[FEM-6-2]杆单元的坐标变换——三维立体空间

目录

• 1 三维空间(3D)杆单元的坐标变换
• • 1.1 单元位移场的表达
  • 1.2 三维空间(3D)多连杆示例
• 参考

1 三维空间(3D)杆单元的坐标变换

三维空间问题中的杆单元如上图所示

1.1 单元位移场的表达

该杆单元在局部坐标系下( o x ox ox)的节点位移依旧为

q e = [ u 1 u 2 ] = [ u 1 u 2 ] T \begin{equation} \begin{aligned} \mathbf{q}^e = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} = \begin{bmatrix} u_1 & u_2 \end{bmatrix}^{T} \end{aligned} \end{equation} qe=[u1​u2​​]=[u1​​u2​​]T​​​

而整体坐标系中( o x y z ‾ \overline{oxyz} oxyz​)的节点位移列阵为

q ‾ e = [ u ‾ 1 v ‾ 1 w ‾ 1 u ‾ 2 v ‾ 2 w ‾ 2 ] = [ u ‾ 1 v ‾ 1 w ‾ 1 u ‾ 2 v ‾ 2 w ‾ 2 ] T \begin{equation} \begin{aligned} \overline{\mathbf{q}}^e = \begin{bmatrix} \overline{u}_1 \\ \overline{v}_1 \\ \overline{w}_1 \\ \overline{u}_2 \\ \overline{v}_2 \\ \overline{w}_2 \end{bmatrix} = \begin{bmatrix} \overline{u}_1 & \overline{v}_1 & \overline{w}_1 & \overline{u}_2 & \overline{v}_2 & \overline{w}_2 \end{bmatrix}^{T} \end{aligned} \end{equation} q​e= ​u1​v1​w1​u2​v2​w2​​ ​=[u1​​v1​​w1​​u2​​v2​​w2​​]T​​​

杆单元轴线在整体坐标系中的方向余弦为

cos ⁡ ( x , x ‾ ) = x ‾ 2 − x ‾ 1 l cos ⁡ ( x , y ‾ ) = y ‾ 2 − y ‾ 1 l cos ⁡ ( x , z ‾ ) = z ‾ 2 − z ‾ 1 l \begin{equation} \begin{aligned} \cos (x,\overline{x}) = \cfrac{\overline{x}_2 - \overline{x}_1}{l} \quad \cos (x,\overline{y}) = \cfrac{\overline{y}_2 - \overline{y}_1}{l} \quad \cos (x,\overline{z}) = \cfrac{\overline{z}_2 - \overline{z}_1}{l} \end{aligned} \end{equation} cos(x,x)=lx2​−x1​​cos(x,y​)=ly​2​−y​1​​cos(x,z)=lz2​−z1​​​​​

其中 ( x ‾ 1 , y ‾ 1 , z ‾ 1 ) (\overline{x}_1, \overline{y}_1, \overline{z}_1) (x1​,y​1​,z1​)和 ( x ‾ 2 , y ‾ 2 , z ‾ 2 ) (\overline{x}_2, \overline{y}_2, \overline{z}_2) (x2​,y​2​,z2​)分别为杆单元的两个端点在整体坐标系中的位置， l l l为杆单元的长度，与平面坐标系中类似， q e \mathbf{q}^e qe与 q ‾ e \overline{\mathbf{q}}^e q​e之间存在以下转换关系

q e = [ u 1 u 2 ] = [ cos ⁡ ( x , x ‾ ) cos ⁡ ( x , y ‾ ) cos ⁡ ( x , z ‾ ) 0 0 0 0 0 0 cos ⁡ ( x , x ‾ ) cos ⁡ ( x , y ‾ ) cos ⁡ ( x , z ‾ ) ] [ u ‾ 1 v ‾ 1 w ‾ 1 u ‾ 2 v ‾ 2 w ‾ 2 ] = T e q ‾ e \begin{equation} \begin{aligned} \mathbf{q}^e &= \begin{bmatrix} u_1 \\ u_2 \end{bmatrix} \\ &= \begin{bmatrix} \cos (x,\overline{x}) & \cos (x,\overline{y}) & \cos (x,\overline{z}) & 0 & 0 & 0 \\ 0 & 0 & 0 & \cos (x,\overline{x}) & \cos (x,\overline{y}) & \cos (x,\overline{z}) \end{bmatrix} \begin{bmatrix} \overline{u}_1 \\ \overline{v}_1 \\ \overline{w}_1 \\ \overline{u}_2 \\ \overline{v}_2 \\ \overline{w}_2 \end{bmatrix} \\ &= \mathbf{T}^{e} \overline{\mathbf{q}}^e \end{aligned} \end{equation} qe​=[u1​u2​​]=[cos(x,x)0​cos(x,y​)0​cos(x,z)0​0cos(x,x)​0cos(x,y​)​0cos(x,z)​] ​u1​v1​w1​u2​v2​w2​​ ​=Teq​e​​​

