midas例题

Example-1
Analysis Type Unit System Dimension
Element Material Boundary Condition
Load Case
2D plane-strain elastic analysis
m, N
Width
10.0 m
Height
10.0 m
Hole diameter
Y
P0
Example-1
10 m 1m
P0
X 1m
10m
Fig. 1.3 Geometry and boundary conditions for plane-strain model – quarter-symmetry
Y
Z
X
Fig. 1.4 Mesh for plane-strain model – quarter-symmetry
0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
Position from the center of hole (m)
(b) Graph of radial and tangential stresses (model 1)
Verification Examples
The axisymmetric model for this problem is shown in Fig. 1.5, and the GTS mesh in Fig. 1.6. The axis of symmetry is along the axis of the hole. By using the axisymmetry model option, the total number of elements is greatly reduced; now, 62 elements are used. The model boundary is at 10 m, which is the same as for the first test. All two models are assigned the same material properties. Two models are subjected to an isotropic compressive stress of 30 MPa.
σθ σr
τ rθ
r
θ
P1
O
a
P2பைடு நூலகம்
Fig. 1.2 Cylindrical hole in an infinite elastic medium
Example-1
The displacements can also be determined assuming conditions of plane strain:
A point located at polar coordinate ( r,θ ) near an opening with radius a (Fig. 1.2) has
stresses σ r ,σθ ,τ rθ given by:
σ= r
p 1
+ 2
p 2
⎡⎣1 − a2
r 2 ⎤⎦ +
p 1
Vrification Manual
Modeling, Integrated Design & Analysis Software
ㅡ
Example-1
Title
Cylindrical Hole in an Infinite Elastic Medium
Description
This problem concerns the determination of stresses and displacements for the case of a
vθ
is the tangential displacement, as
shown in Fig. 1.2. G is the shear modulus, and ν is the Poisson’s ratio.
Verification Examples
GTS MODEL
Two different models were considered for this problem. The first is a plane-strain model with the plane of analysis oriented normal to the axis of the hole. The second is an axisymmetric model with the axis of symmetry aligned with the hole axis.
yy
2
and axisymmetric geometry. The infinite elastic boundary is also tested in this example.
A cylindrical hole with a radius of 1 m exists in an infinite body under a uniform compressive stress of -30 MPa. It is assumed that the problem is symmetric about both the horizontal and vertical axes. Further, it is assumed that the radius of the hole is small compared to the length of the cylinder. This assumption permits the 3D problem to be reduced to a 2D plane-strain problem.
cylindrical hole in an infinite elastic medium subjected to in-situ stress field σ = p ,
xx
1
σ = p . The problem tests the isotropic elastic material model, the plane-strain condition
− 2
p 2
⎡ ⎢⎣1 −
4a2 r2
+
3a 4 r4
⎤ ⎥⎦
cos
2θ
σ θ
=
p 1
+ 2
p 2
⎡⎣1 +
a2
r 2 ⎤⎦ −
p 1
− 2
p 2
⎡ ⎢⎣1 +
3a 4 r4
⎤ ⎥⎦
cos
2θ
p − p ⎛ 2a2 3a4 ⎞
τ =− rθ
1
2
2 ⎜⎝1 + r 2
− r 4 ⎟⎠ sin 2θ
ur vθ
Y Cylindrical hole
Axis of Symmetry
1m 10m
P0
X
Fig. 1.5 Geometry and boundary conditions for axisymmetric mode
Y
Z
X
Fig. 1.6 Mesh for axisymmetric model
MODEL 1
Verification Examples
Results
Fig. 1.7 shows the radial and tangential stress calculated by GTS compared to the analytical solution for σ r and σθ . Fig. 1.8 shows the comparison for radial displacement. These two plots indicate the agreement along a line through the model taken along either the x- or y-axis.
Soil
Cylindrical hole
Fig. 1.1 Cylindrical hole in an infinite elastic medium
1
Verification Examples
Analytical Solution
For a cylindrical hole in an infinite, isotropic, elastic medium under plane-strain conditions, the radial and tangential stress distributions are given by the classical Kirsch solution (e.g., see Jaeger and Cook 1976).
1.0 m
4-node quadrilateral plane-strain element
Modulus of elasticity Poisson’s ratio
E = 6778 MPa ν = 0.2103
Left end
Constrain DX
Bottom end
Constrain DZ
Initial isotropic in-situ compressive stress of -30 MPa. Edge pressure of 30 MPa at right and top ends.
Stress (MPa)
61.05
30.00
(a) Contour of stresses along X-direction (model 1)
60
50
40
30
20
GTS (radial)
10
GTS (tangential)
Analytical (radial)
Analytical (tangential)
For the first model, only a quarter of the problem needs to be analyzed because of the symmetry of the problem. The model and boundary conditions are shown in Fig. 1.4. The mesh is shown in Fig. 1.5 and, as the figure indicates, a radial mesh is produced with increasing mesh size away from the hole. The uniform meshing around the hole provides a more-accurate solution than would a nonuniform mesh. A total 900 elements are used in this mesh. The location of the boundary was varied to evaluate its effect on solution accuracy. The boundary was selected at 10 m (i.e., five hole diameters) from the hole center.
u r
=
p +p
1
2
4G
a2 r
+
p −p
1
2
4G
a2 r
⎡ ⎢⎣4
(1
−
ν
)
−
a2 ⎤ r 2 ⎥⎦ cos 2θ
vθ
=−
p −p
1
2
4G
a2 r
⎡ ⎢⎣2
(1
−
2ν
)
+
a2 ⎤ r 2 ⎥⎦ sin 2θ
in which
u r
is the radial outward displacement, and
MODEL 2
Analysis Type Unit System
Dimension
Element Material Boundary Condition Load Case
2D axisymmetric elastic analysis
m, N
Width
10.0 m
Height
1.0 m
Hole diameter
1.0 m
4-node quadrilateral axisymmetric element
Modulus of elasticity Poisson’s ratio
E = 6778 MPa
ν = 0.2103
Top & Bottom ends
Constrain DZ
Initial isotropic in-situ compressive stress of -30 MPa. Edge pressure of 30 MPa at right end