第四章高等混凝土结构 长安大学 硕士课程

1、Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-1,第四章,受弯截面弹塑性全过程分析,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-2,4.1,弹性分析,b,0,h,h,曲率,y,s,x,0,h,x,M,S,S,A,1,2,s,E,A,截面,应变,应力,换算截面,图,4-1,第阶段计算图式,4.1.1,第阶段

2、（开裂前,计算图式,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-3,一）基本方程,1,平衡条件,0,0,X,M,0,0,0,0,0,0,0,x,h,x,s,s,x,h,x,s,s,bdy,bdy,A,M,y,bdy,y,bdy,A,h,x,4-1,4-2,2,变形条件,0,s,y,h,x,4-3,4-4,根据平截面假定,Department of Bridge Engineering,Changan University,30 Oct, 2006,

3、Advanced Concrete Structures,A-4,s,s,s,E,E,3,物理条件,4-5,4-6,6,个独立方程,13,个参数，即,如果已知其中的任意,7,个，便可解出其余,6,个未知数,0,s,s,s,s,b,h,h,x,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-5,二）中性轴位置,将式,4-3,4-4,分别代入式,4-5,4-6,得,再将式,4-7,4-8,代入式,4-1,得,0,s,s,E,y,E,h,x,4-7,4-8,

4、0,0,0,0,S,x,h,x,s,ybdy,ybdy,h,x,A,即,0,0,0,0,S,x,h,x,y,bdy,y,bdy,A,h,x,4-9,由式,4-9,即可求得,x,的值，也即确定了中性轴的位置,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-6,特别地,对于矩形截面,b,为常数，则由式,4-9,可得,2,2,0,0,0,0,2,2,S,x,h,x,y,y,b,b,A,h,x,2,0,1,2,S,S,bh,A,h,x,bh,A,2,0,0,0

5、,1,2,h,h,x,h,h,h,即,4-10,4-10a,式中,0,s,A,bh,纵向配筋率,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-7,式,4-10a,说明,在第阶段，中性轴位置与,M,无关,且为常数,值略大于而接近于,0.5,中性轴偏于受拉区,若为纯砼截面，则,0,0.5,x,h,三）换算截面,3,0,3,2,0,0,3,3,s,s,A,bh,A,bx,b,h,x,A,h,x,4-12,4-13,其中,x,按,4-10,式确定,0,x,h

6、,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-8,四）弯矩曲率关系,即内力与变形的关系,将式,4-7,式,4-8,代入,4-2,式得,2,2,2,0,0,0,0,x,h,x,s,M,y,bdy,y,bdy,A,h,x,4-14,曲率,的单位是,或者,且,为曲率,半径,1,m,1,mm,1,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concret

7、e Structures,A-9,五）抗弯刚度,定义式,将,4-14,式代入上式得,0,0,B,的单位为,或,2,N,mm,2,KN,m,4-15,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-10,六）弯矩与应力的关系,将式,4-14,变形得,的表达式，再代入,4-7,式得,0,0,y,y,y,4-16,由物理条件，变形条件及,4-14,式得,0,0,0,0,0,s,s,s,h,x,h,x,h,x,4-17,Department of Bridge

8、 Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-11,4.1.2,第阶段末（即将开裂,在第阶段末，截面即将开裂，也就是说，受拉边缘的,砼拉应力,达到其抗拉强度,裂缝即将出现，此时的,弯矩称为,RC,开裂弯矩，记为,可由式,4-16,确定,2,ot,cr,0,2,0,0,cr,cr,ot,y,h,x,则,0,0,cr,ot,cr,h,x,4-18,其中,由,4-10,式确定,cr,x,Department of Bridge Engineering,Changan University,30

9、Oct, 2006,Advanced Concrete Structures,A-12,4.1.3,第阶段（开裂后,计算图式与第阶段相比有所变化,受拉区砼退出工作,b,0,h,h,y,s,截面,应变,应力,图,4-2,第阶段计算图式,s,A,s,E,A,换算截面,x,0,h,x,M,S,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-13,一）基本方程,1,平衡条件,0,0,0,0,0,x,x,s,s,s,s,bdy,A,M,y,bdy,A,h,x,0

