100 Plane Trusses

8/13/2019 100 Plane Trusses

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Lecture 15: PLANE TRUSSES

Plane Truss Joint Stiffness Matrix

e now w s o ou ne e proce ure o ormu a ng e o n s ness ma r x J or a

plane truss structure. Consider an arbitrary member, i. in the generalized plane truss

depicted below:

.

in thex-yplane. The joint translations are the unknown displacements and these

displacements are expressed in terms of theirx andy components.


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The positive directions of the four displacement components (two translations at either

end) of member i are depicted in the figure below

It will be convenient to utilize the direction cosines associated with this arbitrary

member. In terms of the joint coordinates the direction cosines are

L

xx jk

X

=

1

L

yy jk

Y

=

2

( ) ( )22 jkjk yyxxL +=w t


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The beam member stiffness matrix developed in the previous section of notes can be

easily adapted for use in the case of a plane truss. The member stiffness matrix [SM] for

an arbitrar truss member with member axes and Y oriented alon the member and

perpendicular to the member can be obtained by considering Case #1 and Case #7 from

the previous section of notes.

Usin the numberin oint numberin s stem and the member axes de icted in the

following figure

then the member stiffness matrix for

a truss member is as follows


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Note that [SJ] is based on axes oriented to the

structure. Truss member stiffnesses may be

obtained in one of two wa s. Either the

stiffnesses are directly computed using the figure

to the left, or the second method consists of firstobtaining the stiffness matrix relative to the

mem er or en e axes an en mpos ng a

suitable matrix transformation that transforms

these elements to axes relative to the structure.

develop an intuition of how the structure

behaves. Unit displacement in both thex andy

directions are applied at each end of the member.If a unit displacement in thex direction is applied

to thej end of the member, the member shortens

and an axial compression force is induced. The

xx C

L

EA


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The restraint actions at the ends of the truss member in thex andy directions are required.

They are equal to the components of the axial force induced in the member, and are

ent e ere as e ements o t e [SMD] matr x n or er to st ngu s t em rom e ements

of the [SM] matrix. The numbering of these elements are shown in the previous figure.Thus

( )xx

MD

EA

CL

EAS

=

2

11

( )xx

MDMD

yxx

MD

C

EA

SS

CCL

S

==

=

2

1131

21

( )( )yxxMDMD CCL

EASS

== 2141


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In a similar fashion, a unit displacement in they direction at thej end of the member yields

( )( )12 yxxMD CCLEAS

=

( )222

x

yx

MD

CCEA

SS

CL

S

==

=

( )22242 yxMDMD

yx

C

L

EASS

L

==


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In a similar fashion, a unit displacement in thex direction at the kend of the member yields

( )x

x

MD

EA

CL

EAS

=

2

13

( )xxMDMD

yxMD

CL

EASS

L

==

=

21333

23

( )( )yxxMDMD CCL

EASS

== 2343


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In a similar fashion, a unit displacement in they direction at the kend of the member yields

( )( )14 yxxMD CCLEAS

=

( )224

x

yx

MD

CCEA

SS

CL

S

==

=

( )22444 yxMDMD

yx

C

L

EASS

L

==


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We have just developed the four rows of the [SMD] matrix, i.e.,

100 Plane Trusses

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