# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

25 Pages

### nfe

Course: MATH 5587, Fall 2010
School: UCF
Rating:

Word Count: 7731

#### Document Preview

11 Chapter Numerical Methods: Finite Elements In Chapter 5, we introduced the first, the oldest, and in many ways the simplest class of numerical algorithms for approximating the solutions to partial differential equations: finite differences. In the present chapter, we study the second of the two major numerical paradigms: the finite element method. Finite elements are of more recent vintage, having first...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Florida >> UCF >> MATH 5587

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

UCF - MATH - 5587
Chapter 10 A General Framework for Linear Partial Differential EquationsBefore pressing on to the higher dimensional forms of the heat, wave, and Laplace/ Poisson equations, it is worth taking some time to develop a general, abstract, linear algebraic fr
UCF - MATH - 5587
Chapter 12 Partial Differential Equations in SpaceAt last we have reached the ultimate rung of the dimensional ladder (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. As in the one- and
UCF - MATH - 5587
Chapter 4 Separation of VariablesThere are three paradigmatic linear second order partial differential equations that have collectively driven the development of the entire subject. The first two we have already encountered: The wave equation describes v
UCF - MATH - 5587
Chapter 1 What are Partial Differential Equations?Let us begin by specifying our object of study. A differential equation is an equation that relates the derivatives of a (scalar) function depending on one or more variables. For example, d4 u du + u2 = c
University of Florida - CEG - 4012
Lecture 1 Review of Geostatic Stresses Unit WeightsYw = unit weight of water Ym = moist unit weight of unsaturated soil Ysat = unit weight of saturated soil y' = &quot;effective&quot; unit weight of soil = (Ysat - Yw) if soil saturated= Ym if soil not saturated=
University of Florida - CEG - 4012
Geostatic Stresses These self-weight stresses (ay, a'y, ah' a'h) are called geostatic stresses For a level surface there are no shear forces induced by the geostatic stresses, and therefore they are also principal stresses: a, =ay and a3 = ah Karl Terzag
University of Florida - CEG - 4012
using the effective weight concept:= 0&quot; h6'(100 pet + 4'(118-62.4 pet + 6'(126-62.4 pet+4'(120-62.4 pet 230 pst: 1434 pst= 600 pst + 222 pst + 382 pst +=-K (J'y : 0.5 (1434 pst) : 717 pst 14' (62.4 pet): 874 pst U(a' h + u): 717 pst + 874 pst: 15
University of Florida - CEG - 4012
S1 Stresses Changes Due to Surface Loads (Aerv) o Stresses within a soil mass will change as a result of surface loads. The change in total stress spreads and diminishes with distance from the load. Equations and charts are available to calculate both th
University of Florida - CEG - 4012
S2 Pyramid Approximation For Surface Loads: The &quot;2:1 melhod &quot;used for &quot;back of Ihe envelope&quot; solutions (seen on the PE exam). o A=LoadedArea =BxLQ = Tolal Load = q x B x LQr+-&quot;q=unit pressurewhere B = width (short sjde, alwill's) L _ length !loM-si
University of Florida - CEG - 4012
S3 Point Load on the Surface: Boussinesq (1883) - French Physicist(with the parameters as shown at right)Qo Assumotionc:. I. nALF-SPACE - semi-infinite 2. ELASTIC 3. HOMOGENEOUS - same properties throughout 4. ISOTROPIC - same properties in all directi
University of Florida - CEG - 4012
84Circular Surface Load:o The stress increase at a point (A) on the axis beneath the center of a circular loaded area is &quot;easily&quot; determined by integrating the Boussinesq point solution over the loaded area. (same assumptions as before)dav6=30z 5 2
University of Florida - CEG - 4012
S5IPta (m)z(m)~&quot;f.r~;,&lt;,iJa 0r (m)&lt;w, (&quot;,1,;r/a~I,.()OAGy=I x q(kPa)IS ':JI'llC.L.Surface12 12 12 12 12 12 12 1200 0 12 0 12 0 12 2400 1 0 1 0 1 2AB6 612 12 24 24 24C 0E FG0.5 0.5 1 1 2 2 2o'I ()h'f'ltt&quot;,o~f) (,((),
University of Florida - CEG - 4012
r~.lr0.20.3 0.4 0.50.6,86810_ Influence Chart for Vertical Stress Increase Beneath Circular Loaded Area '(From Perlott and Baron)Influence value, I(x 100) 2 3 4 5 60.8 1.020_. 30 40 50 60 I I I I I80 100I11-~V1lllll1Ui .-I'm'N-I'3,!)
