Μεραληθή Πεηξσκάησλ (Rock Mechanics) Μέξνο 1 νλ «Εηζαγσγηθά»
|
|
|
- Ἀγάπιος Κωνσταντίνου
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Μεραληθή Πεηξσκάησλ (Rock Mechanics) Μέξνο 1 νλ «Εηζαγσγηθά» Exadaktylos G. Exadaktylos notes Rock Mechanics part I Slide 1 of 31
2 Η Μετανική ηων Πεηρωμάηων (Rock Mechanics) αζτολείηαι με ηεν μελέηε ηες ζηαηικής και δσναμικής ζσμπεριθοράς ηοσ αρρήκηοσ πεηρώμαηος και ηες βρατόμαδας (ρεγμαηωμένοσ πεηρώμαηος). Η Μετανοηετνία ηων Πεηρωμάηων (Rock engineering) εθαρμόδεηαι για ηον ζτεδιαζμό και ηε δεμιοσργία καηαζκεσών πάνω και μέζα ζηα πεηρώμαηα. Exadaktylos notes Rock Mechanics part I Slide 2 of 31
3 Παξαδείγκαηα θαηαζθεπώλ ζε πεηξώκαηα - Πξαλή (ππαίζξηεο εθκεηαιιεύζεηο) - Υπόγεηα έξγα (ζήξαγγεο, ζάιακνη, θξέαηα, θεθιηκέλα) - Βαζηέο γεσηξήζεηο - Θεκειηώζεηο (θξάγκαηα, γέθπξεο, πάζζαινη, ) -Τνίρνη αληηζηήξημεο Exadaktylos notes Rock Mechanics part I Slide 3 of 31
4 Καηαζθεπέο εληόο ησλ πεηξσκάησλ θαη επί απηώλ. Exadaktylos notes Rock Mechanics part I Slide 4 of 31
5 Η Μεραληθή ησλ Πεηξσκάησλ είλαη ην «εξγαιείν» γηα ηελ πξόβιεςε ηεο κειινληηθήο ζπκπεξηθνξάο ησλ βξαρνκαδώλ. Λ.ρ. ηη ζα ζπκβεί αλ κία ζήξαγγα ή έλαο ζάιακνο νξπρζεί κε έλα ζπγθεθξηκέλν πξνζαλαηνιηζκό θαη ζε έλα νξηζκέλν βάζνο ζε κηα βξαρόκαδα? Τη ζα γίλεη αλ αιιάμεη ν πξνζαλαηνιηζκόο θαη ην βάζνο? Π.ρ. κεηά ηελ θαηάξξεπζε ηνπ ζαιάκνπ ζην Sao Paulo to 2007 κεξηθνί πξόηεηλαλ όηη ππόγεηνη ζηαζκνί ηνπ κεηξό ζε θαηνηθεκέλεο πεξηνρέο πξέπεη λα γίλνληαη ζε βάζε κεγαιύηεξα από 20 m. Oη απαληήζεηο ζε ηέηνηνπ είδνπο εξσηήκαηα απαηηνύληαη ζην rock engineering design. Ο Βξαρνκεραληθόο πξέπεη λα έρεη δπλαηόηεηεο πξόβιεςεο: ρσξίο απηέο δελ ππάξρεη ε βάζε γηα ζπλεθηηθό ζρεδηαζκό! Ο ηξόπνο (κέζνδνο) απάληεζεο ζε ηέηνηνπ είδνπο εξσηήκαηα είλαη θαη ζθνπόο ηνπ καζήκαηνο. Exadaktylos notes Rock Mechanics part I Slide 5 of 31
6 Γηα λα κελ αληηκεησπίζνπκε ιόγσ θαθνύ ζρεδηαζκνύ ηέηνηα πξνβιήκαηα αιιά θαη άιια κεξηθά από ηα νπνία ζα δνύκε θαη ζε άιιεο δηαθάλεηεο. Καηάξξεπζε κεηώπνπ ζηε Σήξαγγα Galgenberg, Aπζηξία 6 Exadaktylos notes Rock Mechanics part I Slide 6 of 31
7 Exadaktylos notes Rock Mechanics part I Slide 7 of 31 Ρσγκέο ζηελ ζθπξνδεηεκέλε επέλδπζε
8 F 1 F 2 Intact rock F 3 Excavation Boundary conditions F n Κύξηνη παξάγνληεο ζρεδηαζκνύ Water flow Fractures Γεσινγία (Geology) Επί ηόπνπ ηάζεηο (Rock stress) Αξξεθην πέηξσκα (Intact rock) Αζπλέρεηεο (Discontinuities, Fractures) Παξάκεηξνη βξαρνκάδαο (Rock mass properties) Τπόγεηα ύδαηα (Water flow) Οξπμε, ππνζηήξημε (Engineering activities) Μνληεινπνίεζε (Modelling) Exadaktylos notes Rock Mechanics part I Slide 8 of 31
9 It is essential for rock engineering to understand the geology at a project site, for example at the JinPing II site Exadaktylos notes Rock Mechanics part I Slide 9 of 31 9
10 Aπνηύπσζε ξεγκάησλ ζηε ζήξαγγα Galgenberg, Aπζηξία Exadaktylos notes Rock Mechanics part I Slide 10 of 31
11 DTM with superimposed faults of the corridor area (corridor=possible path of the tunnel) DTM = Digital Terrain Model Exadaktylos notes Rock Mechanics part I Slide 11 of 31
12 Moξθνινγία θαλνληθώλ ξεγκάησλ (normal faults), La Linea Tunnel, Colombia Exadaktylos notes Rock Mechanics part I Slide 12 of 31
13 Exadaktylos notes Rock Mechanics part I Slide 13 of 31 3-δηάζηαην γεσινγηθν-ηεθηνληθό κνληέιν
14 Input: Γεσινγηθό κνληέιν απαξηηδόκελν από ηα ζηεξεά κνληέια θάζε ιηζνινγηθήο θάζεο Exadaktylos notes Rock Mechanics part I Slide 14 of 31
15 Exadaktylos notes Rock Mechanics part I Slide 15 of 31
16 Xαξαθηεξηζκόο βξαρόκαδαο από ππξήλεο γεσηξήζεσλ (rock cores) 1 2 m Exadaktylos notes Rock Mechanics part I Slide 16 of 31 16
17 Borehole core information 1 m Exadaktylos notes Rock Mechanics part I Slide 17 of 31
18 RQD Rock Quality Designation (RQD) is defined as the percentage of rock cores that have length equal or greater than 10 cm over the total drill length. Deere D. U. Technical description of rock cores for engineering purposes. Rock Mech. Eng. Geol. 1, (1964). Exadaktylos notes Rock Mechanics part I Slide 18 of 31
19 Exadaktylos notes Rock Mechanics part I Slide 19 of 31
20 Πεξηνξηζκνί ζηε ρξήζε ηνπ RQD γηα ηελ πνζνηηθή πεξηγξαθή ηνπ βαζκνύ ξεγκάησζεο Exadaktylos notes Rock Mechanics part I Slide 20 of 31
21 Εθηηκήζαηε ην RQD ησλ ππξήλσλ καξκάξνπ. Exadaktylos notes Rock Mechanics part I Slide 21 of 31
