Combined bending and torsion

Problem description:

• A vertical bar of length \(l\) is subjected to the action of a horizontal force F acting at a distance d from the axis of the bar. Determine the maximum principal stress \(\sigma _{max}\) and the maximum shear stress \(\tau _ {max}\) in the bar.

Reference:

• Timoshenko, Strength of Materials, Part I, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 299, problem 2.

Analysis type(s):

• Static analysis ANTYPE=0

Element type(s):

• Elastic straight pipe element (PIPE16)

• 3D 2 Node pipe element (PIPE288)

Material properties

• \(E = 30 \cdot 10^6 psi\)

• \(u=0.3\)

Geometric properties:

• \(l = 25 l\)

• \(d = 3 ft\)

• Section modulus \((l/c) = 10 in^3\)

• Outer Diameter \(= 4.67017 in\)

• Wall Thickness \(= 2.33508 in\)

Loading:

• \(F = 250 lb\)

• \(M = Fd = 9000 in-lb\)

Analysis assumptions and modeling notes:

• Use consistent length units of inches. Real constants for PIPE16 and section properties for PIPE288 are used to define the pipe Outer Diameter and Wall Thickness. These values are calculated for a solid cross-section from the given section modulus. The offset load is applied as a centroidal force and a moment.

# sphinx_gallery_thumbnail_path = '_static/vm12_setup.png'

from ansys.mapdl.core import launch_mapdl
import pandas

Start MAPDL

Start MAPDL.

mapdl = launch_mapdl(loglevel="WARNING", print_com=True)
mapdl.clear()  # optional as MAPDL just started

Pre-processing with ET PIPE16

mapdl.verify("vm12")
mapdl.prep7()

mapdl.antype("STATIC")
mapdl.et(1, "PIPE16")
mapdl.r(1, 4.67017, 2.33508)  # REAL CONSTANTS FOR SOLID CROSS SECTION
mapdl.mp("EX", 1, 30e6)
mapdl.mp("NUXY", 1, 0.3)
mapdl.n(1)
mapdl.n(2, "", "", 300)
mapdl.e(1, 2)
mapdl.d(1, "ALL")
mapdl.f(2, "MZ", 9000)
mapdl.f(2, "FX", -250)
mapdl.finish()

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

Solve

mapdl.slashsolu()
mapdl.outpr("BASIC", 1)

mapdl.solve()
mapdl.finish()
mapdl.post1()
mapdl.etable("P_STRS", "NMISC", 86)
mapdl.etable("SHR", "NMISC", 88)

STORE SHR      FROM ITEM=NMIS COMP=  88  FOR ALL SELECTED ELEMENTS

Post-processing

p_stress = mapdl.get("P_STRESS", "ELEM", 1, "ETAB", "P_STRS")
shear = mapdl.get("SHEAR", "ELEM", 1, "ETAB", "SHR")
p_trs = shear / 2

# Fill the array with target values
target_p_stress = 7527
target_p_trs = 3777

data = [
    [target_p_stress, p_stress, abs(p_stress / target_p_stress)],
    [target_p_trs, p_trs, abs(p_trs / target_p_trs)],
]
col_headers = ["TARGET", "Mechanical APDL", "RATIO"]
row_headers = ["MAX PRINSTRS psi", "MAX SH STRS psi"]

Verify the results.

print(pandas.DataFrame(data, row_headers, col_headers))

                  TARGET  Mechanical APDL     RATIO
MAX PRINSTRS psi    7527      7526.938960  0.999992
MAX SH STRS psi     3777      3776.921145  0.999979

Pre-processing with ET PIPE288

mapdl.clear("nostart")
mapdl.prep7()
mapdl.run("C***     USING PIPE288")
mapdl.antype("STATIC")
mapdl.et(1, "PIPE288", "", "", "", 2)
mapdl.sectype(1, "PIPE")
mapdl.secdata(4.67017, 2.33508)
mapdl.keyopt(1, 3, 3)  # CUBIC SHAPE FUNCTION
mapdl.mp("EX", 1, 30e6)
mapdl.mp("NUXY", 1, 0.3)
mapdl.n(1)
mapdl.n(2, "", "", 300)
mapdl.e(1, 2)
mapdl.d(1, "ALL")
mapdl.f(2, "MZ", 9000)
mapdl.f(2, "FX", -250)
mapdl.finish()

mapdl.allsel()
mapdl.eplot()

Solve

mapdl.slashsolu()
mapdl.outpr("BASIC", 1)
mapdl.solve()
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

mapdl.post1()
mapdl.set("LAST")
mapdl.graphics("POWER")
mapdl.eshape(1)
mapdl.view(1, 1, 1, 1)

mapdl.show(option="REV", fname="png")
mapdl.plesol("S", 1)
mapdl.show("close")

p_stress = mapdl.get("P_STRESS", "PLNSOL", 0, "MAX")

mapdl.show(option="REV", fname="png")
mapdl.plesol("S", "INT")
mapdl.show("close")

shear = mapdl.get("SHEAR", "PLNSOL", 0, "MAX")
p_trs = shear / 2


# Fill the array with target values
target_p_stress = 7527.0
target_p_trs = 3777.0

data = [
    [target_p_stress, p_stress, abs(p_stress / target_p_stress)],
    [target_p_trs, p_trs, abs(p_trs / target_p_trs)],
]
col_headers = ["TARGET", "Mechanical APDL", "RATIO"]
row_headers = ["MAX PRINSTRS psi", "MAX SH STRS psi"]

Verify the results.

print(pandas.DataFrame(data, row_headers, col_headers))

mapdl.finish()
mapdl.starlist("vm12", "vrt")

                  TARGET  Mechanical APDL     RATIO
MAX PRINSTRS psi  7527.0       7526.93994  0.999992
MAX SH STRS psi   3777.0       3776.92212  0.999979

LISTING OF THE DATA ON FILE

 *** NOTE ***                            CP =       0.000   TIME= 00:00:00
 Unable to open *LIST file vm12.vrt.

Stop MAPDL.

mapdl.exit()