ASPIRE is a quarterly magazine published by PCI in cooperation with the associations of the National Concrete Bridge Council. The editorial content focuses on the latest technology and key issues in the Concrete Bridge Industry.
Issue link: http://www.aspiremagazinebyengineers.com/i/296646
EDITOR'S NOTE If you would like to have a specific provision of the AASHTO LRFD Bridge Design Specifications explained in this series of articles, please contact us at www .aspirebridge.org. 44 | ASPIRE , Winter 2014 A A S H T O L R F D by Dr. Dennis R. Mertz Sectional Design Model: Maximum or Concurrent Force Effects? S ince the discussion in the Fall 2013 issue of ASPIRE TM regarding how cracked concrete carries shear, readers raised questions about the application of several equations in Article 5.8.3, Sectional Design Model, of the AASHTO LRFD Bridge Design Specifications. In several equations, the variables M u and V u appear along with N u and in one case T u . Are these concurrent values, maximum values, or what? Applying the modified-compression field theory model, the net longitudinal tensile strain in the section at the centroid of the tension reinforcement, ɛ s , is estimated using LRFD Specifications Equation 22.214.171.124.2-4, shown below. This strain is then used to determine β, the factor indicating the ability of diagonally cracked concrete to transmit torsion and shear, and θ, the angle of inclination of the diagonal compressive stresses. The variables, β and θ, and their influence on shear resistance, are discussed in the Fall 2013 issue. Equation 126.96.36.199.2-4 includes M u and V u along with N u . Fo r a l l t h e s e c t i o n d e s i g n m o d e l s , i n d e t e r m i n i n g t h e r e q u i r e d l o n g i t u d i n a l reinforcement due to the interaction of the force effects, the variables M u and V u appear again along with N u and in one case T u . This interaction is illustrated in the free-body diagram, adapted from the LRFD Specifications, shown above. Summing moments about point 0 in the figure as detailed in LRFD Specifications Article C188.8.131.52, yields Equation 184.108.40.206-1, shown below, as used for sections not subject to torsion. For sections subject to combined shear and torsion, Equation 220.127.116.11.3-1, shown below, is used. The three equations discussed here represent a single point in time, as shown in the free- body diagram. Thus, the force effects should be concurrent values due to a common load condition. In general, when checking shear, the maximum shear is used with the other c o n c u r r e n t fo r c e e f fe c t s ; w h e n c h e c k i n g moment, the maximum moment is used with the other concurrent force effects. Theoretically, at each section, each of the maximum force effect s should be checked with it s other concurrent forces though this is not typically done. Equation 18.104.22.168.2-4, for ɛ s , is applied at sections along the beam (typically 1/10 th points) to determine the shear resistance of each section. For each section, the maximum shear and the other concurrent force effects at that section are used in the equation. Equations 22.214.171.124-1 and 126.96.36.199.3-1, for the required longitudinal reinforcement, are again applied for sections along the beam to determine the required reinforcement at each section. Again, for each section, the maximum shear and the other concurrent force effects at that section determine the required longitudinal reinforcement. The various strength limit-state force effects (in other words, the sum of the factored force effects from the governing strength limit- state load combination), M u , V u , N u , and T u , in the LRFD equations, do not represent the maximums for the section due to different load conditions. They represent the maximum force effect under consideration along with the other concurrent values for the single governing load condition at each section. ɛ s POINT 0 0.5 0.5 c cot d v d v V s V u N u T V p θ θ cot d v AspireBook_Win14.indb 44 12/10/13 12:54 PM