22 Exadaktylos notes Rock Mechanics part I Slide 22 of 31 Εθηηκήζαηε ην RQD ησλ ππξήλσλ καξκάξνπ.
23 Εθηηκήζαηε ην RQD αλά 2 m ησλ ππξήλσλ καξκάξνπ. Exadaktylos notes Rock Mechanics part I Slide 23 of 31
24 Sampling issues Location of borehole (1. Πξνζαλαηνιηζκόο γεώηξεζεο ζε ζρέζε κε ηνλ πξνζαλαηνιηζκό ησλ ξεγκάησλ θαη ησλ δηαθιάζεσλ) 2. Θέζε ηεο γεώηξεζεο ε ζρέζε κε ηελ ζέζε ησλ ξεγκαηώζεσλ Orientation of borehole Exadaktylos notes Rock Mechanics part I Slide 24 of 31
25 Exadaktylos notes Rock Mechanics part I Slide 25 of 31
26 Exadaktylos notes Rock Mechanics part I Slide 26 of 31
27 Κύξηνη παξάγνληεο ζρεδηαζκνύ Γεσινγία (Geology) Επί ηόπνπ ηάζεηο (Rock stress) Αξξεθην πέηξσκα (Intact rock) Αζπλέρεηεο (Discontinuities, Fractures) Παξάκεηξνη βξαρνκάδαο (Rock mass properties) Τπόγεηα ύδαηα (Water flow) Οξπμε, ππνζηήξημε/ελίζρπζε (Engineering activities) Μνληεινπνίεζε (Modelling) Exadaktylos notes Rock Mechanics part I Slide 27 of 31
28 Stress is a tensor quantity A SCALAR is a quantity with magnitude only. Examples of scalars are temperature, time, mass and pure colour - they are described completely by one value, e.g. degrees, seconds, kilograms and frequency. A VECTOR is a quantity with magnitude and direction. Examples of vectors are force, velocity, and the frequency of fractures encountered along a line in a rock mass - they are described completely by three values, for example, x, y, z components which together specify both direction and magnitude. A TENSOR is a quantity with magnitude, direction and the plane under consideration. Examples of tensors are stress, strain, permeability and moment of inertia-they are described completely by six values, Exadaktylos notes Rock Mechanics part I Slide 28 of 31
29 A solid can sustain a shear force, whereas a liquid or gas cannot. A liquid or gas contains a pressure, i.e. a force per unit area, which acts equally in all directions and hence is a scalar quantity. Rock can sustain a shear force and hence contains stress, which is a tensor quantity. Exadaktylos notes Rock Mechanics part I Slide 29 of 31
30 Οη 3 θύξηεο ηάζεηο Exadaktylos notes Rock Mechanics part I Slide 30 of 31
31 ύκβαζε 1: Θιηπηηθέο ηάζεηο ζεσξνύληαη ζεηηθέο πνζόηεηεο ύκβαζε 2: ζ 1 >ζ 2 >ζ 3 Exadaktylos notes Rock Mechanics part I Slide 31 of 31
32 (c) Previous tectonic forces. There may also be residual stresses from earlier tectonic events: when a fractured rock mass is compressed and then unloaded, stresses can be left locked in the rock mass. An analogue example of this effect is a pre-stressed car windscreen. Παξακέλνπζεο ηάζεηο ιόγσ πξνεγνύκελεο ηεθηνληθήο δξαζηεξηόηεηαο θαη απνθόξηηζεο Exadaktylos notes Rock Mechanics part I Slide 32 of 31
33 A pop-up observed at a quarry site in granite in Southeastern Manitoba and the horizontal stress determined using the USBM Borehole Deformation Gauge in a vertical borehole. (Quarry popup. (b) Measured horizontal stress) Exadaktylos notes Rock Mechanics part I Slide 33 of 31
34 Tεθηνληθέο ηάζεηο θνληά ζε ξήγκα ή αζπλέρεηα Exadaktylos notes Rock Mechanics part I Slide 34 of 31
35 Geological effect Engineering effect A B C F rac tu re , 1 3 = 0 R ock m as s Principal stresses are parallel and perpendicular to open fracture surfaces and excavation surfaces An open fracture will perturb the stress field and cause the principal stresses to be locally parallel and perpendicular to the fracture surface. Free surfaces in a rock mass cause the local principal stresses to be parallel and perpendicular to the free surface. Exadaktylos notes Rock Mechanics part I Slide 35 of 31