式中： T e \mathbf{T}^{e} Te为坐标变换矩阵，即

T e = [ cos ⁡ ( x , x ‾ ) cos ⁡ ( x , y ‾ ) cos ⁡ ( x , z ‾ ) 0 0 0 0 0 0 cos ⁡ ( x , x ‾ ) cos ⁡ ( x , y ‾ ) cos ⁡ ( x , z ‾ ) ] \begin{equation} \begin{aligned} \mathbf{T}^{e} = \begin{bmatrix} \cos (x,\overline{x}) & \cos (x,\overline{y}) & \cos (x,\overline{z}) & 0 & 0 & 0 \\ 0 & 0 & 0 & \cos (x,\overline{x}) & \cos (x,\overline{y}) & \cos (x,\overline{z}) \end{bmatrix} \end{aligned} \end{equation} Te=[cos(x,x)0​cos(x,y​)0​cos(x,z)0​0cos(x,x)​0cos(x,y​)​0cos(x,z)​]​​​

刚度矩阵和节点力的变换与平面情形相同，即

K ‾ e ( 6 × 6 ) = T e T ( 6 × 2 ) K e ( 2 × 2 ) T e ( 2 × 6 ) \begin{equation} \begin{aligned} \underset{(6 \times 6)}{\overline{\mathbf{K}}^e} = \underset{(6 \times 2)}{\mathbf{T}^{eT}} \underset{(2 \times 2)}{\mathbf{K}^e} \underset{(2 \times 6)}{\mathbf{T}^e} \end{aligned} \end{equation} (6×6)Ke​=(6×2)TeT​(2×2)Ke​(2×6)Te​​​​

P ‾ e ( 6 × 2 ) = T e T ( 6 × 2 ) P e ( 2 × 1 ) \begin{equation} \begin{aligned} \underset{(6 \times 2)}{\overline{\mathbf{P}}^e} = \underset{(6 \times 2)}{\mathbf{T}^{eT}} \underset{(2 \times 1)}{\mathbf{P}^e} \end{aligned} \end{equation} (6×2)Pe​=(6×2)TeT​(2×1)Pe​​​​

单元应变场、应力场、势能、刚度方程表达与二维空间中类似，仅需改变坐标变换矩阵 T e \mathbf{T}^{e} Te即可

1.2 三维空间(3D)多连杆示例

上图为整体坐标系中的三维空间三连杆结构，故每个杆件的单元刚度方程分别为

K ( 1 ) ⋅ q ( 1 ) = P ( 1 ) [ K 11 ( 1 ) K 12 ( 1 ) K 13 ( 1 ) K 14 ( 1 ) K 15 ( 1 ) K 16 ( 1 ) K 21 ( 1 ) K 22 ( 1 ) K 23 ( 1 ) K 24 ( 1 ) K 25 ( 1 ) K 26 ( 1 ) K 31 ( 1 ) K 32 ( 1 ) K 33 ( 1 ) K 34 ( 1 ) K 35 ( 1 ) K 36 ( 1 ) K 41 ( 1 ) K 42 ( 1 ) K 43 ( 1 ) K 44 ( 1 ) K 45 ( 1 ) K 46 ( 1 ) K 51 ( 1 ) K 52 ( 1 ) K 53 ( 1 ) K 54 ( 1 ) K 55 ( 1 ) K 56 ( 1 ) K 61 ( 1 ) K 62 ( 1 ) K 63 ( 1 ) K 64 ( 1 ) K 65 ( 1 ) K 66 ( 1 ) ] ⋅ [ u ‾ 1 v ‾ 1 w ‾ 1 u ‾ 2 v ‾ 2 w ‾ 2 ] = [ P ‾ x 1 ( 1 ) P ‾ y 1 ( 1 ) P ‾ z 1 ( 1 ) P ‾ x 2 ( 1 ) P ‾ y 2 ( 1 ) P ‾ z 2 ( 1 ) ] \begin{equation} \begin{aligned} \mathbf{K}^{(1)} \cdot \mathbf{q}^{(1)} &= \mathbf{P}^{(1)} \\ \begin{bmatrix} K_{11}^{(1)} & K_{12}^{(1)} & K_{13}^{(1)} & K_{14}^{(1)} & K_{15}^{(1)} & K_{16}^{(1)}\\ K_{21}^{(1)} & K_{22}^{(1)} & K_{23}^{(1)} & K_{24}^{(1)} & K_{25}^{(1)} & K_{26}^{(1)}\\ K_{31}^{(1)} & K_{32}^{(1)} & K_{33}^{(1)} & K_{34}^{(1)} & K_{35}^{(1)} & K_{36}^{(1)} \\ K_{41}^{(1)} & K_{42}^{(1)} & K_{43}^{(1)} & K_{44}^{(1)} & K_{45}^{(1)} & K_{46}^{(1)} \\ K_{51}^{(1)} & K_{52}^{(1)} & K_{53}^{(1)} & K_{54}^{(1)} & K_{55}^{(1)} & K_{56}^{(1)} \\ K_{61}^{(1)} & K_{62}^{(1)} & K_{63}^{(1)} & K_{64}^{(1)} & K_{65}^{(1)} & K_{66}^{(1)} \end{bmatrix} \cdot \begin{bmatrix} \overline{u}_1 \\ \overline{v}_1 \\ \overline{w}_1 \\ \overline{u}_2 \\ \overline{v}_2 \\ \overline{w}_2 \\ \end{bmatrix} &= \begin{bmatrix} \overline{P}_{x1}^{(1)} \\ \overline{P}_{y1}^{(1)} \\ \overline{P}_{z1}^{(1)} \\ \overline{P}_{x2}^{(1)} \\ \overline{P}_{y2}^{(1)} \\ \overline{P}_{z2}^{(1)} \end{bmatrix} \end{aligned} \end{equation} K(1)⋅q(1) ​K11(1)​K21(1)​K31(1)​K41(1)​K51(1)​K61(1)​​K12(1)​K22(1)​K32(1)​K42(1)​K52(1)​K62(1)​​K13(1)​K23(1)​K33(1)​K43(1)​K53(1)​K63(1)​​K14(1)​K24(1)​K34(1)​K44(1)​K54(1)​K64(1)​​K15(1)​K25(1)​K35(1)​K45(1)​K55(1)​K65(1)​​K16(1)​K26(1)​K36(1)​K46(1)​K56(1)​K66(1)​​ ​⋅ ​u1​v1​w1​u2​v2​w2​​ ​​=P(1)= ​Px1(1)​Py1(1)​Pz1(1)​Px2(1)​Py2(1)​Pz2(1)​​ ​​​​

