Basic Equations ya Centroidal line 80 0 (1)Geometric condition Eci=8o+yop Esi=Eo+ysio (2)Constitutive law Oci =0(Ec)=0(0 P,ya) Osi=Esi)=os(o,P,ysi) 目土本工程幸院
11 (1) Geometric condition ci y si y f ci e si e 0 e si ci Centroidal line e ci = e 0 + ycif e si = e 0 + ysif Basic Equations (2) Constitutive law ( ) ( , , ) ( ) ( , , ) s i s s i s s i ci c ci c ci y y e f e f = = = = 11
Basic Equations (3)Equilibrium equations N=∫o.(ee)dA.+∫o,(e,)dA,=∑oaA+∑onA M=∫o.(e.y.dA+∫o,(e,)y,dA,=∑oyaA+∑oyA Numerical iterative method 1.Given (N,)(N,M)or other conditions 2.From given conditions,iterate basic equations until equilibrium equations are satisfied Suitable for sections of any shape and reinforcement distribution under flexure,eccentric tension/compression,etc 土本工程学院 12
12 = + = + = + = + c c c c s s s s ci ci ci s i s i s i c c c s s s ci ci s i s i M y dA y dA y A y A N dA dA A A e e e e ( ) ( ) ( ) ( ) (3) Equilibrium equations Basic Equations Numerical iterative method 1. Given (N N M , , , or other conditions ) ( ) 2. From given conditions, iterate basic equations until equilibrium equations are satisfied Suitable for sections of any shape and reinforcement distribution under flexure, eccentric tension/compression, etc 12
Charles Witney's Research (1937) cu The compressive coefficients NA of the stress block are given for the following shapes. 000 b→ Asy k3 is ratio of maximum stress strain stress f in the compressive zone of 3 a beam to the cylinder strength f'(0.85 is a typical value for common concrete) k1.0.85 k1=05 k1▣0.67 3=0.425 3=0.333 k3=0.375 (b)Triangle (c)Parabola (a)Concrete 土本工程学院 13
13 13 Charles Witney’s Research (1937) The compressive coefficients of the stress block are given for the following shapes. k3 is ratio of maximum stress fc in the compressive zone of a beam to the cylinder strength fc ’ (0.85 is a typical value for common concrete)
Research of Blume,et al (1961) Computed flexural strength of beams is relatively insensitive to assumed maximum concrete strain but the ultimate curvature is on the contrary. 0.005 1.0 p0.02 0. 0.6 ompressive stresses 0.2 0.001 0.0020.0030.0040.005 0.006 0.007 Concrete strain at top edge ee 土本又程学院 Fin cforinly rinforcd conr bembdocom 14
14 Research of Blume, et al (1961) Computed flexural strength of beams is relatively insensitive to assumed maximum concrete strain but the ultimate curvature is on the contrary. 14
Nonrectangular compressed areas 1.0 =3000p20.7N/mm2 ●e=d at max moment 0.5 o c=0at max moment 0.001 0.00200.003 0.004 0.005 Concrete strains Fig.3.8.EIfect of section shape on the concrete strain at the extreme compression fiber at maximum moment. Mattock and Kriz(1961):flexural strength of beams with nonrectangular compressed areas can be estimated quite accurately using the stress block parameters and extreme fiber strain of rectangular compressed areas. 目土本工程学院 15
15 Nonrectangular compressed areas 15 Mattock and Kriz (1961): flexural strength of beams with nonrectangular compressed areas can be estimated quite accurately using the stress block parameters and extreme fiber strain of rectangular compressed areas