ALPHA COLLEGE OF ENGINEERING
THIRUMAZHISAI, CHENNAI – 600124 DEPARTMENT OF MECHANICAL ENGINEERING
ME 6603 - FINITE ELEMENT ANALYSIS UNIT WISE IMPORTANT FORMULAE UNIT – I (INTRODUCTION) 1. Initial and Bounda! "alu# Po$l#%&' (i) W#n to oot& (% 1* %+) a# #al and un,#-ual T# Co%l#%#nta! /un0tion !() 2 C1#%13 C+#%+ (ii) W#n to oot& (% 1* %+) a# #al and #-ual (%12%+2%) T# Co%l#%#nta! /un0tion !() 2 (C 13 C+) #% (iii) W#n to oot& a# a"in4 #al and i%a4ina! at (56i7) T# Co%l#%#nta! /un0tion !() 2 # 5 (C1Co&73 C+Sin7) +. W#i4t#d R#&idual M#tod& (i) Point Collo0ation %#tod R#&idual(R) 2 8 (ii) Su$,do%ain Collo0ation %#tod 9 R d 2 8 (iii) L#a&t S-ua# M#tod 9 R+ d 2 8 (o) 9 (o) 9 R (dR:da)d 2 8 1
(i") ;al#
=.
Ra!l#i4,Rit> M#tod
Total ot#ntial #n#4! 2 Stain En#4! – Wo< don# $! #t#nal /o0#&
?2U–@
(i)
Fo $#a% o$l#%* U 2 (EI:+) 9 (d+!:d+)+d @ 2 9 ! d (udl load) @ 2 W !%a (oint load) ! 2 a1 Sin (?:l) 3 a+ Sin (=?:l) B.M (M)2 EI (d+!:d+)
(ii) Fo $a Po$l#%* U 2 (EA:+) 9 (du:d)+d @ 2 Fu (o) Pu ! 2 a83a13a++ (iii) Fo Sin4 Po$l#% U2 @ 2 Fu
+
(W##*
2
2 u+,u1)
UNIT – II ONE DIMENSIONAL FINITE ELEMENT ANALSIS 1,D BAR ELEMENT* 1. Lin#a Pol!no%ial E-uation, u = ao+a1x
+. Sa# /un0tion& N1 =
l − x l
x N = l 2
3. St!!"#$$ %&t'(, ❑
[ K ] =∫ [ B ]T [ D ] [ B ] dv V
)*#'#, [B] [D]
→ Strain displacement relationship matrix. →
Elasticity matrix or Stress-strain relationship matrix.
[D] = [E] = E = Young’s modulus. dv = dx
∴
Sti!!ness matrix
[ K ] = AE l
[− − ] 1
1
1
1
4. G#"#'&+ FEA #&t" $, "#$ = [%] "u$
)*#'#,
"#$ is an element !orce &ector ['olumn matrix]. [%] is a sti!!ness matrix [(o) matrix]. "u$ is a nodal displacement ['olumn matrix]. ⇒
{ } [
]{ }
F 1 AE 1 −1 u 1 = F 2 l −1 1 u 2
/. 1D D$+&#%#"t #&t", u= N 1 u 1 + N 2 u 2
6. F'# #t' # t $#+! #5*t, 3
[ Fe ] = ρAl 2
. R#&t" !'#,
{} 1 1
[ R ] =[ K ] {u }−{ F }
7. St'#$$, σ = E
du dx
*here+ E = Young’s modulus
du dx =
u 2 −u 1 l
8. T#%#'&t'# #!!#t, #orce+
{ }
{ F }= EAα ∆ T −1 1
Stress+
σ = E
( )
du − Eα ∆ T dx
*here+ = rea o! cross section o! ,ar element. = emperature di!!erence. / = 'oe!!icient o! thermal expansion.
