Our firm has designed a large pipeline along the edge of a canal in Singapore. Our team has been tasked with designing a structure to support the pipe. Other engineers on the team have designed the truss members for strength. Because we are new to the firm, our supervisor wants to verify our

Question

Our firm has designed a large pipeline along the edge of a canal in Singapore. Our team has been tasked with designing a structure to support the pipe. Other engineers on the team have designed the truss members for strength.

Because we are new to the firm, our supervisor wants to verify our knowledge of the Direct Stiffness Method. A determinate truss is a great way to do that.

We are at the point where we can compile the system of equations for the truss. Following our office's standard convention, we numbered the DOFs moving alphabetically through the nodes. In Example 14.1, we chose the near node for each element; the local x'-direction is shown near the middle of each element.

Accepted Answer

Structural Stiffness Matrix:
Element stiffness matrices with DOFs noted and partitioned into pairs of DOFs:

[latex][K_{AB}] = \begin{matrix} \1 \ 2 \ 3 \ 4 \end{matrix} \begin{array}{r c}\begin{matrix}1&&&&2&&&&3&&&&4&\end{matrix} \\left[\begin{array}{cc:cc}4840 \ \mathrm{kN} / \mathrm{cm} &0& -4840 \ \mathrm{kN} / \mathrm{cm} & 0 \0 & 0 & 0 & 0 \\hdashline-4840 \ \mathrm{kN} / \mathrm{cm} & 0 & 4840 \ \mathrm{kN} / \mathrm{cm} & 0 \0 & 0 & 0 & 0\end{array} ight]\end{array}[/latex]

[latex][K_{AB}] = \begin{matrix} \1 \ 2 \ 7 \ 8 \end{matrix} \begin{array}{ r c}\begin{matrix}1&&&&&&&2&&&&&&&&7&&&&&&8&& &&\end{matrix} \\left[\begin{array}{cc:cc}940.4 \ \mathrm{kN} / \mathrm{cm} & 806.1 \ kN/cm & -940.4 \ \mathrm{kN} / \mathrm{cm} &-860.1 \ kN/cm \860.1 \ kN/cm & 690.9 \ kN/cm & -806.1 \ kN/cm & -690.9 \ kN/cm \\hdashline -940.4 \ \mathrm{kN} / \mathrm{cm} & -806.1 \ kN/cm & 940.4 \ \mathrm{kN} / \mathrm{cm} & 806.1 \ kN/cm \-806.1 \ kN/cm & -690.9 \ kN/cm & 806.1 \ kN/cm & 690.9 \ kN/cm \end{array} ight]\end{array}[/latex]

[latex][K_{BC}] = \begin{matrix} \3 \ 4 \ 5 \ 6 \end{matrix} \begin{array}{r c}\begin{matrix}&3&&&&4&&&5&&&&&6&\end{matrix} \\left[\begin{array}{cc:cc}5647 \ \mathrm{kN} / \mathrm{cm} & 0 & -5647 \ \mathrm{kN} / \mathrm{cm} & 0 \0 & 0 & 0 & 0 \\hdashline -5647 \ kN/cm & 0 & 5647 \ kN/cm & 0\0 & 0 & 0 & 0 \end{array} ight]\end{array}[/latex]

[latex][K_{BD}] = \begin{matrix} \3 \ 4 \ 7 \ 8 \end{matrix} \begin{array}{r c}\begin{matrix} 3&&&&4&&&&7&&&8&&&&\end{matrix}\ \left[\begin{array}{cc:cc}0 & 0 & 0 & 0 \0 & 5647 \ kN/cm & 0 & -5647 \ kN/cm \\hdashline 0 & 0 & 0 & 0 \0 & -5647 \ kN/cm & 0 & 5647 \ kN/cm \end{array} ight]\end{array}[/latex]

[latex][K_{CD}] = \begin{matrix}\ 5 \ 6 \ 7 \ 8 \end{matrix}\begin{array}{r c} \begin{matrix} 5&&&&&&6&&&&&&&&&7&&&&&&8&&&& \end{matrix}\ \left[\begin{array}{cc:cc}886.1 \ kN/cm & -886.1 \ kN/cm & -886.1 \ kN/cm & 886.1 \ kN/cm \-886.1 \ kN/cm & 886.1 \ kN/cm & 886.1 \ kN/cm & -886.1 \ kN/cm \\hdashline -886.1 \ kN/cm & 886.1 \ kN/cm & 886.1 \ kN/cm & -886.1 \ kN/cm \886.1 \ kN/cm & -886.1 \ kN/cm & -886.1 \ kN/cm & 886.1 \ kN/cm \end{array} ight]\end{array}[/latex]

Compile the terms to form the structural stiffness:

Work in 2 x 2 chunks.
Elements AB and AD share DOFs 1 and 2.

