Frame and distributed load 1 Consider the frame shown below

1. Consider the frame shown below. Assume that the modulus of Elasticity for both beam elements is E moment of inertia is I. Assume pure bending. The node at 2 is restricted to move in the vertical direction For this problem use the following matrix inverse identity ed] =ad-bel-c db a Solve the system for the unknowns, Outline the procedure and clearly describe your steps. (30 pts) Find the reactions at node 2 (10 pts)

Solution

Can try Finite element approach

We have a system with 3 nodes ( 1,2, 3) with zero moment at 2 ( pin joint) and displacement only in y direction

Nodeal coordinate matrix nodel 1 1,0

node 2 0, L

node 3 L,L ( origin at lower left corner)

Direction cosines (l,m) for element 1 [ -.707/L , -.707/L]

DC for element 2 [ 1/L, 0 ]

Write stiffness matrices for elements 1 and 2, in terms of l,m

obtain 4X4 matrices for each element. (use bending matrix)

Assemble the Global Stiffness Matrix for the assemblage ( 6x6)

Now Global displacement vector { q1, Q2, q3, Q4, q5,Q6} where q1=0 (fixed for point 1), Q2` =1

q3 (for node 2) ( moves only in vertical and q4=0, no moment)

q5=q6 =0 ( cantilever end)

Applied Force vector is ( f1,f2,f3,f4,f5,f6) where we have zeros except for f3,f4,f5,f6

obtain these from the reactions`at the ends of element 1 and the b.m. at ends of element,=> f4=0

Set up the matrix eqn with given displacements and force moment vector. Using elimination method reduce to 2X2 matrix eqn, involving q2 and q3. (q2 is the displacment and q3 the slope at the moving end).

For a 2X2 matrix can use the invers ematrix identity shown.