$3$ Stiffness Method $-$ Truss. A simple truss is shown in Figure $4.$ The truss supports a sign

weighing $60kN$ at joint $1.$ Joints $2,3,4$ are pin connected to a larger structure. The

length of each truss element is given in the figure. Assume $AE$ is the same for members

$1$ and $3.$ Use $\frac{5}{13}AE$ for the vertical member. $(25$ pts$).$

A Create the member stiffness matrices for the three elements.

B Construct the structural stiffness matrix, displacement vector, and force vector for the

truss.

C Solve for the vertical and horizontal displacements at joint $1.$

D Determine the vertical and horizontal reactions at joints $2-4.$

E If joint $3$ had a vertical settlement of $0\mathrm{.}02m($downward$),$ explain how this would

change the calculation and the results $($no calculations required$).$

Figure $4$ Structural Stiffness Matrix for Truss. Reference Material

k=[AELcos2+12EIL3sin(AEL-12EIL3)cossin-6EIL2sin-(AELcos2+12EIL3sin2)-(AEL-12EIL3)cossin-6EIL2sinAELsin2+12EIL3cos26EIL2cos-(AEL-12EIL3)cossin-(AELsin2+12EIL3cos2)6EIL2cos6EIL2cos4EIL6EIL2sin-6EIL2cos2EIL-(AEL-12EL3)cossin6EIL2sinAELcos2+12EIL3sin2(AEL-12EIL3)cossin6EIL2sin-(AEL-12EIL3)cossin-(AELsin2+12EIL3cos2)-6EIL2cos(AEL-12EIL3)cossinAELsin2+12IL3cos-6EIL2cos

-6EIL2sin61L2cos2EIL6EIL2sin-6EIL2cos4EIL]

$-\frac{6EI}{L^2}sin$

$-(\frac{AE}{L}-\frac{12EI}{L^3})cossin$

$-(\frac{AE}{L}co{s}^2+\frac{12EI}{L^3}si{n}^2)$

$-\frac{6EI}{L^2}sin$

$-(\frac{AE}{L}-\frac{12EI}{L^3})cossin$

$-(\frac{AE}{L}co{s}^2+\frac{12EI}{L^3}si{n}^2)$

$-\frac{6EI}{L^2}sin$

$-\frac{6EI}{L^2}sin$

$-(\frac{AE}{L}-\frac{12EI}{L^3})cossin$

$-(\frac{AE}{L}co{s}^2+\frac{12EI}{L^3}si{n}^2)$

$-\frac{6EI}{L^2}sin$

$(\frac{AE}{L}-\frac{12EI}{L^3})cossin$

$k_{\text{t}\text{r}\text{u}\text{s}\text{s}}=\frac{AE}{L}\left[\begin{array}{ccccccccccccc}co{s}^2& sincos& -co{s}^2& -sincos& si{n}^2& -sincos& -si{n}^2& -sincos& cos2& sincos& -si{n}^2& sincos& si{n}^2\end{array}\right]$

$-sincos$

$-co{s}^2$

$sincos$

$k_{\text{b}\text{e}\text{a}\text{m}}=EI\left[\begin{array}{ccccccccccccc}\frac{12}{L^3}& \frac{6}{L^2}& -\frac{12}{L^3}& \frac{6}{L^2}& \frac{4}{L}& -\frac{6}{L^2}& \frac{2}{L}& -\frac{6}{L^2}& \frac{12}{L^3}& -\frac{6}{L^2}& \frac{2}{L}& -\frac{6}{L^2}& \frac{4}{L}\end{array}\right]$

$\frac{6}{L^2}$

$-\frac{12}{L^3}$

$\frac{6}{L^2}3$ Stiffness Method $-$ Truss. A simple truss is shown in Figure $4.$ The truss supports a sign

weighing $60kN$ at joint $1.$ Joints $2,3,4$ are pin connected to a larger structure. The

length of each truss element is given in the figure. Assume $AE$ is the same for members

$1$ and $3.$ Use $\frac{5}{13}AE$ for the vertical member. $(25$ pts$).$

A Create the member stiffness matrices for the three elements.

B Construct the structural stiffness matrix, displacement vector, and force vector for the

truss.

C Solve for the vertical and horizontal displacements at joint $1.$

D Determine the vertical and horizontal reactions at joints $2-4.$

E If joint $3$ had a vertical settlement of $0\mathrm{.}02m($downward$),$ explain how this would

change the calculation and the results $($no calculations required$).$

Figure $4$ Structural Stiffness Matrix for Truss$3$ Stiffness Method $-$ Truss. A simple truss is shown in Figure $4.$ The truss supports a sign

weighing $60kN$

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