CEE 8442 Advanced Structural Mechanics Lecture 9 Selection and Effects of k

1. CEE 8442Advanced Structural MechanicsLecture 9Selection and Effects of k

2. Topics and Applications • The Coefficient of Subgrade Reaction (CSR), k, foundation modulus • Determination • Application of Superposition, Cooper E-80 Train • Subgrade transition (like your homework)

3. The Coefficient of Subgrade Reaction (CSR): • Standard application • How it typical values are determined • Modeling issues • Current research • Example

4. Winkler Foundation CSR parameter ( ks ) has units of force / length3 Simplest analytical model of continuous elastic foundation When deflection, d is imposed on foundation, it resists with a pressure, q q (x) = ksd (x)

5. Standard Application Estimate the settlement of a footing under a concentrated load Given the concentrated load, allowable bearing pressure and coefficient of subgrade reaction, settlement can be estimated as; D = (4 * qall * B2) / (ks (B+1)2) Quick and efficient way to estimate a fairly straightforward and common situation

6. How It Is Determined • Plate-bearing test • Testing method specified in ASTM D 1196-93 • Result of test is a plot of settlement vs. pressure • Material CSR = slope of the elastic portion • Field plate-bearing tests - time consuming and costly • Representative values correlated with other soil properties

8. Typical Values Das, Principles of Foundation Engineering, 3rd Edition

9. Modeling Issues • Foundation problems complex • Simplifications introduce approximations & error • Typical model = 2-D mathematical expression, k = psi • Real soils capable of load spreading • Stress at point results in settlement at many points • Real world models need to account for stress-deformation characteristics of soil, shape and size of loaded area and magnitude and position of nearby loaded areas

10. Current Research • Center for Geotechnology (CGT) at Manhattan College • Soil Structure Interaction (SSI) Research Project • Project has published a number of papers on modeling and obtaining more accurate subgrade models • http://www.engineering.manhattan.edu/civil/CGT.html

11. In-Class Example • Long strip footing with column load at end • Solution • w(x) = ( 2 P b / k ) e-bx cos bx • M(x) = ( -P / b ) e-bx sin bx • Concentrate on M(x) equation • Vary ks values and check results

12. Example • Dense, dry sand w/ ks = 460 lb / in3 • Dense, dry sand w/ ks = 1380 lb / in3 • Moments for low and high values of ks • Mlow = 160,500 lb-in • Mhigh = 145,100 lb-in • Conclusion: Approximately a 10% difference for moment

13. Conclusion • Winkler Foundation Model is approximately 130 years old • Exceptions have been taken with the model for approximately 60 years • Means of improvement are not new! • Simplicity is the appeal • Previous results (i.e. structure performance) acceptable, or it would have been discarded long ago

14. Analysis of a Cooper E-80 Train on a Single Rail for Different k’s • Theodore Cooper was one of the first engineers to establish live loads for railroads • Presently, AREA, American Railway Engineering Association, recommends the use of the Cooper E-80 train for live load

15. Investigation • Solve for the displacement and moment on an elastic foundation, constant k, due to the Cooper E-80 Train loading • Determine the effects on displacement and moment caused by different values of k

16. Analysis • Infinitely long elastic foundations • Center the train about the origin • Use Superposition and a Green’s function, Kp for a point load on an infinite beam

17. Analysis continued • Deflection and Moment

18. Typical Values • Concrete Sub-grade; k=4000 lbs/in2 • Crushed Stone Sub-grade; k= 1800 lbs/in2 • Soil Sub-grade; k = 300 lbs/in2

19. Results: Deflection

20. Results: Moment

21. Conclusions • Superposition makes this problem feasible • Analysis required only Excel • Increasing k decreases • deflection • moment

22. Railroad Track Configurations Characteristics and Analysis

23. Track Types • Ballasted Tracks • Most Common Type • Direct Fixation Tracks • Long Island Railroad • Embedded Tracks • Turf Tracks

24. Ballasted Track Track Modulus is estimated considering: -crosstie size -depth of ballast and sub ballast -type of ballast rock or stone -crosstie spacing

25. Ballasted Track Ballast: -Constructed from limestone, heavy stone, or granite -(k limestone<k h.stone<k granite) Ties: -Constructed from wood or concrete -(k wood < k concrete)

26. Direct Fixation Track Track Modulus estimated considering the vertical deflection, which can occur in: -rail bending -flexure of slab at subbase materials for at-grade installations This is the standard method of construction for tracks on aerial structures and tunnels.

