Cast-in-Place Concrete Connections for Precast Deck Systems (2011)

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H-i Appendix H Subassemblage Sectional Calculations and Analyses

H-ii H.1: Subassemblage Specimen Design Calculations and Analyses The subassemblage specimens utilized during the current study were primarily considered to provide a comparison of the benefits of particular crack control details relative to one another. To provide an understanding of the expected behavior of each specimen based on its individual measured material properties, geometry, and reinforcement details, a detailed analytical investigation was completed. The relevant calculations are outlined below.

H-1 Sectional calculations for subassemblage specimens Specimen dimensions and variables are defined in plan and elevation views of the subassemblage specimens in Figures H.1 and H.2, respectively. As shown in Figure H.1, each subassemblage specimen was 120 in. “long”, which corresponded with the span length during the load tests while the width of each specimen, which was measured parallel to the longitudinal precast joint, ranged from 62.75 in. to 67.25 in. The origin of each specimen, for reference regarding the instrumentation and reinforcement placement, as well as general documentation, was located on the bottom of the section, directly at the precast joint. The origin corresponded with the end of each specimen nearest the embedded transverse reinforcement that was spaced a clear distance of 3 in. from the face, which was consistent among each of the seven subassemblage specimens, and is also shown in Figure H.2. As illustrated in Figure H.2, the total section depth is composed of the precast member depth, which was either 12 in. or 16 in., plus a 2 in. thick deck for all specimens other than SSMBLG3-HighBars, which had a 2 in. thick deck near the joint region, and tapered to 3-1/2 in. a distance of 15 in. in either direction from the joint. Figure H.1: Plan view of subassemblage specimens

H-2 (a) Subassemblage section view perpendicular to joint (b) Subassemblage section view parallel to joint Figure H.2: Subassemblage specimen section views

H-3 Note: The index number in each array represents the subassemblage specimen number. For example, subassemblage 1 will always be represented as the first entry in any array variable Useful units: kip 1000lbf:= ksi kip in 2 := Specimen range variable: i 1 7..:= Specimen naming includes specimen number - description - maximum distance between perpendicular reinforcement immediately above the precast joint Specimens i "1-Control1-9in." "2-NoCage-18in." "3-HighBars-9in." "4-Deep-9in." "5-No.6Bars-9in." "6-Frosch-4.5in." "7-Control2-9in." := Material Properties Measured material properties have been documented herein. Values were measured on the first day of laboratory testing for each specimen CIP Elastic modulus: tensile strength via 3 point beam test: concrete compressive strength: EcSSi 4740ksi 5633ksi 3933ksi 4703ksi 4112ksi 5270ksi 4194ksi ft i .840ksi .746ksi .678ksi .757ksi .609ksi .806ksi .732ksi : fci 6552psi 6577psi 4726psi 7151psi 5204psi 6898psi 7005psi := Steel elastic modulus: Es 29000ksi:=

H-4 Specimen and reinforcement geometry and layout Ashook equals the area of a single perpendicular hooked bar equals the area of all horizontally oriented legs of the stirrups in the cage at a given level, i.e.: all the horizontal legs of the stirrups at the bottom of the trough Ascage bss equals nominal width, perpendicular to joint, of specimen hcip equals the nominal depth of CIP concrete between top of precast flange at the joint and the top of the specimen dshook equals the nominal depth of the hooks oriented perpendicular to joint, measured from top of specimen to center of reinforcement, also used as depth of lower horizontal leg of cage reinforcement dtoplegofcage equals the nominal depth of the center of the top horizontal leg of the cage reinforcement from the top of the specimen dsdeck equals the nominal depth of the center of the deck reinforcement oriented perpendicular to the joint from the top of the specimen Area of No. 3 mild reinforcement: Ashooki .2in 2 .2in 2 .2in 2 .2in 2 .44in 2 .2in 2 .2in 2 Ascagei .33in 2 0in 2 .33in 2 .33in 2 .33in 2 1.43in 2 .33in 2 A3 0.11in 2 := bss i 62.75in 67.25in 62.75in 62.75in 62.75in 64.0in 62.75in hcipi 11in 11in 11in 15in 11in 11in 11in dshooki 9.75in 9.75in 7in 13.75in 9.6875in 9.75in 9.75in dsdeck 1.5625in:= dtoplegofcage 2.25in:=