University of Florida - CEG - 4012
S11~~trlp and Square FootingsStrips footings have fixed width and infinite (or relatively long) length and are often used for smaller buildings (1-2 story) and walls. Square footings are a special case of the rectangular footing.BqzInfluence charts
University of Florida - CEG - 4012
812o Boussinesq Influence Chart for Strip &amp; Square Footings48382B88I - 1-.a~ I ~/v 28'. l' 1-~'l.O; 1./0-11 VI VI I , /'0.' VI / I ~, l&quot; / / / \ IIV ~ v ~ rtf :\l/I-&quot; ./ /\ \III 11I\.q,.I'.'~) OAq~ ~ ~.-.~ ~ t'. ~0.4;1 ~l\.'~~8 8U!
University of Florida - CEG - 4012
813 Rectangular Surface Loads: (Boussinesq solution - the limn&quot; methodo This method determines the vertical stress ~ay at a point P under the corner of a rectangular loaded area using footing dimensions normalized to the depth z: m = Biz n=Uz (note that
University of Florida - CEG - 4012
o814 Influence Chart for Vertical Stress Increase Beneath Corner of a Rectangular Load (Perloff &amp; Baron)I PI JrI i J I1II i,i J:It- -_ 'j0.23 1i-I--TyO.22~mt!tJ0.21Ffj:fSffi Ff1':11+WM-10,0.20 H+J+.;+J+j 0.190.18=pl&lt; pflm,nl for sq
University of Florida - CEG - 4012
S 15a Example 3: Solution: Find the stress increase at Point T al a depth of 6 1t.qfor &lt;D m = Biz = ':J.)(&quot; 6&gt;~ n = Uz = ~), : cs 0 in chart find I = 0.0(, I for Q) m = BIz = 3/~ v.n) n = Uz = 'I!&quot; t.F) in chart find I = 0,10&lt;.&gt;= 1000 psf : 3'2'T4'
University of Florida - CEG - 4012
I\:._~B:.-_ _I-=-_ c.(j;)A_ _-,~ ,e: .,III~. - - - -,- . ' . - - . - 1&gt;I,pl; -.-- .~ p ,bt;'-~,:-JE~IL_-j. - _.\ ,~( 5'1ftt /'d-+ncfw_;.lj~ 0: = ~ [ &quot;I HP6-rae-tiP] x:~c. - - - 6 - l - - - _ . IiEp
University of Florida - CEG - 4012
S16 Embankment Load: a Another important &quot;shape&quot; is the embankment (e.g. highways, dams, etc.) If the embankment material is soil then the surface load (pressure) at full height is ~ q '1 H.ba=Chart on next page provides influence factors due to 1/2
University of Florida - CEG - 4012
517oInfluence Chart for Vertical Stress Increase Beneath and Embankment (PerloH and Baron)VALUE OF~05'0.4500'0020050102,,.20,e1.2~~1a35 680050.8v. v./45bit ~ LOl./.400.9/ 1/ .4 / / 1/21/40oe07.35\.-1/I I II I I II
University of Florida - CEG - 4012
p1 SETILEMENT IN CLAY SUMMARY Strains due Surface Load: Surface loads cause a change in the stresses within the soil mass and induce two types of soil strain.zt zy/ t y'iIIa,tn/ 't xza,1&quot;Ez,. [ II/ !i1-,/I II.~Y'J- -'J,. IIztII-(
University of Florida - CEG - 4012
p2Immediate Settlement (Pi)-=-TJ].sn~ettlementis g,y,e.to,J;otatjonal strainJ&lt;;fLstortion) within tile soil - not a chan e in volume. (Note that if no shear strain, say a blanket loaa, then no Imme late settleiii8nt.)-QL PirOriginal DistortedImmedi
University of Florida - CEG - 4012
p3 ,lmmedlate-Set1lement Case I: B toading on the Surface aLan ElastiGJ:Ialf SpacJl (e.g. a thiclclayer of ciay)kv,~ ~t.'\C0 .JBIqI.,~'I IIBcircle, square, or rectangular footingL_._-'_. _ 1I IP, = C,qB(1-1&quot;)Ewhere~= =Poisson's ratio
University of Florida - CEG - 4012