TRUSS ELEMENTS, 1. St!!"#$$ %&t'(,
[
l ² lm −l ² −lm lm m ² −lm −m ² [ K ] = AE ¿ −l ² −lm l ² lm −lm −m ² lm m ²
)*#'#, = rea o! the truss element 4
]
E = Young’s modulus o! element le = E0ui&alent length x 2 − xı l=cosθ =
¿
m=sin θ =
y 2− yı
¿
¿= √ ( x 2− x 1 ) +( y 2 − y 1 ) ² 2
2. St'&" #"#'59, U =
1 2
{u }T {u } [ K ]
3. F"t# #+#%#"t 5#"#'&+ #&t", { F }=[ K ] {u } )*#'#, [%] = sti!!ness matrix "1$ = nodal displacement matrix
4. St'#$$,
E σ = [ −l −m l m ]
¿
{} u1 u2 u3 u4
SPRINGS 1. St!!"#$$ %&t'(,
[− ]
[ K ] =k 2. T#"$+# !'#,
1
1
−1 1
T =k . ∆ u
*here+ = spring constant ∆u
= change in de!ormation
∆ u =u 2− u 1
:EAMS 5
1. S*&# !"t"$, 1 N = ( 2 x −3 x + x ) 3
1
N 2= N 3=
1
3
1
2
3
3
3
( x − 2 x + x ) 3
2
2
3
(−2 x +3 x ) 3
1
2
N 4 = 3 ( x − x 3
2
2
)
2. St!!"#$$ %&t'(,
[ K ] = E!
[
12
6
6
4 ²
−12 −6
6 2 ²
3 12 −6 − 12 −6 6 2 ² −6 4 ²
]
3. F"t# #+#%#"t #&t", { F }=[ K ] {u }
⇒
{} [ F 1 y m1 F 2 y m2
E!
=
12
6
6
4 ²
−12 −6
6 2 ²
]{ } d1 y ∅1
3 −12 −6 12 −6 d2 y 6 2 ² −6 4 ² ∅2
)*#'#, 3 = length o! the ,eam element E = Young’s modulus 4 = 5oment o! inertia
LONGITUDINAL ; TRANS
6
ONE DIMENSIONAL HEAT TRANSFER PRO:LEMS 1. F"t# E+#%#"t E&t" F' 1D H#&t C"t" t* !'## #" C"#t"
2. F"t# E+#%#"t E&t" F' 1D H#&t C"t", C"#t" &" I"t#'"&+ H#&t G#"#'&t" 7
UNIT III (+D SCALAR ARIABLE PROBLEMS CONSTANT TRIANGULAR ELEMENT =CST>,
1. S*&# !"t"$, "1 +¿ # x +$ y 1
1
2 A
N 1=¿ "2 +¿ # x +$ y 2
2
2 A
N 2=¿ "3 +¿ # x +$ y 3
3
2 A
N 3=¿
8
*here+
"1= x 2 y 3 − y 2 x 3 "2= x 3 y 1 − y 3 x 1 "3= x 1 y 2 − y 1 x 2 #1 = y 2− y 3 # 2= y 3− y 1 #3 = y 1− y 2 $ 1= x 3− x 2 $ 2= x 1− x 3 $ 3= x 2− x 1
A =
1 2
| | 1 1 1
x 1 y 1 x 2 y 2 x 3 y 3
2. D$+&#%#"t !"t"$,
[
u=
N 1
0
0
N 1
N 2 0
0
N 2
N 3 0
0
N 3
]
{} u1 v1 u2 v2
u3 v3
3. St!!"#$$ %&t'(, [%] = [B] [D] [B] t
4. St'&" $+&#%#"t %&t'(,
9
[ B ]=
1 2 A
