结构力学下多结点力矩分配法

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,多结点的力矩分配法,*基本理论为单结点理论,*反复使用单结点原理,*渐近收敛,单结点弯矩分配法,1,、附加刚臂,，写,固端弯矩,、求,约束力矩,R,P,2,、计算分配系数,3,、,R,P,反号分配,、,传递,4,、统计（合计）,、作图,分配与传递,50,50,固端弯矩,0,0,-45,0,合计,0,50,5,0,2m,4m,60,kN,55,kNm,A,B,C,2m,分配系数与约束力矩,0.5,-100,0.5,50,5,50,50,62.5,多结点力矩分配法源出于单结点，使用时保证放松的结点是单结点即可,例题,1,：用力矩分配法计算下列结构。,EI=,常数,2m,2m,2m,2m,4m,2m,8,kN,16,kN,8,kN/m,多结点的力矩分配,8,kN,16,kN,8,kN/m,16,kNm,解：把悬臂杆等效如下结构,8,kN,16,kN,8,kN/m,16,kNm,R,1P,R,2P,8,kN,16,kN,8,kN/m,16,kNm,R,1P,R,2P,-R,1P,-R,2P,-R,1P,-R,2P,+,-R,1P,R,-R,-R,2P,-R,1,R,1,=R,2P,+R,-R,1P,-R,1,-R,1,右结点可分配了,R,-R,+,把上述过程写在一起,就是如下的步骤,:,R,2P,右结点约束,反号分配左结点,左结点约束,右结点反号分配,此时该结点平衡了,R,2,此时该结点平衡了,此结点又不平衡了,此结点约束力有变化,R,1,=R+R,2P,-R,1P,-R,1,R,R,1P,R,2P,。,如此完成渐,近,过程,-R,2,右结点加约束,左结点反号分配,此时该结点平衡了,此结点约束力有变化,R,3,8,kN,16,kN,8,kN/m,16,kNm,分配系数,0.5,0.5,4/7,3/7,固端弯矩,-4,4,-8,8,-8,16,锁,C,，放,B,2,2,1,1,B,C,锁,B,，放,C,-4/7,-3/7,-2/7,0,锁,C,，放,B,1/7,1/7,1/14,1/14,锁,B,，放,C,-4/98,-3/98,0,-41/14,合计,43/7,-43/7,119/14,-119/14,16,多结点的力矩分配,具体演算过程,弯矩图,2.93,6.14,8.5,16,例题,2,8,kN,12,kN/m,8,kN,4m,4m,4m,2m,3m,EI,EI,5EI/8,EI,多结点的力矩分配,-41/14,43/7,-43/7,119/14,-119/14,16,解：,8,kN,12,kN/m,16,kNm,分配系数,0.2,0.8,8/11,3/11,固端弯矩,16,16,-16,16,8,16,-17.45,-6.55,-8.73,0,分配、传递,1.75,6.98,-1.75,3.49,分配、传递,分配、传递,-2.54,-0.95,-1.27,0,分配、传递,0.25,1.02,-0.25,0.51,分配、传递,-0.37,-0.14,-0.19,0,分配,0.04,0.15,合计,14,18.04,-18.04,-0.36,0.36,16,4m,EI,4m,EI,5m,5EI/8,不思，故有惑；不求，故无得；不问，故无知,弯矩图,14,18.04,0.36,16,单位：,kNm,例题,3,20,kNm,3,kN/m,2m,4m,4m,4m,EI=,常数,不思，故有惑；不求，故无得；不问，故无知,解：,0.5,0.5,20,4/11,4/11,3/11,-2.67,-2.67,2.67,-10,-10,-5,-5,2.79,2.09,2.79,1.40,1.40,-0.7,-0.7,-0.35,-0.35,0.13,0.09,0.13,-5.35,-10.7,-9.3,-2.43,2.18,0.25,4.13,你能画出弯矩图吗？,弯矩图,5.35,10.7,9.3,2.18,2.43,4.13,0.25,单位：,kNm,不思，故有惑；不求，故无得；不问，故无知,例题,4,7.5m,5m,5m,7.5m,5m,5m,20kN/m,20kN/m,解：取半结构计算。