EPS - LDS Calculations - Brigham City Page 1 EPS Allowable Stress Calculations (Rev. 2) 8:11 AM Required Calculate the vertical stress in the top of the EPS block from the live loads (traffic load) for a HL-93 truck loading. Given Pavement and EPS properties given Table 1 Dual axial truck loading shown in Figure 1 Table 1 - Material properties ρ (kg/m 3 ) γ (lb/ft 3 ) E (MPa) v K (MPa) G (MPa) EPS19 18.4 1.15 4.0 0.1 1.67 1.82 Road base 2241 140 400 0.3 333 153.8 Asphalt 1 2321 145 6890 0.3 5742 2650.0 Concrete 2401 150 25000 0.18 13021 10593.2 Pasted from <file:///c:\users\sfbartlett\documents\flac\flac%203d\eps-pavement\material_properties.xls> 1 The asphalt properties in the numerical model were assigned the same properties (Young (E), bulk (K) and shear modulus (G)) as the road base in order to represent a completely cracked and damaged asphalt pavement.
EPS - LDS Calculations - Brigham City Page 2 Fig. 1 - HL-93 Dual Axial Loading - 25 kip per axial with spacing of 4 ft between axels End View Assumptions / Approach Dual Axel Side View Materials behave within the elastic range and the properties in Table 1 are appropriate for this range Tire loading converted to equivalent square loading in an area 0.3 m x 0.3 m = 0.09 m 2 3D model Width of roadway is 13 m with 3 lanes (x-direction) Length of roadway = continuous roadway = approximated by 39 m length (y-direction) Boundary conditions Base = fixed (fixed z) Sides = free (free x, y, z) Ends = fixed (fixed y) For a single 25-kip axial with dual tires, the contact area can be estimated by converting the set of duals into a singular area by assuming that the singular area has an area equal to the contact area of the duals. q = QD / ACD Eq. (1) Table 2 - HL-93 tire loading Dual Tire Loading (single axial/one side) QD = 12.5 kips 55.6 KN q = 90 psi 618 Kpa ACD = 139 in^2 0.09 m 2 The 618 kpa vertical stress from the 12.5 kip loading will be put into the numerical modeling to determine the stress redistribution in the pavement section and into the EPS. This stress will be applied over a circular area representing the dual tire at the top of the pavement is 0.17 m.
EPS - LDS Calculations - Brigham City Page 3 Numerical model and boundary conditions end nodes fixed (y) free (x and y) free (x and y) z y x end nodes fixed (y) all bottom nodes fixed (z) Nodal spacing x-direction (width) = 13 m / 52 nodes = 0.25 m per node y-direction (length) = 39 m / 130 nodes = 0.3 m per node x-direction (height) = 3 m / 20 nodes = 0.15 m per node Material Property assignment (see Table 1) top 3 layer = road base (red) 4th layer = load distribution slab (reinforced concrete) (green) 5th to 20th layer = EPS19
EPS - LDS Calculations - Brigham City Page 4 Model Properties
EPS - LDS Calculations - Brigham City Page 5 Load Application (12,5 kip tire loads) 12.5 k vertical loadings applied to the model 12.5 k 12.5 k 12.5 k truck 1 12.5 k
