Module 9: Matrix Stiffness Method - Examples

A bar has $A=2000\text{ mm}^2$, $E=200000\text{ N/mm}^2$, and $L=4000\text{ mm}$. Find $AE/L$.

Using $AE/L=100\text{ kN/mm}$, write the local two-degree stiffness matrix.

A planar truss element runs from $(0,0)$ to $(4,3)\text{ m}$. Find its direction cosines.

Two axial springs with stiffnesses $k_1=30\text{ kN/mm}$ and $k_2=50\text{ kN/mm}$ both resist the same scalar nodal displacement. What is the assembled diagonal stiffness at that DOF?

For the local axial-bar matrix

show that a rigid translation produces no resisting force.

The bar in Example 5 has node 1 fixed, so $d_1=0$, while node 2 is free. What reduced stiffness remains for $d_2$?

Solve

with stiffness in $\text{kN/mm}$ and force in $\text{kN}$.

For a restrained coordinate, the reaction equation is $Q_r=K_{rf}D_f+K_{rr}D_r$. Let $K_{rf}=-25\text{ kN/mm}$, $D_f=3.0\text{ mm}$, and $D_r=0$. Find $Q_r$.

A one-free-DOF partition has $K_{ff}=40\text{ kN/mm}$, $K_{fr}=-10\text{ kN/mm}$, applied free load $Q_f=60\text{ kN}$, and prescribed support displacement $D_r=-2.0\text{ mm}$. Find $D_f$.

A prismatic frame element of length $L=6.0\text{ m}$ carries $w=12\text{ kN/m}$ over the full span. Find the magnitude of each fixed-end moment used when forming consistent member-load effects.

A restrained frame model still produces a singular $K_{ff}$. One beam-to-column joint has rotational releases on every connected member, and no diagonal brace or rigid moment path restrains story sidesway. Identify the likely structural issue.