MAE 4700/ 5700/ 5701 HW 7 Due: December 4, 2015

Esther Koo efk28

Chen Shen cs829

Brian Chantrupon bc566

Esther Ansys & Writing 10 hours

Brian Ansys & Writing 12 hours

Chen MATLAB & Writing 12 hours

Sources: None

2) One quarter of the plate was taken, so the new geometry became width = 0.15 and length = 0.5 . The quarter-plate was meshed using rectangular elements. MeshGeometry.m

For symmetry boundary conditions, the second surface (left) had no horizontal displacement

and no tangential traction. The bottom surface was broken up into two surfaces: the first

surface (bottom) and the fifth surface (crack). The first surface had symmetry conditions with

no vertical displacement and no tangential traction. The fifth surface was treated as a free

surface (no tangential or normal tractions). The only natural boundary condition entered in

InputData.m was = 30 at the third surface (top). Since the plate was thin, the plane stress condition was used.

% Choose a meshing method % "1" for mesh directly generated from Matlab "BoxGrid_2D" function % "2" for mesh generated from Ansys and loaded from "loadFromGridFile" % function LoadChoice = 1;

% Plot instruction PlotInstructions.plot_mesh = 'no'; % What to plot. For big meshes, PlotInstructions.plot_node = 'no'; % it is better not to plot node and

% vector. PlotInstructions.plot_boundary = 'no'; % Change this information as

% appropriate PlotInstructions.plot_contour = 'yes'; PlotInstructions.plot_vector = 'yes'; PlotInstructions.plot_deformed = 'yes'; PlotInstructions.plot_fringes = 'yes';

nnpe = 4; % number of nodes per element. % currently T3 and Q4 elements are supported

nsd = 2; % number of spatial dimensions if LoadChoice==1 % use LoadChoice =1 only when the geometry is a

% rectangular % Set this information for each problem

xl = 0.0; % left location of the range in the x direction xr = 0.15; % right location of the range in the x direction yb = 0.0; % bottom location of the range in the y direction yt = 0.5; % top location of the range in the y direction

nx = 12; % number of elements in x direction ny = 12; % number of elements in y direction

InputData.m

The crack was located at the last third of the bottom surface, so boundStruct was redefined

in MeshGeometry.m by giving the last third of the elements and associated nodes of the

bottom surface to the fifth surface and the remaining elements and nodes of the bottom

surface to the first surface.

MeshGeometry.m

The Von Mises stress for a planar plate was defined as3

2(

2 + 2 + 2

2 ). The Von Mises

scalar stress field and all stress components were plotted.

% Define the essential BCs % boundStruct.SurfEssV = []; boundStruct.SurfEssV = [2 1 0 1 2 0]; % e.g. [4 2 20] means all nodes on surface

% # 4, % degree of freedom #2 (y direction), has

% a value of 20.

% Define the natural BCs % The natural boundary condition is defined in tangential and normal % direction (rather than global x and y direction),outer normal is % positive. boundStruct.SurfNat = [3 0 30e7]; % e.g. [3 10 -10] means surface # 3 has % a constantly distributed tangential

% traction % 10 and normal traction (pointing in)

% 10. % Define material properties E =2e11; % Young's Modulus nu =0.35; % Poisson's Ratio PlaneStress='yes';% 'yes' for plane stress, 'no' for plane strain

% Use the provided box grid generator [nCoords,elCon,boundStruct]=BoxGrid_2D(nsd,xl,xr,yb,yt,nnpe,nx,ny);

BoundaryNodes(1).Nodes = [1:2/3*nx+1]; % The first boundary (bottom)

% The fifth boundary (bottom) BoundaryNodes(5).Nodes = [2/3*nx+1:nx+1]; % The first boundary (bottom) BoundaryElems(1).Elems = [1:2/3*nx]; BoundaryElems(1).SurfaceIndicator = -2*ones(2/3*nx,1); % The fifth boundary (bottom) BoundaryElems(5).Elems = [2/3*nx+1:nx]; BoundaryElems(5).SurfaceIndicator = -2*ones(1/3*nx,1);

PostProcessor.m

% Calculate Von Mises stress von_mises = sqrt(3/2*(sigma_xx.^2 + sigma_yy.^2 + 2*sigma_xy.^2));

