User:Eml5526.s11.team2.brenner/hwk3

=Problem 3.2: =

Given
For the spring system given in the figure.



Find
a) Number the elements and nodes. b) Assemble the global stiffness and force matrix. c) Partition the system and solve for the nodal displacements. d) Compute the reaction forces.

(a)
The elements and nodes can be represented in the following fashion.

(b)
Figure (3) shows a Free Body Diagram with the forces and reactions acting at each node.



Observing the FBD a system of equations can be defined in terms of the balance of forces for each element and the reactions and externals forces experienced by the system as the following column vectors; where $$ \displaystyle \mathbf{F^{(i)}} \quad i=1,\ldots,4\ $$ represents the appropriate element in the system, $$ \displaystyle \mathbf{f} $$ represents the external forces acting on the system and $$ \displaystyle  \mathbf{r} $$ represents the reactions of the system.

The column vectors are as follows

Substituing the above column vectors into (3.2.b.1) porvides the following system of equations.

The Forces experienced in a 2 node element can be defined with the following 2 equations in matrix form

where k is defined by the Area, Young's modulus and length of the element $$ \displaystyle k = AE/l $$

Employing the element stiffness found in (3.2.b.3) and aligning the element nodes to their respective global nodes as defined in Fig 3.

One begins by augmenting the force and displacement matrices by adding zeroes. The local element node number is equal to the global element node number by Fig 3. One performs this on all elements of the system. This provides each element of the system with respect to the global matrix of the system.

Substituting (3.2.b.4), (3.2.b.5), (3.2.b.6), (3.2.b.7) into (3.2.b.2) and performing matrix addtion provides the System of Equations governing the system, where the first matrix term on the LHS of (3.2.b.8) represents the global stiffness matrix and the RHS of (3.2.b.8) represents the global force matrix.

(c)
Reviewing Fig. 1 the springs are attached to the wall. It can be assumed that element 1 and element 3 will have no displacement pass global node 1 & 4 and global node 4 & 2, respectively. Therefore $$ \displaystyle u_1 = u_2 = 0 $$ which modifies 3.2.b.8 to the following

Doing a sum of forces at Node 3 and using (3.2.b.5) & (3.2.b.7) provides a relation between the global displacements experienced between at 3 & 4 and since $$ \displaystyle  u_1 = 0 $$

This reduces the nodal displacements to the following

(d)
Solving for the reaction forces using (3.2.b.9) & (3.2.b.11) leads to the following