EPS - LDS Calculations - Brigham City Page 6 Load Application (sleeper slab) length of approach slab = 25.5 feet = 7.77 m width of approach slab = 42 feet 10 inches = 13.05 m width of sleeper slab = 5 feet = 1.5 m stress from sleeper slab at base of footing = 150 k / 5 feet / 42.83 ft = 700.4 psf = 33.6 kpa (from 150 k tire loads only) FLAC3D 3.10 2006 Itasca Consulting Group, Inc. Step 21515 Model Perspective 07:51:39 Tue Dec 31 2013 Center: X: 6.500e+000 Y: 1.950e+001 Z: 0.000e+000 Rotation: X: 15.000 Y: 0.000 Z: 20.000 Dist: 9.089e+001 Mag.: 1.25 Ang.: 22.500 Plane Origin: X: 6.500e+000 Y: 1.950e+001 Z: 2.990e+000 Plane Normal: X: 0.000e+000 Y: 0.000e+000 Z: 1.000e+000 Contour of SZZ Plane: on Magfac = 0.000e+000 Live mech zones shown Gradient Calculation -3.3590e+004 to -3.0000e+004-3.0000e+004 to -2.5000e+004-2.5000e+004 to -2.0000e+004-2.0000e+004 to -1.5000e+004-1.5000e+004 to -1.0000e+004-1.0000e+004 to -5.0000e+003-5.0000e+003 to 0.0000e+000 0.0000e+000 to 7.6496e+002 Interval = 5.0e+003 33.6 kpa stress applied over 1.5 m strip sleeper slab Itasca Consulting Group, Inc. Minneapolis, MN USA
EPS - LDS Calculations - Brigham City Page 7 Results - 12 kip tire load - top of EPS Plan view of plane cut horizontally through model at top of EPS layer (z = 2.4 m). The maximum stress from the tire loading is -1.7e4 Pa (negative means acting downward). This converts to: 1.74e4 Pa / 1000 Pa / Kpa / 96 kpa / 1 tsf * 2000 psf / tsf / 144 psf / psi = 2.5 psi
EPS - LDS Calculations - Brigham City Page 8 Results - 12 kip tire load - X-section view Top of EPS layer vertical stress between -1.0e4 and -2.0e4 Pa X-sectional view of plane cut vertical through model at rear axial load (i.e., y - 18.75 m). EPS layer is the 5th layer (top to bottom). The maximum stress from the tire loading is between 10 and 20 kpa.
EPS - LDS Calculations - Brigham City Page 9 2:32 PM Results - 150 k tire load on sleeper slab - results on top of EPS FLAC3D 3.10 2006 Itasca Consulting Group, Inc. Step 21515 Model Perspective 08:16:51 Tue Dec 31 2013 Center: X: 6.500e+000 Y: 1.950e+001 Z: 1.500e+000 Rotation: X: 15.000 Y: 0.000 Z: 20.000 Dist: 9.089e+001 Mag.: 1.95 Ang.: 22.500 Plane Origin: X: 6.500e+000 Y: 1.950e+001 Z: 2.400e+000 Plane Normal: X: 0.000e+000 Y: 0.000e+000 Z: 1.000e+000 Contour of SZZ Plane: on Magfac = 0.000e+000 Live mech zones shown Gradient Calculation -2.4071e+004 to -2.2500e+004-2.2500e+004 to -2.0000e+004-2.0000e+004 to -1.7500e+004-1.7500e+004 to -1.5000e+004-1.5000e+004 to -1.2500e+004-1.2500e+004 to -1.0000e+004-1.0000e+004 to -7.5000e+003-7.5000e+003 to -5.0000e+003-5.0000e+003 to -2.5000e+003-2.5000e+003 to 0.0000e+000 Itasca Consulting Group, Inc. Minneapolis, MN USA Plan view of plane cut horizontally through model at top of EPS layer (z = 2.4 m). The maximum stress from the tire loading is -1.5e4 Pa (negative means acting downward). This converts to: 1.5e4 Pa / 1000 Pa / Kpa / 96 kpa / 1 tsf * 2000 psf / tsf / 144 psf / psi = 2.17 psi ( from 150 k tire load only)
EPS - LDS Calculations - Brigham City Page 10 Results - 150 k load applied to sleeper slab - X-section view Top of EPS layer vertical stress between -1.0e4 and -1.5e4 Pa Top of EPS layer vertical stress between X-sectional view of plane cut vertical through middle of sleeper slab footing (i.e., y = 7.75 m). EPS layer is the 5th layer (top to bottom). The maximum stress from the tire loading is between 10 and 20 kpa.
