1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
|
{
"metadata": {
"name": "",
"signature": "sha256:bed4afe1f989fb55f213b8e274b88cbf3d61242768410ad03d4984a97542fc14"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 8:Transformation of stress and strain and Yield and Fracture criteria "
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 8.1 page number 405 "
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"o = 22.5 #degrees , The angle of infetisimal wedge \n",
"A = 1 #mm2 The area of the element \n",
"A_ab = 1*(math.cos(radians(o))) #mm2 - The area corresponds to AB\n",
"A_bc = 1*(math.sin(radians(o))) #mm2 - The area corresponds to BC\n",
"S_1 = 3 #MN The stresses applying on the element \n",
"S_2 = 2 #MN\n",
"S_3 = 2 #MN\n",
"S_4 = 1 #MN \n",
"F_1 = S_1*A_ab # The Forces obtained by multiplying stress by their areas \n",
"F_2 = S_2*A_ab\n",
"F_3 = S_3*A_bc\n",
"F_4 = S_4*A_bc\n",
"#sum of F_N = 0 equilibrim in normal direction \n",
"N = (F_1-F_3)*(math.cos(radians(o))) + (F_4 - F_2)*(math.sin(radians(o)))\n",
"\n",
"#sum of F_s = 0 equilibrim in tangential direction \n",
"\n",
"S = (F_2-F_4)*(math.cos(radians(o))) + (F_1 - F_3)*(math.sin(radians(o)))\n",
"\n",
"Stress_Normal = N/A #Mpa - The stress action in normal direction on AB\n",
"Stress_tan = S/A #Mpa - The stress action in tangential direction on AB\n",
"print \"The stress action in normal direction on AB\",round(Stress_Normal,2),\"Mpa\"\n",
"print \"The stress action in tangential direction on AB\",round(Stress_tan,2),\"Mpa\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The stress action in normal direction on AB 1.29 Mpa\n",
"The stress action in tangential direction on AB 2.12 Mpa\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 8.2 page number 413"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"o = -22.5 #degrees , The angle of infetisimal wedge \n",
"A = 1 #mm2 The area of the element \n",
"A_ab = 1*(math.cos(radians(o))) #mm2 - The area corresponds to AB\n",
"A_bc = 1*(math.sin(radians(o))) #mm2 - The area corresponds to BC\n",
"S_1 = 3.0 #MN The stresses applying on the element \n",
"S_2 = 2.0 #MN\n",
"S_3 = 2.0 #MN\n",
"S_4 = 1.0 #MN\n",
"#Caliculations \n",
"\n",
"F_1 = S_1*A_ab # The Forces obtained by multiplying stress by their areas \n",
"F_2 = S_2*A_ab\n",
"F_3 = S_3*A_bc\n",
"F_4 = S_4*A_bc\n",
"#sum of F_N = 0 equilibrim in normal direction \n",
"N = (F_1-F_3)*(math.cos(radians(o))) + (F_4 - F_2)*(math.sin(radians(o)))\n",
"\n",
"#sum of F_s = 0 equilibrim in tangential direction \n",
"\n",
"S = (F_2-F_4)*(math.cos(radians(o))) + (F_1 - F_3)*(math.sin(radians(o)))\n",
"\n",
"Stress_Normal = N/A #Mpa - The stress action in normal direction on AB\n",
"Stress_tan = S/A #Mpa - The stress action in tangential direction on AB\n",
"print \"a) The stress action in normal direction on AB\",round(Stress_Normal,2),\"Mpa\"\n",
"print \"a) The stress action in tangential direction on AB\",round(Stress_tan,2),\"Mpa\"\n",
"\n",
"#Part- b\n",
"\n",
"S_max = (S_4+S_1)/2 + (((((S_4-S_1)/2)**2) + S_3**2)**0.5) #Mpa - The maximum stress\n",
"S_min = (S_4+S_1)/2.0 - (((((S_4-S_1/2))**2) + S_3**2)**0.5) #Mpa - The minumum stress\n",
"k = 0.5*math.atan(S_3/((S_1-S_4)/2)) #radians The angle of principle axis\n",
"k_1 = math.degrees(k)\n",
"k_2 = k_1+90 #The principle plane angles\n",
"print \"b) The principle stress \",round(S_max,1),\"Mpa tension\"\n",
"print \"b) The principle stress \",round(S_min,2),\"Mpa compression\"\n",
"print \"b) The principle plane angles are\",round(k_1,0),\",\",round(k_2,0),\"degrees\"\n",
"\n",
"#part-c\n",
"#The maximum shear stress case\n",
"t_xy = (((((S_4-S_1)/2)**2) + S_3**2)**0.5) #Mpa - The maximum shear stress case\n",
"K = 0.5*math.atan((-(S_1-S_4)/(2*S_3))) #radians The angle of principle axis\n",
"K_0 = math.degrees(K)\n",
"if K_0<0:\n",
" K_1 = K_0+90\n",
"else:\n",
" K_1 = K_0\n",
"K_2 = K_1+90 #PRinciple plain angles\n",
"T_xy = -((S_1-S_4)/2)*(math.sin(radians(2*K_1))) + ((S_4+S_1)/2)*(math.cos(radians(2*K_1))) # Shear stress\n",
"print \"c) The maximum shear is \",round(T_xy,2),\"Mpa\" \n",
"S_mat_a = array([round(S_max,1),round(S_min,1),0]) #MPa maximum stress matrix\n",
"S_mat_b = array([(S_4+S_1)/2,round(T_xy,2),round(T_xy,2),(S_4+S_1)/2]) #MPa maximum stress matrix at maximum shear\n",
"print \"a)\",S_mat_a,\"Mpa\"\n",
"print \"b)\",S_mat_b,\"Mpa\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"a) The stress action in normal direction on AB 4.12 Mpa\n",
"a) The stress action in tangential direction on AB 0.71 Mpa\n",
"b) The maximum stress 4.2 Mpa tension\n",
