Problem Description

Figure below shows the beam model. Linear static analysis is performed on the beam. The maximum bending stress at mid span (point B) is determined. All dimensions are in inches.
Solution Type: Linear Static
Unit: inch/pound/second
Model Geometry: Length: L = 50 in; Width: w = 2 in; Height: h1 = 3 in and h2 = 9 in;
Material Properties:
• Young’s Modulus: E = 30.0 E+6 psi
• Poisson’s Ratio: nu = 0.2
Boundary Conditions:
The deep end (CD) is fixed in all translations and all rotations. All the other nodes are constrained in the Z-translation and X and Y-rotations. A load P = 4,000 lb is applied at the free end of the cantilever in the negative Y-direction
References:
[1]. Crandall, S. H. and Dahl, N. C., An Introduction to the Mechanics of Solids, 3rd Edition. New York: McGraw-Hill, Inc., 1959.

APDL CODE

 FINI
 /CLEAR,START
 /PREP7
 ! ELEMENT TYPE: BEAM188
 ET, 1, BEAM188
  
 ! +Cross section ID 1,Rectangular, h = 3, w = 2
 SECTYPE, 1, BEAM, RECT
 SECOFFSET, CENT 
 SECDATA, 2, 3
  
 ! +Cross section ID 2,Rectangular, h = 9, w = 2
 SECTYPE, 2, BEAM, RECT
 SECOFFSET, USER, 0, 3
 SECDATA, 2, 9
  
  
 ! +Variable cross section ID 3,VARIABLE from ID 1 and ID 2,L = 50
 SECTYPE,3,TAPER, ,VARIABLE  
 SECDATA, 1, , , ,
 SECDATA, 2, 50, , , 
  
 ! MATERIAL PROPERTIES
 ! + Young’s Modulus: 30e6 psi
 MP, EX, 1, 30E6
 ! + Poisson’s Ratio: 0.2
 MP, PRXY, 1, 0.2
  
 ! MODEL GEOMETRY 
 ! + Keypoints: K, npt, x, y, z
 K, 1, 0, 0, 0
 K, 2, 50, 0, 0
 K, 3, 0, 1, 0  ! orientation keypoint
 ! + Lines: L, npt 1, npt 2
 L, 1, 2
  
 ! MESHING
 ! +Specifies the divisions and spacing ratio on unmeshed lines, n = 6
 LESIZE,1,,,6 ! 
 ! +Associates element attributes with the selected, unmeshed lines
 LATT,1, ,1, ,3, ,3
 ! +Mesh
 LMESH, 1 
 /ESHAPE,1.0  ! view element shape, SCALE = 1
 EPLOT
  
 ! BOUNDARY CONDITIONS
 DK,2,ALL
  
 ! LOADS
 FK,1,FY,-4000
  
 ! SOLVE
 /SOLU
 SOLVE
  
 ! RESULT,Post Processing
 /POST1
 ! + Plot results on nodes (N), S: Stress, EQV: von-Mises stress
 ! U : Displacement
 PLNSOL,U,SUM 
 PLNSOL,S,EQV 
 ! + Print results on Elements (E), S: Stress
 PRESOL,S,PRIN 

RESULTS

COMPARISON

Description Theory [1] ANYSYS
Maximum Bending Stress at Mid-Span (psi) 8333   8328.38