5.10.4. CALCULATIONS FOR THE MARSHALL MIX DESIN OF BITUMINOUS MIXTURES 1. Scope. This method covers the formulas used to compute the various values used in the Marshall Mix Design of % virgin aggregate mixtures or reclaimed material and virgin aggregate mixtures. 2. Procedure. 2.1. Aggregate Specific ravity. 2.1.1. Virgin Aggregate: Measure the Bulk (Dry) Specific ravity of the coarse and fine aggregate fractions (KT-6, Procedure I & II). 2.1.2. Reclaimed Material: KT-6, as indicated aove can e used on extracted aggregate if a sufficient quantity is availale. If sufficient extracted aggregate is not availale, conduct KT-39 on the reclaimed material. Otain the specific gravity of the extracted asphalt and calculate the specific gravity of the comined aggregate in the reclaimed material. 2.2. Determine the specific gravity of the asphalt cement. 2.3. Calculate the Bulk Specific ravity of the aggregate comination in the paving mixture. ( s ) 2.4. Measure the Maximum Specific ravity of the loose paving mixture. (KT-39) ( mm ) 2.5. Measure the Bulk Specific ravity of the compacted paving mixture. (KT-15) ( m ) 2.6. Calculate the Effective Specific ravity of the aggregate. ( se ) 2.7. Calculate asphalt asorption of the aggregate. (P a ) 2.8. Calculate the effective asphalt content of the mixture. (P e ) 2.9. Calculate the percent voids in the mineral aggregate in the compacted paving mixture. (VMA) 2.10. Calculate the percent air voids in the compacted mixture. (P a ) 2.11. Calculate the percent voids filled with asphalt in the compacted mixture. (VFA) 2.12. Calculate the limiting asphalt content at the tentative design asphalt content. (P' max) 2.13. Calculate the Bearing Capacity for each asphalt content. 3. Equations. Equations for the aove calculations follow in this text and their application may e expedited y use of the appropriate worksheet (Figure 1). Also, there is a computer program availale which will complete these calculations. For purposes of illustration, the following data is presumed "known" for a ituminous mixture under evaluation. Page 1/14 5.10.4 04-12
3.1. BASIC DATA FOR SAMPLE OF PAVIN MIXTURE 3.1.1. Constituents: Material Specific ravity Test Method Mixture Composition Percent y wt. of Apparent Bulk Dry Total Mix Dry Aggr. Asph. Cement Coarse Aggr. Fine Aggr. 1.010 2.759 2.905 1 2.606 2 2.711 T-228 KT-6-I KT-6-II P 6.96 P 1 51.45 P 2 41.59 P' 7.48 P' 1 55.30 P' 2 44.70 Tale 1 3.1.2. Paving Mixture: Bulk specific gravity of compacted paving mixture sample. m (KT-15) = 2.344 Maximum specific gravity of paving mixture sample. mm (KT-39) = 2.438 NOTE: The calculations are simplified y converting from percent y dry weight of aggregates to percent y total weight of mixture. This is accomplished y use of the following formulas: P = ( P x ) ( + P ) 1 P 1 = ( P x ) ( + P ) P = Percent asphalt, total mixture asis. P' = Percent asphalt, dry weight asis. P 1 = Percent Coarse Aggr., total mixture asis. P' 1 = Percent Coarse Aggr., dry weight asis. Example: P = (7.48 x ) = 6.96% (asphalt) ( + 7.48) P 1 = (55.30 x ) = 51.45% (coarse aggr.) ( + 7.48) P 2 and P 3 are calculated in the same manner as P 1. 3.2. Bulk Specific ravity of Aggregate: When the total aggregate consists of separate fractions of coarse aggregate, fine aggregate, and mineral filler, all having different specific gravities, the ulk and apparent specific gravities for the total aggregate are calculated as follows: Page 2/14 5.10.4 04-12
