Solution Manual Vol 7

# 2 together with the failure envelope from which 0 44

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Unformatted text preview: ope. Thus from Figure Q4.4 uf 36 kN=m2 Figure Q4.4 24 Shear strength 4.5 3 (kN/m2) 150 300 450 600 1 3 (kN/m2) 103 202 305 410 1 (kN/m2) 253 502 755 1010 u (kN/m2) 82 169 252 331 03 (kN/m2) 68 131 198 269 01 (kN/m2) 171 333 503 679 The Mohr circles and failure envelope are drawn in Figure Q4.5, from which c0 0 and 0 25 1/2 . Figure Q4.5 The principal stress difference at failure depends only on the value of all-round pressure under which consolidation took place, i.e. 250 kN/m2. Hence, by proportion, the expected value of (1 3)f 170 kN/m2. 4.6 03 (kN/m2) 200 400 600 V/V0 0.061 0.086 0.108 l/l0 0.095 0.110 0.124 Area (mm2) 1177 1165 1155 Load (N) 565 1015 1321 1 3 (kN/m2) 480 871 1144 01 (kN/m2) 680 1271 1744 The average cross-sectional area of each specimen is obtained from Equation 4.10; the original values of A, l and V are: A0 1134 mm2, l0 76 mm, V0 86 200 mm3. The Mohr circles are drawn in Figure Q4.6(a) and (b). From (a) the secant parameters are measured as 34 , 31.5 and 29 . The failure envelope, shown in (b), exhibits a curvature and between 300 and 500 kN/m2 is approximated to a straight line, from which c0 20 kN/m2 and 0 31 . Shear strength 25 29 (a) 34 (kN/m2) 400 31.5 0 400 800 (kN/m2) 31 1200 1600 (b) (kN/m2) 300 400 500 Failure envelope 20 (kN/m2) 0 400 800 (kN/m2) 1200 1600 Figure Q4.6 4.7 The torque required to produce shear failure is given by T dh cu cu d 2 2 Z 0 d=2 2r drcu r Z d=2 d 2h 4cu r2 dr 2 0 2 d h d3 cu 2 6 Then, 2 5 10 53 35 cu 103 2 6 ; cu 76 kN=m3 26 Shear strength 4.8 The relevant stress values are calculated as follows: 3 600 kN=m2 1 3 1 u 01 03 1 (1 3 ) 2 1 0 ( 03 ) 2 1 1 (1 3 ) 2 0 600 200 400 400 0 400 600 80 680 229 451 371 40 411 640 158 758 277 481 323 79 402 679 214 814 318 496 282 107 389 707 279 879 388 491 212 139 351 739 319 919 433 486 167 159 326 759 The stress paths are plotted in Figure Q4.8. The initial points on the effective and total stress paths are separated by the value of the back pressure (us 200 kN/m2). 433 200 0:73 Af 319 Figure Q4.8 4.9 B u3 144 0:96 3 350 200 1 1 3 (kN/m2) 0 201 252 275 282 283 u1 (kN/m2) 0 100 96 78 68 65 A u1 =1 0.50 0.38 0.28 0.24 0.23 &quot;a (%) 0 2 4 6 8 10 Shear strength 27 Figure Q4.9 The variation of A with axial strain is plotted in Figure Q4.9. At failure, A 0.23. Chapter 5 Stresses and displacements 5.1 Vertical stress is given by z Q 5000 Ip 2 Ip z2 5 Values of Ip are obtained from Table 5.1. r (m) 0 1 2 3 4 5 7 10 r/z 0 0.2 0.4 0.6 0.8 1.0 1.4 2.0 Ip 0.478 0.433 0.329 0.221 0.139 0.084 0.032 0.009 z (kN/m2) 96 87 66 44 28 17 6 2 The variation of z with radial distance (r) is plotted in Figure Q5.1. Figure Q5.1 Stresses and displacements 29 5.2 Below the centre load (Figure Q5.2): r 0 for the 7500-kN load z ; Ip 0:478 r 5 1:25 for the 10 000- and 9000-kN loads z 4 ; Ip 0:045 Then, z X Q z I 2 p 7500 0:478 10 000 0:045 9000 0:045 42 42 42 224 28 25 277 kN=m2 Figure Q5.2 5.3 The vertical stress under a corner of a rectangular area is given by z qIr where values of Ir are ob...
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