stress analysis of a rectangular plate with a hole in the center - pdf
DESCRIPTION
TRANSCRIPT
FINITE ELEMENT
STRUCTURAL ANALYSIS
PROJECT – 1
(INDIVIDUAL PROBLEMS)
SASI BEERA
35763829
Aim:
To analyze a symmetric plate with a hole having a unit thickness for two load cases,
1. Tensile load at the two ends of the plate
2. Bending loads at the two ends. Applying varying tensile pressure at the top half and
varying compressive load at the bottom half simulates bending effect.
Assumptions:
The following assumptions are made in the analysis,
1. The analysis is linear static
2. The material is linear isotropic
3. The problem is considered to be plane stress. I.e. the stresses in the Z direction are
ignored.
4. Geometric symmetry is exploited
Units:
Geometrical dimensions : mm
Pressure : MPa
Geometry:
The dimensions for the geometry are calculated from the following set of equations based
on the person number. Only the quarter symmetry is analyzed for the tensile load
problem
L/H = α
R/H=β
Where α = (K+1)/2 = 1.5
And β = 1/ (l+3) = 1/12 Calculated from the person number
Nu = 0.4
Assuming R = 1 mm
From the above,
L = 18mm, H = 12mm, Nu = 0.4
BCs and loads:
Constrained in x direction Pressure
of 6MPa
Constrained in y direction
Ansys Macro For Tensile Load Case:
!!! Macro for a plate with a Hole!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Finish
/clear
/title,plate_with_hole
!!############
!!Total 8 key points are required to create quarter symm model
!!############
k,1,0,0 !! Hole centre point
k,2,2.5,0 !! Hole first point
k,3,25,0
k,4,25,25
k,5,0,25
k,6,0,2.5 !! Hole last point
k,7,5,0
k,8,0,5
!---------------------------------------------------------------------
larc,2,6,1,2.5 creation of arc line for hole
larc,7,8,1,5
!--------------------------------------------------------
lstr,2,7
lstr,7,3 !! creation of straight line
lstr,3,4 ! creation of straight line
lstr,4,5 ! creation of straight line
lstr,5,8 ! creation of straight line
Pressure load of 6MPa
Constrained in X direction
Constrained in Y direction
lstr,8,6
!------------------------------------------------------
!al,all !!creation of area by line
al,p !!creation of area by graphical picking
!-----------------------------------------------------
et,1,42 !! element type definition 4 noded
!et,1,82 !! element type defenition 8 noded
!keyopt,1,3,3 !! setting keyoption to "plane stress with thickness, this is to include
thickness
keyopt,1,3,0 !!plane stress
!r,1,2 !! defining thickness as real constant, thickness values assumed as 2mm
!mp,ex,1,210000 !! material prop definition
!mp,nuxy,1,0.35
mp,ex,1,289580 !! Young's Modulus for Berillium Alloy
mp,nuxy,1,0.1 !! Poisson's ration for Berillium Alloy
type,1 !! Activate the defined element type ( required element for meshing the current
selected area)
real,1 !! Activate real constant
mat,1 !! Activate material property
!lesize,all,,,12 !! define line size
lesize,p
mshkey,0 !! activate mesh type like mapped or free etc
amesh,all
!!------------------------------------------------------
!! now to apply BC's
!!------------------------------------------------------
nsel,s,loc,x,0.001,-0.001 !! select nodes at the symmetry section
d,all,ux,0 !! X - constarints
nsel,s,loc,y,0.001,-0.001 !! select nodes at the symmetry section
d,all,uy,0 !! Y - constarints
!nsel,s,loc,x,9.9999,10.001 !! select nodes for load application
nsel,s,loc,x,24.9999,25.001
sf,all,pres,-6 !! Apply a tensile pressure load of 6 MPa
!! Solution
alls !! do an all selection before entering the solution :)
save,platewithhole,db !! save db, this db is loaded with mesh, Mat props and BC's :)
solve
!!------------END----------------------
Results for Tensile Plane 42
Iteration 1:
Max stress = 7.483 MPa
Iteration 2:
Max stress = 7.2288 MPa
Iteration 3:
Max stress = 11.467 MPa
Iteration 4:
Max stress = 15.918 MPa
Consolidated Results:
Plate under Tension (Plane 42 element)
No of
elements
No of
Nodes
Displacement
(mm)
Stress
(MPa)
Strain
Energy
(N-mm)
Stress
Concentrati
on Factor
% Error
in Energy
norms
11 37 0.246E-04 7.4831 0.04956 1.643 3.456
24 69 0.258E-04 7.2288 0.04975 2.437 1.945
40 119 0.243E-04 11.467 0.04977 2.681 1.421
258 940 0.28809E-04 15.918 0.04977 2.702 0.829
The solution converged for the iteration 4 in terms of stress and strain energy.