% Plot stresses if strcmp(PlotInstructions.plot_contour,'yes') % Here are two options for plotting stress. Try MakePatchPlot and % PlotContour to see which you prefer. % range=max(max(abs(sig))); % MakePatchPlot(sigma_xx,range,meshStruct, 'xx stress '); % MakePatchPlot(sigma_yy,range,meshStruct, 'yy stress '); % MakePatchPlot(sigma_xy,range,meshStruct, 'xy stress '); plotcontour(meshStruct,sigma_xx, 'xx stress ',15,'area','on'); plotcontour(meshStruct,sigma_yy, 'yy stress ',15,'area','on'); plotcontour(meshStruct,sigma_xy, 'xy stress ',15,'area','on'); plotcontour(meshStruct,von_mises, 'Von Mises Stress ',15,'area','on'); end

Because there was a cusp between the first and fifth surfaces, stress at that node (and

therefore maximum stress) would go to infinity as the number of elements in the mesh

increased. The y displacement of the rightmost node (the free end of the crack in the quarter-

plate and center of the crack in the whole plate) was used to determine convergence of the

solution as seen in Table 1.

Table 1. Convergence values

Number of Elements in x-direction Number of Elements in y-direction Displacement (m)

12 12 1.3373e-04

30 30 1.5071e-04

90 90 1.5758e-04

120 120 1.5839e-04

240 240 1.5959e-04

360 360 1.5998e-04

At an increasing number of elements in the x and y directions, the displacement value

converged to 1.6e-04 m, so the solution overall has converged.

3) The cantilever beam was modeled using 1D, 2D, and 3D elements using a variety of

different mesh sizes in ANSYS. Displacement, DOF, and stress distributions were

determined for each combination of mesh size and type of element, and are summarized in

Table 2 and Table 3 below.

Table 2. Total degrees of freedom and displacements

Mesh DOF (1D) v1 (1D) [m] DOF (2D) v1 (2D) [m] DOF (3D) v1 (3D) [m]

2x2x2 3 -0.005133 18 -0.0050773 81 -0.0050354

4x4x4 5 -0.005133 50 -0.0051191 375 -0.0050931

8x8x8 9 -0.005133 162 -0.0051237 2187 -0.0050996

16x16x8 17 -0.005133 578 -0.0051249 7803 -0.0051013

64x64x8 65 -0.005133 8450 -0.0051254 114075 -0.0051024

Table 3. Maximum values of xx at x = 2m and x = 0m

Mesh Max xx at

x=2m (1D)

[MPa]

Max xx at

x=2m (2D)

[MPa]

Max xx at

x=2m (3D)

[MPa]

Max xx at

x=0m (1D)

[MPa]

Max xx at

x=0m (2D)

[MPa]

Max xx at

x=0m (3D)

[MPa]

2x2x2 2.3176 2.3863 2.292 4.6352 4.7451 4.853

4x4x4 2.3176 2.3304 2.3465 4.6352 4.7426 4.8387

8x8x8 2.3176 2.3095 2.3145 4.6352 4.7727 4.9219

16x16x8 2.3176 2.3165 2.3166 4.6352 4.9234 5.308

64x64x8 2.3176 2.3176 2.3176 4.6352 5.9295 6.9953

a) For 1D elements, 3 degrees of freedom are sufficient to achieve accuracy of three

significant digits for both displacement and stress. A 1D element is also easy to

model, and thus requires a lower number of degrees of freedom to converge. For 2D

elements, 50 degrees of freedom are needed to achieve an accurate displacement and

162 degrees of freedom are required to achieve an accurate stress value to three

rounded significant digits. Lastly, 3D elements require a minimum of 2187 degrees

of freedom for an accurate displacement and stress.

b) The following comparison is done for the 64x64x8 mesh, which is the most refined

mesh implemented on the beam. The displacement value between the 1D and 2D

elements differs by 0.15%. Between the 2D and 3D elements, the value differs by

0.45%, while the difference in the displacement between 1D and 3D elements is

greatest with a value of 0.60%. However overall, the percent difference between each

type of element is below 1% and can be considered to be very low. The values of

stress at x = 2m for all three element types are the same up to five significant digits

and thus agree very well.

c) Displacement converges faster than stress when refining the mesh because stress is a

second derivative of displacement and thus takes longer to reach an exact solution, if

it is able to reach a solution at all.

d) For 1D, 2D, and 3D elements, the displacement results display very little effects from

stress singularities because they are mainly concentrated on the fixed end. As seen in

the results, all displacement values ultimately converge at three significant digits as

the mesh is refined. Stress is calculated as force divided by area. When a beam is

loaded at one end, the four sharp corners of the fixed end of the beam and the location

of the point source can have an infinitely small area. As the mesh is refined, the area

becomes smaller, resulting in a value of stress that becomes larger. The maximum

stress at the fixed end of the beam modeled with 1D elements does not exhibit stress

singularities because the stress values stay the same as the mesh is refined. However

for 2D elements, when the mesh is refined from 16x16x8 to 64x64x8, the maximum

stress value suddenly increases by 18.5%. A similar trend is shown in 3D elements

but at an even earlier point during mesh refinement: as the mesh changes from 8x8x8

to 16x16x8, the maximum stress value increases by 7.55%; and when the mesh

changes from 16x16x8 to 64x64x8, the stress value increases by 27.4%.