[
#1
0
#2
0
#3
0
$1
0
$2
0
$1
#1 $ 2
#2
$3 # 3
0
$3
]
/. St'#$$ - $t'&" %&t'( " 5#"#'&+ 2D !'%,
D =
E
( 1 + v ) ( 1 −2 v )
[
−v
v
1
v
v v
−v
1
v
0
0
0
0
0
0
0
0
0
1
1
D=
E
−v
1
2
[ ] v
0
v
1
0
0
0
−v
1
2
P+&"# $t'&" "t",
D=
E
( 1 + v ) ( 1 −2 v )
[
−v
v
0
−v
0
1
v 0
1
0
]
−2 v
1
2
6. E+#%#"t $t'#$$, { σ } =[ D ] [ B ] {u }
⇒
{}
0
0
0
0
−2 v 2
P+&"# $t'#$$ "t", 1
0
0
−v
v
0
{} u1 v1
σ x u σ y =[ D ] [ B ] 2 v2 % xy u3 v3
10
1
0
0
0
0
−2 v
0
2
0
0
0
−v
1
2
]
M&(%% $t'#$$, σ x + σ y σ − σ σ m&x ¿ σ = + x y + % xy 1
√(
2
)
2
2
2
M"%% $t'#$$, σ x + σ y σ −σ σ m'( ¿ σ = − x y + % xy 2
√(
2
)
2
2
2
. P'"+# &"5+#, tan 2 θ "
=
2 % xy
σ x −σ y
7. E+#%#"t $t'&", { e }=[ B ] { u }
TEMPETATURE EFFECT OF CST ELEMENT, 1. I"t&+ $t'&", P+&"# $t'#$$,
{ }
α ) T { e 0 }= α ) T 0
P+&"# $t'&",
{ }
α ) T { e 0 }=(1 +v ) α ) T 0
2. E+#%#"t t#%#'&t'# !'#, { F }=[ B ]T [ D ] {e } A* 0
*here+ t = thicness = area o! the element.
2D HEAT TRANSFER PRO:LEMS St!!"#$$ M&t'( !' ?t* C"t" &" C"#t"
11
UNIT I< =2D
1
2 A
N 1=¿ α 2+¿ +
2
$+ , 2 -
2 A
N 2=¿ α 3+ ¿ +
3
$+ , 3 -
2 A
N 3 =¿
)*#'#, α 1=$ 2 - 3− $ 3 - 2 α 2=$ 3 - 1− $ 1 - 3
12
α 3=$ 1 - 2− $ 2 - 1 + 1= - 2− - 3 + 2= - 3− - 1 + 3= - 1− - 2 , 1=$ 3− $ 2 , 2=$ 1− $ 3 , 3=$ 2− $ 1
A =
1 2
| | 1 1 1
$ 1 - 1 $ 2 - 2 $ 3 - 3
2. St'&" $+&#%#"t %&t'(, + 1 0
+ 2
+3
0
0
α 1+¿ + $ +, 1
1
$
¿ α 2+¿ + $ +, -
¿0 ¿
α 3+ ¿ +
3
$
$+ , 3 -
2
2
0
$
¿0
¿
¿ 0
, 1
, 1 0 , 2 0 , 3 + 1 , 2 + 2 , 3 + 3
¿ ¿ [ B ] = 1 ¿ 2 A
3. St'#$$ $t'&" '#+&t" 13
[ D ] =
E
( 1 + v ) ( 1−2 v )
[
(1− v ) v v
v
0
0
v
v ( 1− v ) v
( 1− v ) 0 1− 2 v
0
2
0 0
]
4. St!!"#$$ %&t'(, [ K ] =2 .$A [ B ]T [ D ] [ B ]
TEMPETATURE EFFECT OF A($9%%#t' E+#%#"t, 3. I"t&+ $t'&",
{ e }= 0
{ } α ) T α ) T 0
α)T
4. E+#%#"t t#%#'&t'# !'#, { F }=[ B ]T [ D ] {e }2 $A 0
*here+ t = thicness = area o! the element.