H-5 Location of uncracked and cracked neutral axis Method of transformed sections is utilized to calculate neutral axis (NA) location, the modular ratio is the ratio of the elastic modulus of the steel to the elastic modulus of the concrete n i Es EcSSi := n 6.1 5.1 7.4 6.2 7.1 5.5 6.9                   = Depth of neutral axis is measured from top surface of each specimen, as follows: Black box in upper right corner of equation indicates that the equation is not evaluated; only shown for illustration of method of calculation Atotal x  ⋅ i n 1−( )As ds⋅ Aconcrete dconcrete⋅+ ∑ Uncracked xuncracked i bss i hcipi     2 ⋅ 2 n i 1−( ) 8Ashooki Ascagei+  dshooki⋅ Ascagei dtoplegofcage⋅+ 5 A3⋅ dsdeck⋅+ ...      ⋅+ bss i hcipi ⋅ n i 1−( ) 8Ashooki 2Ascagei+ 5A3+ ⋅+ := xuncracked 5.54 5.53 5.5 7.54 5.61 5.54 5.54                   in⋅=

H-6 Cracked Neutral axis depth is calculated as above; the depth of the concrete in compression is equal to the neutral axis depth, therefore a quadratic equation must be solved. The coefficients of the quadratic equation are as follows: The cracked transformed section is shown in Figure H.3: Figure H.3: Cracked transformed section, with neutral axis, x, measured from top of section The area of the deck reinforcement should be subtracted from the total area of the section by multiplying by (n-1) if x is greater than dsdeck , but should not be subtracted if x less than dsdeck . Assume that the neutral axis is larger than the depth of the deck reinforcement: quadratic equation for neutral axis depth (x): Ax2 Bx+ C+ 0 A bss i 2 B n i 8Ashooki 2Ascagei +    ⋅ n i 1−( ) 5⋅ A3⋅+ C n i 8Ashooki Ascagei +    ⋅ dshooki ⋅ n i Ascagei ⋅ dtoplegofcage⋅+ ni 1−( ) 5⋅ A3⋅ dsdeck⋅+ −

H-7 The polyroots function is used to evaluate the parabolic function for each specimen. The polyroots function takes the coefficients A,B, and C and returns, for a parabolic function, two roots. Let the matrix Coeffs be the matrix of coefficients, with each column representing a subassemblage specimen: Coeffs M 0← M 3 i, bss i 2 ← M 2 i, ni 8Ashooki 2Ascagei +    ⋅ n i 1−( ) 5A3( )⋅+  1 in ⋅← M 1 i, ni 8Ashooki Ascagei +    dshooki ⋅    ⋅ n i Ascagei dtoplegofcage⋅   ⋅+ n i 1−( ) 5⋅ A3 dsdeck⋅+ ...    − 1 in 2 ⋅← i 1 7..∈for Mreturn := <-- constant term Coeffs 124.1− 16.6 31.4 83.9− 10.5 33.6 110.6− 20.2 31.4 172.7− 16.8 31.4 273.5− 32.8 31.4 184.1− 27 32 140.3− 18.9 31.4         in⋅= <-- linear term <-- parabolic term Evaluating the polyroots function returns two roots, the second root, denoted by the subscript 2 after the polyroots function, is the positive root for each case, which is saved to the variable xcracked xcracked i polyroots Coeffs i 〈 〉( )2 in⋅:= xcracked 1.74 1.43 1.58 2.09 2.48 2.01 1.84                   in⋅= Check assumption that the depth of the deck reinforcement is less than the depth of the neutral axis: NAcheck i "OK" xcracked i dsdeck>if "NG" xcracked i dsdeck<if := NAcheck "OK" "NG" "OK" "OK" "OK" "OK" "OK"                   = dsdeck 1.563 in⋅=