Unfortunately for 2D and 3D elements, a refined mesh leads to stress singularities and

lowers the likelihood of convergence at the fixed end of the beam. The extent of

stress singularities can be seen when comparing stress values between all element

types at a refinement level of 64x64x8. 2D elements lead to a 24.5% increase in

stress value compared to 1D elements, while 3D elements lead to a 40.5% increase in

stress value from 1D elements.

e) In order to determine displacements, 3D elements provide the most accurate result.

However efficiency is decreased because at least 7803 degrees of freedom are

required to obtain a converged solution. When stresses are determined, 1D elements

are sufficient to obtain the stress within the beam and at the free end of the beam.

The stress value obtained with 1D, 2D, and 3D elements is ultimately the same,

except that 1D elements reach the solution earlier with a coarser mesh. As a result,

1D elements for calculating the stress at the free end of the beam and within the beam

are accurate and efficient. During calculation of the stress at the fixed end of the

beam, 1D elements do not exhibit stress singularities as the mesh is refined.

Although efficiency is maintained, accuracy decreases. As a result, 2D elements and

3D elements provide more detail of stress behavior at the fixed end of the beam

because the refined mesh allows for a more accurate stress value.

4)

a) The bracket was meshed in a 2D modeller as seen in Figure 1.

Figure 1. Mesh plot of the bracket in ANSYS

b) The geometric parameters used for the initial bracket are shown below in Table 4 and

Figure 2. The initial total volume of the bracket was calculated to be 73.376 m3.

Analysis of the bracket using this initial configuration resulted in the von Mises

equivalent stress plot shown in Figure 3.

Table 4. Geometric parameters, initial conditions, and design variables employed

Parameter: Initial Value:

[m]

Lower Limit:

[m]

Upper

Limit: [m]

10-H 10 5 10

30-W 30 25 30

R1 1 0.05 5

R2 2 1.5 4.95

Though the thickness, T was also specified as a design parameter to be varied in the

initial problem, it was instead set to constant value 0.25 m (the lower limit of the given

design variable) to minimize the volume. This assumption was taken due to the fact

the thickness was constant throughout the body and would not affect the final analysis.

Additionally, due to the fact that ANSYS cannot model dimensions of 0, the

parameters for H and W where changed to 10-H and 30-W respectively, while R1 and

R2 were varied such that its values would not necessarily be zero or make any other

dependent dimensions be zero though it was close enough.

Figure 2. Bracket dimensions

Figure 3. Initial von Mises equivalent stress plot

c) The response surfaces for (10-H,R1), (10-H,R2), and (W,H) are shown in the below.

Though the response surface for (R2, T) was requested, it is not shown because T was

made to not be a varying parameter which is what was advised by the Professor.

Figure 4. 3D response surface plot of the H and R1 parameter vs. the maximum von Mises

stress

Figure 5. 3D response surface plot of the H and R2 parameter vs. the maximum von Mises

stress

Figure 6. 3D response surface plot of the W and H parameters vs. the maximum von Mises

stress

d) Using the response surfaces shown in Figure 4, Figure 5, and Figure 6, Ansys will

find candidate points below the maximum stress constraint that is input into the

optimization window. Figure 7 shows the candidate points generated by ANSYS

using the 3D response surfaces. The candidate point is chosen that has been verified

to be below the stress limit of 100 kPa and has minimized the volume the most. In this

case, Candidate Point 1 is the best solution, with a 10-H value of 5.7525m (meaning H

is 4.2475m), R1 value of 2.0827, R2 value of 2.4956, and 30-W value of 4.949m

(meaning W is 25.051m). The total volume of the bracket using these parameters is

50.424 m3, and the complete geometric configuration is detailed in Figure 8. Figure 9

shows the von Mises equivalent stress plot for the optimal configuration.

Figure 7. Candidate points and their verified values as generated by ANSYS optimization for the

support bracket

Figure 8. The optimized support bracket according to Candidate Point 1 parameters. The

dimensions are shown estimated to 1 significant figure.

Figure 9. Von Mises equivalent stress plot for the optimal configuration