10、,0,X,M,4-19,4-20,2,变形条件,0,s,y,h,x,4-3,4-4,3,物理条件,s,s,s,E,E,4-5,4-6,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-14,二）中性轴的位置,将变形条件，物理条件代入式,4-19,得,0,0,0,S,x,y,bdy,A,h,x,4-21,由式,4-21,即可求得中性轴的位置,x,2,0,0,0,2,S,x,y,b,A,h,x,2,0,1,0,2,s,bx,h,x,可知,x,仍为常数，与,M

11、,无关,由式,4-22,求得的,x,远小于式,4-10a,求得的,x,说明受拉区砼,开裂时，中性轴突然上升，往受压区移动，使得,x,减小，这是一个,非常重要的物理现象,对于矩形截面,b,为常数，则由式,4-21,可得,4-22,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-15,三）换算截面,0,2,2,0,0,0,s,x,s,b,x,y,bdy,h,x,4-23,4-24,对于矩形截面,b,为常数，则,3,2,0,0,3,s,bx,h,x,4-24

12、a,其中,x,按,4-22,确定,四,s,与前式,4-14,4-15,4-16,4-17,相同,只不过式中,I,0,应按式,4-24,计算,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-16,4.1.4,第阶段末（纵筋屈服,4,17,s,y,由于,由式,得,0,0,0,0,y,y,s,y,h,x,h,x,0,0,y,y,y,h,x,4-25,其中,按式,4-21,计算,y,x,Department of Bridge Engineering,Chan

13、gan University,30 Oct, 2006,Advanced Concrete Structures,A-17,4.2,受弯截面弹塑性分析,平衡条件，变形条件与弹性分析时相同，物理,关系不用虎克定律的线弹性模型，而采用与实验结,果较为吻合的本构关系，即砼本构关系（拉、压,采用,非线性模式,二次抛物线,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-18,4.2.1,第阶段（开裂前,计算图式,b,0,h,h,y,s,x,0,h,x,M,S,S

14、,A,截面,应变分布,应力分布,换算截面,图,4-3,弹塑性分析第阶段的计算图式,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-19,一）基本方程,1,平衡条件,0,0,X,M,0,0,0,0,0,0,0,x,h,x,x,h,x,s,s,s,s,bdy,bdy,A,M,y,bdy,y,bdy,A,h,x,4-1,4-2,2,变形条件,0,s,y,h,x,4-3,4-4,Department of Bridge Engineering,Changan

15、University,30 Oct, 2006,Advanced Concrete Structures,A-20,3,物理条件,A,B,0,0,0,u,2,0,0,0,2,a,砼受压,二次抛物线,4-26,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-21,A,B,0,0t,0,ut,2,2,ot,ot,ot,b,砼受拉,二次抛物线,4-27,Department of Bridge Engineering,Changan University,3

16、0 Oct, 2006,Advanced Concrete Structures,A-22,A,B,0,y,y,u,c,钢筋受拉（压）（理想弹塑性,s,s,s,s,y,s,y,y,s,u,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-23,二,中性轴位置,将变形（相容）条件代入物理条件得,2,0,0,0,2,y,y,2,2,ot,ot,ot,y,y,0,s,s,h,x,压区砼,拉区砼,拉区钢筋,4-28,4-29,4-30,Department of

17、 Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-24,再将式,4-28,4-29,4-30,代入式,4-1,得,2,2,0,0,0,0,0,0,0,2,2,x,h,x,ot,ot,ot,s,s,y,y,y,y,dy,bdy,h,x,A,3,3,2,2,0,0,0,0,3,3,0,ot,ot,ot,s,h,x,b,x,b,x,h,x,h,x,s,4-31,上式为,x,的一元三次方程，已知材料性能,几何尺寸,b,h,h,0,As,曲率,则可求解受压区高度,s,0,0,ot,ot,x

