VM140

VM140
Stretching, Twisting and Bending of a Long Shaft

Overview

Reference:	S. Timoshenko, Strength of Material, Part I, Elementary Theory and Problems, 3rd Edition, D. Van Nostrand Co., Inc., New York, NY, 1955, pg. 296, article 65.
Analysis Type(s):	Static Analysis (ANTYPE = 0)
Element Type(s):	Axisymmetric-Harmonic 8-Node Structural Solid Elements (PLANE83)
Input Listing:	vm140.dat

Test Case

A long solid circular shaft of length and diameter d is built-in at one end and loaded at the other end by a twisting moment, an axial force, and a vertical force as shown. Determine the maximum shear stress τ at the wall due to the moment. Determine the maximum normal stress σ_y at the wall and at one inch from the wall due to the forces. Also determine the maximum combined stress σ₁ at the wall due to both the moment and the forces.

Figure 194: Shaft Problem Sketch

Material Properties

Geometric Properties

E = 30 x 10⁶ psi

υ = 0.0

= 24 in

d = 1 in

F_v = -25 lb

M = -200 in-lb

F_a = 100 lb

Analysis Assumptions and Modeling Notes

The loads are applied at only one node (node 75) for convenience since the cross section of interest (at the wall) is far from the load. Nodal forces are applied on full circumference basis and are calculated for symmetric mode 1 as follows:

FZ =	-2M/d	= 400
FX =	2_Fv	= -50 (see Figure 5.4: Bending and Shear Loading (ISYM = 1) in the Element Reference)
FY =	F_a	= 100

Poisson's ratio is taken as zero to avoid the stress concentration at the built-in end due to the axial force. The nonaxisymmetric loading capability of this element type (PLANE83) is used to model the bending effect. POST1 is used to report maximum stresses at the wall.

Results Comparison

Maximum Stresses	Target	Mechanical APDL	Ratio
TORSION shear stress, psi	1018.6	1018.6[1]	1.000
AXIAL + BENDING Stress_y, psi (at y = 0)	6238.9	6239.9[1]	1.000
COMBINED Stress₁ , psi	6401.0	6402.1[1]	1.000

POST1 nodal stresses at node 51