36 Principal stresses occur on planes with no shear stress acting on them. Hence Principal stresses parallel to excavation surface 1 Principal stresses parallel perpendicular to excavation to surface excavation surface 1 Principal stress perpendicular to This is for the case where there are excavation no tensile surface stresses in the rock, i.e. the 3 value of zero is the lowest of the three principal stresses. This is for the case where there are no tensile stresses in the rock, i.e. the 3 value of zero is the lowest of the three principal stresses = 0 3 = 0 Exadaktylos notes Rock Mechanics part I Slide 36 of 31
37 In situ stress In the majority of rock mechanics models, the in situ rock stress is required as a boundary condition input to the model. However, it is difficult to validate the in situ stress assumption locally, both in terms of the principal stress values/directions and their consistency through the fractured rock mass. In situ stress picture Cornwall hydrofrac? World stress map? Mueller, B., Reinecker, J., Heidbach, O. and Fuchs, K. (2000): The 2000 release of the World Stress Map (available online at 37 Exadaktylos notes Rock Mechanics part I Slide 37 of 31
38 Predicted European tectonic stresses based on mid-atlantic ridge push and one type of model (from Gölke and Coblentz (1996). Επί ηόπνπ ηάζεηο ιόγσ θίλεζεο ιηζνζθαηξηθώλ ή ηεθηνληθώλ πιαθώλ (θειπθώλ). H θίλεζε δείρλεηαη κε ηα βέιε ελώ ηα ηξίγσλα ζηα ζύλνξα ηεο πιάθαο (θάησ θαη δεμηά) ζπκβνιίδνπλ αθίλεηα ζύλνξα. Exadaktylos notes Rock Mechanics part I Slide 38 of 31 38
39 Επί ηόπνπ ηάζεηο ιόγσ θίλεζεο ιηζνζθαηξηθώλ πιαθώλ (θειπθώλ) Predicted European tectonic stresses based on mid-atlantic ridge push and one type of model (from Gölke and Coblentz, 1996) MPa Σξη-δηάζηαην κνληέιν γηα ηνλ ππνινγηζκό ησλ ηάζεσλ Figure 2. Example of 3DEC modelling of the influence of brittle deformation zones on rock stresses, Laxemar site, Sweden, horizontal section, ~ 7 km across (prepared for SKB, by Eva Hakami, Itasca, Sweden). Exadaktylos notes Rock Mechanics part I Slide 39 of 31 39
40 Ρεγκαησκέλε δώλε πιεξσκέλε κε άξγηιν Επίδξαζε ηεο ηνπνγξαθίαο θαη πξνϋπαξρόλησλ ξεγκάησλ ζηηο επί ηόπνπ ηάζεηο. Exadaktylos notes Rock Mechanics part I Slide 40 of 31
41 z z z Σάζεηο ιόγσ βαξύηεηαο ζε πξαλέο ζ 1 = γz, γ=0.027 ΜPa/m Exadaktylos notes Rock Mechanics part I Slide 41 of 31
42 Kαηαλνκή θύξηαο επί ηόπνπ ηάζεο ζ 1 ιόγσ βαξύηεηαο ζε πξαλέο Kύξηεο ηάζεηο Exadaktylos notes Rock Mechanics part I Slide 42 of 31
43 Kαηαλνκή ειάρηζηεο θύξηαο επί ηόπνπ ηάζεο ζ 3 ιόγσ βαξύηεηαο ζε πξαλέο ζ 3 = Κ x ζ 1, Κ = ζπληειεζηήο πιεπξηθήο ηάζεο Exadaktylos notes Rock Mechanics part I Slide 43 of 31
44 Slide from: The Recent Tectonic Stress Field and Strong Earthquakes in China Xie Furen Institute of Crustal Dynamics, CEA Dynamic regions of China and adjacent areas 欧亚板块 北美板块 太平洋板块 印度板块 菲律宾板块 Σεθηνληθέο ηάζεηο Exadaktylos notes Rock Mechanics part I Slide 44 of 31
45 Exadaktylos notes Rock Mechanics part I Slide 45 of 31 Σεθηνληθό κνληέιν ηεο Βξεηαλίαο κε ηελ ζρεηηθή θίλεζε ησλ ηεκαρώλ κεηαμύ ησλ ξεγκάησλ (θαίλεηαη κε βέιε) γηα ηελ πξόβιεςε ησλ πεξηνρώλ πνπ κπνξνύλ λα «δώζνπλ» ζεηζκό (κε κπιε ρξώκα).
