Note

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Out-of-Plane Bending of a Curved Bar#

Problem description:

A portion of a horizontal circular ring, built-in at A, is loaded by a vertical (Z) load F applied at the end B. The ring has a solid circular cross-section of diameter d. Determine the deflection \(\delta\) at end B, the maximum bending stress \(\sigma_{Bend}\) , and the maximum torsional shear stress τ.

Reference:

S. Timoshenko, Strength of Materials, Part I, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 412, eq. 241.

Analysis type(s):

Static Analysis ANTYPE=0

Element type(s):

Elastic Curved Pipe Element (PIPE18)
3-D 3 Node Pipe Element (PIPE289)

Material properties:

\(E = 30 \cdot 10^6 psi\)
\(\mu = 0.3\)

Geometric properties:

\(r = 100 in\)
\(d = 2 in\)
\(\theta = 90°\)

Loading:

\(F = 50 lb\)

Analysis Assumptions and Modeling Notes:

Node 10 is arbitrarily located on the radius of curvature side of the element to define the plane of the elbow when PIPE18 elements are used. The wall thickness is set to half the diameter for a solid bar. Since the section has no hole in the middle, ovalization cannot occur and PIPE289 elements can be used to determine the deflection and stresses.

# sphinx_gallery_thumbnail_path = '_static/vm18_setup1.png'

# Importing the `launch_mapdl` function from the `ansys.mapdl.core` module
from ansys.mapdl.core import launch_mapdl
import numpy as np
import pandas as pd

# Launch MAPDL with specified settings
mapdl = launch_mapdl(loglevel="WARNING", print_com=True, remove_temp_dir_on_exit=True)

# Clear the existing database
mapdl.clear()

# Run the FINISH command to exists normally from a processor
mapdl.finish()

# Set the ANSYS version
mapdl.com("ANSYS MEDIA REL. 2022R2 (05/13/2022) REF. VERIF. MANUAL: REL. 2022R2")

# Run the /VERIFY command
mapdl.run("/VERIFY,VM18")

# Set the title of the analysis
mapdl.title("VM18 OUT-OF-PLANE BENDING OF A CURVED BAR")

# Enter the model creation /Prep7 preprocessor
mapdl.prep7()

*****MAPDL VERIFICATION RUN ONLY*****
     DO NOT USE RESULTS FOR PRODUCTION

          ***** MAPDL ANALYSIS DEFINITION (PREP7) *****

Define element type and real properties#

Use Elastic Curved Pipe element (PIPE18) and set KEYOPT(6)=2 for printing member forces.

mapdl.et(1, "PIPE18", "", "", "", "", "", 2)

# Define geometry parameters (OD, wall thickness, radius) using "r" command (real constant)
mapdl.r(1, 2, 1, 100)

REAL CONSTANT SET          1  ITEMS   1 TO   6
    2.0000       1.0000       100.00       0.0000       0.0000       0.0000

Define material#

Set up the material and its type (a single material), Young’s modulus of 30e6 and Poisson’s ratio NUXY of 0.3 is specified.

mapdl.mp("EX", 1, 30e6)
mapdl.mp("NUXY", 1, 0.3)

MATERIAL          1     NUXY =  0.3000000

Define geometry#

Set up the nodes and elements. This creates a mesh just like in the problem setup.

# Define nodes
mapdl.n(1, 100)
mapdl.n(2, "", 100)
mapdl.n(10)

# Define element
mapdl.e(1, 2, 10)

Define boundary conditions and load#

Fix all dofs at node 1. Specify nodal force F = -50 lb along Z direction at node 2. Then exit prep7 processor.

mapdl.d(1, "ALL")  # Define boundary conditions
mapdl.f(2, "FZ", -50)  # Define load

# Selects all entities
mapdl.allsel()
# Element plot
mapdl.eplot(vtk=False)

# Finish preprocessing processor
mapdl.finish()

***** ROUTINE COMPLETED *****  CP =         0.000

Solve#

Enter solution mode and solve the system.

mapdl.slashsolu()

# Set the analysis type to STATIC
mapdl.antype("STATIC")

# Set output options
mapdl.outpr("BASIC", 1)

# Perform the solution
mapdl.solve()
# exists solution processor
mapdl.finish()

FINISH SOLUTION PROCESSING

 *** NOTE ***                            CP =       0.000   TIME= 00:00:00
 Distributed parallel processing has been reactivated.