[latex]K_{11} = k_{11}^{A B} + k_{11}^{A D} = +4840 \mathrm{kN} / \mathrm{cm} + 940.4 \mathrm{kN} / \mathrm{cm} = 5780 \mathrm{kN} / \mathrm{cm}[/latex]

[latex]K_{12} = k_{12}^{A B} + k_{12}^{A D} = 0 + 806.1 \mathrm{kN} / \mathrm{cm} = 806.1 \mathrm{kN} / \mathrm{cm}[/latex]

[latex]K_{21} = k_{21}^{A B} + k_{21}^{A D} = 0 + 806.1 \mathrm{kN} / \mathrm{cm} = 806.1 \mathrm{kN} / \mathrm{cm}[/latex]

[latex]K_{22} = k_{22}^{A B} + k_{22}^{A D} = 0 + 690.9 \mathrm{kN} / \mathrm{cm} = 690.9 \mathrm{kN} / \mathrm{cm}[/latex]

Only element AB has DOF pairs 1 and 2 with 3 and 4.

[latex]K_{13} = k_{13}^{A B} = -4840 \mathrm{kN} / \mathrm{cm}[/latex]

[latex]K_{14} = k_{14}^{A B} = 0[/latex]

[latex]K_{23} = k_{23}^{A B} = 0[/latex]

[latex]K_{24} = k_{24}^{A B} = 0[/latex]

No element has D O F pairs 1 and 2 with 5 and 6.

[latex]K_{15} = K_{16} = K_{25} = K_{26} = 0[/latex]

Only element AD has DOF pairs 1 and 2 with 7 and 8.

[latex]K_{17} = k_{17}^{A D} = -940.4 \mathrm{kN} / \mathrm{cm}[/latex]

[latex]K_{18} = k_{18}^{A D} = -806.1 \mathrm{kN} / \mathrm{cm}[/latex]

[latex]K_{27} = k_{27}^{A D} = -806.1 \mathrm{kN} / \mathrm{cm}[/latex]

[latex]K_{28} = k_{28}^{A D} = -690.9 \mathrm{kN} / \mathrm{cm}[/latex]

All three elements share DOFs 3 and 4.

[latex]K_{33} = k_{33}^{A B} + k_{33}^{B C} + k_{33}^{B D} = 4840 \mathrm{kN} / \mathrm{cm} + 5647 \mathrm{kN} / \mathrm{cm} + 0 = 10,487 \mathrm{kN} / \mathrm{cm}[/latex]

[latex]K_{34} = k_{34}^{A B} + k_{34}^{B C} + k_{34}^{B D} = 0 + 0 + 0 = 0[/latex]

[latex]K_{43} = k_{43}^{A B} + k_{43}^{B C} + k_{43}^{B D} = 0 + 0 + 0 = 0[/latex]

[latex]K_{44} = k_{44}^{A B} + k_{44}^{B C} + k_{44}^{B D} = 0 + 0 + 5647 \mathrm{kN} / \mathrm{cm} = 5647 \mathrm{kN} / \mathrm{cm}[/latex]

Continue the process for all 64 entries in [K].