"b) The minumum stress -0.06 Mpa compression\n",
"b) The principle plane angles are 32.0 , 122.0 degrees\n",
"c) The maximum shear is -2.24 Mpa\n",
"a) [ 4.2 -0.1 0. ] Mpa\n",
"b) [ 2. -2.24 -2.24 2. ] Mpa\n"
]
}
],
"prompt_number": 58
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 8.3 page number 421"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"S_x = -2 #Mpa _ the noraml stress in x direction\n",
"S_y = 4 #Mpa _ the noraml stress in Y direction\n",
"c = (S_x + S_y)/2 #Mpa - The centre of the mohr circle \n",
"point_x = -2 #The x coordinate of a point on mohr circle\n",
"point_y = 4 #The y coordinate of a point on mohr circle\n",
"Radius = pow((point_x-c)**2 + point_y**2,0.5) # The radius of the mohr circle\n",
"S_1 = Radius +1#MPa The principle stress\n",
"S_2 = -Radius +1 #Mpa The principle stress\n",
"S_xy_max = Radius #Mpa The maximum shear stress\n",
"print \"The principle stresses are\",S_1 ,\"Mpa\",S_2,\"Mpa\"\n",
"print \"The maximum shear stress\",S_xy_max,\"Mpa\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The principle stresses are 6.0 Mpa -4.0 Mpa\n",
"The maximum shear stress 5.0 Mpa\n"
]
}
],
"prompt_number": 61
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 8.4 page number 423"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"import math \n",
"S_x = 3.0 #Mpa _ the noraml stress in x direction\n",
"S_y = 1.0 #Mpa _ the noraml stress in Y direction\n",
"c = (S_x + S_y)/2 #Mpa - The centre of the mohr circle \n",
"point_x = 1 #The x coordinate of a point on mohr circle\n",
"point_y = 3 #The y coordinate of a point on mohr circle\n",
"#Caliculations \n",
"\n",
"Radius = pow((point_x-c)**2 + point_y**2,0.5) # The radius of the mohr circle\n",
"#22.5 degrees line is drawn \n",
"o = 22.5 #degrees \n",
"a = 71.5 - 2*o #Degrees, from diagram \n",
"stress_n = c + Radius*math.sin(math.degrees(o)) #Mpa The normal stress on the plane \n",
"stress_t = Radius*math.cos(math.degrees(o)) #Mpa The tangential stress on the plane\n",
"print \"The normal stress on the 221/2 plane \",round(stress_n,2),\"Mpa\"\n",
"print \"The tangential stress on the 221/2 plane \",round(stress_t,2),\"Mpa\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The normal stress on the 221/2 plane 4.82 Mpa\n",
"The tangential stress on the 221/2 plane 1.43 Mpa\n"
]
}
],
"prompt_number": 84
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 8.7 page number 437"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"e_x = -500 #10-6 m/m The contraction in X direction\n",
"e_y = 300 #10-6 m/m The contraction in Y direction\n",
"e_xy = -600 #10-6 m/m discorted angle\n",
"centre = (e_x + e_y)/2 #10-6 m/m \n",
"point_x = -500 #The x coordinate of a point on mohr circle\n",
"point_y = 300 #The y coordinate of a point on mohr circle\n",
"Radius = 500 #10-6 m/m - from mohr circle\n",
"e_1 = Radius +centre #MPa The principle strain\n",
"e_2 = -Radius +centre #Mpa The principle strain\n",
"k = math.atan(300.0/900) # from geometry\n",
"k_1 = math.degrees(k)\n",
"print \"The principle strains are\",e_1,\"um/m\",e_2,\"um/m\"\n",
"print \"The angle of principle plane\",round(k_1,2) ,\"degrees\"\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The principle strains are 400 um/m -600 um/m\n",
"The angle of principle plane 18.43 degrees\n"
]
}
],
"prompt_number": 87
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 8.8 page number 441"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Given\n",
"e_0 = -500 #10-6 m/m \n",
"e_45 = 200 #10-6 m/m \n",
"e_90 = 300 #10-6 m/m\n",
"E = 200 #Gpa - youngs modulus of steel \n",
"v = 0.3 # poissions ratio \n",
"#Caliculations \n",
"\n",
"e_xy = 2*e_45 - (e_0 +e_90 ) #10-6 m/m from equation 8-40 in text\n",
"# from example 8.7\n",
"e_x = -500 #10-6 m/m The contraction in X direction\n",
"e_y = 300 #10-6 m/m The contraction in Y direction\n",
"e_xy = -600 #10-6 m/m discorted angle\n",
"centre = (e_x + e_y)/2 #10-6 m/m \n",
"point_x = -500 #The x coordinate of a point on mohr circle\n",
"point_y = 300 #The y coordinate of a point on mohr circle\n",
"Radius = 500 #10-6 m/m - from mohr circle\n",
"e_1 = Radius +centre #MPa The principle strain\n",
"e_2 = -Radius +centre #Mpa The principle strain\n",
"\n",
"stress_1 = E*(10**-3)*(e_1+v*e_2)/(1-v**2) #Mpa the stress in this direction \n",
"stress_2 = E*(10**-3)*(e_2+v*e_1)/(1-v**2) #Mpa the stress in this direction \n",
"print\"The principle stresses are \",round(stress_1,2),\"Mpa\",round(stress_2,2),\"MPa\" "
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The principle stresses are 48.35 Mpa -105.49 MPa\n"
]
}
],
"prompt_number": 91
}
],
"metadata": {}
}
]
}
|
