User:Eml4500.f08.delta 6.krueger/Notes Monday September 15th

Global stiffness matrix K.


 * {| style="text-align:center; border-collapse:collapse;" cellpadding="4"


 * rowspan="6" | K  =
 * rowspan="6" | $$\left.\begin{align} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{align} \right($$
 * title="stiffness matrix from element one" style="border-left: 1px solid black; border-top: 1px solid black;" | $$k_{11}^{(1)}$$
 * title="stiffness matrix from element one" style="border-top: 1px solid black;" | $$k_{12}^{(1)}$$
 * title="stiffness matrix from element one" style="border-top: 1px solid black;" | $$k_{13}^{(1)}$$
 * title="stiffness matrix from element one" style="border-top: 1px solid black; border-right: 1px solid black;" | $$k_{11}^{(1)}$$
 * 0
 * 0
 * rowspan="6" | $$\left)\begin{align} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \end{align} \right.$$
 * title="stiffness matrix from element one" style="border-left: 1px solid black;" | $$k_{21}^{(1)}$$
 * title="stiffness matrix from element one" | $$k_{22}^{(1)}$$
 * title="stiffness matrix from element one" | $$k_{23}^{(1)}$$
 * title="stiffness matrix from element one" style="border-right: 1px solid black;" | $$k_{24}^{(1)}$$
 * 0
 * 0
 * title="stiffness matrix from element one" style="border-left: 1px solid black;" | $$k_{31}^{(1)}$$
 * title="stiffness matrix from element one" style="border-right: 1px solid black;" | $$k_{32}^{(1)}$$
 * title="stiffness matrix from element one and element two" style="border-top: 1px solid black;" | $$k_{33}^{(1)} + k_{11}^{(2)}$$
 * title="stiffness matrix from element one and element two" style="border-right: 1px solid black; border-top: 1px solid black;" | $$k_{34}^{(1)} + k_{12}^{(2)}$$
 * title="stiffness matrix from element two" style="border-top: 1px solid black;" | $$k_{13}^{(2)}$$
 * title="stiffness matrix from element two" style="border-right: 1px solid black; border-top: 1px solid black;" | $$k_{14}^{(2)}$$
 * title="stiffness matrix from element one" style="border-left: 1px solid black; border-bottom: 1px solid black;" | $$k_{41}^{(1)}$$
 * title="stiffness matrix from element one" style="border-right: 1px solid black; border-bottom: 1px solid black;" | $$k_{41}^{(1)}$$
 * title="stiffness matrix from element one and element two" style="border-bottom: 1px solid black;" | $$k_{43}^{(1)} + k_{21}^{(2)}$$
 * title="stiffness matrix from element one and element two" style="border-right: 1px solid black; border-bottom: 1px solid black;" | $$k_{44}^{(1)} + k_{22}^{(2)}$$
 * title="stiffness matrix from element two" | $$k_{23}^{(2)}$$
 * title="stiffness matrix from element two" style="border-right: 1px solid black;" | $$k_{24}^{(2)}$$
 * 0
 * style="border-right: 1px solid black;" | 0
 * title="stiffness matrix from element two" | $$k_{31}^{(2)}$$
 * title="stiffness matrix from element two" | $$k_{32}^{(2)}$$
 * title="stiffness matrix from element two" | $$k_{33}^{(2)}$$
 * title="stiffness matrix from element two" style="border-right: 1px solid black;" | $$k_{34}^{(2)}$$
 * 0
 * style="border-right: 1px solid black;" | 0
 * title="stiffness matrix from element two" style="border-bottom: 1px solid black;" | $$k_{41}^{(2)}$$
 * title="stiffness matrix from element two" style="border-bottom: 1px solid black;" | $$k_{42}^{(2)}$$
 * title="stiffness matrix from element two" style="border-bottom: 1px solid black;" | $$k_{43}^{(2)}$$
 * title="stiffness matrix from element two" style="border-right: 1px solid black; border-bottom: 1px solid black;" | $$k_{44}^{(2)}$$
 * }
 * 0
 * style="border-right: 1px solid black;" | 0
 * title="stiffness matrix from element two" | $$k_{31}^{(2)}$$
 * title="stiffness matrix from element two" | $$k_{32}^{(2)}$$
 * title="stiffness matrix from element two" | $$k_{33}^{(2)}$$
 * title="stiffness matrix from element two" style="border-right: 1px solid black;" | $$k_{34}^{(2)}$$
 * 0
 * style="border-right: 1px solid black;" | 0
 * title="stiffness matrix from element two" style="border-bottom: 1px solid black;" | $$k_{41}^{(2)}$$
 * title="stiffness matrix from element two" style="border-bottom: 1px solid black;" | $$k_{42}^{(2)}$$
 * title="stiffness matrix from element two" style="border-bottom: 1px solid black;" | $$k_{43}^{(2)}$$
 * title="stiffness matrix from element two" style="border-right: 1px solid black; border-bottom: 1px solid black;" | $$k_{44}^{(2)}$$
 * }
 * title="stiffness matrix from element two" style="border-bottom: 1px solid black;" | $$k_{43}^{(2)}$$
 * title="stiffness matrix from element two" style="border-right: 1px solid black; border-bottom: 1px solid black;" | $$k_{44}^{(2)}$$
 * }
 * }

The box, within the matrix on the top left, is the stiffness matrix from element one. The box to the lower right is the stiffness matrix from element two.

Where capital k (K) is the global stiffness coefficient and lowercase k (k) is the element stiffness coefficient. Below is the relationship between the global stiffness coefficients and the element stiffness coefficients for the global stiffness matrix above.

K11 = k11(1) ; K12 = k12(1) ; K13 = k13(1) ; K14 = k14(1) ; K15 = 0 ; K16 = 0

K21 = k21(1) ; K22 = k22(1) ; K23 = k23(1) ; K24 = k24(1) ; K25 = k25(1) ; K26 = k26(1)

K31 = k31(1) ; K32 = k32(1) ; K33 = k33(1) + k11(2) ; K34 = k34(1) + k12(2); K13 = k13(2) ; K14 = k14(2)

K41 = k41(1) ; K42 = k42(1) ; K43 = k43(1) + k21(2) ; K44 = k44(1) + k22(2); K23 = k23(2) ; K24 = k24(2)

K51 = 0 ; K52 = 0 ; K53 = k31(2) ; K54 = k32(2) ; K55 = k33(2) ; K56 = k34(2)

K61 = 0 ; K62 = 0 ; K63 = k41(2) ; K64 = k42(2) ; K65 = k43(2) ; K66 = k44(2)

Below are selected sample calculations of the global stiffness matrix.

$$K_{11}=\frac{9}{16}$$

$$K_{12}=\frac{3\sqrt{3}}{16}$$

$$K_{33}=K_{33}^{\left(1\right)}+K_{11}^{\left(2\right)}=\frac 9{16}+\frac 52=\frac{49}{16}=3.0625$$

$$K_{34}=K_{33}^{\left(1\right)}+K_{11}^{\left(2\right)}=\frac{3\sqrt 3}{16}+\left(-\frac 52\right)\approx -2.1752$$

$$K_{44}=K_{44}^{\left(1\right)}+K_{22}^{\left(2\right)}=\frac 3{16}+\frac 52=2.6875$$

4) Elimination of known degrees of freedom (dofs).
This Process reduces the global force displacement relationship by eliminating the first and last two columns of the global stiffness matrix.

d1 = d2 = d5 = d6 = 0

The displacements above are zero. Therefore, columns 1, 2, 5 and 6 of the global stiffness matrix go to zero. This reduces the matrix to a 6x2.