EPS - LDS Calculations - Brigham City Page 11 Live load calculations - Controlling case LL tire = 2.5 psi (from FLAC for minimal pavement thickness section) LL Lane load = 3 lanes x 10 ft per line / 39 * 0.65 klf /10 ft wide lane = 0.05 ksf or 50 psf or 0.35 psi Dead load calculations ρ γ E v K G thickness v. stress (kg/m3) (lb/ft3) (MPa) (MPa) (MPa) (ft) Clayey Soil 1900 118.7 20 0.25 13.33 8.00 0 0 Backfill 2100 131.25 40 0.25 26.67 16.00 2.5 328.125 EPS19 18.4 1.15 4.0 0.1 1.67 1.82 0 0 EPS22 21.6 1.35 5.0 0.1 2.08 2.27 0 0 EPS29 28.8 1.80 7.5 0.1 3.13 3.41 0 0 Roadbase 2241 140 400 0.3 333 153.8 0.33 46.666667 Asphalt 2321 145 6890 0.3 5742 2650.0 0.83 120.83333 Concrete 2401 150 25000 0.18 13021 10593.2 0.5 75 sum = 570.625 3.96 psf psi Pasted from <file:///c:\users\sfbartlett\documents\flac\flac%203d\eps-pavement\material_properties.xls>
EPS - LDS Calculations - Brigham City Page 12 Allowable Stress Calculation (controlling case) 12-k tire loads placed off the approach slab - combination of live and dead loads 12-k tire loadings placed off the approach slab are the controlling case Allowable stress in EPS typically is typically taken at 1 percent strain Compressive resistance of EPS22 at 1 percent strain = 7.3 psi (ASTM D6817) Applied Stress = LL tire + LL lane + DL = 2.5 + 0.35 + 4.0 = 6.85 psi (controls) Thus, EPS with a compressive resistance of 6.85 psi, or higher, is required.
EPS - LDS Calculations - Brigham City Page 13 Friday, January 10, 2014 2:32 PM The project team wants to explore the case where embankment support for the approach slab is removed by settlement and the approach slab is supported on one end by the bridge abutment and on the other end by the sleeper slab footing. This scenario is unlikely for EPS embankment because the sleeper slab footing is a shallow foundation supported by the embankment, thus as the foundation soils and embankment settle, the sleeper slab must settle correspondingly. The contact remains between the EPS and approach at least in the area near the footing. In addition, the elastic compression of the EPS in the area immediately under the sleeper footing creates downward differential movement of the footing, which also aids in preventing a gap to form between the base of the approach slab and the EPS supported embankment. Nonetheless, the "hypothetical complete lost of contact between the slab and EPS supported embankment will be analyzed as requested. The assumption made is that the approach slab is simply supported by the sleeper slab footing and the bridge abutment with 50 percent of the approach slab weight transferred to the sleeper slab footing. Footing loads for simply supported approach slab weight of sleeper slab [[(60" x 9") + (12" x 13")]/144 in^2/ft^2 x 42.83 ft] x 150lb/ft^3 = 31051 lb = 31.05 kips weight of 1/2 of approach slab (simply supported on both ends) 42.83 ft x 25.42/2 ft x 1.083 ft x 150 lb/ft^3 = 88433 lbs = 88.433 kips truck axial loadings = 150 kips (previous) total force at base of footing 31.05+88.433+150=269.483 kips total stress at base of footing 269.483/(42.83*5)=1.2584 ksf (1.2584/2)*96=60.4032 kpa (apply this to the numerical model)