46 Characterizing rock stress at a site F 1 F 2 F 3 Intact rock Water flow Excavation Boundary conditions F n N Trend 16 MPa Plunge 10 MPa 7 MPa Fractures Natural stress: the in situ stress which exists prior to engineering. Gravitational stress: the stress state caused by the weight of the rock above. Induced stress: the natural stress state as perturbed by engineering. Tectonic stress: the stress state caused by tectonic plate movement. Residual stress: the stress state caused by previous tectonic activity. Thermal stress: the stress state caused by temperature change. Paleostress: a previous natural stress that is no longer acting. Near-field stress: the stress state in the region of an engineering perturbation. Far-field stress: the stress state beyond the near-field. Local stress: the stress state in a region of interest. Exadaktylos notes Rock Mechanics part I Slide 46 of 31 46
47 Exadaktylos notes Rock Mechanics part I Slide 47 of 31
48 Exadaktylos notes Rock Mechanics part I Slide 48 of 31
49 Exadaktylos notes Rock Mechanics part I Slide 49 of 31 P b = πίεζε ζξαύζεο (breakdown or opening pressure) P s = shut-in pressure < P b γηαηί δελ ρξεηάδεηαη λα ππεξληθεζεί ε εθειθπζηηθή αληνρή ηνπ πεηξώκαηνο
50 Exadaktylos notes Rock Mechanics part I Slide 50 of 31
51 Exadaktylos notes Rock Mechanics part I Slide 51 of 31
52 Exadaktylos notes Rock Mechanics part I Slide 52 of 31 ύκβαζε πξνζήκνπ ηάζεσλ: Θιηπηηθέο ηάζεηο > 0, Εθειθπζηηθέο ηάζεηο <0
53 ζζ/ζh 2 1 R 1 ( H h) 1 ( H h) r 2 ( ) 2( )cos2 P H h H h 3R 4 r 4 cos2 P R r 2 2 rr compression tension Circular hole (ζh/ζh=2/3) ζ [degrees] P/sigmaH=0 P/sigmaH=0.4 P/sigmaH=0.8 P/sigmaH=1.2 P/sigmaH=1.4 P/sigmaH=1.6 the highest stress concentration occurs always at the points of the contour corresponding to ζ=0 o, 180 o, i.e. along the direction of the minimum principal stress. Ιncreasing the borehole pressure P results in a decrease of the tangential stress at ζ=0 o, 180 o and increase of tension at ζ=90 ν, 270 ν Exadaktylos notes Rock Mechanics part I Slide 53 of 31
54 Exadaktylos notes Rock Mechanics part I Slide 54 of 31 sin ) ( 2 1 cos2 3 1 ) ( ) ( 2 1 cos ) ( ) ( ) ( ) ( ) ( r R r R r R r R r R r R r R h H o r h H h H o h H h H o r 0,, 2 2 ) ( 2 2 ) ( r i i r r R P r R P Kirsch s (1898) stress solution assuming isotropic linear elasticity Stability of deep boreholes b h H b H h T h H h H h H h H R r i o P T P T P P 3 3 ) 2( ) ( )cos180 2( ) ( 0 ) ( Τπέξζεζε (ιόγσ γξακκηθόηεηαο) -Σ = εθειθπζηηθή αληνρή ηνπ πεηξώκαηνο (<0) Πίεζε πόξσλ = 0, Αιιηώο έλλνηα «ελεξγήο ηάζεο» θαηά Terzaghi
55 Exadaktylos notes Rock Mechanics part I Slide 55 of 31 p b h H p b p h p H P P T P P T P P 3 3 Πίεζε πόξσλ = Pp, έλλνηα «ελεξγήο ηάζεο» θαηά Terzaghi
56 Exadaktylos notes Rock Mechanics part I Slide 56 of 31
57 Exadaktylos notes Rock Mechanics part I Slide 57 of 31 Issue of this journal devoted to rock stress estimation which contains the four International Society for Rock Mechanics Suggested Methods and 17 papers on experiences of stress measurement
58 Exadaktylos notes Rock Mechanics part I Slide 58 of 31
59 Exadaktylos notes Rock Mechanics part I Slide 59 of 31
60 εκαζία ηνπ κέηξνπ ειαζηηθόηεηαο θαη ηνπ ιόγνπ ηνπ Poisson ζην ζρεδηαζκό πδξαπιηθήο ζξαύζεο. Mεηά ην πξόβιεκα ηνπ Kirsch, ην πξόβιεκα ηνπ ειιεηπηηθνύ αλνίγκαηνο είλαη ην πην ζεκαληηθό θαη ιύζεθε από ηνλ Inglis (1913). Exadaktylos notes Rock Mechanics part I Slide 60 of 31
61 εκαζία ηνπ κέηξνπ ειαζηηθόηεηαο θαη ηνπ ιόγνπ ηνπ Poisson ζηελ ζπκπεξηθνξά ηεο ζεκειίσζεο Exadaktylos notes Rock Mechanics part I Slide 61 of 31
62 Exadaktylos notes Rock Mechanics part I Slide 62 of 31
63 (plane Ox 2 x 3 parallel to the Earth s surface) Ιζόηξνπν πέηξσκα Νόκνο ηνπ Hooke ij 1 E 1, (i, j 1,2,3 ) ij kk ij Exadaktylos notes Rock Mechanics part I Slide 63 of 31
64 a 1 a 1 a , a 33 11, a a a E Θέηνπκε 11 v z , h Καη βξίζθνπκε h K, K v 1 Nα απνδεηρζεί από εζάο! Exadaktylos notes Rock Mechanics part I Slide 64 of 31
65 Sheorey (1994) based on an elasto-static thermal model of the earth (curvature of the crust, variation of elastic constants, density and thermal expansion coefficients through the crust and mantle), gave the eqn K E h Πσο ζα ζρνιηάδαηε απηό ην ζρήκα? 1 z Exadaktylos notes Rock Mechanics part I Slide 65 of 31
66 Hoek and Brown collated stress data Exadaktylos notes Rock Mechanics part I Slide 66 of 31
67 Due to the action of significant geostatic and tectonic forces that are encountered at great depths, deep boreholes suffer from severe instabilities, breakouts and exfoliations that can become critical for the progress of the drilling process and may interrupt or even stop energy production. Breakouts lead in general to progressive deterioration of the borehole. Θιηπηηθή ζξαύζε γεσηξήζεσλ (borehole breakouts) Exadaktylos notes Rock Mechanics part I Slide 67 of 31
68 (the main cause of failure: stress concentration around holes) Exadaktylos notes Rock Mechanics part I Slide 68 of 31
69 Introductory remarks Theoretical approach: The ability to predict fracture and stress orientations is becoming increasingly important in development of production wells. Experimental approach: Wireline borehole scanner data to identify the borehole breakouts and its orientation to maximum stress zones. Exadaktylos notes Rock Mechanics part I Slide 69 of 31