Stress concentration Factor = Max Stress/Nominal Stress
Nominal Stress = Force/ Original Cross sectional Area
The results converged with the 4th iteration in terms of stress and strain energy.
Results for Tensile Plane 82
Iteration 1:
Max stress = 16.139 MPa
Iteration 2:
Max stress = 17.456 MPa
Iteration 3:
Max stress = 17.828 MPa
Iteration 4:
Max stress = 18.091 MPa
Consolidated Results:
Plate under Tension (Plane 82 element)
No of
elements
No of
Nodes
Displacement
(mm)
Stress
(MPa)
Strain
Energy
(N-mm)
Stress
Concentrati
on Factor
% Error
in Energy
norms
158 706 0.270E-04 16.139 0.04975 2.467 5.1250
638 1937 0.298E-04 17.456 0.04978 2.685 0.5714
2064 7578 0.348E-04 17.828 0.04977 2.708 0.2721
18426 64405 0.348E-04 18.091 0.04977 2.717 0.021
From the above consolidated tables, it can be clearly seen that, with a higher order
element where in more number of nodes are present for the same element density gives a
better accuracy in the results. Hence, it is better to use higher order elements. Only the
snag with higher order elements in bigger FEA problems is that the computational time
required is enormously high.
Bending condition
The same macro file shown above can be used for this case too with minor modification.
Geometry and Loading:
Variable tensile loading of 6 MPa
Results for Bending (Plane 42)
Max stress = 6.012 MPa
Consolidated results:
Plate under Bending (Plane 42 element)
No of
elements
No of
Nodes
Displacement
(mm)
Stress
(MPa)
Strain
Energy (N-
mm)
% Error in
Energy
norms
61 75 6.246E-04 6.121 0.02478 3.916
225 241 6.258E-04 6.019 0.02489 2.1259
502 554 6.243E-04 6.012 0.02583 1.4391
848 973 6.288E-04 6.012 0.02583 1.0130
The results converged for the fourth iteration in terms of the stress and the strain energy.
The bending stresses for this case are seen on the top and the bottom fibers. Hence, the
presence of the hole does not contribute to the bending stresses. Therefore the stress
concentration factor will turn out to be 1 for all the cases.
Bending Stress = (M*y)/I
Where M = Bending moment
Y = Distance to the max stress location
I = Area moment of inertia
Results for Bending (Plane 82)
Max stress = 6.009 MPa
Consolidated Results:
Plate under Bending (Plane 82 element)
No of
elements
No of
Nodes
Displacement
(mm)
Stress
(MPa)
Strain
Energy (N-
mm)
% Error in
Energy
norms
57 209 6.046E-04 6.118 0.02477 3.916
222 774 6.058E-04 6.009 0.02489 2.1259
513 1422 6.063E-04 6.009 0.02489 1.4391
The solution converged for the iteration 3 in terms of stress and strain energy. From the
above table it can be clearly seen that, with a higher order element where in more number
of nodes are present for the same element density gives a better accuracy in the results
and with less number of iterations. Hence, it is better to use higher order elements. Only
the snag with higher order elements in bigger FEA problems is that the computational
time required is enormously high.
Conclusions:
1. A refined mesh gives accurate results. Once the convergence is achieved, further
refinement of the mesh will produce the same results. This is illustrated by the
graph below.
2. Use of higher order elements yields accurate results with less number of
iterations, as is seen from the discussion in the report.