2024-05-10 18:43:47
浏览数 (5)
在上篇几何非线性| 应变张量，得到拉格朗日应变表达式为
mathbf E = frac{1}{2}(mathbf H mathbf H^T mathbf H^T mathbf H)
用指标记法
E_{ij}=frac{1}{2}(frac{partial u_i }{partial X_j} frac{partial u_j }{partial X_i}
frac{partial u_k }{partial X_j}frac{partial u_k }{partial X_i}) quad (1)
对于杆系结构，有
epsilon_{x}=frac{partial u }{partial x} frac{1}{2}(frac{partial u }{partial x})^2
frac{1}{2}(frac{partial v }{partial x})^2 frac{1}{2}(frac{partial w }{partial x})^2 quad (2)
拉格朗日应变适用于描述几何非线性。
▲图1
如图1所示的桁架单元，局部坐标下的位移插值
begin{split}
u(x) &=[1- frac{x}{l},0,frac{x}{l},0]begin{Bmatrix}
u_1 \
v_1 \
u_2 \
v_2 \
end{Bmatrix} \
v(x) &=[0,1- frac{x}{l},0,frac{x}{l}]begin{Bmatrix}
u_1 \
v_1 \
u_2 \
v_2 \
end{Bmatrix}
end{split}
mathbf u = mathbf Nmathbf q^e quad (3)
其中，
mathbf q^e 是单元节点位移矩阵。
begin{split}
frac{partial u }{partial x} &=u^{'}\
&= frac{1}{l}begin{bmatrix}
-1 & 0 & 1 & 0 \
end{bmatrix}
begin{Bmatrix}
u_1 \
v_1 \
u_2 \
v_2 \
end{Bmatrix} \
&= mathbf C mathbf q^e
end{split}
begin{split}
frac{partial v }{partial x} &=v^{'}\
&= frac{1}{l}begin{bmatrix}
0 & -1 & 0 & 1 \
end{bmatrix}
begin{Bmatrix}
u_1 \
v_1 \
u_2 \
v_2 \
end{Bmatrix} \
&= mathbf D mathbf q^e
end{split}
拉格朗日应变
epsilon = mathbf C mathbf q^e frac{1}{2} {mathbf q^e}^T {mathbf C}^T mathbf C mathbf q^e frac{1}{2} {mathbf q^e}^T {mathbf D}^T mathbf D mathbf q^e quad (4)
虚位移
delta mathbf u = mathbf N delta mathbf q^e quad (5)
虚应变
begin{split}
delta epsilon &= frac{partial epsilon }{partial mathbf q^e }delta mathbf q^e \
&= mathbf B(mathbf q^e) delta mathbf q^e
end{split} quad (6)
这里，(6)用到了变分运算公式
delta f = frac{partial f }{partial x }delta x frac{partial f }{partial y }delta y
内力虚功为
begin{split}
delta W_i & = int_V delta boldsymbol {epsilon}^T boldsymbol{sigma} dV \
& = delta mathbf q^{eT} int_V mathbf B^T boldsymbol
{sigma} dV
end{split} quad (7)
记
f_i= int_V mathbf B^T boldsymbol
{sigma} dV quad (8)
则
delta W_i = delta mathbf q^{eT} f_i quad (9)
由(6)可得
begin{split}
bf B &= frac{partial epsilon }{partial mathbf q^{e}}\
&= frac{partial epsilon }{partial u^{'}}frac{partial u^{'} }{partial mathbf q^{e}} frac{partial epsilon }{partial v^{'}}frac{partial v^{'} }{partial mathbf q^{e}}\
&= (1 u^{'}) mathbf {C} v^{'} mathbf D \
&= mathbf C mathbf q^{eT} mathbf C^Tmathbf C mathbf q^{eT} mathbf D^Tmathbf D
end{split} quad (10)
应力
begin{split}
sigma &= E(u^{'} frac{1}{2}u^{'2} frac{1}{2}v^{'2})\
&= E(mathbf C mathbf q^e frac{1}{2} {mathbf q^e}^T {mathbf C}^T mathbf C mathbf q^e frac{1}{2} {mathbf q^e}^T {mathbf D}^T mathbf D mathbf q^e)
end{split} quad (11)
由(8)(10)(11)可得
f_i= ((1 u^{'}) mathbf C^T v^{'} mathbf D^T )sigma Al quad (12)
▲图2
如图2所示的非线性迭代过程，当某一迭代步
i 达到收敛标准时，可以认为处于平衡状态，即
f_i(mathbf q^i) = f_e(mathbf q^i) quad (13)