70 (World stress map (Mediterranean region) Approximately 19% of the stress orientation indicators in the World Stress Map (WSM) database have been determined from borehole breakouts. Exadaktylos notes Rock Mechanics part I Slide 70 of 31
71 The danger of damage to the tunnel can be reduced by, if possible, orientating the tunnel so that it is parallel to the major principal stress 1 perpendicular to tunnel axis 1 parallel to tunnel axis 1 is vertical Exadaktylos notes Rock Mechanics part I Slide 71 of 31
72 The problem of the possible proximity of faults: causing major alterations to the stress field outside the overall variation Horizontal stress Without faults Exadaktylos notes Rock Mechanics part I Slide 72 of 31
73 Horizontal stress With fault Exadaktylos notes Rock Mechanics part I Slide 73 of 31
74 Conclusion: Cannot predict these local macroperturbations but they will be relatively rare and so will have to be treated as extreme values Exadaktylos notes Rock Mechanics part I Slide 74 of 31
75 Αλ ζέιαηε λα θαηαζθεπάζεηε έλα ππόγεην ζάιακν πσο ζα ζρεδηάδαηε ηε δηαηνκή ηνπ (H/W??) ιακβάλνληαο ππόςηλ ην επί ηόπνπ πεδίν ηάζεσλ? Η Αλαπηύμαηε ηε κέζνδν ζαο. W Exadaktylos notes Rock Mechanics part I Slide 75 of 31
76 Will rock spalling occur? Pre-existing natural rock stress The strength of the rock is compared to the local stress values Spalling can start to occur at about half the uniaxial compressive strength Tunnels parallel to the maximum in situ principal stress are subject to less stress concentration Excavationinduced stresses around the excavation Stress diagram from work by Dr Erik Johansson, Saanio and Riekkola, Finland Exadaktylos notes Rock Mechanics part I Slide 76 of 31 76
77 Exadaktylos notes Rock Mechanics part I Slide 77 of 31 Spalling in a shaft
78 Exadaktylos notes Rock Mechanics part I Slide 78 of 31 Worst case scenario spalling in the South African gold mines
79 Επίδξαζε ηεο αληνρήο ηεο βξαρνκάδαο ζηελ έθηαζε ηεο δώλεο κηθξνξσγκάησζεο γύξσ από ππόγεηνπο ζαιάκνπο 1. Πσο κπνξεί λα εθηηκεζεί ε έθηαζε ηεο δώλεο κηθξνξσγκάησζεο γύξσ από ππόγεην άλνηγκα? 2. Πνηεο είλαη νη παξάκεηξνη πνπ επεξεάδνπλ ηελ έθηαζε ηεο? Exadaktylos notes Rock Mechanics part I Slide 79 of 31
80 Exadaktylos notes Rock Mechanics part I Slide 80 of 31 Επί ηόπνπ ηάζεηο? (κάιινλ ην μέραζε ν πνηεηήο!)
81 Gravity-driven failure Stress-driven failure Exadaktylos notes Rock Mechanics part I Slide 81 of 31
Strain gauge and rosettes
Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified
Dr. D. Dinev, Department of Structural Mechanics, UACEG
Lecture 4 Material behavior: Constitutive equations Field of the game Print version Lecture on Theory of lasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACG 4.1 Contents
Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Mechanical Behaviour of Materials Chapter 5 Plasticity Theory
Mechanical Behaviour of Materials Chapter 5 Plasticity Theory Dr.-Ing. 郭瑞昭 Yield criteria Question: For what combinations of loads will the cylinder begin to yield plastically? The criteria for deciding
Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
MECHANICAL PROPERTIES OF MATERIALS
MECHANICAL PROPERTIES OF MATERIALS! Simple Tension Test! The Stress-Strain Diagram! Stress-Strain Behavior of Ductile and Brittle Materials! Hooke s Law! Strain Energy! Poisson s Ratio! The Shear Stress-Strain
Chapter 7 Transformations of Stress and Strain
Chapter 7 Transformations of Stress and Strain INTRODUCTION Transformation of Plane Stress Mohr s Circle for Plane Stress Application of Mohr s Circle to 3D Analsis 90 60 60 0 0 50 90 Introduction 7-1
CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Macromechanics of a Laminate. Textbook: Mechanics of Composite Materials Author: Autar Kaw
Macromechanics of a Laminate Tetboo: Mechanics of Composite Materials Author: Autar Kaw Figure 4.1 Fiber Direction θ z CHAPTER OJECTIVES Understand the code for laminate stacing sequence Develop relationships
Introduction to Theory of. Elasticity. Kengo Nakajima Summer
Introduction to Theor of lasticit Summer Kengo Nakajima Technical & Scientific Computing I (48-7) Seminar on Computer Science (48-4) elast Theor of lasticit Target Stress Governing quations elast 3 Theor
Chapter 2. Stress, Principal Stresses, Strain Energy
Chapter Stress, Principal Stresses, Strain nergy Traction vector, stress tensor z z σz τ zy ΔA ΔF A ΔA ΔF x ΔF z ΔF y y τ zx τ xz τxy σx τ yx τ yz σy y A x x F i j k is the traction force acting on the
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Homework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
[1] P Q. Fig. 3.1
![[1] P Q. Fig. 3.1](/thumbs/79/80362156.jpg "[1] P Q. Fig. 3.1")
1 (a) Define resistance....... [1] (b) The smallest conductor within a computer processing chip can be represented as a rectangular block that is one atom high, four atoms wide and twenty atoms long. One
Second Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Stresses in a Plane. Mohr s Circle. Cross Section thru Body. MET 210W Mohr s Circle 1. Some parts experience normal stresses in
ME 10W E. Evans Stresses in a Plane Some parts eperience normal stresses in two directions. hese tpes of problems are called Plane Stress or Biaial Stress Cross Section thru Bod z angent and normal to
Approximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Section 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
EE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
the total number of electrons passing through the lamp.