p4 Ilmme(llate Sell ement Case II: tpading-on tb-e Sufface of a Com-p-ressible Soil (Clay) Onaerlain By a Rigid Bounaary (Roc or Dense Sand) BqHE,fl Rigid Boundary~Table P2 provides C's value under the center of flexible footings. (If the clay layer
University of Florida - CEG - 4012
p5 'Imme&lt;llatE! Settlement Case III:~oaaing-on the-Surface of a Stiff baye Underlain oy a Less Rigl&lt;'FEayer of Great Ihicknes.sFairly common for upper crust due to desiccation etc..HS.qE III &quot;Table P3 gives the C&quot;s values under the center of flex
University of Florida - CEG - 4012
p6~_Approximate Solutions: Using various combinations of Tables P1, P2 and P3 can get &quot;ballpark&quot; estimate of immediate settlement for many cases not covered. Always start withclosest initial approximation, then &quot;correct&quot; as necessary. Examples:1.Rigi
University of Florida - CEG - 4012
p7Table P1 Values of Shape and Rigidity Factor Cs at Various Points of Elastic Half-Space SurfaceMiddle of Short Side Middle of Long Side'-'Shape Circle (flexible) Circle (rigid) Square (flexible) Square (rigid) UB . 1.5 Rectangle (flexible) 2.0 UB= 3
University of Florida - CEG - 4012
p8SupplementThese are the ,?omplete interpolation cales for the Immediate Settlement Case II example on (p. p4). Since the boundary characteristics aren't defined we need to consider both cases (i.e., and u = 0). For each case, interpolate the C's for t
University of Florida - CEG - 4012
p9 o Review of Consolidation (Oedometer) Test Objective to determine the following parameters: Cc CR~~ ~-Lle I Ll log cr'v -M I Ll log cr'v Co (!1xpansion)~~ ~-M Ilog (p,/pil -M Ilog (p,/p,) Cs(~well)3.PI\:OflSO f,J.,4w\cr'c = Pc(F I L')4. (o
University of Florida - CEG - 4012
p10 4. Plot dial gage readings R vs. log time for each load and fit a smooth curve for analysis. Calculate Cv and Ca using the Casagrande construction:Ro-_.J-&quot;-a _:&quot;:a1 11c V -.:1 11T5~ (~/,J) &quot;l/cfw_su(rJ:1.O./~ ( fI/l) 2/i~d (7)1 Jo. t 1a Jf.
University of Florida - CEG - 4012
p11 6. Calculate void ratio e from R lOo for each load p and plot e-log p curve to find preconsoJidation pressure Pc using Casagrande constwction-:void ratio,e,\J, ~('tfJ(,lJ;(V (log P- a um curvalure (minimum radius) on reloaQ.Rortioll.ot. a) Find
University of Florida - CEG - 4012
p12 Obtain Field Curve from Lab Curveo Disturbance Effects- -. '&quot;'&quot;&quot;void ratio,ellc ~'&quot; lloU, -.: (;clJ yl&quot;/&quot; ,.~ .&quot; , ,'.&quot;. ,'A', , , , , , '. ' '. ,,curvature caused by stress removal, disturbance due to sampling and handling, and test a
University of Florida - CEG - 4012
p13oConstruction of NC Field Curve using Schmertmann Methodvoid ratio,er . itJ ~ffj&quot;11(OY-o.(tf l &lt;' ,;0~ 'tv'J [c.log PoLConstruction of OC Field Curve using Schmertmann Method~Recompression Slope CReoLab Virgin Compression void ratio,eo
University of Florida - CEG - 4012
p14Different ways to look at data from a Consolidation Test: (data from San Francisco Bay Mud, Holtz &amp; Kovacs, 1981)Stress, kPa Straint-t-t- -.- -0.35-~;:-0.40 +-~i-r-t-+-l-:l:J10 20 30 40 50 60 70 80 90 1000.000 0.013 0.031 0.138 0.235 0.315 0.