EPS - LDS Calculations - Brigham City Page 14 Friday, January 10, 2014 Load Application (sleeper slab with loss of support and with cracked concrete length of approach slab = 25.5 feet = 7.77 m width of approach slab = 42 feet 10 inches = 13.05 m width of sleeper slab = 5 feet = 1.5 m stress from weight of sleeper slab, 50 percent weight of approach slab and 150 k tire loadings = 60.4 kpa (previous page) FLAC3D 3.10 2006 Itasca Consulting Group, Inc. Step 13534 Model Perspective 18:15:11 Fri Jan 10 2014 Center: X: 6.500e+000 Y: 1.950e+001 Z: 1.500e+000 Rotation: X: 10.000 Y: 0.000 Z: 20.000 Dist: 5.605e+001 Mag.: 0.8 Ang.: 22.500 Plane Origin: X: 6.500e+000 Y: 1.950e+001 Z: 2.990e+000 Plane Normal: X: 0.000e+000 Y: 0.000e+000 Z: 1.000e+000 Contour of SZZ Plane: on Magfac = 0.000e+000 Live mech zones shown Gradient Calculation -6.0964e+004 to -6.0000e+004-6.0000e+004 to -5.0000e+004-5.0000e+004 to -4.0000e+004-4.0000e+004 to -3.0000e+004-3.0000e+004 to -2.0000e+004-2.0000e+004 to -1.0000e+004-1.0000e+004 to 0.0000e+000 0.0000e+000 to 1.4345e+003 Interval = 1.0e+004 60.4 kpa stress applied over 1.5 m strip sleeper slab Itasca Consulting Group, Inc. Minneapolis, MN USA
EPS - LDS Calculations - Brigham City Page 15 Friday, January 10, 2014 2:32 PM Calculation of adjusted elastic modulus for load distributions slab for use in continuum model Eadj. model = Econcrete x Icracked/ Iuncracked Eadj. model = adjusted elastic modulus for numerical model Econcrete = Young's modulus of concrete for 4000 psi strenth Icracked = cracked moment of inertia Iuncracked = uncracked moment of inertia Assume simple 1-way beam action of load distribution slab Iuncracked = 1/12 b h 3 (for beam) (12*6*6*6)/12=216 in 4 Icracked = b = width = 12 inches (unit width) As = area of steel (2#5bars) = (2)(0.31) = 0.62in 2 n = Esteel/Econcrete = 29000000psi/(57000*(4000) 0.5 ) psi = 8.04 d = depth to steel in tension = 5 inches y = depth to neutral axis 8.04*0.62*(((1+2*12*5)/(8.04*0.62))^0.5-1)/12=1.6312 y = 1.63 inches Icracked (12*1.63^3)/3+8.04*0.62*(5-1.63)^2=73.9349 in 4 Eadj. model 57000*(4000)^0.5*(73.9349/216)=1.2339E6 psi 1.2339E6*(144/2000)*96=8.5287E6 kpa or 8.529E3 MPa (use this)
EPS - LDS Calculations - Brigham City Page 16 Friday, January 10, 2014 2:32 PM Results - 150 k tire load + sleeper slab ftg. + 1/2 approach slab - results on top of EPS FLAC3D 3.10 2006 Itasca Consulting Group, Inc. Step 13955 Model Perspective 21:54:27 Fri Jan 10 2014 Center: X: 6.500e+000 Y: 1.950e+001 Z: 1.500e+000 Rotation: X: 15.000 Y: 0.000 Z: 20.000 Dist: 9.089e+001 Mag.: 1.5 Ang.: 22.500 Plane Origin: X: 6.500e+000 Y: 1.950e+001 Z: 2.690e+000 Plane Normal: X: 0.000e+000 Y: 0.000e+000 Z: 1.000e+000 Contour of SZZ Plane: on Magfac = 0.000e+000 Live mech zones shown Gradient Calculation -3.7345e+004 to -3.5000e+004-3.5000e+004 to -3.0000e+004-3.0000e+004 to -2.5000e+004-2.5000e+004 to -2.0000e+004-2.0000e+004 to -1.5000e+004-1.5000e+004 