3.2.1. Bulk Specific ravity, s. s P 1,P 2,P n 1, 2, n s = P 1 + P 2 +... P n (Eq. 1, Figure 1) P1 + P2 1 2 +... Pn n = Bulk dry specific gravity of the total aggregate = Percentages y weight of aggregates, 1,2,n; = Bulk specific gravities of aggregates 1,2,n. The ulk specific gravity of mineral filler is difficult to determine accurately at the present time. However, if the apparent specific gravity of the filler is used instead, the error is usually negligile. Calculation using the data from Tale 1. s = 51.450 + 51.450 2.606 + 41.590 41.590 2.711 93.040 19.743 + 15.341 = 2.652 3.3. Effective Specific ravity of Aggregate: When ased on the maximum specific gravity of a paving mixture, mm, the effective specific gravity of the aggregate, se, includes all void spaces in the aggregate particles except those that asor asphalt. It is determined as follows se = P mm - P (Eq. 2, Figure 1) Pmm P - mm se = Effective specific gravity of aggregate P mm = total loose mixture = % P = asphalt, percent y total weight of mixture mm = maximum specific gravity of paving mixture (no air voids), (KT-39) = specific gravity of asphalt Calculation using the data from Tale 1. 6960. se 6960. 2. 438 1010. 93040. 2. 726 34126. NOTE: The volume of asphalt inder asored y an aggregate is almost invarialy less than the volume of water asored. Consequently, the value for the effective specific gravity of an aggregate ( se ) should Page 3/14 5.10.4 04-12
e etween its ulk ( s ) and apparent specific gravities ( sa ). When the effective specific gravity falls outside these limits, its value must e assumed to e incorrect. If this occurs; the calculations, the maximum specific gravity of the total mix (KT-39) and the composition of the mix should then e rechecked for the source of the error. If the apparent specific gravity of the coarse aggregate is 2.759 and the apparent specific gravity of the fine aggregate is 2.905 (see Tale 1.), the apparent specific gravity, sa, of the total aggregate can e calculated y the same formula as the ulk specific gravity y using the apparent specific gravity of each aggregate constituent. For this example, then, the calculated apparent specific gravity, sa, is; sa = 51.450 + 41.590 = 93.040 = 2 ( 51.450 2.759 ) + ( 41.590 2.905 ) 32.965 In the example, the three specific gravities are as follows:.822 Bulk Specific ravity s = 2.652 Effective Specific ravity se = 2.726 Apparent Specific ravity sa = 2.822 3.4. Maximum Specific ravities of Mixtures with Different Asphalt Contents: In designing a paving mixture with a given aggregate, the maximum specific gravities, mm, at different asphalt contents are needed to calculate the percentage of air voids for each asphalt content. While the same maximum specific gravity can e determined for each asphalt content y KT-39, the precision of the test is est when the mixture has close to the optimum asphalt content. Also, it is preferale to measure the maximum specific gravity in duplicate or triplicate. After averaging the results from these tests and calculating the effective specific gravity of the aggregate, the maximum specific gravity for any other asphalt content can e otained. For all practical purposes, the effective specific gravity of the aggregate is constant ecause the asphalt asorption does not vary apprecialy with variations in asphalt content. mm Pmm Ps P se (Eq. 3, Figure 1) mm = maximum specific gravity of paving mixture (no air voids) P mm = total loose mixture = % P s = aggregate, percent y total weight of mixture (P 1 + P 2 + P 3 + P n ) P = asphalt, percent y total weight of mixture. se = effective specific gravity of aggregate. = specific gravity of asphalt. Calculation using specific gravity data from Tale 1., effective specific gravity, se, determined in section 3.3. and an asphalt content, P of 6.96.% Page 4/14 5.10.4 04-12
