRSECT, PLSECT, or FSSECT commands) uses a path

defined by two nodes (with the PPATH command). The section is defined by a path consisting of two

end points (nodes N1 and N2) as shown in Figure 19.4 (p. 1016) (nodes) and 47 intermediate points

(automatically determined by linear interpolation in the active display coordinate system (DSYS). Nodes

N1 and N2 are normally both presumed to be at free surfaces.

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of ANSYS, Inc. and its subsidiaries and affiliates.

POST1 - Stress Linearization

Initially, a path must be defined and the results mapped onto that path as defined above. The logic for

most of the remainder of the stress linearization calculation depends on whether the structure is

axisymmetric or not, as indicated by the value of (input as RHO on PRSECT, PLSECT, or FSSECT

commands). For = 0.0, the structure is not axisymmetric (Cartesian case); and for nonzero values of ,

the structure is axisymmetric. The explicit definition of , as well as the discussion of the treatment of

axisymmetric structures, is discussed later.

Figure 19.4 Coordinates of Cross Section

N

1

N

2

t/2

t

X

s

19.4.1. Cartesian Case

Refer to Figure 19.5 (p. 1017) for a graphical representation of stresses. The membrane values of the stress

components are computed from:

(1918) im

i st

t

tdx=

12

2

where:

im

= membrane value of stress component i

t = thickness of section, as shown in Figure 19.4 (p. 1016)

i = stress component i along path from results file (`total' stress)

xs = coordinate along path, as shown in Figure 19.4 (p. 1016)

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Chapter 19: Postprocessing

Figure 19.5 Typical Stress Distribution

i1

i1

b

i2

b

i2

p

i

m

i2

i

Node 1

Stress

Node 2

x

s

t

2

+

i1

p

t

2

-

The subscript i is allowed to vary from 1 to 6, representing x, y, z, xy, yz, and xz, respectively. These

stresses are in global Cartesian coordinates. Strictly speaking, the integrals such as the one above are

not literally performed; rather it is evaluated by numerical integration:

(1919)

im i i

i jj

= + +

=

1

48 2 2

1 49

2

47, ,

,

where:

i,j = total stress component i at point j along path

The integral notation will continue to be used, for ease of reading.

The bending values of the stress components at node N1 are computed from:

(1920) ib

i s st

t

tx dx

1 2 2

26=

where:

ib1 = bending value of stress component i at node N1




The bending values of the stress components at node N2 are simply

(1921) ib

ib

2 1=

where:

ib2 = bending value of the stress component i at node N2

The peak value of stress at a point is the difference between the total stress and the sum of the

membrane and bending stresses. Thus, the peak stress at node N1 is:

(1922) ip

i im

ib

1 1 1=

where:

ip1 = peak value of stress component i at node N1

i1 = value of total stress component i at node N1

Similarly, for node N2,

(1923) ip

i im

ib

2 2 2=

At the center point (x = 0.0)

(1924) icp

ic im=

where:

icp

= peak value of stress component i at center

ic = computed (total) value of stress component i at center

19.4.2. Axisymmetric Case (General)

The axisymmetric case is the same, in principle, as the Cartesian case, except for the fact that there is

more material at a greater radius than at a smaller radius. Thus, the neutral axis is shifted radially outward

a distance xf, as shown in Figure 19.6 (p. 1019). The axes shown in Figure 19.6 (p. 1019) are Cartesian, i.e.,

the logic presented here is only valid for structures axisymmetric in the global cylindrical system. As

stated above, the axisymmetric case is selected if 0.0. is defined as the radius of curvature of the

midsurface in the X-Y plane, as shown in Figure 19.7 (p. 1019). A point on the centerplane of the torus

has its curvatures defined by two radii: and the radial position Rc. Both of these radii will be used in

the forthcoming development. In the case of an axisymmetric straight section such as a cylinder, cone,

or disk, = , so that the input must be a large number (or -1).



Figure 19.6 Axisymmetric Cross-Section

Neutral Surface

t

x

x

f

y

Y

R

2

R

c

R

1

N

1

N

2

x,R

t

2

Figure 19.7 Geometry Used for Axisymmetric Evaluations

Torus

Cylinder ( = )

Y

x,R

x

y

R

c

Each of the components for the axisymmetric case needs to be treated separately. For this case, the

stress components are rotated into section coordinates, so that x stresses are parallel to the path and

y stresses are normal to the path.

Starting with the y direction membrane stress, the force over a small sector is:




(1925)F R dxy ytt= 2

2

where:

Fy = total force over small sector

y = actual stress in y (meridional) direction

R = radius to point being integrated

= angle over a small sector in the hoop direction

t = thickness of section (distance between nodes N1 and N2)

The area over which the force acts is:

(1926)A R ty c=

where:

Ay = area of small sector

RR R

c =+1 2

2

R1 = radius to node N1R2 = radius to node N2

Thus, the average membrane stress is:

(1927)

ym y

y

yt

t

c

F

A

Rdx

R t= =

22

where:

ym

= y membrane stress

To process the bending stresses, the distance from the center surface to the neutral surface is needed.