H-8 Recalculate the depth of the neutral axis with the area of the deck reinforcement not subtracted from the total area of the section when the deck reinforcement is below the neutral axis depth Coeffs M 0← M 3 i, bss i 2 ← M 2 i, ni 8Ashooki 2Ascagei +    ⋅ n i 5⋅ A3( ) xcracked i dsdeck<if n i 1−( ) 5⋅ A3⋅  xcracked i dsdeck>if +        1 in ⋅← M 1 i, ni 8Ashooki Ascagei +    dshooki ⋅    ⋅ n i Ascagei dtoplegofcage⋅   ⋅+ n i 5⋅ A3⋅ dsdeck⋅( ) xcracked i dsdeck<if n i 1−( ) 5⋅ A3⋅ dsdeck⋅  xcracked i dsdeck>if + ...            − 1 in 2 ⋅← i 1 7..∈for Mreturn := <-- constant term Coeffs 124.1− 16.6 31.4 84.7− 11.1 33.6 110.6− 20.2 31.4 172.7− 16.8 31.4 273.5− 32.8 31.4 184.1− 27 32 140.3− 18.9 31.4         in⋅= <-- linear term <-- parabolic term Evaluating the polyroots function returns two roots, the second root, denoted by the subscript 2 after the polyroots function, is the positive root for each case, which is saved to the variable xcracked xcracked i polyroots Coeffs i 〈 〉( )2 in⋅:= xcracked 1.74 1.43 1.58 2.09 2.48 2.01 1.84                   in⋅=

H-9 Moment of Inertia The moment of inertia of the section is the moment of inertia of the individual components about centroid of component plus area of component multiplied by the distance to centroid of section squared (parallel axis theorem). Assume reinforcement contributes only the portion defined by the parallel axis theorem: Isection components Icomponent∑ Acomponent distance2⋅+ Uncracked Iuncracked i 1 12 bss i hcipi     3 ⋅ bss i hcipi ⋅ hcipi 2 xuncracked i −       2 ⋅+ n i 1−( ) 8Ashooki dshooki xuncracked i−  2 Ascagei dshooki xuncracked i −    2 + Ascagei dtoplegofcage xuncracked i −    2 ⋅ 5 A3⋅ dsdeck xuncracked i −    2 ⋅++ ...        + ...:= Iuncracked 7199 7614 7064 18184 7433 7450 7236                   in 4 ⋅=

H-10 Cracked Icracked i 1 12 bss i xcracked i     3 bss i xcracked i ⋅ xcracked i 2 xcracked i −       2 ⋅+ n i 8Ashooki dshooki xcracked i −    2 ⋅ Ascagei dshooki xcracked i −    2 ⋅+ Ascagei dtoplegofcage xcracked i −    2 ⋅+ ...        ⋅+ ... n i 5⋅ A3⋅ dsdeck xcracked i −    2 ⋅    xcracked i dsdeck<if n i 1−( ) 5⋅ A3⋅ dsdeck xcracked i−  2 ⋅    xcracked i dsdeck>if + .. := Icracked 868 636 502 1810 1732 1173 966                   in 4 ⋅= Curvature Mcracki ft i Iuncracked i ⋅ hcipi xuncracked i − := Mcrack 1107 1038 870 1845 840 1099 970                   in kip⋅⋅= Flexural capacity of specimen. Because the transverse hooks are not continuous into both adjacent precast sections, consider only half of the transverse hooks and none of the reinforcing cage Calculate β1 for each specimen, β1 is a function of the concrete strength, rounded to the nearest thousand: rounded_fci round fci 1 psi ⋅ 3−,     := rounded_fci 7000 7000 5000 7000 5000 7000 7000                   =