mm 9304. 6960. 2. 726 1010. 2. 438 41022. 3.5. Asphalt Asorption: Asorption is expressed as a percentage y weight of aggregate rather than as a percentage y total weight of mixture. Asphalt, P a, asorption is determined as follows: se s P a = ( - ) ( s x se ) (Eq. 4, Figure 1) P a = asored asphalt, percent y weight of aggregate. se = effective specific gravity of aggregate. s = ulk specific gravity of aggregate. = specific gravity of asphalt. Calculation using ulk and effective gravities determined in sections 3.2. and 3.3. and asphalt specific gravity from Tale 1. P a = x (2.726-2.652 ) (2.652 x 2.726) x 1.010 = x 0.074 7.229 x 1.010 = 1.03 3.6. Effective Asphalt Content of a Paving Mixture: The effective asphalt content, P e, of a paving mixture is the total asphalt content minus the quantity of asphalt lost y asorption into the aggregate particles. It is the portion of the total asphalt content that remains as a coating on the outside of the aggregate particles, and is the asphalt content on which service performance of an asphalt paving mixture depends. The formula is: a P e = P - ( P s ) P (Eq. 5, Figure 1) P e = effective asphalt content, percent y total weight of mixture. P = asphalt, percent y total weight of mixture. P a = asored asphalt, percent y weight of aggregate. P s = aggregate, percent y total weight of mixture. Calculation using data from Tale 1 and section 3.5. P e = 6.96 - ( 1.03 ) X 93.04 = 6.96-0.9 = 6 6. 00 3.7. Percent VMA in Compacted Paving Mixture: The voids in the mineral aggregate, VMA, are defined as the intergranular void space etween the aggregate particles in a compacted paving mixture, that includes the air voids and the effective asphalt content, expressed as a percent of the total volume. The VMA is calculated on the asis of the ulk specific gravity of the aggregate and is expressed as a percentage of the ulk volume of the compacted paving mixture. Therefore, the VMA can e calculated y sutracting the volume of the aggregate determined y its ulk specific gravity from the ulk volume of the compacted paving mixture. The method of calculation is illustrated as follows: Page 5/14 5.10.4 04-12
If mix composition is determined as percent y weight of the total mixture: VMA = - m Ps s (Eq. 6, Figure 1) VMA = voids in mineral aggregate (percent of ulk vol.) s = ulk specific gravity of aggregate. m = ulk specific gravity of compacted mixture. (KT-15) P s = aggregate, percent y total dry weight of mixture. Calculation using data from Tale 1 and section 3.2. VMA = - (2.344 x 93.04) 2.652 = - 82.23 = 17.77 Calculation using data from Tale 1 and section 3.2. P s = [ + P ] x P' = asphalt, percent y dry weight of aggregate. VMA = - [ 2.344 2.651 x + 7.48 ()] = - 82.27 = 17.73 3.8. Calculation of Percent Air Voids in Compacted Mixture: The air voids, P a, in a compacted paving mixture consist of the small air spaces etween the coated aggregate particles. The percentage of air voids in a compacted mixture can e determined y the following equation: mm m P a = [ - mm ] (Eq. 7, Figure 1) P a = air voids in compacted mixture, percent of total volume. mm = maximum specific gravity of paving mixture (as determined in section 3.4. or as determined directly for a paving mixture y KT-39) m = ulk specific gravity of compacted mixture. Calculation using data from Tale 1. P a = 2.438-2.344 2.438 = 9.400 2.438 = 3.85 Page 6/14 5.10.4 04-12
3.9. Calculation of Percent Voids Filled with Asphalt: The voids filled with asphalt, VFA, in a compacted paving mixture consists of that portion of the initial voids in the aggregate, VMA, which has een filled with the non-asored, or effective asphalt. The percentage of voids filled with asphalt in a compacted specimen can e determined y the following equation: P e VFA VMA m or (Eq. 8, Figure 1) VMA P VFA VMA a VMA = voids in mineral aggregate, percent of ulk vol. P e = effective asphalt content. = specific gravity of asphalt. m = ulk specific gravity of compacted mixture. VFA = x ( 5.98 1.01 ) ( 17.73 2.344 ) = x 5.921 7.564 = 78.3% or VFA = 17.73-3.85 17.73 = 13.88 = 78.3% 17.73 Page 7/14 5.10.4 04-12
