Chap 5 Development of Beam Equations: Review Simple Beam Theory Two Methods To Develop Beam Equations

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Chapter 5 Development of Beam Equations
Chap 5 Development of Beam Equations

Review Simple beam theory Two methods to develop beam equations o Direct Stiffness Method o !otential ener"y approach
Use relationship between stress and strain and prin iple of minimum potential energ% to reate element equilibrium equations! "e tion 5#&$5#6 plus se tion 5#1 ex ept step '# "e tion 5#&$5#5 and most part of 5#1 are for both methods (rgani)e "e tions! 5#1* 5#6* 5#&$5#5
Use available relationship between internal and external for es to reate element equilibrium equations! "e tion 5#1$5#5
Sec# 5#$ Beam Stiffness
Review Mechanics of Materials Coordinates and si"n conventions see next page
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Beam in pure bendin"
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Curvature of a "eneral bent beam
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%eometrical relationships
+ x, =
1
+"ee the figure in previous page,
-pproximate for small deformation!

dv . + x, dx .
& &
. dv dx
Deformation relationships +"ee a figure in page &,

x =
y
= y
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!hysical &material' equations ((()sed in !otential Ener"y method /or small deformation* linear elasti materials /or one dimensional stress ondition +pure bending,!
x = E =
Ey
= Ey
Equilibrium equations based on 0e hani s of 0aterials ((()sed in Direct Stiffness Method /or pure bending!
+ x, =
1
M + x, EI
or
!rocedures to *btain the %lobal Equations by Direct Stiffness Method in +E,

6
dv M + x, . + x, = dx EI .
& &
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Step $# Select Element Type "imple Beam Elements! 1&D beam element2 ompare with 3D beam elements in 3D frames! % d1% d&% 1% 1x &% &x x 1) d1x &) d&x d1) 1 d&) & ) "ele t &D beam element! 3gnore axial for e* torsion* and out of plane shear and bending# ,ssumptions for -D Beam element "traight bar
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Constant ross se tion 3sotropi material 5wo nodes at ends &D element! Ea h node two D(/! di* i Ea h element! four D(/! d1* 1* d&* & Basic un.nowns! D(/ of displa ements! dy/ 0 +no dx* d)* x* %, 6o axial for e* torsion* and out of plane shear and bending and related displa ements# 1ariables and local coordinates 2/ y &or v' 7ateral displa ements! v+x, Distributed lateral load! w+x,
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Step -# Select a Displacement +unction
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E2press the displacement function in terms of interpolatin" Shape +unctions in matri2 form
.< = ; N + x , v + x, = ; N :=d
1
N + x,
&
N + x,
3
/ield displa ement fun tion

1 1y & 1 3
?redefined "hape /un tion for ea h un@nown D(/# 6i is onl% fun tion of x.
&y
. d . N + x ,: . d .
'
1y
&y
&
Un@nown D(/2s for ea h element
. + N + x , . . + N + x ,d v + x , = N + x,d
. + N + x ,
'
&
%eometric minin" of these four Shape +unctions for four D*+3s

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Step 4# Define the Strain5Displacement and Stress5Strain Relationships Direct Stiffness Method! Based on the Simple Beam Theory! 5he relationships between the internal for es and deformations are! 0oment! M&2' 6 E7 &d-v5d2-' "hear! 1&2' 6 E7 &d4v5d24' 0+x, and A+x, are depend on displa ement v+x, -pproximate displa ement fun tion v+x,!
are fun tion of shape fun tions and un@nown nodal displa ements
61+x,* 6&+x,* 63+x,* 6'+x, are fun tions of x d1 * 1 * d & * &
are fixed un@nown values

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Step 8# Derive the Element Stiffness Matri2 and Equations By Direct Stiffness Method
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1'
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Sec# 5#9 !otential Ener"y ,pproach to Derive Beam Element Equations

Concepts
( Discrete +inite Element Method ( Strain5Displacement of Stress5Strain Relations for simple beams d v . d v . = y = Ey . . or dx . dx . ( +unctional and 1ariational Methods 0inimum ?otential Energ% 0ethod! +page '8,p 6 ) :
& & x & x &
(f all the displa ements that satisf% the given boundar% onditions of a stru ture* those that satisf% the equations of equilibrium are distinguishable b% a stationar% value of the potential energ%# 3f the stationar% value is a minimum* the equilibrium state is stable#
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( ;umerical 7nterpolation Methods ;6:
!otential Ener"y ,pproach for a Beam Element

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-# ,ssume the displacement pattern to vary with a finite set of

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undetermined parameters ( un.nown nodal displacements