This distance is shown in Figure 19.6 (p. 1019) and is:

(1928)xt cos

Rf

c

=2

12

The derivation of Equation 1928 (p. 1020) is the same as for yf given at the end of SHELL61 - Axisymmetric-

Harmonic Structural Shell (p. 594). Thus, the bending moment may be given by:



(1929)M x x dFftt= ( )2

2

or

(1930)M x x R dxf ytt= ( ) 2

2

The moment of inertia is:

(1931)I R t R t xc c f= 1

12

3 2

The bending stresses are:

(1932)b Mc

I=

where:

c = distance from the neutral axis to the extreme fiber

Combining the above three equations,

(1933)yb fM x x

I11=( )

or

(1934) y

b f

c f

f yt

tx x

R tt

x

x x Rdx11

2 22

2

12

=

( )

where:

yb1 = y bending stress at node N1

Also,

(1935)yb fM x x

I22=( )

or




(1936)

yb f

c f

f yt

tx x

R tt

x

x x Rdx2

2

2 22

2

12

=

( )

where:

yb

2 = y bending stress at node N2

x represents the stress in the direction of the thickness. Thus, x1 and x2 are the negative of the

pressure (if any) at the free surface at nodes N1 and N2, respectively. A membrane stress is computed

as:

(1937) xm

xt

t

tdx=

12

2

where:

xm

= the x membrane stress

The treatment of the thickness-direction "bending" stresses is controlled by KB (input as KBR on PRSECT,

PLSECT, or FSSECT commands). When the thickness-direction bending stresses are to be ignored (KB= 1), bending stresses are equated to zero:

(1938)xb1 0=

(1939)xb

20=

When the bending stresses are to be included (KB = 0), bending stresses are computed as:

(1940) xb

x xm

1 1=

(1941) xb

x xm

2 2=

where:

xb1 = x bending stress at node N1

x1 = total x stress at node N1

xb

2 = x bending stress at node N2x2 = total x stress at node N2



and when KB = 2, membrane and bending stresses are computed using Equation 1927 (p. 1020), Equa-

tion 1934 (p. 1021), and Equation 1936 (p. 1022) substituting x for y.

The hoop stresses are processed next.

(1942)

hm h

h

ht

tF

A

x dx

t= =

+

( )2

2

where:

hm

= hoop membrane stress

Fh = total force over small sector

= angle over small sector in the meridional (y) direction

h = hoop stress

Ah = area of small sector in the x-y plane

r = radius of curvature of the midsurface of the section (input as RHO)

x = coordinate thru cross-section

t = thickness of cross-section

Equation 1942 (p. 1023) can be reduced to:

(1943) hm

ht

t

t

xdx= +

11

2

2

Using logic analogous to that needed to derive Equation 1934 (p. 1021) and Equation 1936 (p. 1022), the

hoop bending stresses are computed by:

(1944)

hb h

h

h ht

tx x

tt

x

x xx

dx11

22

2

2

12

1=

+

( )

and

(1945)

hb h

h

h ht

tx x

tt

x

x xx

dx11

22

2

2

12

1=

+

( )

where:




(1946)xt

h =2

12

for hoop-related calculations of Equation 1944 (p. 1023) and Equation 1945 (p. 1023).

An xy membrane shear stress is computed as:

(1947) xym

cxyt

t

R tRdx=

12

2

where:

xym

= xy membrane shear stress

xy = xy shear stress

Since the shear stress distribution is assumed to be parabolic and equal to zero at the ends, the xy

bending shear stress is set to 0.0. The other two shear stresses (xz, yz) are assumed to be zero if KB =

0 or 1. If KB = 2, the shear membrane and bending stresses are computing using Equation 1927 (p. 1020),

Equation 1934 (p. 1021), and Equation 1936 (p. 1022) substituting xy for y

All peak stresses are computed from

(1948) iP

i im

ib=

where:

iP = peak value of stress component i

i = total value of stress of component i

19.4.3. Axisymmetric Case

(Specializations for Centerline)

At this point it is important to mention one exceptional configuration related to the y-direction membrane

and bending stress calculations above. For paths defined on the centerline (X = 0), Rc = 0 and cos =

0, and therefore Equation 1927 (p. 1020), Equation 1928 (p. 1020), Equation 1934 (p. 1021), and Equa-

tion 1936 (p. 1022) are undefined. Since centerline paths are also vertical ( = 90), it follows that R = Rc,

and Rc is directly cancelled from stress Equation 1927 (p. 1020), Equation 1934 (p. 1021), and Equa-

tion 1935 (p. 1021). However, xf remains undefined. Figure 19.8 (p. 1025) shows a centerline path from N1to N2 in which the inside and outside wall surfaces form perpendicular intersections with the centerline.



Figure 19.8 Centerline Sections

r

N

1

'

N

2

N

1

N

2

'

R

c

For this configuration it is evident that cos = Rc/ as approaches 90 (or as N N1 2 approaches N1

- N2). Thus for any paths very near or exactly on the centerline, Equation 1928 (p. 1020) is generalized to

be:

(1949)x

t cos

RR

t

tR

tf

cc

c

=

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Membrane Bending