WORKSHEET FOR ANALYSIS OF COMPACTED PAVIN MIXTURE Contract #: Date: Project Numer: La No.: Eq. % AC y Wt. Aggr. 7.48 P % AC y Wt. Mix 6.96 P =A % Aggr. y Wt. Mix 93.04 P s =B S.. of Asphalt 1.010 =C S.. of Aggr. (ulk) 2.652 1 s =D Max. S.. Mix (KT-39) 2.438 3 mm (computed) E Theo. Max. K/M 3 F (SI)=F Eff. S.. of Aggr. 2.726 2 se = % As. Asphalt 1.05 4 P a =H Eff. Asph. Content 5.98 5 P e =I % V. M. A. 17.73 6 VMA=J S.. of plugs (KT-15) 2.344 m =K La Plugs K/M 3 L (SI)=L % Air Voids 3.85 7 P a =M % Voids Filled 78.3 8 VFA=N P Max O Eff. Film Thickness P Filler/Binder Ratio Q P * COMBINED AREATE SP. R. P = A = APPARENT BULK % + P +4.75 mm 2.759 2.606 55.30-4.75 mm 2.905 2.711 44.70 P s = B = - A Theo. Com. 2.822 F = 0E B se = = A - E C L = 0K - D P = M = * P a = H = * * C E D * I H * P e = I = A - * B C VFA = N= J K * B K VMA = J = - D E - K Figure 1. SI Example Page 8/14 5.10.4 04-12
WORKSHEET FOR ANALYSIS OF COMPACTED PAVIN MIXTURE Contract #: Date: Project Numer: La No.: Eq. % AC y Wt. Aggr. 7.48 P % AC y Wt. Mix 6.96 P =A % Aggr. y Wt. Mix 93.04 P s =B S.. of Asphalt 1.010 =C S.. of Aggr. (ulk) 2.652 1 s =D Max. S.. Mix (KT-39) 2.438 3 mm (computed) E Theo. Max. K/M 3 F (SI)=F Eff. S.. of Aggr. 2.726 2 se = % As. Asphalt 1.05 4 P a =H Eff. Asph. Content 5.98 5 P e =I % V. M. A. 17.73 6 VMA=J S.. of plugs (KT-15) 2.344 m =K La Plugs K/M 3 L (SI)=L % Air Voids 3.85 7 P a =M % Voids Filled 78.3 8 VFA=N P Max O Eff. Film Thickness P Filler/Binder Ratio Q P * COMBINED AREATE SP. R. P = A = APPARENT BULK % + P +4 2.759 2.606 55.30-4 2.905 2.711 44.70 P s = B = - A Theo. Com. 2.822 F = E * 62.4 B se = = L = K * 62.4 A - E C - D P = M = * P a = H = * * C E D * I H * P e = I = A - * B C VFA = N= J K * B K VMA = J = - D E - K Figure 1. English Example Page 9/14 5.10.4 04-12
4. Preliminary Calculations for Recycled Mixtures: When designing recycled mixtures, certain calculations must e made to otain the correct values for P ', and s prior to entering the "Worksheet for Analysis of Compacted Paving Mixtures" (Figure 1). The following sequence of equations may e followed to otain the needed values. 4.1. Reclaimed Roadway (RAP). iven: P BRAP = Percent asphalt in RAP from extractions = 6.2 P BN = Percent added asphalt or recycling agent. AC-RAP = Specific gravity of reclaimed asphalt = 1.025 BN = Specific gravity of added asphalt or recycling agent = 1.008 RA = Specific gravity of reclaimed agg. = 2.522 SBN = Specific gravity of new aggregates = 2.651 P RAP = Percent reclaimed in aggregate lend = 40% P VA = Percent new aggregate in lend = 60% 4.1.1. % asphalt in RAP (Wet Weight Basis) P BRAP BRAP (1 + P = ) 6.2 1 + ( 6.2 ) = 5.84 4.1.2. % aggregate in RAP (Wet Weight Basis) - a. = - 5.84 = 94.16 4.1.3. Weight of asphalt in RAP portion/ g of aggregate lend (a. )( P RAP ) = (5.84)(40) = 2.34 4.1.4. Weight of aggregate in RAP portion/ g of aggregate lend (. )( P RAP ) = (94.16)(40) = 37.66 4.1.5. Weight of reclaimed material/ g of aggregate lend c. + d. = 2.34 + 37.66 = 40.00 4.1.6. Weight of new aggregate/ g of aggregate lend - e. = - 40.00 = 60.00 Page 10/14 5.10.4 04-12
COMBINED MATERIALS % % % % g. % additive - Dry wt. asis 2.0 2.5 3.0 3.5 h. Additive wt./ g sample = (P BN ) 2.0 2.5 3.0 3.5 c. Asphalt wt. in RAP/ g sample 2.34 2.34 2.34 2.34 i. Revised wt of itumen (c. + h.) 4.34 4.84 5.34 5.84 d. Aggr. wt. in RAP/ g sample 37.66 37.66 37.66 37.66 f. New aggr. wt./ g sample 60.00 60.00 60.00 60.00 j. Revised wt. of aggr. = (d + f) 97.66 97.66 97.66 97.66 k. Total weight = (i + j) 102.0 102.5 103.0 103.5 l. % asphlt dry wt of aggr (i/j)() 4.44 4.96 5.47 5.98 Enter as P ' on Fig. 1 worksheet m. Specific gravity - comined ituminous material 1.017 1.016 1.015 1.015 i h c + BN AC-RAP Enter as on Fig. 1 worksheet n. Specific gravity - Comined aggregates (Bulk Dry) 2.600 2.600 2.600 2.600 d RA j + f SBN Enter as SB on Fig. 1 