7/23/2019 Summary of Element Transformation Matrices

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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 t"e follo#ing relations"ips:

Element nodal force transformation: { }   [ ] { } F F ′

  = Γ 

Element nodal displacement transformation: { }   [ ] { }′∆ = Γ ∆

Element stiffness matri$ transformation: [ ] [ ] [ ] [ ]T 

k k ′= Γ Γ 

%n t"ese e&uations, primed &uantities are in element 'local( coordinates, #"ile unprimed&uantities are in glo)al 'structure, o*erall( coordinates+ "e transformation matrices for

 perpendicular coordinate are orthogonal, t"at is, [ ] [ ]1   T −

Γ = Γ  regardless of #"et"er [! is

s&uare or rectangular+

-or a frame element at orientation φ  'measured countercloc.#ise from t"e  x a$is to t"e  x/

a$is( t"e specialied forms of [! for *arious elements studied in t"is course are:

2- truss element:[ ] cos sin 0 0

0 0 cos sin2 4

φ φ 

φ φ 

Γ    = 

×  

4- truss element:[ ]

cos sin 0 0

sin cos 0 0

0 0 cos sin4 40 0 sin cos

φ φ 

φ φ 

φ φ φ φ 

−Γ  =

× −

4- )eam element: ot applica)le 'or [%!( )ecause )eams "a*e colinear  x and  x/+

5- )eamcolumn element:[ ]

cos sin 0 0 0 0

sin cos 0 0 0 0

0 0 1 0 0 0

0 0 0 cos sin 05 5

0 0 0 sin cos 0

0 0 0 0 0 1

φ φ 

φ φ 

φ φ 

φ φ 

− Γ 

=  ×

−

 ote t"at eac" of t"ese transformation matrices for straig"t t#onode elements can )e partitioned and #ritten in t"e general form:

[ ]  [ ] [ ]

[ ] [ ]

0

0

γ  

γ  

Γ = 

in #"ic" [γ! is t"e ort"ogonal nodal transformation for a single node and [0! is a null matri$+