K ( 2 ) ⋅ q ( 2 ) = P ( 2 ) [ K 44 ( 2 ) K 45 ( 2 ) K 46 ( 2 ) K 47 ( 2 ) K 48 ( 2 ) K 49 ( 2 ) K 54 ( 2 ) K 55 ( 2 ) K 56 ( 2 ) K 57 ( 2 ) K 58 ( 2 ) K 59 ( 2 ) K 64 ( 2 ) K 65 ( 2 ) K 66 ( 2 ) K 67 ( 2 ) K 68 ( 2 ) K 69 ( 2 ) K 74 ( 2 ) K 75 ( 2 ) K 76 ( 2 ) K 77 ( 2 ) K 78 ( 2 ) K 79 ( 2 ) K 84 ( 2 ) K 85 ( 2 ) K 86 ( 2 ) K 87 ( 2 ) K 88 ( 2 ) K 89 ( 2 ) K 94 ( 2 ) K 95 ( 2 ) K 96 ( 2 ) K 97 ( 2 ) K 98 ( 2 ) K 99 ( 2 ) ] ⋅ [ u ‾ 2 v ‾ 2 w ‾ 2 u ‾ 3 v ‾ 3 w ‾ 3 ] = [ P ‾ x 2 ( 2 ) P ‾ y 2 ( 2 ) P ‾ z 2 ( 2 ) P ‾ x 3 ( 2 ) P ‾ y 3 ( 2 ) P ‾ z 3 ( 2 ) ] \begin{equation} \begin{aligned} \mathbf{K}^{(2)} \cdot \mathbf{q}^{(2)} &= \mathbf{P}^{(2)} \\ \begin{bmatrix} K_{44}^{(2)} & K_{45}^{(2)} & K_{46}^{(2)} & K_{47}^{(2)} & K_{48}^{(2)} & K_{49}^{(2)}\\ K_{54}^{(2)} & K_{55}^{(2)} & K_{56}^{(2)} & K_{57}^{(2)} & K_{58}^{(2)} & K_{59}^{(2)}\\ K_{64}^{(2)} & K_{65}^{(2)} & K_{66}^{(2)} & K_{67}^{(2)} & K_{68}^{(2)} & K_{69}^{(2)} \\ K_{74}^{(2)} & K_{75}^{(2)} & K_{76}^{(2)} & K_{77}^{(2)} & K_{78}^{(2)} & K_{79}^{(2)} \\ K_{84}^{(2)} & K_{85}^{(2)} & K_{86}^{(2)} & K_{87}^{(2)} & K_{88}^{(2)} & K_{89}^{(2)} \\ K_{94}^{(2)} & K_{95}^{(2)} & K_{96}^{(2)} & K_{97}^{(2)} & K_{98}^{(2)} & K_{99}^{(2)} \end{bmatrix} \cdot \begin{bmatrix} \overline{u}_2 \\ \overline{v}_2 \\ \overline{w}_2 \\ \overline{u}_3 \\ \overline{v}_3 \\ \overline{w}_3 \\ \end{bmatrix} &= \begin{bmatrix} \overline{P}_{x2}^{(2)} \\ \overline{P}_{y2}^{(2)} \\ \overline{P}_{z2}^{(2)} \\ \overline{P}_{x3}^{(2)} \\ \overline{P}_{y3}^{(2)} \\ \overline{P}_{z3}^{(2)} \end{bmatrix} \end{aligned} \end{equation} K(2)⋅q(2) ​K44(2)​K54(2)​K64(2)​K74(2)​K84(2)​K94(2)​​K45(2)​K55(2)​K65(2)​K75(2)​K85(2)​K95(2)​​K46(2)​K56(2)​K66(2)​K76(2)​K86(2)​K96(2)​​K47(2)​K57(2)​K67(2)​K77(2)​K87(2)​K97(2)​​K48(2)​K58(2)​K68(2)​K78(2)​K88(2)​K98(2)​​K49(2)​K59(2)​K69(2)​K79(2)​K89(2)​K99(2)​​ ​⋅ ​u2​v2​w2​u3​v3​w3​​ ​​=P(2)= ​Px2(2)​Py2(2)​Pz2(2)​Px3(2)​Py3(2)​Pz3(2)​​ ​​​​