,A,B,C,D,E,F,41.67,2/7,1/7,4/7,0,1/3,1/3,1/3,41.67,-41.67,0,0,0,0,0,41.67,-41.67,-11.9,-23.8,-,5.95,-11.9,5.95,3.97,3.97,3.97,1.99,1.99,1.99,-,0.57,-,1.14,-,0.28,0.28,-12.47,18.42,-6.23,6.23,-49.98,45.83,-39.6,3.97,1.99,12.47,6.23,18.42,49.98,45.83,39.6,3.98,1.99,例,5,8,kN,16,kN,6,kN/m,R,1,R,2,8,kN,16,kN,6,kN/m,2,2,m,4m,2,2,m,EI=,常数,-R,1,-R,2,+,8,kN,16,kN,3,kN/m,分配系数,0.5,0.5,0.8,0.2,固端弯矩,-4,4,-8,8,-16,-8,分配与传递,6.4,1.6,-1.6,3.2,分配与传递,0.4,0.4,0.2,0.2,分配与传递,-0.16,-0.04,0.04,-0.08,分配与传递,0.04,0.04,0.02,合计,3.78,4.44,-4.44,14.44,-14.44,-9.56,例,5,用力矩分配法计算下列结构。杆长都是,L,。,P,P,EI,1,=,EI,EI,EI,A,B,C,D,E,解：,（,1,）力矩分配法是计算无侧移刚架，,但对剪力静定杆也可使用。,因为在,B,结点附加刚臂后：,BC,杆为一端固定，另端可滑动杆，,AB,杆成为,A,端固定铰，另端可滑动杆。,(2),关于分配系数,P,P,EI,1,=,EI,EI,EI,A,B,C,D,E,B,B,A,C,A,B,支座,B,转动时无弯矩产生,位移图,不思，故有惑；不求，故无得；不问，故无知,A,B,E,C,P,PL,P,1/4,3/4,0,-PL/2,-PL/2,-2PL,-5PL/2,5PL/8,15PL/8,-,5PL/8,-9PL/8,PL/8,15PL/8,-2PL,9PL/8,PL/8,2PL,15PL/8,例题,6,用力矩分配法计算具有剪力静定杆结构,10kN,10kN,5,kN/m,4m,5m,4m,1m,i=,常数,20,50,50,10,A,B,20,不思，故有惑；不求，故无得；不问，故无知,-20,-20,-50,-50,10,3/4,1/4,1/5,1/5,3/5,-20,-60,12,-12,36,12,-12,24,8,-8,1.6,-1.6,4.8,1.6,-1.6,1.2,0.4,-0.4,0.08,0.24,0.08,25.2,-25.2,-14.72,51.04,-36.32,-63.53,0,0,0,25.2,51.04,63.53,36.32,14.72,小结,：,多结点力矩分配时，先附加刚臂求出固端弯矩，然后,放松若干结点，保证每个结点都是单结点，分配，传递后，再锁,紧这些结点，然后放松先前锁紧的结点，进行分配与传递，然后,锁紧，两组结点轮换进行。,例如下图。,1,2,3,4,先放松,1,，,3,结点，锁紧,2,，,4,结点，则,1,，,3,各是单结点分配,然后，锁紧,1,，,3,结点，放松,2,，,4,结点，则,2,，,4,结点也为单结点分配,1,2,3,4,对下列结构，可进行如下的结点分组。,练习题,8,kN,3,kN/m,2m,2m,4m,EI,EI,1,2,EI=,常数,2m,2m,2m,2m,2m,30,kN/m,8,kN,3,kN/m,2m,2m,4m,EI,EI,答案,分配系数,0.75,0.25,7.5,2.5,固端弯矩,0,6,-16,-8,-2.5,0,0,13.5,-13.5,-10.5,3/11,4/11,4/11,1/3,1/3,1/3,1/4,1/4,1/4,1/4,1/4,1/4,1/4,1/4,15,-10,10,-10,10,-1.36,-1.82,-1.82,-0.91,-0.91,0.23,0.23,0.23,0.23,0.3,0.3,0.3,0.12,0.15,0.12,0.12,0.15,-0.07,-0.07,-0.07,0.07,-0.035,-0.035,13.64,-1.82,-11.82,-0.68,9.4,0.3,-9.7,10.15,0.08,0.12,0.23,0.23,0.12,0.23,0.19,-0.07,-0.07,-0.035,-0.035,13.64,11.82,9.4,9.7,10.15,1.82,0.68,0.3,0.08,0.23,0.12,0.12,0.035,0.07,0.035,0.19,0.23,作业：,0.3,0.3,0.4,-12,0.32,0.36,0.32,4.5,0.4,0.3,0.3,7.5,-12,12,-7.5,7.5,3.6,3.6,4.8,-2.25,-3,-2.25,2.4,-1.5,5.4,-1.728,-1.944,-1.728,-0.86,-0.86,0.258,0.258,0.344,0.344,0.258,0.258,0.172,0.172,-0.11,-0.11,-0.124,3.86,3.86,-7.72,12.73,-2.06,3.98,-10.67,-1.99,-1.99,3.86,3.86,-7.72,12.73,-2.06,3.98,-10.67,-1.99,-1.99,