UNIT ,ISOPARAMETRIC FORMULATION 14
I$ P&'&%#t' &'+&t#'&+ E+#%#"t 1. S*&# !"t"$, 1 N = [ 1−/ ] [ 1−0 ] 1
4
N 2= N 3= N 4 =
1
[ 1+ / ] [ 1 −0 ]
4 1
[ 1 +/ ] [1 + 0 ]
4 1
[ 1 −/ ] [ 1 + 0 ]
4
2. St'&" $+&#%#"t %&t'(,
[ B ]=
1 4
|1 |
[
1 22 0
−1
21
−1
0
0
− 1
12
0
1 11
1 22
][
1 11 2
21
1 12
−[ 1− 0 ] −[ 1−/ ]
3. D$+&#%#"t !"t", R#t&"5+&' #+#%#"t,
0 0
{} u1 v1
u=
[
N 1
0
0
N 1
N 2 0
0
N 2
N 3 0
0 N
4
N 3
0
0
]
u2 v2
N 4 u3 v3 u4 v4
u= N 1 u1 + N 2 u 2+ N 3 u3 + N 4 u 4 v = N 1 v 1+ N 2 v 2 + N 3 v 3 + N 4 v 4
15
[ 1− 0 ] 0 [ 1 +0 ] 0 −[ 1 +0 ] 0 −[ 1 + / ] 0 [ 1 + / ] 0 [ 1− / ] 0 0 −[ 1 −0 ] 0 [ 1−0 ] 0 [ 1 + 0 ] 0 −[ 1+ 0 ] −[ 1− / ] 0 −[ 1+ / ] 0 [ 1+ / ] 0 [ 1− / ] 0
]
{} x 1 y 1
u=
[
N 1
0
0
N 1
N 2 0
N 3
0
N 2
0
0 N
4
N 3
0
0
]
x 2 y 2
N 4 x 3
y 3 x 4 y 4
x = N 1 x 1 + N 2 x 2 + N 3 x3 + N 4 x 4 y = N 1 y 1+ N 2 y 2 + N 3 y 3+ N 4 y 4
4. B&?&" %&t'(,
[ 1 ] =
1 11 1 21
1 12 1 22 1
1 11= {− [ 1 −0 ] x 1 + [ 1−0 ] x2 + [ 1 + 0 ] x3 −[ 1 + 0 ] x 4 }
*here6
4
1 12 =
1 21 =
1
{−[ 1 −0 ] y +[ 1− 0 ] y +[ 1+ 0 ] y −[1 + 0 ] y }
4
1
2
3
4
1
{−[ 1 −/ ] x −[ 1 + / ] x +[ 1+ / ] x +[ 1 +/ ] x }
4
1
2
1
3
4
1 22 = {−[ 1−/ ] y 1−[ 1 + / ] y 2+ [ 1 + / ] y 3 + [ 1 + / ] y 4 } 4
/. F'# #t',
{}
{ F }e = [ N ]T F x
F y
*here+ 16
/ 3 0 = natural co-ordinates [B] = strain-displacement relationship matrix [D] = stress strain relationship matrix 7 = shape !unction
F x
= load or !orce on x direction
F y
= !orce on y direction
6. E+#%#"t $t'#$$, { σ } =[ D ] [ B } {u }
G&$$&" &'&t'# =O'> N%#'&+ I"t#5'&t" =>
F' 2 "t &'&t'#
1
∫ 4 ( x ) dx =5 1 4 ( x 1 ) +5 2 4 ( x 2 ) −1
*here+
1 2 1 &" (1 13, (2 -13
=>
F' 3 "t &'&t'#
1
∫ 4 ( x ) dx =5 1 4 ( x 1 ) +5 2 4 ( x 2 )+ 5 3 4 ( x 3 ) −1
*here+
1 3 /8, 2 78 &" (1 3/, (2 0, (3 -3/
=>
F' ?+# I"t#5'&t", 4 ( x 3 y ) dxdy =¿ 1
∬¿
12!=(1,91> 12!=(1,92> 21!=(2,91> 22!=(2,92>
−1
17
18