式中
f_i 是结构内力，
f_e 是外荷载，
mathbf q^i 是
i 迭代步时的节点位移。
i 1 迭代步时的内力用一阶泰勒展开
f_i(mathbf q^{i 1}) approx f_i(mathbf q^i) frac{partial f_i }{partial mathbf q} Delta mathbf q^{i} quad (14)
由(13)(14)得
frac{partial f_i }{partial mathbf q} Delta mathbf q^{i} = f_e(mathbf q^{i 1})-f_i(mathbf u^i) quad (15)
记
mathbf K_T = frac{partial f_i }{partial mathbf q} quad (16)
其中，
mathbf K_T 叫做切线刚度矩阵，(15)可写成
mathbf K_T Delta mathbf q^{i} = f_e(mathbf q^{i 1})-f_i(mathbf q^i) quad (17)
mathbf K_T 是内力的导数，
f_e(mathbf q^{i 1}) 是新的荷载步下的外荷载。
begin{split}
mathbf K_T &= int_V mathbf B^T frac{partial boldsymbol
{sigma} }{partial boldsymbol {epsilon}}
frac{partial boldsymbol {epsilon}} {partial mathbf q}
dV int_V frac{partial mathbf B^T }{partial mathbf q} boldsymbol {sigma}dV\
&= int_V mathbf B^T E mathbf B dV int_V frac{partial mathbf B^T }{partial mathbf q} boldsymbol {sigma}dV\
&= mathbf K_{mathbf q} mathbf K_{boldsymbol
{sigma}}\
end{split} quad (17)
其中
mathbf K_{mathbf q} 叫做初始刚度矩阵，
mathbf K_{sigma} 叫做几何刚度矩阵。对于桁架单元
begin{split}
mathbf K_{mathbf q} &= [(1 u^{'}) mathbf {C}^T v^{'} mathbf D^T][(1 u^{'}) mathbf {C} v^{'} mathbf D]EAl\
&= [(1 u^{'})^2mathbf {C}^Tmathbf {C} v^{'}(1 u^{'})(mathbf {D}^Tmathbf {C} mathbf {C}^Tmathbf {D} ) v^{'2}mathbf {D}^Tmathbf {D}] EAl\
end{split} quad (18)
几何刚度矩阵
begin{split}
mathbf K_{boldsymbol
{sigma}} &= frac{partial }{partial u^{'} }(1 u^{'}) mathbf {C}^T frac{partial u^{'} }{partial mathbf q } frac{partial }{partial v^{'} }(v^{'}) mathbf {D}^T frac{partial v^{'} }{partial mathbf q } boldsymbol
{sigma}Al \
&= (mathbf {C}^Tmathbf {C} mathbf {D}^Tmathbf {D}) boldsymbol
{sigma}Al
end{split} quad (19)
其中
begin{split}
mathbf {C}^Tmathbf {C} &= frac{1}{l^2}
begin{bmatrix}
-1 \
0 \
1 \
0 \
end{bmatrix}
begin{bmatrix}
-1 & 0 & 1 & 0 \
end{bmatrix} \
&= frac{1}{l^2}
begin{bmatrix}
1 & 0 & -1 & 0 \
0 & 0 & 0 & 0 \
-1 & 0 & 1 & 0 \
0 & 0 & 0 & 0 \
end{bmatrix} \
end{split}
同理
begin{split}
mathbf {D}^Tmathbf {D} &= frac{1}{l^2}
begin{bmatrix}
0 & 0 & 0 & 0 \
0 & 1 & 0 & -1 \
0 & 0 & 0 & 0 \
0 & -1 & 0 & 1 \
end{bmatrix} \
end{split}
最终得到局部坐标下的切线刚度矩阵为
begin{split}
mathbf K_T &= frac{EA}{l}
begin{bmatrix}
(1 u^{'})^2 & (v^{'} v^{'}u^{'}) & -(1 u^{'})^2 & -(v^{'} v^{'}u^{'}) \
(v^{'} v^{'}u^{'}) & v^{'2} & -(v^{'} v^{'}u^{'}) & -v^{'2} \
-(1 u^{'})^2 & -(v^{'} v^{'}u^{'}) & (1 u^{'})^2 & (v^{'} v^{'}u^{'}) \
-(v^{'} v^{'}u^{'}) & -v^{'2} & (v^{'} v^{'}u^{'}) & v^{'2} \
end{bmatrix} \
&quad frac{sigma A}{l}
begin{bmatrix}
1 & 0 & -1 & 0 \
0 & 1 & 0 & -1 \
-1 & 0 & 1 & 0 \
0 & -1 & 0 & 1 \
end{bmatrix} \
end{split} quad (20)
局部坐标和整体坐标下的节点位移转换关系
mathbf q^e = mathbf T mathbf q^g
整体坐标下的切线刚度矩阵
mathbf K_T^g = mathbf T^T mathbf K_Tmathbf T
