第2章3_单元刚度方程和单元刚度矩阵

第三节单元刚度方程和单元刚度矩阵单元的杆端力和杆端位移之间的关系是通过单元刚度方程反映出来的，本节重点掌握单元刚度矩阵中每个刚度系数的物理意义，由此求得不同杆单元的刚度矩阵。（1）单元刚度方程单元的刚度方程给出了单元的杆端位移δ(e)与杆端力F(e)之间的关系.其中矩阵K(e)称为单元刚度矩阵。单元刚度矩阵是一个方阵.它的阶数和内容视单元而定。如杆端位移δ(e)和杆端力F(e)为6阶向量，则K(e)为6X6方阵。)()()(eeeδKF单元的刚度方程：654321666564636261565554535251464544434241363534333231262524232221161514131211654321uuuuuukkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkFFFFFF单元刚度矩阵物理意义利用矩阵乘法,展开可得：如：单元刚度矩阵中第i列的元素表示第i号位移为一单位值(ui=1，其它为0)时引起的六个杆端力。单元刚度矩阵中的每一个元素称为刚度系数，刚度系数表示一个力。矩阵中第r行s列的元素krs，表示第s号位移为一单位值时引起沿第r个杆端力。由反力互等定理可知krs=ksr。所以单元刚度矩阵是一个对称矩阵。它的每一个元素的值都可由结构力学中位移法的刚度方程中获得。666565464363262161665655545435325215156465454443432421414636535434333232131362652542432322212126165154143132121111ukukukukukukFukukukukukukFukukukukukukFukukukukukukFukukukukukukFukukukukukukF(2)平面桁架单元平面桁架单元只有轴向变形,杆端力也只有轴力；yxFNjiFNiuilyFNiilxFNjjjujyxFNjiFNiuilyFNiilxFNjjjujjjiiNjNivuvulEAlEAlEAlEAFF00000000000000lEAKe0000010100000101)(单元的杆端力向量可表示为:F(e)={FNi0FNj0}T单元杆端位移向量可表示为：δ(e)={uiviujvj}T根据单元刚度矩阵的物理意义,由得单元的刚度方程为:则刚度矩阵：ulEAFEAlFuNN（3）平面两端刚结点梁单元平面两端刚节点梁单元在一般情况下单元上作用着杆端力：轴力、剪力和弯矩，单元的刚度方程为：根据单元的刚度矩阵的物理意义,由梁单元受力和变形及前面等截面直杆的刚度方程可以列出平面两端刚节点梁单元的单元刚度矩阵为:)()()(eeeδKFFQjjMiFNiFQiiyMjFNjxTjjjiiievuvu)(δ则：TjQjNjiQiNieMFFMFF)(FTjjjiiieMQNMQN)(F或：注意：杆端力与内力的符号规定不尽相同。lyx2lylxvi1l21EIlEIluiEI1lEIlyxluj6EI6EI12EI3ll312EIy6EIl2l312EI12EIl3xl26EIlvj1lyll2lx4EI2EI6EI6EI2l0i126EI4EI2EIl2ll6EI1llyx0jvi=1lyx2lylxvi1l21EIlEIluiEI1lEIlyxluj6EI6EI12EI3ll312EIy6EIl2l312EI12EIl3xl26EIlvj1lyll2lx4EI2EI6EI6EI2l0i126EI4EI2EIl2ll6EI1llyx0jθi=1lyx2lylxvi1l21EIlEIluiEI1lEIlyxluj6EI6EI12EI3ll312EIy6EIl2l312EI12EIl3xl26EIlvj1lyll2lx4EI2EI6EI6EI2l0i126EI4EI2EIl2ll6EI1llyx0jvj=1lyx2lylxvi1l21EIlEIluiEI1lEIlyxluj6EI6EI12EI3ll312EIy6EIl2l312EI12EIl3xl26EIlvj1lyll2lx4EI2EI6EI6EI2l0i126EI4EI2EIl2ll6EI1llyx0jθj=1lEAlEA00000000lEAlEA00312lEI312lEI26lEI26lEI00312lEI312lEI26lEI26lEI0026lEI26lEIlEI4lEI20026lEI26lEIlEI4lEI2ui=1vi=1θi=1uj=1vj=1θj=1平面梁单元的单元刚度矩阵iNiQiMjNjQjMui=1lyx1EAlEAluiuj=1EA1lEAlyxluj分别填写在ui=1，vi=1，θi=1，uj=1，vj=1，θj=1作用下，杆左右端截面的轴力、剪力、弯矩及右端截面的轴力、剪力、弯矩。