[latex][K] = \begin{matrix} \1 \ 2 \3 \ 4 \ 5 \ 6 \ 7 \ 8 \end{matrix}\begin{array}{r c}\begin{matrix}1&&&&&2&&&&&3&&&&4&&&&&5&&&&&6&&&&&7&&&&&8&& \end{matrix} \ \left[\begin{array}{cccccccc}5780 \frac{\mathrm{kN}}{\mathrm{cm}} &806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -4840 \frac{\mathrm{kN}}{\mathrm{cm}} &0& 0 & 0 & -940.4 \frac{\mathrm{kN}}{\mathrm{cm}}& -806.1 \frac{\mathrm{kN}}{\mathrm{cm}} \806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 690.9 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 0 & 0 & 0 & -806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -690.9 \frac{\mathrm{kN}}{\mathrm{cm}} \-4840 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 10487 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & -5647 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 0 & 0 \0 & 0 & 0 & 5647 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 0 & 0 & -5647 \frac{\mathrm{kN}}{\mathrm{cm}} \0 & 0 & -5647 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 6533 \frac{\mathrm{kN}}{\mathrm{cm}} & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} \0 & 0 & 0 & 0 & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} \-940.4 \frac{\mathrm{kN}}{\mathrm{cm}} & -806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 0 & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 1826.5 \frac{\mathrm{kN}}{\mathrm{cm}} & -80.0 \frac{\mathrm{kN}}{\mathrm{cm}} \-806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -690.9 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & -5647 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -80.0 \frac{\mathrm{kN}}{\mathrm{cm}} & 7224 \frac{\mathrm{kN}}{\mathrm{cm}}\end{array} ight]\end{array}[/latex]

Evaluation of Results:
Satisfied Fundamental Principles?
The matrix is symmetric about the diagonal, as expected. ✓

Load Vector:
Three unknown reactions:

[latex][Q] = \begin{matrix} 1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \end{matrix}\begin{bmatrix} 0 \ -300 kN & \ 0 \ B_y \ C_x \ C_y \ 0 \ 0 \end{bmatrix}[/latex]

Deformation Vector:
Five unknown displacements:

[latex][D] = \begin{matrix} 1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \end{matrix}\begin{bmatrix} \Delta_{Ax} \ \Delta_{Ay} \ \Delta_{Bx} \ 0 \ 0 \ 0 \ \Delta_{Dx} \ \Delta_{Dy} \end{bmatrix}[/latex]

System of Equations:

[latex]\begin{matrix} 1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \end{matrix}\begin{bmatrix} 0 \ -300 kN & \ 0 \ B_y \ C_x \ C_y \ 0 \ 0 \end{bmatrix} =\begin{array}{r c}\begin{matrix}&&1&&&&&2&&&&3&&&&4&&&&&5&&&&&6&&&&&7&&&&8&&&&&&& \end{matrix}\\left[\begin{array}{cccccccc}5780 \frac{\mathrm{kN}}{\mathrm{cm}} &806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -4840 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 &0 & 0 & -940.4\frac{\mathrm{kN}}{\mathrm{cm}} & -806.1 \frac{\mathrm{kN}}{\mathrm{cm}} \806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 690.9 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 0 & 0 & 0 & -806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -690.9 \frac{\mathrm{kN}}{\mathrm{cm}} \-4840 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 10487 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & -5647 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 0 & 0 \0 & 0 & 0 & 5647 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 0 & 0 & -5647 \frac{\mathrm{kN}}{\mathrm{cm}} \0 & 0 & -5647 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 6533 \frac{\mathrm{kN}}{\mathrm{cm}} & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} \0 & 0 & 0 & 0 & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} \-940.4 \frac{\mathrm{kN}}{\mathrm{cm}} & -806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & 0 & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & 1826.5 \frac{\mathrm{kN}}{\mathrm{cm}} & -80.0 \frac{\mathrm{kN}}{\mathrm{cm}} \-806.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -690.9 \frac{\mathrm{kN}}{\mathrm{cm}} & 0 & -5647 \frac{\mathrm{kN}}{\mathrm{cm}} & 886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -886.1 \frac{\mathrm{kN}}{\mathrm{cm}} & -80.0 \frac{\mathrm{kN}}{\mathrm{cm}} & 7224 \frac{\mathrm{kN}}{\mathrm{cm}}\end{array} ight] \begin{bmatrix} \Delta_{Ax} \ \Delta_{Ay} \ \Delta_{Bx} \ 0 \ 0 \ 0 \ \Delta_{Dx} \ \Delta_{Dy} \end{bmatrix}\end{array}[/latex]

Question 14.2: Our firm has designed a large pipeline along the edge of a c...

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