H-11 β1i 0.85 rounded_fci 4000≤if 0.85 0.05 rounded_fci 4000−    1000 ⋅−       rounded_fci 4000>if 0.65 rounded_fci 8000>if := β1 0.70 0.70 0.80 0.70 0.80 0.70 0.70                   = Mflexurei 4Ashooki 60⋅ ksi dshooki β1i xcracked i ⋅ 2 −       ⋅:= Mflexure 439 444 306 625 918 434 437                   in kip⋅⋅= φ uncracked i Mcracki EcSSi Iuncracked i ⋅ := φ uncracked 0.000032 0.000024 0.000031 0.000022 0.000027 0.000028 0.000032                   1 in ⋅= The same moment is applied immediately before and after cracking: φ cracked i Mcracki EcSSi Icracked i ⋅ := φ cracked 0.000269 0.00029 0.000441 0.000217 0.000118 0.000178 0.00024                   1 in ⋅=

H-12 Compare results, normalized by the Control 1 specimen: xuncracked i xuncracked1 1.000 0.998 0.993 1.362 1.013 1.000 1.001                   = xcracked i xcracked1 1.000 0.822 0.909 1.203 1.422 1.157 1.054                   = Iuncracked i Iuncracked1 1.000 1.058 0.981 2.526 1.032 1.035 1.005                   = Icracked i Icracked1 1.000 0.732 0.578 2.084 1.995 1.351 1.112                   = φ uncracked i φ uncracked1 1.000 0.746 0.966 0.665 0.847 0.863 0.986                   = φ cracked i φ cracked1 1.000 1.077 1.641 0.806 0.438 0.661 0.891                   =

H-13 µε εsSW 1.881 10 3 × 1.995 10 3 × 3.198 10 3 × 1.633 10 3 × 1.128 10 3 × 1.105 10 3 × 2.134 10 3 ×                           = The yield strength of a representative piece of the mild steel reinforcement was measured in the laboratory to be approximately 70 ksi. Therefore the predicted steel stresses are below the yield and are assumed to be linear elastic. q .15 kip ft 3 := wi q bss i ⋅ hcipi 3in+    ⋅:= (3" is added for flange depth) Steel members used for clamping apparatus; 2 of each member were used in the setup L8x8x1-1/18 wL8 56.9 lbf ft := LL8 84in:= WL8 2 wL8⋅ LL8⋅ 796.6 lbf=:= W8x35 LW8x35 68.5in:= WW8x35 2 35⋅ lbf ft LW8x35⋅ 399.583 lbf=:= SW = self weight MSW 158.0 166.9 158.0 193.4 158.0 160.4 158.0                   in kip⋅⋅= MSWi w i Lspan 2 ⋅ 8 WL8 WW8x35+( ) Lspan⋅ 4 +:= Because the instrumentation utilized during the subassemblage specimens tests was zeroed at the start of each load step, the effects of self weight was not included in the relative strains measured with the embedment resistive gages. The predicted increase in the transverse stress in the hooked bars was calculated as shown below: εsSWi n i MSWi ⋅ dshooki xcracked i −    ⋅ EcSSi Icracked i 10 6 ⋅:=

H-14 The strain in the transverse hooks due to self weight and the weight of the clamping system was between 1105 and 3198µε. The largest transverse strain measured in the hooks was approximately 1455µε in SSMBLG1-Control1, in which case the state of strain near that gage was expected to be closer to 3336µε. The onset of strain hardening was measured to occur at a strain of approximately 6000µε, which suggested that strain hardening was not expected in the reinforcement at the locations that were instrumented, however because the locations of the strain gages on the steel bars were localized over the joint, and were unlikely to coincide with the location of cracking, the strain in the reinforcement at the location of the crack may have been larger.