由此可得单元的刚度方程：平面梁单元的单元的刚度方程为:jjjiiijjjiiivuvulEIlEIlEIlEIlEIlEIlEIlEIlEAlEAlEIlEIlEIlEIlEIlEIlEIlEIlEAlEAMQNMQN460260612061200000260460612061200000222323222323平面两端刚节点梁单元的单元刚度矩阵为:单元刚度矩阵常用子块形式表示:lEIlEIlEIlEIlEIlEIlEIlEIlEAlEAlEIlEIlEIlEIlEIlEIlEIlEIlEAlEAe460260612061200000260460612061200000222323222323)(K)()()()()(ejjejieijeiieKKKKK其中每个都是3×3的方阵，子块K(e)ij表示杆端j作用一单位位移时,杆i端引起的杆端力。（4）一端刚结点另一端铰结点的梁单元铰支端一般只有两个位移需计算.铰结点的转角位移可认为它是不独立的而不予考虑.这样单元的杆端位移向量及杆端力向量都只有五阶.单元刚度矩阵为5×5:BF1F2F3CAF4EF5CETjjiiievuvu}{)(δTQjNjiQiNieFFMFF)(F如梁右端为铰结点，则：TjjiiieQNMQN)(F或：)()()(eeeδKF根据单元的刚度矩阵的物理意义,由梁单元受力和变形可以列出该单元的单元刚度矩阵为:lEAlEA000000lEAlEA0033lEI33lEI23lEI0033lEI33lEI23lEI0023lEI23lEIlEI3ui=1vi=1θi=1uj=1vj=1平面一端刚结点另一端铰结点梁单元的单元刚度矩阵iNiQiMjNjQvi=13EI33EIll23EI2ly3EIly123EIlvi3EIl3l3EI3lvj1ll0i13EI2lx3EIl3xθi=13EI33EIll23EI2ly3EIly123EIlvi3EIl3l3EI3lvj1ll0i13EI2lx3EIl3xvj=13EI33EIll23EI2ly3EIly123EIlvi3EIl3l3EI3lvj1ll0i13EI2lx3EIl3x分别填写在ui=1，vi=1，θi=1，uj=1，vj=1，作用下，杆左右端截面的轴力、剪力、弯矩及右端截面的轴力、剪力。由此可得单元的刚度方程：若单元i端为铰结点，j端为刚结点,同样可建立起单元刚度矩阵:32322323)(303300003033030330000lEIlEIlEIlEAlEAlEIlEIlEIlEIlEIlEIlEAlEAKelEIlEIlEIlEIlEIlEIlEAlEAlEIlEIlEIlEAlEAKe33030330300003303000022233233)(若单元i端为刚结点，j端为铰结点,则单元刚度矩阵为:(5)空间桁架单元空间桁架单元每个节点具有x、y、z方向的三个位移分量。00000000000000000000000000000000lEAlEAlEAlEAKe)(单元的杆端力向量可表示为:单元杆端位移向量可表示为：单元的刚度方程为：根据单元刚度矩阵的物理意义得:Tjjjiiiewvuwvu)(TjNiNeFFF0000)()()()(eeeKF(6)空间刚架单元空间刚架单元每个节点具有应有6个自由度，即沿三个坐标轴方向的线位移及分别绕三个坐标轴的转角。杆端位移和杆端力向量均为12阶。单元的杆端力向量可表示为:)()()(eeeKF单元杆端位移向量可表示为：单元的刚度方程为：Tjzjyjxjjjiziyixiiiewvuwvu)(TjzjyjxjzQjyQjNiziyixizQiyQiNeMMMFFFMMMFFFF)(则单元刚度矩阵为12×12阶。可根据单元刚度矩阵中的各系数的物理意义求得空间刚架单元的刚度矩阵。lEIlEIlEIlEIlEIlEIlEIlEIlGJlGJlEIlEIlEIlEIlEIlEIlEAlEAlEIlEIlEIlEIlGJlEIlEIlEAKzzzzYyyyzxyyyzzzzzYzzyze40006020006040600020600000000001200060120012060001200000040006040600000120012022223233232233对称)(空间刚架单元刚度矩阵返回目录