1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy
b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Parametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
Capacitors - Capacitance, Charge and Potential Difference
Capacitors - Capacitance, Charge and Potential Difference Capacitors store electric charge. This ability to store electric charge is known as capacitance. A simple capacitor consists of 2 parallel metal
Potential Dividers. 46 minutes. 46 marks. Page 1 of 11
Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Higher Derivative Gravity Theories
Higher Derivative Gravity Theories Black Holes in AdS space-times James Mashiyane Supervisor: Prof Kevin Goldstein University of the Witwatersrand Second Mandelstam, 20 January 2018 James Mashiyane WITS)
Mechanics of Materials Lab
Mechanics of Materials Lab Lecture 9 Strain and lasticity Textbook: Mechanical Behavior of Materials Sec. 6.6, 5.3, 5.4 Jiangyu Li Jiangyu Li, Prof. M.. Tuttle Strain: Fundamental Definitions "Strain"
3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Mean bond enthalpy Standard enthalpy of formation Bond N H N N N N H O O O
Q1. (a) Explain the meaning of the terms mean bond enthalpy and standard enthalpy of formation. Mean bond enthalpy... Standard enthalpy of formation... (5) (b) Some mean bond enthalpies are given below.
(1) Describe the process by which mercury atoms become excited in a fluorescent tube (3)
Q1. (a) A fluorescent tube is filled with mercury vapour at low pressure. In order to emit electromagnetic radiation the mercury atoms must first be excited. (i) What is meant by an excited atom? (1) (ii)
Math 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
D Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Reminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Written Examination. Antennas and Propagation (AA ) April 26, 2017.
Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ
4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a
5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
ADVANCED STRUCTURAL MECHANICS
VSB TECHNICAL UNIVERSITY OF OSTRAVA FACULTY OF CIVIL ENGINEERING ADVANCED STRUCTURAL MECHANICS Lecture 1 Jiří Brožovský Office: LP H 406/3 Phone: 597 321 321 E-mail: jiri.brozovsky@vsb.cz WWW: http://fast10.vsb.cz/brozovsky/
Technical Data for Profiles. α ( C) = 250 N/mm 2 (36,000 lb./in. 2 ) = 200 N/mm 2 (29,000 lb./in 2 ) A 5 = 10% A 10 = 8%
91 500 201 0/11 Aluminum raming Linear Motion and Assembly Technologies 1 Section : Engineering Data and Speciications Technical Data or Proiles Metric U.S. Equivalent Material designation according to
Assalamu `alaikum wr. wb.
LUMP SUM Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. Assalamu `alaikum wr. wb. LUMP SUM Wassalamu alaikum wr. wb. LUMP SUM Lump sum lump sum lump sum. lump sum fixed price lump sum lump
9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Second Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
DuPont Suva 95 Refrigerant
Technical Information T-95 ENG DuPont Suva refrigerants Thermodynamic Properties of DuPont Suva 95 Refrigerant (R-508B) The DuPont Oval Logo, The miracles of science, and Suva, are trademarks or registered
Section 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
University of Waterloo. ME Mechanical Design 1. Partial notes Part 1
University of Waterloo Department of Mechanical Engineering ME 3 - Mechanical Design 1 Partial notes Part 1 G. Glinka Fall 005 1 Forces and stresses Stresses and Stress Tensor Two basic types of forces
derivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Fractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 24/3/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Όλοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα μικρότεροι του 10000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Αν κάπου κάνετε κάποιες υποθέσεις
1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
PhysicsAndMathsTutor.com 1
PhysicsAndMathsTutor.com 1 Q1. The magnetic flux through a coil of N turns is increased uniformly from zero to a maximum value in a time t. An emf, E, is induced across the coil. What is the maximum value
Homework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ
ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΝΟΣΗΛΕΥΤΙΚΗΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΨΥΧΟΛΟΓΙΚΕΣ ΕΠΙΠΤΩΣΕΙΣ ΣΕ ΓΥΝΑΙΚΕΣ ΜΕΤΑ ΑΠΟ ΜΑΣΤΕΚΤΟΜΗ ΓΕΩΡΓΙΑ ΤΡΙΣΟΚΚΑ Λευκωσία 2012 ΤΕΧΝΟΛΟΓΙΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΣΧΟΛΗ ΕΠΙΣΤΗΜΩΝ
CRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
ΠΣΤΥΙΑΚΗ ΔΡΓΑΙΑ. Μειέηε Υξόλνπ Απνζηείξσζεο Κνλζέξβαο κε Τπνινγηζηηθή Ρεπζηνδπλακηθή. Αζαλαζηάδνπ Βαξβάξα
ΣΔΥΝΟΛΟΓΙΚΟ ΔΚΠΑΙΓΔΤΣΙΚΟ ΙΓΡΤΜΑ ΘΔΑΛΟΝΙΚΗ ΥΟΛΗ ΣΔΥΝΟΛΟΓΙΑ ΣΡΟΦΙΜΩΝ & ΓΙΑΣΡΟΦΗ ΣΜΗΜΑ ΣΔΥΝΟΛΟΓΙΑ ΣΡΟΦΙΜΩΝ ΠΣΤΥΙΑΚΗ ΔΡΓΑΙΑ Μειέηε Υξόλνπ Απνζηείξσζεο Κνλζέξβαο κε Τπνινγηζηηθή Ρεπζηνδπλακηθή Αζαλαζηάδνπ Βαξβάξα
UDZ Swirl diffuser. Product facts. Quick-selection. Swirl diffuser UDZ. Product code example:
UDZ Swirl diffuser Swirl diffuser UDZ, which is intended for installation in a ventilation duct, can be used in premises with a large volume, for example factory premises, storage areas, superstores, halls,
ARISTOTLE UNIVERSITY OF THESSALONIKI FACULTY OF FORESTRY AND NATURAL ENVIRONMENT Institute of Mountainous Water Management and Control
ARISTOTLE UNIVERSITY OF THESSALONIKI FACULTY OF FORESTRY AND NATURAL ENVIRONMENT Institute of Mountainous Water Management and Control Torrent Basin, Mountainous Watershed Management Dr. Panagiotis Stefanidis
1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r(t) = 3cost, 4t, 3sint
1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P, π, rt) cost, t, sint ). b) 5 points) Find curvature of the curve at the point P. Solution: a) r t) sint,,
Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
ΑΚΑ ΗΜΙΑ ΕΜΠΟΡΙΚΟΥ ΝΑΥΤΙΚΟΥ ΜΑΚΕ ΟΝΙΑΣ ΣΧΟΛΗ ΜΗΧΑΝΙΚΩΝ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ
ΑΚΑ ΗΜΙΑ ΕΜΠΟΡΙΚΟΥ ΝΑΥΤΙΚΟΥ ΜΑΚΕ ΟΝΙΑΣ ΣΧΟΛΗ ΜΗΧΑΝΙΚΩΝ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΘΕΜΑ :ΤΥΠΟΙ ΑΕΡΟΣΥΜΠΙΕΣΤΩΝ ΚΑΙ ΤΡΟΠΟΙ ΛΕΙΤΟΥΡΓΙΑΣ ΣΠΟΥ ΑΣΤΡΙΑ: ΕΥΘΥΜΙΑ ΟΥ ΣΩΣΑΝΝΑ ΕΠΙΒΛΕΠΩΝ ΚΑΘΗΓΗΤΗΣ : ΓΟΥΛΟΠΟΥΛΟΣ ΑΘΑΝΑΣΙΟΣ 1 ΑΚΑ
ΠΟΛΥΤΕΧΝΕΙΟ ΚΡΗΤΗΣ ΣΧΟΛΗ ΜΗΧΑΝΙΚΩΝ ΠΕΡΙΒΑΛΛΟΝΤΟΣ
ΠΟΛΥΤΕΧΝΕΙΟ ΚΡΗΤΗΣ ΣΧΟΛΗ ΜΗΧΑΝΙΚΩΝ ΠΕΡΙΒΑΛΛΟΝΤΟΣ Τομέας Περιβαλλοντικής Υδραυλικής και Γεωπεριβαλλοντικής Μηχανικής (III) Εργαστήριο Γεωπεριβαλλοντικής Μηχανικής TECHNICAL UNIVERSITY OF CRETE SCHOOL of
Figure 3 Three observations (Vp, Vs and density isosurfaces) intersecting in the PLF space. Solutions exist at the two indicated points.
φ φ φ φ Figure 1 Resampling of a rock-physics model from velocity-porosity to lithology-porosity space. C i are model results for various clay contents. φ ρ ρ δ Figure 2 Bulk modulus constraint cube in
DuPont Suva. DuPont. Thermodynamic Properties of. Refrigerant (R-410A) Technical Information. refrigerants T-410A ENG
Technical Information T-410A ENG DuPont Suva refrigerants Thermodynamic Properties of DuPont Suva 410A Refrigerant (R-410A) The DuPont Oval Logo, The miracles of science, and Suva, are trademarks or registered
Numerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Technical Information T-9100 SI. Suva. refrigerants. Thermodynamic Properties of. Suva Refrigerant [R-410A (50/50)]
![Technical Information T-9100 SI. Suva. refrigerants. Thermodynamic Properties of. Suva Refrigerant [R-410A (50/50)]](/thumbs/89/98928029.jpg "Technical Information T-9100 SI. Suva. refrigerants. Thermodynamic Properties of. Suva Refrigerant [R-410A (50/50)]")
d Suva refrigerants Technical Information T-9100SI Thermodynamic Properties of Suva 9100 Refrigerant [R-410A (50/50)] Thermodynamic Properties of Suva 9100 Refrigerant SI Units New tables of the thermodynamic
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΧΗΜΙΚΩΝ ΜΗΧΑΝΙΚΩΝ Τομέας ΙV: Τομέας Σύνθεσης και Ανάπτυξης Βιομηχανικών Διαδικασιών Διπλωματική Εργασία: Επίδραση της Προσθήκης Αιθανόλης στην Τάση Ατμών της Βενζίνης
Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Matrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Section 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:
Seria : 0. T_ME_(+B)_Strength of Materia_9078 Dehi Noida Bhopa Hyderabad Jaipur Luckno Indore une Bhubanesar Kokata atna Web: E-mai: info@madeeasy.in h: 0-56 CLSS TEST 08-9 MECHNICL ENGINEERING Subject
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
DETERMINATION OF FRICTION COEFFICIENT
EXPERIMENT 5 DETERMINATION OF FRICTION COEFFICIENT Purpose: Determine μ k, the kinetic coefficient of friction and μ s, the static friction coefficient by sliding block down that acts as an inclined plane.