5he on ept of "hape /un tion is the same as that in "e # 5#1!
v+x, B a1x3Ca&x&Ca3xCa' B ;6:=d< B ;61 6& 63 6':;d1% 1 d&% &:5 B 61d1%C6&1C63d&%C6'& B +1D73,+&x3$3x&7C73,d1%C+1D73,+x37$&x&7&Cx73, 1 C+1D73,+$&x3C3x&7,d&%C+1D73,+x37$x&7&, & B v +x* d1%* 1*d&%* &,
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4# E2press the total potential ener"y in accordance with the displacement pattern
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8# Minimi0e the total potential ener"y with respect to un.nown nodal displacements to obtain the element equations and stiffness matri2es
;@:=d<B=f<
&>
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Step 5# ,ssemble the Element Equations to *btain the %lobal Equations and 7ntroduce Boundary Conditions Example! elements* D(/2s* oordinates* element stiffness matrix!
Sec# 5#- E2ample of ,ssembla"e of Beam Stiffness Matrices
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Elobal equilibrium equations* boundar% onditions* final ompressed equilibrium equations!
&&
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Sec# 5#4 E2amples of Beam ,nalysis )sin" the Direct Stiffness +E, Method
"ee Example 5#1 on page 159! ? 1 7 & 7 3 E3 onstant
Element "tiffness 0atrix! developed b% dire t stiffness method or b% potential energ% approa h is the same# -ssumable element stiffness matrixes into global stiffness matrix are the same as Example in "e # 5#&# (nl% load and boundar% onditions are not the same! Use the load /1%B $?* and boundar% onditions! d&%B>* d3%B>* 3B>#
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"olve equilibrium equations* get global un@nown nodal 4 PL 3PL PL = = displa ements! d = 1& EI ' EI ' EI Determine global nodal for es and moments b% "lobal full stiffness matrix and equilibrium equations in global oor2s!
3 & 1y &
&'
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Determine element for es b% element stiffness matrix and equilibrium equations in local oordinates!
&5
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"ign onvention for element for es and internal for es! 3nterpreter internal for e results!
&6
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5#8 Distributed <oadin" ( Equivalent ;odal +orces

Fe all "overnin" equations of stiffness method! =>? @dA 6 @+A $ Basi un@nowns! ;odal Displacement @dA $ 0eaning of @+A ;odal +orces
Conversion! Distributed <oadin" ;odal +orces Static equivalent Dire t "tiffness 0ethod Bor. equivalent ?otential Energ% 0ethod
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1# "tati Equivalent $ +i2ed(end Reaction Method!
$ /ixed$end Fea tions! B 5he rea tions at the ends of an element if the ends of the element are assumed to be fixed# $ "tati Equivalent /or es! B 5he nodal for es have the same effe t on the element as the a tual distributed load# $ "tati Equivalent /or es B $ +/ixed$end Fea tions, $ "ee tabulate stati equivalent for es in -ppendix D
3# Gor@ Equivalent $ 1irtual Bor. Method!
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$ Concept
?urpose! 5o repla e a distributed load b% a set of dis rete loads at nodal D(/2s# Concept of Equivalent 1irtual wor. of the actual distributed load is equal to that of the discrete load repla ement for arbitrary virtual nodal displa ement#
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$ Method
Fe all Gor@ done b% -pplied 7oad in ?otential Energ% 0ethod! Gor@ done b% the distributed load!
W
distributed
= w+ x . ,v .+ x . , dx .
>
/or uniform w, w+x,=w* and
here
v .+ x . , = ; N :=d < = ; N + x ,
1
N + x,
&
N + x,
3
/or wor@ done b% Discrete load 3t needs

W
discrete
. d . N + x ,: . d .
'
1y
&y
&
. * . * . *d ., = W +d
1y 1 &y & 1 1y 1 1
or
Wdis rete B
.d . +m .d . . +f f .
&
&y
. +m .
&
&
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. * . * . *d . in Wdistributed B% reorgani)ing the oeffi ients of d Eet equivalent for es

1y 1 &y &
Equivalent forces
. f 1
.1 m
. f &
.& m
8# %eneral +ormulation for %lobal ;odal +orce C Equivalent ;odal +orces +see page 14&,
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5#5 Beam Element with ;odal Din"e E Modified Stiffness Matri2
Step $ Startin" with the "enerali0ed unreleased beam stiffness matri2
3&
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Step - 7mpose the hin"e condition
m . =>
&
&
. . Condense out the de"ree of freedom associated with m . Eliminate the de"ree of freedom by partitionin"
&
&
=>
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Step 4 *btain the element equations &force(displacement equations' with the hin"e at node - and the element stiffness matri2
3'
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5#F %aler.in3s Method to Derive Beam Element Equations
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Chap 5 Development of Beam Equations: Review Simple Beam Theory Two Methods To Develop Beam Equations

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Chapter 5 Development of Beam Equations