The $3-$element, $4-$node frame structure shown in the figure below is to be analyzed, as per our discussion in class. All members of the structure are made of the same material $($E$)$ and have the same geometric properties A and I. The problem data is: A $=0\mathrm{.}01$ m$2,$ E $=760$ MPa, I $=8\mathrm{.}75\backslash$times $106$ m$4$ The elements are welded at nodes $1,2$ and $3.$ But at node $4,$ there is a pin joint. The linearly varying distributed force on element $3$ is acting normal to that element. The point forces of $50$ kN and $20$ kN are horizontal and vertical, respectively. $1.$ Compute the unknown generalized nodal displacements: U$5,$ U$6,$ U$12.$ Solution Hint: U$4=9\mathrm{.}57$ mm $2.$ Compute the unknown generalized reaction forces: Q$1,$ Q$2,$ Q$3,$ Q$8,$ Q$9,$ Q$10,$ Q$11.$ Solution Hint: Q$7=75063\mathrm{.}2$ N $3.$ Compute the bending moment at the midpoint location of element $3.$ The $3-$element, $4-$node frame structure shown in the figure below is to be analyzed, as per our discussion in class. All members of the structure are made of the same material $($E$)$ and have the same geometric properties A and I. The problem data is: A$=0\mathrm{.}01$ m$^2,$ E$=760$ MPa, I$=8\mathrm{.}75\backslash$times $10^-6$ m$^4$ The elements are welded at nodes $1,2$ and $3.$ But at node $4,$ there is a pin joint. The linearly varying distributed force on element $3$ is acting normal to that element. The point forces of $50$ kN and $20$ kN are horizontal and vertical, respectively.

The $3-$element, $4-$node frame structure shown in the figure below is to be analyzed, as per our discussion in class. All members of the structure are made of the same material $($ E $)$ and have the same geometric properties A and I. The problem data is:

$A=0\mathrm{.}01m^2,E=760$MPa,$I=8\mathrm{.}75{10}^{-6}{m}^4$

The elements are welded at nodes $1,2$ and $3.$ But at node $4,$ there is a pin joint. The linearly varying distributed force on element $3$ is acting normal to that element. The point forces of $75$ kN and $20$ kN are horizontal and vertical, respectively.

Problem $1$ Compute the unknown generalized nodal displacements: $U_5,U_6,U_{12}.$