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CEE 371: Modeling of Structural Systems Spring 2004
PROBLEM SET NUMBER 8a
Date due: Wednesday, April 14, 2004, at the beginning of
lecture.
NOTES
1. If you receive help on a Problem Set, acknowledge this
assistance near the top of the firstpage of your written submission
with a statement such as Assistance received from: [listof
names].
READING ASSIGNMENTAC10:5 and Lecture Notes on the Stiffness
Method
LEARNING OBJECTIVES1. Students should understand the need for
local and global coordinate systems in structural
analysis.2. Students should understand the concept of a
transformation matrix, and be able to
transform stiffness equations for an axial force element written
in an element local systeminto a global coordinate system.3.
Students should understand the equilibrium basis of the structural
stiffness equations
{P} = [K]{}.4. Students should understand the Direct Stiffness
Method of structural analysis and be
familiar with the vocabulary and definitions that describe this
method.5. Students should know how to assemble the global stiffness
matrix, [K], and the global
load matrix {P} from contributions from each element.6. Students
should know how to partition matrix equations used in the Direct
Stiffness
Method in particular transformation equations and, to isolate
known and unknownquantities for solution, the structural stiffness
equations.
7. Students should be able to use the Direct Stiffness Method to
analyze truss structures with
a few kinematic unknowns by hand.
ASSIGNMENT1. Study the attached handout on Summary of Element
Transformation Matrices for the
Stiffness Method, which covers the transformation matrices
applicable to the elementstiffnesses you derived in Problem Set 7.
Use the definition of the inverse of a matrix, [A]-1[A]= [I], to
demonstrate that the transformation matrix for a beam-column
element is anorthogonal matrix, that is, that []-1= []T. Also show
that to demonstrate such orthogonalityis it sufficient to show that
the submatrix [] is orthogonal.
2. Evaluate numerically {Pf} and [Kff] for the three-bar truss
problem depicted on the next page.Use consistent units. Cross
sectional areas shown with each element are in mm2, and Eis
200kN/mm2 for all elements. You do not need to assemble the other
submatrices of the globalequilibrium equation, andyou do not need
to solve the stiffness equations. {Bonus question:Without doing any
calculations, what is the vertical reaction at node c?} [Hint: To
ease yourwork, for each element, use node a as the origin ofx'.
Then evaluate only as many kij's of eachstiffness matrix as you
will need for this limited assembly. Use the general form of the 4
by 4truss element stiffness matrix in global coordinates.] [MGZ
Problem 3.1(a)]
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CEE 371: Modeling of Structural Systems Spring 2004
SUMMARY OF ELEMENT TRANSFORMATION MATRICES
FOR THE STIFFNESS METHOD
Element transformation matrices, [], appear in the following
relationships:
Element nodal force transformation: { } [ ] { }F F
=
Element nodal displacement transformation: { } [ ] { } =
Element stiffness matrix transformation: [ ] [ ] [ ] [ ]T
k k=
In these equations, primed quantities are in element (local)
coordinates, while unprimedquantities are in global (structure,
overall) coordinates. The transformation matrices for
perpendicular coordinate are orthogonal, that is, [ ] [ ]1 T
= regardless of whether [] is
square or rectangular.
For a frame element at orientation (measured counterclockwise
from the x axis to the x'
axis) the specialized forms of [] for various elements studied
in this course are:
2-DOF truss element:[ ] cos sin 0 0
0 0 cos sin2 4
=
4-DOF truss element:[ ]
cos sin 0 0
sin cos 0 0
0 0 cos sin4 40 0 sin cos
=
4-DOF beam element: Not applicable (or [I]) because beams have
co-linearx and x'.
6-DOF beam-column element:[ ]
cos sin 0 0 0 0
sin cos 0 0 0 0
0 0 1 0 0 0
0 0 0 cos sin 06 6
0 0 0 sin cos 0
0 0 0 0 0 1
=
Note that each of these transformation matrices for two-node
elements can be partitioned andwritten in the general form:
[ ][ ] [ ]
[ ] [ ]
0
0
=