worksheet 5. Supplemental Calculations (Dry Aggregate Basis): 5.1. Calculation of Film Thickness: The film thickness is a calculated or theoretical value and does not denote a measurale property of the mix. It is, however, useful in the analysis of ituminous mixtures when evaluating the effect of gradation changes. Higher film thickness is generally desirale (within practical considerations) since thicker films are etter ale to withstand asphalt hardening. Film thickness is calculated for other purposes such as slurry seal design. Convert the gradation (percent retained) to percent passing and enter into a worksheet (similar to the tale elow). Then complete the calculations for Surface Area (S.A.). Page 11/14 5.10.4 04-12
DETERMINATION OF SURFACE AREA Sieve Size Percent x S.A. Factor # = Surface Area Passing m 2 /kg (ft 2 /l) m 2 /kg (ft 2 /l) 19.0 mm (3/4 in.) * 0.0041(0.02) 0.410(2.0) 12.5 mm (1/2 in.) 93 * 9.5 mm (3/8 in.) 81 * 4.75 mm (No. 4) 65 0.0041(0.02) 0.266(1.30) 2.36 mm (No. 8) 48 0.0082(0.04) 0.393(1.92) 1.18 mm (No. 16) 38 0.0164(0.08) 0.623(3.04) 600 µm (No. 30) 27 0.0287(0.14) 0.774(3.78) 300 µm (No. 50) 13 0.0614(0.30) 0.799(3.90) 150 µm (No. ) 8 0.1229(0.60) 0.983(4.80) 75 µm (No. 200) 6.8 0.3277(1.60) 2.228(10.88) Total Surface Area = 6.476 m 2 /kg (31.62 ft 2 /l) #Surface area factors shown are applicale only when all the aove-listed sieves are used in the sieve analysis. *Surface area factor is 0.41 m 2 /kg (2 ft 2 /l) for any material retained aove the 4.75 mm (No. 4) sieve. Thus, the surface area equals 0.410 (2.0) for all material aove the 4.75 mm (No. 4) sieve. 5.2. Calculation of Bitumen Index: The itumen index is the ratio of effective asphalt content to the aggregate surface area. SI: English: SI Example : English Example : P e = effective asphalt content, see section 3.6. 5.3. Conversion of Bitumen Index to Film Thickness (Microns): Film Thickness (FT) = Bitumen Index x 4.87 = 1.891 x 4.87 = 9.2 Microns 6. Maximum Percent of Asphalt: Determine the maximum percentage of asphalt (P ' max) which can e placed in a mix and still maintain a safe level of air voids under conditions of post-construction traffic and elevated mat temperature. 6.1. The maximum volume of asphalt which a mix can contain is the volume of the voids in mineral aggregate (VMA) plus the volume of asphalt asored into the aggregate. The volume of the asored asphalt (VP a ) may e determined y the following formula: Page 12/14 5.10.4 04-12
a s VP a = P x = 1.05 x 2.651 1.01 = 2.76 P a = asored asphalt, section 3.5. s = ulk dry specific gravity of the total aggregate, section 3.2.1. = apparent specific gravity of asphalt, section 3.1.1. 6.2. Total Safe Volume at 74 o C (165 o F.) represents the availale volume minus 2% to allow for plant fluctuation. The Total Safe Volume (TSV) is determined y the following equation: TSV = VMA + VP - 2.0 a = 17.73 + 2.76-2.0 = 18.49 VMA = voids in mineral aggregates, section 3.7. 6.3. Conversion of TSV to percentage of asphalt, (P '), is accomplished in the following steps: P = TSV( ) (0.9638) max s = 18.49 x 1.01 x 0.9638 2.651 = 6.79% P = P max( + P max ) max = 6.79(106.79) = 7.25% P max = maximum percent asphalt y weight of total mix P 'max = maximum percent asphalt y weight of dry aggregate Page 13/14 5.10.4 04-12
7. Compute Bearing Capacity, kpa(psi), as follows: Bearing Capacity, kpa = SI: 1.55 Staility ( Flow Staility in Newtons(N) Flow in 0.25mm SI Example : 120 - Flow ) 1.55(7784) 120-12 Bearing Capacity, kpa= [ ] =1086kPa 12 English Example: Bearing Capacity, psi = Staility 120 - Flow x Flow = 1750 120-12 x = 157psi 12 8. Compute Filler - Binder Ratio as follows : Percent of -75m (-No.200) fraction (see susection (f)(1)) F / B = Percent of asphalt (dry weight asis)(see susection (c)(1)) = 6.8 7.48 = 0.91 Page 14/14 5.10.4 04-12