Right Rear Door. Let's now finish the door hinge saga with the right rear door
Right Rear Door Let's now finish the door hinge saga with the right rear door You may have been already guessed my steps, so there is not much to describe in detail. Old upper one file:///c /Documents
Απνηειέζκαηα Εξσηεκαηνινγίνπ 2o ηεηξάκελν 2011-12
Απνηειέζκαηα Εξσηεκαηνινγίνπ 2o ηεηξάκελν 11-12 Project 6: Ταμίδη κε ηε Μεραλή ηνπ Φξόλνπ Υπεύζπλνη Καζεγεηέο: Ε. Μπηιαλάθε Φ. Αλησλάηνο Δρώηηζη 3: Πνηα από ηα παξαθάησ ΜΜΕ ηεξαξρείηε από πιεπξάο ζεκαζίαο;
PARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
The Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
DuPont Suva 95 Refrigerant
Technical Information T-95 SI DuPont Suva refrigerants Thermodynamic Properties of DuPont Suva 95 Refrigerant (R-508B) The DuPont Oval Logo, The miracles of science, and Suva, are trademarks or registered
ΓΕΩΜΕΣΡΙΚΗ ΣΕΚΜΗΡΙΩΗ ΣΟΤ ΙΕΡΟΤ ΝΑΟΤ ΣΟΤ ΣΙΜΙΟΤ ΣΑΤΡΟΤ ΣΟ ΠΕΛΕΝΔΡΙ ΣΗ ΚΤΠΡΟΤ ΜΕ ΕΦΑΡΜΟΓΗ ΑΤΣΟΜΑΣΟΠΟΙΗΜΕΝΟΤ ΤΣΗΜΑΣΟ ΨΗΦΙΑΚΗ ΦΩΣΟΓΡΑΜΜΕΣΡΙΑ
ΕΘΝΙΚΟ ΜΕΣΟΒΙΟ ΠΟΛΤΣΕΧΝΕΙΟ ΣΜΗΜΑ ΑΓΡΟΝΟΜΩΝ-ΣΟΠΟΓΡΑΦΩΝ ΜΗΧΑΝΙΚΩΝ ΣΟΜΕΑ ΣΟΠΟΓΡΑΦΙΑ ΕΡΓΑΣΗΡΙΟ ΦΩΣΟΓΡΑΜΜΕΣΡΙΑ ΓΕΩΜΕΣΡΙΚΗ ΣΕΚΜΗΡΙΩΗ ΣΟΤ ΙΕΡΟΤ ΝΑΟΤ ΣΟΤ ΣΙΜΙΟΤ ΣΑΤΡΟΤ ΣΟ ΠΕΛΕΝΔΡΙ ΣΗ ΚΤΠΡΟΤ ΜΕ ΕΦΑΡΜΟΓΗ ΑΤΣΟΜΑΣΟΠΟΙΗΜΕΝΟΤ
k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +](/thumbs/73/69566903.jpg "k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +")
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
«ΑΓΡΟΤΟΥΡΙΣΜΟΣ ΚΑΙ ΤΟΠΙΚΗ ΑΝΑΠΤΥΞΗ: Ο ΡΟΛΟΣ ΤΩΝ ΝΕΩΝ ΤΕΧΝΟΛΟΓΙΩΝ ΣΤΗΝ ΠΡΟΩΘΗΣΗ ΤΩΝ ΓΥΝΑΙΚΕΙΩΝ ΣΥΝΕΤΑΙΡΙΣΜΩΝ»
I ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΣΧΟΛΗ ΝΟΜΙΚΩΝ ΟΙΚΟΝΟΜΙΚΩΝ ΚΑΙ ΠΟΛΙΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΟΙΚΟΝΟΜΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗΝ «ΔΙΟΙΚΗΣΗ ΚΑΙ ΟΙΚΟΝΟΜΙΑ» ΚΑΤΕΥΘΥΝΣΗ: ΟΙΚΟΝΟΜΙΚΗ
ΚΥΠΡΙΑΚΟΣ ΣΥΝΔΕΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY 21 ος ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ Δεύτερος Γύρος - 30 Μαρτίου 2011
Διάρκεια Διαγωνισμού: 3 ώρες Απαντήστε όλες τις ερωτήσεις Μέγιστο Βάρος (20 Μονάδες) Δίνεται ένα σύνολο από N σφαιρίδια τα οποία δεν έχουν όλα το ίδιο βάρος μεταξύ τους και ένα κουτί που αντέχει μέχρι
6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
ΕΘΝΙΚΗ ΣΧΟΛΗ ΔΗΜΟΣΙΑΣ ΔΙΟΙΚΗΣΗΣ ΙΓ' ΕΚΠΑΙΔΕΥΤΙΚΗ ΣΕΙΡΑ
ΕΘΝΙΚΗ ΣΧΟΛΗ ΔΗΜΟΣΙΑΣ ΔΙΟΙΚΗΣΗΣ ΙΓ' ΕΚΠΑΙΔΕΥΤΙΚΗ ΣΕΙΡΑ ΤΜΗΜΑ ΤΟΠΙΚΗΣ ΑΥΤΟΔΙΟΙΚΗΣΗΣ ΚΑΙ ΠΕΡΙΦΕΡΕΙΑΚΗΣ ΑΝΑΠΤΥΞΗΣ ΤΕΛΙΚΗ ΕΡΓΑΣΙΑ: ΠΕΡΙΒΑΛΛΟΝ ΚΑΙ ΑΝΑΠΤΥΞΗ: ΠΡΟΣΕΓΓΙΣΗ ΜΕΣΩ ΔΕΙΚΤΩΝ Επιβλέπων: Αθ.Δελαπάσχος
Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων. Εξάμηνο 7 ο
Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων Εξάμηνο 7 ο Procedures and Functions Stored procedures and functions are named blocks of code that enable you to group and organize a series of SQL and PL/SQL
Παιχνίδι γλωζζικής καηανόηζης με ζχήμαηα!
Cpyright 2013 Λόγος & Επικοινωνία // All rights Reserved Παιχνίδι γλωζζικής καηανόηζης με ζχήμαηα! Αυηό ηο παιχνίδι έχει ζηόχους: 1. ηελ εθγύκλαζε ηεο αθνπζηηθήο κλήκεο ησλ παηδηώλ 2. ηελ εμάζθεζε ζηελ
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
10.7 Performance of Second-Order System (Unit Step Response)
Lecture Notes on Control Systems/D. Ghose/0 57 0.7 Performance of Second-Order System (Unit Step Response) Consider the second order system a ÿ + a ẏ + a 0 y = b 0 r So, Y (s) R(s) = b 0 a s + a s + a
5.0 DESIGN CALCULATIONS
5.0 DESIGN CALCULATIONS Load Data Reference Drawing No. 2-87-010-80926 Foundation loading for steel chimney 1-00-281-53214 Boiler foundation plan sketch : Figure 1 Quantity Unit Dia of Stack, d 6.00 m
Partial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
BEHAVIOR OF MASSIVE EARTH RETAINING WALLS UNDER EARTHQUAKE SHAKING Comparisons to EC-8 Provisions
UNIVERSITY OF PATRAS DEPARTMENT OF CIVIL ENGINEERING Laboratory of Geotechnical Engineering BEHAVIOR OF MASSIVE EARTH RETAINING WALLS UNDER EARTHQUAKE SHAKING Comparisons to EC-8 Provisions Prof. G. Athanasopoulos