University of Florida - CEG - 4012
p15Consolidation Settlement: Recall that the settle. e esive soii under load IS mo':!ly due to consoli atlon:P=p;lfl,.le~p==-J-Time._._- .-wherepPitotal settlementimmediate settlement1-:;:]:.p, p,= (primary) consolidation settlement = seco
University of Florida - CEG - 4012
p16 o Graphical presentation of Fluid-Filled Cylinder with Spring Analogy: Depends on Opening Size Depends on Soil Properties~. .;.z._.~. .Pressure, - -PSPRINGpForce,o,&quot;, .ITime-AnalogyPWATER, ,,,- ., . -~cr'v_!i.crvSoil lIu (exces
University of Florida - CEG - 4012
p17NC Magnitude of Consolidation, pc: Normally consolidated (NC) clay (saturated)for NC Clay . cr'o = de (or Pc) i.e. the present effective overburden pressure, a'o. is the maximum pressure that the soilhas ever experienced,(j'C(or Pc)=preconsolida
University of Florida - CEG - 4012
p18 o Phase Diagrams: Initial StateVvo = ~'&quot; Av = HAIFinal StateIv = HAllH177:777.777771-t-VVF=H.rASubtracting:&quot;'e = (e, _ eo) = (H VFAlso:Hs-H,) = (- &quot;'H)Hsor(A)H= Hs + Hv = Hs + ~: ) = Hs (1 + eo)(1orH -s - 1+ eo( H)(8)Combinin
University of Florida - CEG - 4012
p19NC Settlement Example 1:Find Pc under center of mat foundation Sand Ym = 15.7 kNlm' Y,ot = 19.1 kN/m' 3 'water = 9.81 kN/m NC Clay (OCR = 1) ,&quot;I = 18.6 kNlm' Cc = 0.28, eo = 0.90Solution: calculate pc for the clay layer. Use point P at mid-height of
University of Florida - CEG - 4012
p20 NC Settlement Example 2: Find pc under tank center: Circular Tank, 30 ft Diameter, 20 ft High, Filled with Oil, SG = 0.91, , ~D1(b&quot;,-&quot; 0,'11) =1136pst Sand: 'i'Mo'&quot; = 100 pct, Ys&quot; = 1 05 pet Smectite Clay: Cc= 0.4, NC, eo= 1.4, Y= 120 pcSub-Iaye.!&quot; ~
University of Florida - CEG - 4012
p21 OC Magnitude of Consolidation, pc: Over-consolidated (OC) ciay (saturated)I for OC Clay cr'o &lt; cr'c (or Pc) i.e. the present effective overburden pressure, cr'o, is less than the maximum past pressure the soil has experienced, cr'c (or Pc) = preconso
University of Florida - CEG - 4012
p22Highly Over-consolidated (HOC):I (cr'O + !J.crv ) S; cr' cVirgin Curve with Slope Cc (= Compression index) Void Ratio,e6_81&quot;]:f\t\i . l 'rfl;Sh.f&lt;./ fl&quot;j't. ( .\$1.1(.RE.I~.Jeo eF-f~:~_-:_:-:_'!-;&quot;:~\~I t.(;.\(~lth_l'rtH'&quot; ~.)~.I4r I&quot;
University of Florida - CEG - 4012
p23Lightly Over-consolidated (LOC):Virgin Curve with Slope Cc (= Compression index) Void Ratio,efl.-c.kJ '-I 51'/e c.(:L,(~f&quot;&lt;~):! - ~Jt-&gt; )f -1-1-I ~f~_c _,'II'-~-'-~-t.ov!.ILog Vertical Effective Stress, o'v0'0Consolidation occurs in tw
University of Florida - CEG - 4012
p24OC Settlement Example 1:Find Pc under center of mat foundationSolution: calculate Pc for the clay layer. Use point Pat mid-height of layer to represent stress change in layer.(Pc in sand = 0)OC Clay (OCR = 1.4) 3 'Ysat = 18.6 kN/m Cc = 0.28, eo =
University of Florida - CEG - 4012
p25 OC Settlement Example 2:Find Pc under tank center: Circular Tank, 30 ft Diameter, 20 ft High, Filled with Oil, SG =0.91, q = 20 (62.4 x 0.91) = 1136 psf7ft 5ft15 ft 12 ftUse 3 sub-layers in upper clay, 2 sub-layers in lower clay. Use Boussinesq f
University of Florida - CEG - 4012
p26Secondary Compression:P = Pi+ Pc . Pswhere: pPi p,p,= total settlement p = immediate settlement = (primary) consolidation settlement = secondary compression (creep)&quot;-~pco Secondary compression is the portion of timedependent settlement that o
University of Florida - CEG - 4012
p27 SUMMARY OF SETTLEMENT CALCULATIONS Define Initial Stresses: a Total Stress, Pore Water Pressure, Effective Stress a Must define the state of stress prior to loading a The behavior of soils is governed by effective stress (thank you Karl Terzaghi) Def
University of Florida - CEG - 4012