K ( 3 ) ⋅ q ( 3 ) = P ( 3 ) [ K 77 ( 3 ) K 78 ( 3 ) K 79 ( 3 ) K 710 ( 3 ) K 711 ( 3 ) K 712 ( 3 ) K 87 ( 3 ) K 88 ( 3 ) K 89 ( 3 ) K 810 ( 3 ) K 811 ( 3 ) K 812 ( 3 ) K 97 ( 3 ) K 98 ( 3 ) K 99 ( 3 ) K 910 ( 3 ) K 911 ( 3 ) K 912 ( 3 ) K 107 ( 3 ) K 108 ( 3 ) K 109 ( 3 ) K 1010 ( 3 ) K 1011 ( 3 ) K 1012 ( 3 ) K 117 ( 3 ) K 118 ( 3 ) K 119 ( 3 ) K 1110 ( 3 ) K 1111 ( 3 ) K 1112 ( 3 ) K 127 ( 3 ) K 128 ( 3 ) K 129 ( 3 ) K 1210 ( 3 ) K 1211 ( 3 ) K 1212 ( 3 ) ] ⋅ [ u ‾ 3 v ‾ 3 w ‾ 3 u ‾ 4 v ‾ 4 w ‾ 4 ] = [ P ‾ x 3 ( 3 ) P ‾ y 3 ( 3 ) P ‾ z 3 ( 3 ) P ‾ x 4 ( 3 ) P ‾ y 4 ( 3 ) P ‾ z 4 ( 3 ) ] \begin{equation} \begin{aligned} \mathbf{K}^{(3)} \cdot \mathbf{q}^{(3)} &= \mathbf{P}^{(3)} \\ \begin{bmatrix} K_{77}^{(3)} & K_{78}^{(3)} & K_{79}^{(3)} & K_{710}^{(3)} & K_{711}^{(3)} & K_{712}^{(3)}\\ K_{87}^{(3)} & K_{88}^{(3)} & K_{89}^{(3)} & K_{810}^{(3)} & K_{811}^{(3)} & K_{812}^{(3)}\\ K_{97}^{(3)} & K_{98}^{(3)} & K_{99}^{(3)} & K_{910}^{(3)} & K_{911}^{(3)} & K_{912}^{(3)} \\ K_{107}^{(3)} & K_{108}^{(3)} & K_{109}^{(3)} & K_{1010}^{(3)} & K_{1011}^{(3)} & K_{1012}^{(3)} \\ K_{117}^{(3)} & K_{118}^{(3)} & K_{119}^{(3)} & K_{1110}^{(3)} & K_{1111}^{(3)} & K_{1112}^{(3)} \\ K_{127}^{(3)} & K_{128}^{(3)} & K_{129}^{(3)} & K_{1210}^{(3)} & K_{1211}^{(3)} & K_{1212}^{(3)} \end{bmatrix} \cdot \begin{bmatrix} \overline{u}_3 \\ \overline{v}_3 \\ \overline{w}_3 \\ \overline{u}_4 \\ \overline{v}_4 \\ \overline{w}_4 \end{bmatrix} &= \begin{bmatrix} \overline{P}_{x3}^{(3)} \\ \overline{P}_{y3}^{(3)} \\ \overline{P}_{z3}^{(3)} \\ \overline{P}_{x4}^{(3)} \\ \overline{P}_{y4}^{(3)} \\ \overline{P}_{z4}^{(3)} \end{bmatrix} \end{aligned} \end{equation} K(3)⋅q(3) ​K77(3)​K87(3)​K97(3)​K107(3)​K117(3)​K127(3)​​K78(3)​K88(3)​K98(3)​K108(3)​K118(3)​K128(3)​​K79(3)​K89(3)​K99(3)​K109(3)​K119(3)​K129(3)​​K710(3)​K810(3)​K910(3)​K1010(3)​K1110(3)​K1210(3)​​K711(3)​K811(3)​K911(3)​K1011(3)​K1111(3)​K1211(3)​​K712(3)​K812(3)​K912(3)​K1012(3)​K1112(3)​K1212(3)​​ ​⋅ ​u3​v3​w3​u4​v4​w4​​ ​​=P(3)= ​Px3(3)​Py3(3)​Pz3(3)​Px4(3)​Py4(3)​Pz4(3)​​ ​​​​