27. Long Island Railroad Concrete Slab Track • During the 1980’s the LIRR undertook the nine year task of replacing all of its ballasted track with direct fixation track. • The slab track system consists of a concrete slab supported on a subgrade of sand and a subbase of asphalt. • The change from a ballasted track to direct fixation track presented several advantages: • ballast, ties, and the associated maintenance are eliminated • less maintenance, means less traffic disruptions • load is distributed more uniformly on the subgrade, thus settlement is reduced • when combined with welded rail, ride quality and operational speeds improve

28. Embedded Track Track Modulus: -difficult to determine -rail deflections are extremely small -field measurements estimate k = 2,000,000 psi

29. Embedded Track Turf Track: Another Type of Embedded Track • Spawned from European light rail systems desire to blend the transit track and system into the landscape. • Developed for selected purposes: • -reduce the visual effect of ballasted track • -reduce the noise from trams as much as possible • -provide year-round greenery in the vicinity of the track

30. Transition from Low Modulus to High Modulus Track Winkler Base Analysis of Track Transition 2 D.E.’s: 1.) EI*wliv(x)-kl*wl(x), (-inf. < x < 0) 2.) EI* wriv(x)-kr*wr(x), (0 < x < inf.) 2 B.C’s: 1.&2.) LIM(x->-inf.) {wl,wli} -> finite 3.&4.) LIM(x->inf.) {wr, wri} -> finite 4 M.C’s: 5.) wl(0)=wr(0) 6.)wli(0)=wri(0) 7.) wlii(0)=wrii(0) 8.)wliii(0)+wriii(0)= (P/EI)

31. Transition from Low Modulus to High Modulus Track - damage can be done to both track and vehicle in areas of abrupt track modulus change

32. Transition from Ballasted to Embedded Track Displacement

33. Transition from Ballasted to Embedded Track Moment

34. Transition from Ballasted to Direct Fixation Track Displacement

35. Transition from Ballasted to Direct Fixation Track Moment

36. Cooper Results for Deflection Varying k

37. Cooper Results for Moment Varying k

38. Transition Zones • Approach Slabs: • -extend from embedded track slab a min. of 20ft. into ballasted section • slab typically located @ 1ft. below the ties immediately adjacent to stiffer track. • replace more compressible subballast with a stiffer base • Reduced spacing between ties Plan View Elevation View

39. Direct Fixation to Ballasted Track Fastener design continues to improve. New fastener spring rates allow modulus to decrease Lower track modulus allows easier transition Embedded to Ballasted Track Continues to evolve and improve Rail deflections are hard to match to ballasted track Differential in track modulus may be too large to overcome simply through flexible rail Transition Zones These transitions require more maintenance than most track sections. The bending forces in each transition will not eliminate all damage.

40. Conclusion • Ballasted Track • Most commonly used Track type • Composed of Crosstie, Ballast, and Fastenings • k value based mainly on Crosstie spacing and Ballast composition • k value for can range from 1500 – 5000 psi for wood X-ties and 5000 – 8000 psi for concrete.

41. Conclusion • Direct Fixation Track • Most commonly used on bridges and in tunnels • Ballastless track in which rail is directly fastened to a concrete slab • k value is easily determined from the amount of vertical deflection within the fasteners. • k value commonly around 15,000 psi

42. Conclusion • Embedded Track • Most commonly used for light rail in urban business centers • Track is embedded within a concrete slab and only portion showing is the rail head • k value is very large, though very hard to establish, because deflections are so small • k value is assumed to be 2,000,000 psi

43. Conclusion • Transition Zone: • Immediate transition causes damage • This damage is greater in the transition from ballast to embedded than it is in ballast to direct fixation due to the large variation in k • Approach slabs are used to ease transition from low to high modulus track

44. The Topic Transition Zone:How can Winkler Foundations Be Related to Fracture Mechanics? Fracture Specimen for Composite Material Application of a Beam on an Elastic Foundation to a Double Cantilever Beam

45. 3 Modes of Failure Opening Sliding Tearing

46. Critical Configuration • Mode I fracture has been found to have the lowest critical strain energy release rate • If load, P, is applied to a specimen in Mode I, Mode II, and Mode III … the Mode I case will govern.

47. Specimen Used in Composite Materials P Comparable to a fracture toughness specimen This type of specimen is used to determine the interlaminar fracture toughness P a

48. Analytical Model P a c x L

49. Formulation • 2 DE’s: EI wcIV + k wc = 0 0 < x < c EI waIV = 0 -a < x < c • 4 BC’s: 1. waII(-a) = 0 2. waIII(-a) = P/EI 3. wcII(c) = 0 4. wcIII(c) = 0 • 4 MC’s: 1. wc(0) = wa(0) 2. wIc(0) = wIa(0) 3. wIIc(0) = wIIa(0) 4. wIIIc(0) = wIIIa(0)

50. Next Week • Introduction to Fracture • Factors Effecting Fracture • Material Toughness • Linear Elastic Fracture Mechanics • Keep working on your projects!