p29 ~Time Rate of Consolidation:During the consolidation process: What %Pc will have occurred at a given time? How much time is required for given %Pc to occur?TimepOne-Dimensional Consolidation: Theory presented by Karl Terzaghi in 1925 ENR.Assump
University of Florida - CEG - 4012
p30 av _ a(vv + Vs ) _ a(eVs + Vs ) _ avs v ae avs I aso-8-+ s-+at at at at at at and since avs = 0 at and V s=1+e oVo=(dXdYdZ)1+e owhere eo= initial void ratio and Va = the initial volumeae = ( -1-) 1+e o at (3)av then -0 otae ae = Vs -0 =(dXdY
University of Florida - CEG - 4012
op31 Using the relationship between T and U we can answer the initial questions about %Pc: 1. What %Pc will have occurred at a given time?0%Le. know t, cV, H, N =&gt; what is U?S&quot;lv&lt;'(.-lit -_ c., t . J-LH /rJ)u1100%+ _LT_-=:=:=L.U = % Consolidat
University of Florida - CEG - 4012
p32o Symmetry in Time Factor Table:\-, Why do Cases 1a, 3 and 4 work for both double and single drainage, but Cases 1band 2 do not? The answer is symmetry. Think about what happens to the excess pore water pressure. If the initial excess pore pressure d
University of Florida - CEG - 4012
p33 Consolidation Rate Example Pc = 7 cm H = 10m (clay) Sand top &amp; bottom, N = 2 k = 1x1Q7 cm/s a, = 3.06x1Q&quot; m'/kN eo = 1.50 u constant with depth (Case 1) Yw = 9.81 kN/m'a) How much settlement after 1 yr? b) How long for 5 cm of settlement?a) with t,
University of Florida - CEG - 4012
p34 Degree of Consolidation (U) vs. Depth (z): Terzaghi's 1-0 consolidation equation: cv ~ =a'u at aumay be solved numerically for given initial and boundary conditions to find the excess pore pressure U,.I (read &quot;u as a function of the depth z and time
University of Florida - CEG - 4012
p34With Z we can use a dimensionless plot of isochrones as shown below for a uniform initial pore water pressure distribution (from Perloff &amp; Baron, 1976). Note that each isochrone at a given time t represents a time factor T and an average degree of con
University of Florida - CEG - 4012
p35Degree of Consolidation Example:A surcharge load is applied to a clay layer. At a depth of 21 ft after 4 months find: a) b) c) d) the degree of consolidation excess pore pressure remaining total pore pressure vertical effective stress uniform surchar
University of Florida - CEG - 4012
pas1FOUNDATION REQUIREMENTS (ALLOWABLE SETTLElV,IEtrn:What is a foundation? Two definitions: r.\ N&quot; I : ! r~&quot;f/- ,I o St'.uctu.a/ . I&quot;,1\I ,.el&gt;' l.-t. '-' If&quot;l. r,'t'for!J II'/: s&quot;hd.rt (11_.r; pit/'f&quot; .' ., .&gt;11'&quot; \' .,' o Geo&quot;Lt.chn,ca/. \$&quot;, J_ Sf'!'
University of Florida - CEG - 4012
pas2 ~Criteria For A Satisfactory Foundation:1. Consider any future Influences such as construction, drainage, excavation, etc. 2. Provide adequate bearing capacity to avoid catastrophic bearing or punching failure. 3. Control settlement to prevent str
University of Florida - CEG - 4012
pas3 Maximum depth (inches) of frost penetration (Sowers &amp; Sowers, 1967 - published in P&amp;B).,Note, Depths in inchesSeismic probability map (National Academy of Sciences, 1969 - published in P&amp;B),_'i -e: !Iiio\,SEISMIC RISK MAP OF THE UNITED STA
University of Florida - CEG - 4012
pas4 3. Allowable Settlement Of Structures a Soil subjected to a load will settle. The settlement mayor may not be harmful to the structure. All structures will settle. Anticipate and design for settlement. a a For foundations on soils that settle slowly,
University of Florida - CEG - 4012
pas5a Tilting SettlementAngular distortion~ ~/L= (pm&quot; - Pm;,) / LL is distance adjacent columns/supports. Possible problems are tilting into adjacent building, aesthetic ~ eye very sensitive to tilting (1:100), rolling, tilting of bridge piers. Tilt