18、,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-25,三,关系,将变形条件代入物理条件，再代入式,4-2,得,2,2,0,0,0,0,0,2,0,2,2,x,h,x,ot,ot,ot,s,s,y,y,y,y,M,y,bdy,y,bdy,h,x,A,4,4,3,3,2,0,2,2,3,4,3,4,ot,o,ot,ot,o,o,s,s,h,x,b,x,b,x,h,x,h,x,4-32,式中,由式,4-31,确定,x,由式,4-32,可知,为,非线性,关系

19、,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-26,四）抗弯刚度,由,将式,4-32,代入该式得,4,4,3,3,2,0,2,2,3,4,3,4,ot,o,s,s,ot,ot,o,o,h,x,b,x,b,x,h,x,h,x,4-33,B,不再是常数，与,及,有关，且为,的,非线性,式,x,x,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concr

20、ete Structures,A-27,五）钢筋和砼的应力,按,4-28,30,进行计算,2,0,0,0,2,y,y,2,2,ot,ot,ot,y,y,0,s,s,h,x,压区砼,拉区砼,拉区钢筋,4-28,4-29,4-30,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-28,4.2.2,第阶段末（即将开裂,计算图式,b,0,h,h,s,图,4-4,弹塑性分析第阶段末的计算图式,s,A,0,cr,h,x,s,S,A,ot,c,r,h,x,cr,x,

21、cr,ut,ot,cr,M,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-29,1,曲率,ut,cr,cr,h,x,4-34,2,中性轴位置,将式,4-31,类比变形，用,代替式,4-31,中的,代替式,4-31,中的,得,cr,cr,x,x,3,3,2,2,0,3,3,0,cr,ut,ut,cr,ot,ut,ut,o,cr,cr,cr,cr,ot,cr,cr,ot,o,o,ut,s,cr,s,cr,h,x,b,x,b,x,h,x,h,x,h,x,h

22、,x,h,x,h,x,h,x,4-35,由式,4-35,可解得到,迭代解,x,的三次方程,cr,x,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-30,可由式,4-32,变形改写得到,用,代替,代替,得,cr,x,3,cr,cr,x,4,2,4,3,2,0,2,3,4,2,3,4,ut,ut,cr,o,cr,cr,cr,cr,o,o,cr,ot,ut,ut,cr,ot,cr,cr,ot,ut,s,cr,s,cr,b,x,x,h,x,h,x,h,x,b

23、,h,x,h,x,h,x,h,x,h,x,4-36,4,抗弯刚度,B,cr,cr,cr,4-37,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-31,4.2.3,第阶段（开裂后,不计中性轴以下受拉砼的作用,b,0,h,h,s,图,4-5,弹塑性分析第阶段计算图式,s,A,0,h,x,s,S,A,ot,y,x,M,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advan

24、ced Concrete Structures,A-32,一）基本方程,1,平衡条件,0,0,X,M,0,0,0,0,0,x,s,s,x,s,s,bdy,A,M,y,bdy,A,h,x,4-38,4-39,2,变形条件,0,s,y,h,x,4-3,4-4,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-33,3,物理条件,a,受压区砼,2,0,0,0,2,4,26,o,b,受拉钢筋,4,27,s,s,s,s,y,Department of Bridge

25、 Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-34,二）中性轴的位置,3,2,0,0,0,0,0,3,s,s,b,x,x,h,x,4,2,2,0,2,3,4,o,s,s,o,o,b,x,x,h,x,三,关系,4-40,4-41,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-35,b,0,h,h,y,图,4-6,弹塑性分析第阶段末计算图式,s,A,

26、0,y,h,x,y,s,A,y,y,y,x,y,M,曲三角,4.2.4,第阶段末（纵筋屈服,计算图式,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-36,1,曲率,y,y,y,h,x,2,中性轴位置,将式,4-31,中,用,代替得,x,y,x,3,2,0,0,0,0,0,0,3,y,y,y,y,y,s,y,y,b,x,x,h,x,h,x,4-43,4-42,Department of Bridge Engineering,Changan Univer

27、sity,30 Oct, 2006,Advanced Concrete Structures,A-37,3,弯矩,纵筋屈服弯矩,y,4,2,0,0,0,2,3,4,y,y,y,o,y,y,y,s,y,y,y,o,o,b,x,x,h,x,h,x,h,x,4,抗弯刚度,y,y,y,4-44,4-45,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-38,4.2.5,第阶段（纵筋屈服后,计算图式,b,0,h,h,s,y,图,4-7,弹塑性分析第阶段计算图式,