 ***** ROUTINE COMPLETED *****  CP =         0.000

Post-processing#

Enter post-processing. Compute deflection and stress quantities.

mapdl.post1()

# Set the current results set to the last set to be read from result file
mapdl.set("LAST")

# Get displacement results at node 2 in the Z direction
def_z = mapdl.get("DEF", "NODE", 2, "U", "Z")

# Create an element table for bending stresses using ETABLE command
strs_ben = mapdl.etable("STRS_BEN", "NMISC", 91)

# Create an element table for shear stresses using ETABLE command
strs_shr = mapdl.etable("STRS_SHR", "LS", 4)

# Get bending stresses (ETAB: STRS_BEN) for element 1
strss_b = mapdl.get("STRSS_B", "ELEM", 1, "ETAB", "STRS_BEN")

# Get shear stresses (ETAB: STRS_SHR) for element 1
strss_t = mapdl.get("STRSS_T", "ELEM", 1, "ETAB", "STRS_SHR")

Verify the results.#

# Set target values
target_val = [-2.648, 6366, -3183]

# Fill result values
sim_res = [def_z, strss_b, strss_t]

col_headers = ["TARGET", "Mechanical APDL", "RATIO"]
row_headers = ["Deflection (in)", "Stress_Bend (psi)", "Shear Stress (psi)"]

data = [target_val, sim_res, np.abs(target_val) / np.abs(sim_res)]

title = f"""

------------------- VM18 RESULTS COMPARISON ---------------------

PIPE18:
-------
"""

print(title)
print(pd.DataFrame(np.transpose(data), row_headers, col_headers))

------------------- VM18 RESULTS COMPARISON ---------------------

PIPE18:
-------

                      TARGET  Mechanical APDL     RATIO
Deflection (in)       -2.648        -2.649729  0.999348
Stress_Bend (psi)   6366.000      6366.197750  0.999969
Shear Stress (psi) -3183.000     -3183.098880  0.999969

Finish the post-processing processor.#

mapdl.finish()

EXIT THE MAPDL POST1 DATABASE PROCESSOR

 ***** ROUTINE COMPLETED *****  CP =         0.000

Clears the database without restarting.#

mapdl.run("/CLEAR,NOSTART")

CLEAR MAPDL DATABASE AND RESTART

 Ansys Mechanical Enterprise Academic Student

Set a new title for the analysis#

mapdl.title("VM18 OUT-OF-PLANE BENDING OF A CURVED BAR Using PIPE289 ELEMENT MODEL")

TITLE=
 VM18 OUT-OF-PLANE BENDING OF A CURVED BAR Using PIPE289 ELEMENT MODEL

Switches to the preprocessor (PREP7)#

mapdl.prep7()

*****MAPDL VERIFICATION RUN ONLY*****
     DO NOT USE RESULTS FOR PRODUCTION

          ***** MAPDL ANALYSIS DEFINITION (PREP7) *****

Define element type and section properties#

Use 3-D 3-Node Pipe element (PIPE289) and set KEYOPT(4)= 2 Thick pipe theory.

mapdl.et(1, "PIPE289", "", "", "", 2)
mapdl.sectype(1, "PIPE")  # Set section type PIPE
mapdl.secdata(2, 1, 16)  # Set section data (OD, wall thickness)

SECTION ID NUMBER:             1
   PIPE SECTION NAME IS:
   PIPE SECTION DATA SUMMARY:
    Outside Diameter     =  2.0000
    Thickness            =  1.0000
    Area                 =  3.1414
    Iyy                  = 0.78529
    Torsion Constant     =  1.5706
    Shear Correction-yy  = 0.86083
    Num of Circum Cells  =  16