to -1.0000e+004-1.0000e+004 to -5.0000e+003-5.0000e+003 to 0.0000e+000 0.0000e+000 to 9.6349e+002 Interval = 5.0e+003 Itasca Consulting Group, Inc. Minneapolis, MN USA Plan view of plane cut horizontally through model at top of EPS layer (z = 2.7 m). The maximum stress from the loading combination is -3.0e4 Pa (negative means acting downward). This converts to: 3.0e4 Pa / 1000 Pa / Kpa / 96 kpa / 1 tsf * 2000 psf / tsf / 144 psf / psi = 4.34psi This stress does not include the weight of the road base under the sleeper slab and the weight of the load distribution slab. Treat these as 1D dead loads road base weight = 8 inches /12 * 140 lb/ft^3 = 93.3 psf or 0.648 psi load distribution slab = 6 inches / 12 * 150 lb/ft^3 = 75 psf or 0.521 psi Total stress from all components on sleeper slab = 4.34 + 0.648 + 0.521 = 5.51 psi LL Lane load = 3 lanes x 10 ft per line / 39 * 0.65 klf /10 ft wide lane = 0.05 ksf or 50 psf or 0.35 psi Total stress with lane load on sleeper slab = 5.86 psi
EPS - LDS Calculations - Brigham City Page 17 FLAC Code (For 12.5 kip Tire loadings) set mechanical ratio 1e-5 gen zone brick size 52 130 20 p0 0,0,0 p1 13,0,0 p2 0,39,0 p3 0,0,3; hor = 0.3 ver = 0.15 model elas prop bulk 1.67e6 shear 1.82e6 range z -.1,2.40 x -.1,13.1 y -.1,39.1 ; EPS19 prop bulk 13021e6 shear 10953e6 range z 2.41,2.55 x -.1,13.1 y -.1,39.1 ; LDS prop bulk 333e6 shear 154e6 range z 2.56,3.01 x -.1,13.1 y -.1,39.1 ; base ; fix z range z -.1.1 ; fixes base ;fix x range x -.1.1 ; fixes left side ;fix x range x 3.9 4.1 ;fixes right side fix y range y -.1.1 ; fixes front face fix y range y 38.9 39.1 ; fixes back face ; ;set gravity 9.81 ;solve apply szz -618e3 range z 2.9,3.1 x 1.79,2.11 y 18.59,18.91 ; dual tire 1 apply szz -618e3 range z 2.9,3.1 x 3.61,3.86 y 18.59,18.91 ; dual tire 2 apply szz -618e3 range z 2.9,3.1 x 1.79,2.11 y 19.79,20.11 ; dual tire 3 apply szz -618e3 range z 2.9,3.1 x 3.61,3.86 y 19.79,20.11 ; dual tire 4 apply szz -618e3 range z 2.9,3.1 x 5.39,5.71 y 18.59,18.91 ; dual tire 5 apply szz -618e3 range z 2.9,3.1 x 7.19,7.51 y 18.59,18.91 ; dual tire 6 apply szz -618e3 range z 2.9,3.1 x 5.39,5.71 y 19.79,20.11 ; dual tire 7 apply szz -618e3 range z 2.9,3.1 x 7.19,7.51 y 19.79,20.11 ; dual tire 8 apply szz -618e3 range z 2.9,3.1 x 8.99,9.31 y 18.59,18.91 ; dual tire 9 apply szz -618e3 range z 2.9,3.1 x 10.49,10.81 y 18.59,18.91 ; dual tire 10 apply szz -618e3 range z 2.9,3.1 x 8.99,9.31 y 19.79,20.11 ; dual tire 11 apply szz -618e3 range z 2.9,3.1 x 10.49,10.81 y 19.79,20.11 ; dual tire 12 ; hist unbal ;step 100 solve
EPS - LDS Calculations - Brigham City Page 18 plot create GravV plot set color On plot set caption On plot set caption left plot set caption size 26 plot set title On plot set title top plot set foreground black plot set background white plot set window position (0.00,0.00) size(1.00,0.89) plot set plane normal (0.000,1.000,0.000) plot set plane origin (6.5000e+000,1.9500e+001,0.0000e+000) plot set mode model plot set center (6.5000e+000,1.9500e+001,1.5000e+000) plot set rotation (0.00, 0.00, 0.00) plot set distance 9.0895e+001 plot set angle 22.50 plot set magnification 1.95e+000 plot add cont szz plane ; plot create EPSV plot set color On plot set caption On plot set caption left plot set caption size 26 plot set title On plot set title top plot set foreground black plot set background white plot set window position (0.00,0.00) size(1.00,0.89) plot set plane normal (0.000,0.000,1.000) plot set plane origin (6.5000e+000,1.9500e+001,0.0000e+000) plot set mode model plot set center (6.5000e+000,1.9500e+001,1.5000e+000) plot set rotation (15.00, 0.00,20.00) plot set distance 9.0895e+001 plot set angle 22.50 plot set magnification 1.95e+000 plot add cont szz plane ; save EPS-pavement.sav
EPS - LDS Calculations - Brigham City Page 19 Friday, January 10, 2014 2:32 PM FLAC Code (For 150 kip sleeper slab + approach slab loading) set mechanical ratio 1e-5 gen zone brick size 52 130 20 p0 0,0,0 p1 13,0,0 p2 0,39,0 p3 0,0,3; hor = 0.3 ver = 0.15 model elas prop bulk 1.67e6 shear 1.82e6 range z -.1,2.67 x -.1,13.1 y -.1,39.1 ; EPS19 prop bulk 4442e6 shear 3614e6 range z 2.68,2.83 x -.1,13.1 y -.1,39.1 ; LDS cracked prop bulk 333e6 shear 154e6 range z 2.84,3.01 x -.1,13.1 y -.1,39.1 ; base ; fix z range z -.1.1 ; fixes base ;fix x range x -.1.1 ; fixes left side ;fix x range x 3.9 4.1 ;fixes right side fix y range y -.1.1 ; fixes front face fix y range y 38.9 39.1 ; fixes back face ; ;set gravity 9.81 ;solve apply szz -60.4e3 range z 2.9,3.1 x -0.1,13.1 y 5.9,7.6 ; sleeper slab ; hist unbal ;step 1000 solve
EPS - LDS Calculations - Brigham City Page 20 Friday, January 10, 2014 2:32 PM plot create GravV plot set color On plot set caption On plot set caption left plot set caption size 26 plot set title On plot set title top plot set foreground black plot set background white plot set window position (0.00,0.00) size(1.00,0.89) plot set plane normal (0.000,1.000,0.000) plot set plane origin (6.5000e+000,1.9500e+001,0.0000e+000) plot set mode model plot set center (6.5000e+000,1.9500e+001,1.5000e+000) plot set rotation (0.00, 0.00, 0.00) plot set distance 9.0895e+001 plot set angle 22.50 plot set magnification 1.95e+000 plot add cont szz plane ; plot create EPSV plot set color On plot set caption On plot set caption left plot set caption size 26 plot set title On plot set title top plot set foreground black plot set background white plot set window position (0.00,0.00) size(1.00,0.89) plot set plane normal (0.000,0.000,1.000) plot set plane origin (6.5000e+000,1.9500e+001,0.0000e+000) plot set mode model plot set center (6.5000e+000,1.9500e+001,1.5000e+000) plot set rotation (15.00, 0.00,20.00) plot set distance 9.0895e+001 plot set angle 22.50 plot set magnification 1.95e+000 plot add cont szz plane ; save EPS-sleeper.sav
EPS - LDS Calculations - Brigham City Page 21 Design Inputs and Drawings Information Sleeper Slab Detail Geofoam Typical Section
EPS - LDS Calculations - Brigham City Page 22 Design Input Information Approach Slab
EPS - LDS Calculations - Brigham City Page 23 Blank 2:32 PM