Graphical methods in reinforced concrete design . eds 2by the tensile stress in the steel equals R bd . M = Resisting moment of beam or slab as determined 2by the compressive stress in concrete equals R bd c The formulas which are deduced in the followingdiscussion to express the conditions of stress in areinforced concrete beam under flexure, are based oncertain assumed qualities and methods of behavior ofthe component materials of the beam. These assump-tions are three in number, and are given below. num-bers 1 & 3 are almost universally accepted, but eventhey are not susceptible to rigid pr
Graphical methods in reinforced concrete design . eds 2by the tensile stress in the steel equals R bd . M = Resisting moment of beam or slab as determined 2by the compressive stress in concrete equals R bd c The formulas which are deduced in the followingdiscussion to express the conditions of stress in areinforced concrete beam under flexure, are based oncertain assumed qualities and methods of behavior ofthe component materials of the beam. These assump-tions are three in number, and are given below. num-bers 1 & 3 are almost universally accepted, but eventhey are not susceptible to rigid proof. Humber 2 isonly assumed as an approximation. Formulas based onan assumption more nearly approaching actual conditionswill be given later. (1). Plane sections before flexure remain plane sec-tions during flexure; or, in other words, the deforma-tion in the fibers is proportional to their distancefrom the neutral axis. (2). The stress in the concrete varies as the strain. (3). The concrete has no strength in tension. fl- /yeufr&//?x/s. ^ >> (5) The strain diagram, , shows the conditionsaccording to assumption (1). This condition is expressed £sa ^^5i t£ (2) ec ft $ r Fs &c Substituting this expression in Eq.(l)/h /-r> C) f?fc /itS —ft Eq.(3) gives the stress in the steel, fs, in terms of the stress in the concrete, ffw? £h £ x. /?£ C4) 7?(4) gives the position of the neutral axis in terms of the stresses fs and fc. Equating the horizontal stresses acting on the section, or £pj6c/=£, £& fa (&) If 0 Substituting for fs its value from Eq.(2) or g/7/of/-/i)=/f* II I, Solving the quadratic for Z /i= & or?-on*-or?. (6) Also from Eq.(5) Substituting for K from Eq.(4) <?. $f- 0) Eq.(6) gives the position of the neutral axis, K, in (6) terms of the percentage of steel. Eq.(8) gives the percent of steel in terms of the stress-es in the concrete and steel. // iThe lever arm of resisting couple, jd, equals the distance between the center
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