28、s,A,0,h,x,1,x,s,s,y,s,A,A,M,1,x,2,x,1,x,2,x,0,s,s,y,s,A,A,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-39,一）基本方程,1,平衡条件,0,0,X,M,0,0,0,0,0,x,y,s,x,y,s,bdy,A,M,y,bdy,A,h,x,4-38,4-39,2,变形条件,4-3,纵筋屈服，只有砼,y,Department of Bridge Engineering,Changan Univer

29、sity,30 Oct, 2006,Advanced Concrete Structures,A-40,3,物理条件（砼受压,a,受压区砼,2,0,0,0,2,b,受拉钢筋,s,y,4-46,4-47,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-41,二）中性轴位置,3,2,0,3,o,y,s,o,o,b,x,x,4-48,求解,x,的一个一元三次方程,4,2,0,2,3,4,o,o,o,y,s,b,x,x,h,x,三,关系,4-49,Depart

30、ment of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-42,4.2.5,第阶段末（砼压坏,计算图式,b,0,h,h,y,图,4-8,弹塑性分析第阶段末计算图式,s,A,0,u,h,x,y,s,A,p,u,y,u,x,0,0,u,M,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-43,1,曲率,u,u,u,h,x,2,中性轴位置

31、,4-50,0,0,0,1,1,3,u,y,s,u,u,o,o,x,h,bh,4-51,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-44,3,截面破坏时所承受的弯矩,2,0,2,3,4,u,o,u,u,u,y,s,u,o,o,b,x,h,x,4-52,将式,4-51,代入,4-52,，并整理得,0,0,2,0,0,0,0,1,1,4,3,1,1,3,u,y,s,u,y,s,u,u,h,bh,4-52a,Department of Bridge En

32、gineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-45,3,截面破坏时所承受的弯矩,如取,0.002,0.0033,代入,4-51,4-52,得,o,u,0,1.347,u,y,o,x,h,4-51b,0,0,1,0.586,y,u,y,s,h,4-52b,4,刚度,u,u,u,4-53,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-46,u,y,发展,4

33、.3,受弯截面的延性,概念,截面的延性是指从纵筋屈服直到砼压坏，截面的变,形能力,定义式,即受弯截面的延性等于第阶段末与第,阶段末的曲率的比值，为一无量纲的数,讨论,延性大就是塑性大，延性小表示脆性大，延性在,超静定结构的内力重分布及结构的抗震设计中有重要的意,义。（抗震的延性设计,倒塌分析,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-47,4.4,受弯截面的塑性分析,图,4-9,计算图式,b,0,h,h,s,A,y,s,A,u,M,X,X,u,a

34、,R,0,h,计算图式,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-48,1,物理条件,s,y,u,受拉区钢筋,受压区砼,4.4.1,基本公式,2,平衡条件,0,0,X,M,0,0,2,u,y,s,u,u,y,s,b,x,A,x,M,b,x,A,h,x,4,54,4,55,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Struct

35、ures,A-49,y,s,u,x,0,y,u,x,h,将式,4-56,代入式,4-55,得,0,1,0.5,y,u,y,s,u,M,A,h,3,中性轴位置,由式,4-54,得,或,4-56,4-56a,4-57,Department of Bridge Engineering,Changan University,30 Oct, 2006,Advanced Concrete Structures,A-50,0,0,0,cr,ot,ot,cr,W,h,x,2,0,1,2,s,cr,s,bh,h,x,bh,3,3,3,0,0,3,3,cr,cr,s,cr,bx,b,h,x,h,x,4-10,4-13,4-18,4.5,塑性系数法（针对即将开裂截面,该法为另一种即将开裂时的简化计算方法,方法原理,对受弯截面作弹性分析，然后乘以塑性系数,r,m,即得,开裂弯矩,M,cr,公式,按弹性分析得,式中,x,cr,和,I,0,分别按式,4-10,4-13,计算，则有,Department of Bridge Engine