组装成整体刚度方程为

K ( A l l ) ⋅ q ( A l l ) = P ( A l l ) [ K 11 ( 1 ) K 12 ( 1 ) K 13 ( 1 ) K 14 ( 1 ) K 15 ( 1 ) K 16 ( 1 ) 0 0 0 0 0 0 K 21 ( 1 ) K 22 ( 1 ) K 23 ( 1 ) K 24 ( 1 ) K 25 ( 1 ) K 26 ( 1 ) 0 0 0 0 0 0 K 31 ( 1 ) K 32 ( 1 ) K 33 ( 1 ) K 34 ( 1 ) K 35 ( 1 ) K 36 ( 1 ) 0 0 0 0 0 0 K 41 ( 1 ) K 42 ( 1 ) K 43 ( 1 ) K 44 ( 1 ) + K 44 ( 2 ) K 45 ( 1 ) + K 45 ( 2 ) K 46 ( 1 ) + K 46 ( 2 ) K 47 ( 2 ) K 48 ( 2 ) K 49 ( 2 ) 0 0 0 K 51 ( 1 ) K 52 ( 1 ) K 53 ( 1 ) K 54 ( 1 ) + K 54 ( 2 ) K 55 ( 1 ) + K 55 ( 2 ) K 56 ( 1 ) + K 56 ( 2 ) K 57 ( 2 ) K 58 ( 2 ) K 59 ( 2 ) 0 0 0 K 61 ( 1 ) K 62 ( 1 ) K 63 ( 1 ) K 64 ( 1 ) + K 64 ( 2 ) K 65 ( 1 ) + K 65 ( 2 ) K 66 ( 1 ) + K 66 ( 2 ) K 67 ( 2 ) K 68 ( 2 ) K 69 ( 2 ) 0 0 0 0 0 0 K 74 ( 2 ) K 75 ( 2 ) K 76 ( 2 ) K 77 ( 2 ) + K 77 ( 3 ) K 78 ( 2 ) + K 78 ( 3 ) K 79 ( 2 ) + K 79 ( 3 ) K 710 ( 3 ) K 711 ( 3 ) K 712 ( 3 ) 0 0 0 K 84 ( 2 ) K 85 ( 2 ) K 86 ( 2 ) K 87 ( 2 ) + K 87 ( 3 ) K 88 ( 2 ) + K 88 ( 3 ) K 89 ( 2 ) + K 89 ( 3 ) K 810 ( 3 ) K 811 ( 3 ) K 812 ( 3 ) 0 0 0 K 94 ( 2 ) K 95 ( 2 ) K 96 ( 2 ) K 97 ( 2 ) + K 97 ( 3 ) K 98 ( 2 ) + K 98 ( 3 ) K 99 ( 2 ) + K 99 ( 3 ) K 910 ( 3 ) K 911 ( 3 ) K 912 ( 3 ) 0 0 0 0 0 0 K 107 ( 3 ) K 108 ( 3 ) K 109 ( 3 ) K 1010 ( 3 ) K 1011 ( 3 ) K 1012 ( 3 ) 0 0 0 0 0 0 K 117 ( 3 ) K 118 ( 3 ) K 119 ( 3 ) K 1110 ( 3 ) K 1111 ( 3 ) K 1112 ( 3 ) 0 0 0 0 0 0 K 127 ( 3 ) K 128 ( 3 ) K 129 ( 3 ) K 1210 ( 3 ) K 1211 ( 3 ) K 1212 ( 3 ) ] ⋅ [ u ‾ 1 v ‾ 1 w ‾ 1 u ‾ 2 v ‾ 2 w ‾ 2 u ‾ 3 v ‾ 3 w ‾ 3 u ‾ 4 v ‾ 4 w ‾ 4 ] = [ P ‾ x 1 ( 1 ) P ‾ y 1 ( 1 ) P ‾ z 1 ( 1 ) P ‾ x 2 ( 1 ) + P ‾ x 2 ( 2 ) P ‾ y 2 ( 1 ) + P ‾ y 2 ( 2 ) P ‾ z 2 ( 1 ) + P ‾ z 2 ( 2 ) P ‾ x 3 ( 2 ) + P ‾ x 3 ( 3 ) P ‾ y 3 ( 2 ) + P ‾ y 3 ( 3 ) P ‾ z 3 ( 2 ) + P ‾ z 3 ( 3 ) P ‾ x 4 ( 3 ) P ‾ y 4 ( 3 ) P ‾ z 4 ( 3 ) ] [ K 11 ( 1 ) K 12 ( 1 ) K 13 ( 1 ) K 14 ( 1 ) K 15 ( 1 ) K 16 ( 1 ) 0 0 0 0 0 0 K 21 ( 1 ) K 22 ( 1 ) K 23 ( 1 ) K 24 ( 1 ) K 25 ( 1 ) K 26 ( 1 ) 0 0 0 0 0 0 K 31 ( 1 ) K 32 ( 1 ) K 33 ( 1 ) K 34 ( 1 ) K 35 ( 1 ) K 36 ( 1 ) 0 0 0 0 0 0 K 41 ( 1 ) K 42 ( 1 ) K 43 ( 1 ) K 44 ( 1 ) + K 44 ( 2 ) K 45 ( 1 ) + K 45 ( 2 ) K 46 ( 1 ) + K 46 ( 2 ) K 47 ( 2 ) K 48 ( 2 ) K 49 ( 2 ) 0 0 0 K 51 ( 1 ) K 52 ( 1 ) K 53 ( 1 ) K 54 ( 1 ) + K 54 ( 2 ) K 55 ( 1 ) + K 55 ( 2 ) K 56 ( 1 ) + K 56 ( 2 ) K 57 ( 2 ) K 58 ( 2 ) K 59 ( 2 ) 0 0 0 K 61 ( 1 ) K 62 ( 1 ) K 63 ( 1 ) K 64 ( 1 ) + K 64 ( 2 ) K 65 ( 1 ) + K 65 ( 2 ) K 66 ( 1 ) + K 66 ( 2 ) K 67 ( 2 ) K 68 ( 2 ) K 69 ( 2 ) 0 0 0 0 0 0 K 74 ( 2 ) K 75 ( 2 ) K 76 ( 2 ) K 77 ( 2 ) + K 77 ( 3 ) K 78 ( 2 ) + K 78 ( 3 ) K 79 ( 2 ) + K 79 ( 3 ) K 710 ( 3 ) K 711 ( 3 ) K 712 ( 3 ) 0 0 0 K 84 ( 2 ) K 85 ( 2 ) K 86 ( 2 ) K 87 ( 2 ) + K 87 ( 3 ) K 88 ( 2 ) + K 88 ( 3 ) K 89 ( 2 ) + K 89 ( 3 ) K 810 ( 3 ) K 811 ( 3 ) K 812 ( 3 ) 0 0 0 K 94 ( 2 ) K 95 ( 2 ) K 96 ( 2 ) K 97 ( 2 ) + K 97 ( 3 ) K 98 ( 2 ) + K 98 ( 3 ) K 99 ( 2 ) + K 99 ( 3 ) K 910 ( 3 ) K 911 ( 3 ) K 912 ( 3 ) 0 0 0 0 0 0 K 107 ( 3 ) K 108 ( 3 ) K 109 ( 3 ) K 1010 ( 3 ) K 1011 ( 3 ) K 1012 ( 3 ) 0 0 0 0 0 0 K 117 ( 3 ) K 118 ( 3 ) K 119 ( 3 ) K 1110 ( 3 ) K 1111 ( 3 ) K 1112 ( 3 ) 0 0 0 0 0 0 K 127 ( 3 ) K 128 ( 3 ) K 129 ( 3 ) K 1210 ( 3 ) K 1211 ( 3 ) K 1212 ( 3 ) ] ⋅ [ u ‾ 1 v ‾ 1 w ‾ 1 u ‾ 2 v ‾ 2 w ‾ 2 u ‾ 3 v ‾ 3 w ‾ 3 u ‾ 4 v ‾ 4 w ‾ 4 ] = [ P ‾ x 1 P ‾ y 1 P ‾ z 1 P ‾ x 2 P ‾ y 2 P ‾ z 2 P ‾ x 3 P ‾ y 3 P ‾ z 3 P ‾ x 4 P ‾ y 4 P ‾ z 4 ] \begin{equation} \begin{aligned} \mathbf{K}^{(All)} \cdot \mathbf{q}^{(All)} &= \mathbf{P}^{(All)} \\ \begin{bmatrix} K_{11}^{(1)} & K_{12}^{(1)} & K_{13}^{(1)} & K_{14}^{(1)} & K_{15}^{(1)} & K_{16}^{(1)} & 0 & 0 & 0 & 0 & 0 & 0 \\ K_{21}^{(1)} & K_{22}^{(1)} & K_{23}^{(1)} & K_{24}^{(1)} & K_{25}^{(1)} & K_{26}^{(1)} & 0 & 0 & 0 & 0 & 0 & 0 \\ K_{31}^{(1)} & K_{32}^{(1)} & K_{33}^{(1)} & K_{34}^{(1)} & K_{35}^{(1)} & K_{36}^{(1)} & 0 & 0 & 0 & 0 & 0 & 0 \\ K_{41}^{(1)} & K_{42}^{(1)} & K_{43}^{(1)} & K_{44}^{(1)} + K_{44}^{(2)} & K_{45}^{(1)} + K_{45}^{(2)} & K_{46}^{(1)} + K_{46}^{(2)} & K_{47}^{(2)} & K_{48}^{(2)} & K_{49}^{(2)} & 0 & 0 & 0 \\ K_{51}^{(1)} & K_{52}^{(1)} & K_{53}^{(1)} & K_{54}^{(1)} + K_{54}^{(2)} & K_{55}^{(1)} + K_{55}^{(2)} & K_{56}^{(1)} + K_{56}^{(2)} & K_{57}^{(2)} & K_{58}^{(2)} & K_{59}^{(2)} & 0 & 0 & 0 \\ K_{61}^{(1)} & K_{62}^{(1)} & K_{63}^{(1)} & K_{64}^{(1)} + K_{64}^{(2)} & K_{65}^{(1)} + K_{65}^{(2)} & K_{66}^{(1)} + K_{66}^{(2)} & K_{67}^{(2)} & K_{68}^{(2)} & K_{69}^{(2)} & 0 & 0 & 0 \\ 0 & 0 & 0 & K_{74}^{(2)} & K_{75}^{(2)} & K_{76}^{(2)} & K_{77}^{(2)} + K_{77}^{(3)} & K_{78}^{(2)} + K_{78}^{(3)} & K_{79}^{(2)} + K_{79}^{(3)} & K_{710}^{(3)} & K_{711}^{(3)} & K_{712}^{(3)} \\ 0 & 0 & 0 & K_{84}^{(2)} & K_{85}^{(2)} & K_{86}^{(2)} & K_{87}^{(2)} + K_{87}^{(3)} & K_{88}^{(2)} + K_{88}^{(3)} & K_{89}^{(2)} + K_{89}^{(3)} & K_{810}^{(3)} & K_{811}^{(3)} & K_{812}^{(3)} \\ 0 & 0 & 0 & K_{94}^{(2)} & K_{95}^{(2)} & K_{96}^{(2)} & K_{97}^{(2)} + K_{97}^{(3)} & K_{98}^{(2)} + K_{98}^{(3)} & K_{99}^{(2)} + K_{99}^{(3)} & K_{910}^{(3)} & K_{911}^{(3)} & K_{912}^{(3)} \\ 0 & 0 & 0 & 0 & 0 & 0 & K_{107}^{(3)} & K_{108}^{(3)} & K_{109}^{(3)} & K_{1010}^{(3)} & K_{1011}^{(3)} & K_{1012}^{(3)} \\ 0 & 0 & 0 & 0 & 0 & 0 &K_{117}^{(3)} & K_{118}^{(3)} & K_{119}^{(3)} & K_{1110}^{(3)} & K_{1111}^{(3)} & K_{1112}^{(3)} \\ 0 & 0 & 0 & 0 & 0 & 0 &K_{127}^{(3)} & K_{128}^{(3)} & K_{129}^{(3)} & K_{1210}^{(3)} & K_{1211}^{(3)} & K_{1212}^{(3)} \end{bmatrix} \cdot \begin{bmatrix} \overline{u}_1 \\ \overline{v}_1 \\ \overline{w}_1 \\ \overline{u}_2 \\ \overline{v}_2 \\ \overline{w}_2 \\ \overline{u}_3 \\ \overline{v}_3 \\ \overline{w}_3 \\ \overline{u}_4 \\ \overline{v}_4 \\ \overline{w}_4 \end{bmatrix} &= \begin{bmatrix} \overline{P}_{x1}^{(1)} \\ \overline{P}_{y1}^{(1)} \\ \overline{P}_{z1}^{(1)} \\ \overline{P}_{x2}^{(1)} + \overline{P}_{x2}^{(2)} \\ \overline{P}_{y2}^{(1)} + \overline{P}_{y2}^{(2)} \\ \overline{P}_{z2}^{(1)} + \overline{P}_{z2}^{(2)} \\ \overline{P}_{x3}^{(2)} + \overline{P}_{x3}^{(3)} \\ \overline{P}_{y3}^{(2)} + \overline{P}_{y3}^{(3)} \\ \overline{P}_{z3}^{(2)} + \overline{P}_{z3}^{(3)} \\ \overline{P}_{x4}^{(3)} \\ \overline{P}_{y4}^{(3)} \\ \overline{P}_{z4}^{(3)} \end{bmatrix} \\ \begin{bmatrix} K_{11}^{(1)} & K_{12}^{(1)} & K_{13}^{(1)} & K_{14}^{(1)} & K_{15}^{(1)} & K_{16}^{(1)} & 0 & 0 & 0 & 0 & 0 & 0 \\ K_{21}^{(1)} & K_{22}^{(1)} & K_{23}^{(1)} & K_{24}^{(1)} & K_{25}^{(1)} & K_{26}^{(1)} & 0 & 0 & 0 & 0 & 0 & 0 \\ K_{31}^{(1)} & K_{32}^{(1)} & K_{33}^{(1)} & K_{34}^{(1)} & K_{35}^{(1)} & K_{36}^{(1)} & 0 & 0 & 0 & 0 & 0 & 0 \\ K_{41}^{(1)} & K_{42}^{(1)} & K_{43}^{(1)} & K_{44}^{(1)} + K_{44}^{(2)} & K_{45}^{(1)} + K_{45}^{(2)} & K_{46}^{(1)} + K_{46}^{(2)} & K_{47}^{(2)} & K_{48}^{(2)} & K_{49}^{(2)} & 0 & 0 & 0 \\ K_{51}^{(1)} & K_{52}^{(1)} & K_{53}^{(1)} & K_{54}^{(1)} + K_{54}^{(2)} & K_{55}^{(1)} + K_{55}^{(2)} & K_{56}^{(1)} + K_{56}^{(2)} & K_{57}^{(2)} & K_{58}^{(2)} & K_{59}^{(2)} & 0 & 0 & 0 \\ K_{61}^{(1)} & K_{62}^{(1)} & K_{63}^{(1)} & K_{64}^{(1)} + K_{64}^{(2)} & K_{65}^{(1)} + K_{65}^{(2)} & K_{66}^{(1)} + K_{66}^{(2)} & K_{67}^{(2)} & K_{68}^{(2)} & K_{69}^{(2)} & 0 & 0 & 0 \\ 0 & 0 & 0 & K_{74}^{(2)} & K_{75}^{(2)} & K_{76}^{(2)} & K_{77}^{(2)} + K_{77}^{(3)} & K_{78}^{(2)} + K_{78}^{(3)} & K_{79}^{(2)} + K_{79}^{(3)} & K_{710}^{(3)} & K_{711}^{(3)} & K_{712}^{(3)} \\ 0 & 0 & 0 & K_{84}^{(2)} & K_{85}^{(2)} & K_{86}^{(2)} & K_{87}^{(2)} + K_{87}^{(3)} & K_{88}^{(2)} + K_{88}^{(3)} & K_{89}^{(2)} + K_{89}^{(3)} & K_{810}^{(3)} & K_{811}^{(3)} & K_{812}^{(3)} \\ 0 & 0 & 0 & K_{94}^{(2)} & K_{95}^{(2)} & K_{96}^{(2)} & K_{97}^{(2)} + K_{97}^{(3)} & K_{98}^{(2)} + K_{98}^{(3)} & K_{99}^{(2)} + K_{99}^{(3)} & K_{910}^{(3)} & K_{911}^{(3)} & K_{912}^{(3)} \\ 0 & 0 & 0 & 0 & 0 & 0 & K_{107}^{(3)} & K_{108}^{(3)} & K_{109}^{(3)} & K_{1010}^{(3)} & K_{1011}^{(3)} & K_{1012}^{(3)} \\ 0 & 0 & 0 & 0 & 0 & 0 &K_{117}^{(3)} & K_{118}^{(3)} & K_{119}^{(3)} & K_{1110}^{(3)} & K_{1111}^{(3)} & K_{1112}^{(3)} \\ 0 & 0 & 0 & 0 & 0 & 0 &K_{127}^{(3)} & K_{128}^{(3)} & K_{129}^{(3)} & K_{1210}^{(3)} & K_{1211}^{(3)} & K_{1212}^{(3)} \end{bmatrix} \cdot \begin{bmatrix} \overline{u}_1 \\ \overline{v}_1 \\ \overline{w}_1 \\ \overline{u}_2 \\ \overline{v}_2 \\ \overline{w}_2 \\ \overline{u}_3 \\ \overline{v}_3 \\ \overline{w}_3 \\ \overline{u}_4 \\ \overline{v}_4 \\ \overline{w}_4 \end{bmatrix} &= \begin{bmatrix} \overline{P}_{x1} \\ \overline{P}_{y1} \\ \overline{P}_{z1} \\ \overline{P}_{x2} \\ \overline{P}_{y2} \\ \overline{P}_{z2} \\ \overline{P}_{x3} \\ \overline{P}_{y3} \\ \overline{P}_{z3} \\ \overline{P}_{x4} \\ \overline{P}_{y4} \\ \overline{P}_{z4} \end{bmatrix} \end{aligned} \end{equation} K(All)⋅q(All) ​K11(1)​K21(1)​K31(1)​K41(1)​K51(1)​K61(1)​000000​K12(1)​K22(1)​K32(1)​K42(1)​K52(1)​K62(1)​000000​K13(1)​K23(1)​K33(1)​K43(1)​K53(1)​K63(1)​000000​K14(1)​K24(1)​K34(1)​K44(1)​+K44(2)​K54(1)​+K54(2)​K64(1)​+K64(2)​K74(2)​K84(2)​K94(2)​000​K15(1)​K25(1)​K35(1)​K45(1)​+K45(2)​K55(1)​+K55(2)​K65(1)​+K65(2)​K75(2)​K85(2)​K95(2)​000​K16(1)​K26(1)​K36(1)​K46(1)​+K46(2)​K56(1)​+K56(2)​K66(1)​+K66(2)​K76(2)​K86(2)​K96(2)​000​000K47(2)​K57(2)​K67(2)​K77(2)​+K77(3)​K87(2)​+K87(3)​K97(2)​+K97(3)​K107(3)​K117(3)​K127(3)​​000K48(2)​K58(2)​K68(2)​K78(2)​+K78(3)​K88(2)​+K88(3)​K98(2)​+K98(3)​K108(3)​K118(3)​K128(3)​​000K49(2)​K59(2)​K69(2)​K79(2)​+K79(3)​K89(2)​+K89(3)​K99(2)​+K99(3)​K109(3)​K119(3)​K129(3)​​000000K710(3)​K810(3)​K910(3)​K1010(3)​K1110(3)​K1210(3)​​000000K711(3)​K811(3)​K911(3)​K1011(3)​K1111(3)​K1211(3)​​000000K712(3)​K812(3)​K912(3)​K1012(3)​K1112(3)​K1212(3)​​ ​⋅ ​u1​v1​w1​u2​v2​w2​u3​v3​w3​u4​v4​w4​​ ​ ​K11(1)​K21(1)​K31(1)​K41(1)​K51(1)​K61(1)​000000​K12(1)​K22(1)​K32(1)​K42(1)​K52(1)​K62(1)​000000​K13(1)​K23(1)​K33(1)​K43(1)​K53(1)​K63(1)​000000​K14(1)​K24(1)​K34(1)​K44(1)​+K44(2)​K54(1)​+K54(2)​K64(1)​+K64(2)​K74(2)​K84(2)​K94(2)​000​K15(1)​K25(1)​K35(1)​K45(1)​+K45(2)​K55(1)​+K55(2)​K65(1)​+K65(2)​K75(2)​K85(2)​K95(2)​000​K16(1)​K26(1)​K36(1)​K46(1)​+K46(2)​K56(1)​+K56(2)​K66(1)​+K66(2)​K76(2)​K86(2)​K96(2)​000​000K47(2)​K57(2)​K67(2)​K77(2)​+K77(3)​K87(2)​+K87(3)​K97(2)​+K97(3)​K107(3)​K117(3)​K127(3)​​000K48(2)​K58(2)​K68(2)​K78(2)​+K78(3)​K88(2)​+K88(3)​K98(2)​+K98(3)​K108(3)​K118(3)​K128(3)​​000K49(2)​K59(2)​K69(2)​K79(2)​+K79(3)​K89(2)​+K89(3)​K99(2)​+K99(3)​K109(3)​K119(3)​K129(3)​​000000K710(3)​K810(3)​K910(3)​K1010(3)​K1110(3)​K1210(3)​​000000K711(3)​K811(3)​K911(3)​K1011(3)​K1111(3)​K1211(3)​​000000K712(3)​K812(3)​K912(3)​K1012(3)​K1112(3)​K1212(3)​​ ​⋅ ​u1​v1​w1​u2​v2​w2​u3​v3​w3​u4​v4​w4​​ ​​=P(All)= ​Px1(1)​Py1(1)​Pz1(1)​Px2(1)​+Px2(2)​Py2(1)​+Py2(2)​Pz2(1)​+Pz2(2)​Px3(2)​+Px3(3)​Py3(2)​+Py3(3)​Pz3(2)​+Pz3(3)​Px4(3)​Py4(3)​Pz4(3)​​ ​= ​Px1​Py1​Pz1​Px2​Py2​Pz2​Px3​Py3​Pz3​Px4​Py4​Pz4​​ ​​​​

整体刚度矩阵的组合方式如下图所示

参考

1. 曾攀. 有限元分析基础教程[M]. 北京: 清华大学出版社, 2008.