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1In this paper, a strut-and-tie model approach is presented for calculating the strength of 2 reinforced concrete pile caps. The proposed method employs constitutive laws for cracked 3 reinforced concrete and considers strain compatibility. This method is used to calculate the load 4 carrying capacity of 116 pile caps that have been tested to failure in structural research 5 laboratories. This method is illustrated to provide more accurate estimates of behavior and 6 capacity than the special provisions for slabs and footings of 1999 American Concrete Institute 7 (ACI) code, the pile cap provisions in the 2002 CRSI Design Handbook, and the strut-and-tie 8 model provisions in either 2005 ACI code or the 2004 Canadian Standards Association (CSA) 9 A23.3. The comparison shows that the proposed method consistently well predicts the strengths 10 of pile caps with shear span-to-depth ratios ranging from 0.49 to 1.8 and concrete strengths less 11 than 41 MPa. The proposed approach provides valuable insight into the design and behavior of 12 pile caps. 13

The paper presents the development of an adaptable strut-and-tie model that can be applied to the design or analysis of four-pile caps that support axial compression and biaxial flexure from a supported rectangular column. Due to an absence of relevant test data, the model is validated using non-linear finite element analyses (NLFEA). The results indicate that the use of the proposed model would lead to safe and economical designs. The proposed model can be easily extended to any number of piles, providing a rational procedure for the design of wide range of pile caps.

Our objective was to determine the effects of dry matter intake (DMI), body weight (BW), and diet characteristics on total tract digestibilities of dry matter, neutral detergent fiber, and starch (DMD, NDFD, and StarchD, respectively) in high-producing dairy cows. Our database was composed of 1,942 observations from 662 cows in 54 studies from Michigan, Ohio, and Georgia. On average, cows ate 23 ± 4.5 kg of dry matter/d, weighed 669 ± 79 kg, and produced 38 ± 10 kg of milk/d. Diets were 31 ± 5% neutral detergent fiber, 27 ± 6% starch, 2.6 ± 1.2% fatty acids, and 17 ± 1.4% crude protein. Digestibility means were 66 ± 6, 42 ± 11, and 93 ± 5% for DMD, NDFD, and StarchD, respectively. Forage sources included corn silage, alfalfa, and grasses. Corn source was classified by its ruminal fermentability. Data were analyzed using a mixed effects model, including diet chemical composition, forage source, and corn source, all expressed as percentage of dry matter, except for DMI, which was expressed as percentage of BW (DMI%BW); location and 2-way interactions were fixed effects. Cow, block, period, treatment, and study were included as random effects. Best fitting candidate models were generated using backward and stepwise regression methods. Additionally, the simplest model was generated using only DMI and location as fixed effects and all random effects. Candidate models were cross-validated across studies, and the resulting predictive correlation coefficients across studies (PC) and root mean square error of prediction (RMSEP) were compared by t-test. For each nutrient, the digestibility model that resulted in the highest PC and lowest RMSEP was determined to be the best fitting model. We observed heterogeneous coefficients among the different locations, suggesting that specific location factors influenced digestibilities. The overall location-averaged best fitting prediction equations were: DMD = 69 - 0.83 × DMI%BW (PC = 0.22, RMSEP = 5.39); NDFD = 53 + 0.26 × grass %DM - 0.59 × starch %DM + 3.06 × DMI%BW - 0.46 × DMI%BW (PC = 0.53, RMSEP = 9.70); and StarchD = 96 + 0.19 × HFERM%DM - 0.12 × starch %DM - 1.13 × DMI%BW (PC = 0.34, RMSEP = 4.77); where HFERM%DM is highly-fermentable corn source as percentage of DM. Our results confirm that digestibility is reduced as DMI increases, albeit at a lower rate than that reported in National Research Council. Furthermore, dietary starch depresses NDFD. Whereas DMD can be predicted based on DMI only, the best predictions for NDFD and StarchD require diet characteristics in addition to DMI.

Our objective was to model dry matter intake (DMI) by Holstein dairy cows based on milk energy (MilkE), body weight (BW), change in BW (ΔBW), body condition score (BCS), height, days in milk (DIM), and parity (primiparous and multiparous). Our database included 31,631 weekly observations on 2,791 cows enrolled in 52 studies from 8 states of the United States, mostly in the Upper Midwest. The means ± standard deviations of these variables were 24 ± 5 kg of DMI, 30 ± 6 Mcal of MilkE/d, 624 ± 83 kg of BW, 0.24 ± 1.50 kg of ΔBW/d, 3.0 ± 0.5 BCS, 149 ± 6 cm height, and 102 ± 45 DIM. Data analysis was performed using a mixed-effects model containing location, study within location, diet within study, and location and cow within study as random effects, whereas the fixed effects included the linear effects of the covariates described previously and all possible 2-way interactions between parity and the other covariates. A nonlinear (NLIN) mixed model analysis was developed using a 2-step approach for computational tractability. In the first step, we used a linear (LIN) model component of the NLIN model to predict DMI using only data from mid-lactation dairy cows (76-175 DIM) without including information on DIM. In the second step, a nonlinear adjustment for DIM using all data from 0 to 368 DIM was estimated. Additionally, this NLIN model was compared with an LIN model containing a fourthorder polynomial for DIM using data throughout the entire lactation (0-368 DIM) to assess the utility of an NLIN model for the prediction of DMI. In summary, a total of 8 candidate models were evaluated as follows: 4 ways to express energy required for maintenance (BW, BW 0.75 , BW adjusted for a BCS of 3, and BW 0.75 adjusted for a BCS of 3) × 2 modeling strategies (LIN vs. NLIN). The candidate models were compared using a 5-fold across-studies cross-validation approach repeated 20 times with the best-fitting model chosen as the proposed model. The metrics used for evaluation were the mean bias, slope bias, concordance correlation coefficient (CCC), and root mean squared error of prediction (RMSEP). The proposed prediction equation was DMI (kg/d) = [(3.7 + parity × 5.7) + 0.305 × MilkE (Mcal/d) + 0.022 × BW (kg) + (−0.689 + parity × −1.87) × BCS] × [1-(0.212 + parity × 0.136) × exp (−0.053 × DIM) ] (mean bias = 0.021 kg, slope bias = 0.059, CCC = 0.72, and RMSEP = 2.89 kg), where parity is equal to 1 if the animal is multiparous and 0 otherwise. Finally, the proposed model was compared against the Nutrient Requirements of Dairy Cattle (2001) prediction equation for DMI using an independent data set of 9,050 weekly observations on 1,804 Holstein cows. The proposed model had smaller mean bias and RMSEP and higher CCC than the Nutrient Requirements of Dairy Cattle equation to predict DMI and has potential to improve diet formulation for lactating dairy cows.