BEAMS: STATICALLY INDETERMINATE Statically Indeterminate ...
Analysis of Statically Indeterminate Structures by the …fast10.vsb.cz/koubova/DSM_truss.pdf ·...
Transcript of Analysis of Statically Indeterminate Structures by the …fast10.vsb.cz/koubova/DSM_truss.pdf ·...
Analysis of Statically
Indeterminate
Structures by the
Displacement Method
Displacement method
1. Slope-Deflection Method
� In this method it is assumed that all deformations
are due to bending only.
� Deformations due to axial forces are neglected.
2. Direct Stiffness Method
� Deformations due to axial forces Deformations due to axial forces Deformations due to axial forces Deformations due to axial forces are notnotnotnot neglectedneglectedneglectedneglected.
Displacement method
1. Slope-Deflection Method
� The continuous beam is kinematically indeterminate to
second degree.
� Two unknown joint rotation ϕϕϕϕb, ϕϕϕϕc
2. Direct Stiffness Method
� The continuous beam is kinematically indeterminate to
fourth degree.
� Two unknown joint rotation ϕϕϕϕb, ϕϕϕϕc. and two translations ub, uc
a
q
b c
F1 F2
d
Direct Stiffness Method:
Plane Truss
Truss Analysis
� Plane trusses are made up of short thin members
interconnected at hinges to form triangulated patterns
� A hinge connection can onlyonlyonlyonly transmittransmittransmittransmit forcesforcesforcesforces from one
member to another member but notnotnotnot thethethethe momentmomentmomentmoment
� For analysis purpose, the truss is loadedloadedloadedloaded atatatat thethethethe jointsjointsjointsjoints
� Hence, a truss member is subjected to onlyonlyonlyonly axialaxialaxialaxial forcesforcesforcesforces
and the forces remain constant along the length of the
member
� The forces in the member at its two ends must be of the
same magnitude but act in the opposite directions for
equilibrium
Truss Analysis
� Consider a truss member having crosscrosscrosscross sectionalsectionalsectionalsectional areaareaareaarea A,
Young’sYoung’sYoung’sYoung’s modulusmodulusmodulusmodulus of material E, lengthlengthlengthlength of the member l
� Let the member be subjected to axial tensile force F
� Under the action of constant axial force, applied at each
end, the member gets elongationelongationelongationelongation u
� The force-displacement relation for the truss member
may be written:
� k is the stiffnessstiffnessstiffnessstiffness ofofofof thethethethe trusstrusstrusstruss membermembermembermember and is defined as
the force required for unit deformation of the structure
EA
Flu =F F
u
ukul
EAF ⋅=⋅=
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1
F2 = 20 kN
F1 = 10 kN
a b
c
EA = const.
3 3
4
1. Degrees of freedom
� Truss is kinematically
indeterminate to
second degree.
� Two translations are
unknown – horizontal uc
and vertical wc
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1 F2 = 20 kN
F1 = 10 kN
a b
c
3 3
4
2. Express normal forces
due to elongation:
bcbc
cbbc
acac
caac
abab
baab
ul
EANN
ul
EANN
ul
EANN
⋅==
⋅==
=⋅== 0
( )
5
3cos
5
4sin
m543 22
==
=+==
αα
bcac ll
α α
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1
3. Equilibrium equations (write one equilibrium equation
for each unknown translation)
0sinsin
0coscos
0sinsin
0coscos
0
0
2
1
2
1
=⋅⋅+⋅⋅+
=⋅⋅+⋅⋅−
=⋅+⋅+
=⋅+⋅−
=
=
∑
∑
αα
αα
αα
αα
bcbc
acac
bcbc
acac
cbca
cbca
z
x
ul
EAu
l
EAF
ul
EAu
l
EAF
NNF
NNF
F
F
F2
F1
c
Nca
αα
Ncb
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1
3. Equilibrium equations
205
4
55
4
5
105
3
55
3
5
sinsin
coscos
2
1
−=⋅⋅+⋅⋅
−=⋅⋅+⋅⋅−
−=⋅⋅+⋅⋅
−=⋅⋅+⋅⋅−
bcac
bcac
bcbc
acac
bcbc
acac
uEA
uEA
uEA
uEA
Ful
EAu
l
EA
Ful
EAu
l
EA
αα
αα
EAu
EAu
EAuu
EAuu
bc
ac
bcac
bcac
⋅−=
⋅−=
⋅−=+
⋅−=+−
6
625
6
125
4
500
3
250
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1
kN833,206
625
5
kN167,46
125
5
−=
⋅−⋅=⋅==
−=
⋅−⋅=⋅==
EA
EAu
l
EANN
EA
EAu
l
EANN
bcbc
cbbc
acac
caac
4. Normal forces(After evaluating elongations, substitute it to evaluate normal
forces.)
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1
kN333,35
4167,4
sin:0
kN5,25
3167,4
cos:0
=⋅+=
⋅−==
=⋅+=
⋅−==
∑
∑
az
caazza
ax
caaxxa
R
NRF
R
NRF
α
α
a
Nac = Nca
αNab = 0
Rax
Raz
b
Nbc = Ncb
αNba = 0
Rbx
Rbz
5. Reactions
kN667,165
4833,20
sin:0
kN5,125
3833,20
cos:0
=⋅+=
⋅−==
=⋅+=
⋅−==
∑
∑
bz
cbbzzb
bx
cbbxxb
R
NRF
R
NRF
α
α
Direct Stiffness Method:
Plane Truss
Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Direct Stiffness Method, Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Local and Global CoLocal and Global CoLocal and Global CoLocal and Global Co----ordinate Systemordinate Systemordinate Systemordinate System
� All members are oriented at
different directions
� It is required to transform
member displacements and
forces from the locallocallocallocal cocococo----
ordinateordinateordinateordinate systemsystemsystemsystem x*z* to
globalglobalglobalglobal cocococo----ordinateordinateordinateordinate systemsystemsystemsystem xz
� So that a global load-
displacement relation may
be written for the complete
truss
F2
F1 x
z
x*
z*
Direct Stiffness Method, Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Transformation matrix Tab
⋅
=
⇒+=+=
ba
ba
ab
ab
ba
ab
bababa
ababab
Z
X
Z
X
X
X
ZXX
ZXX
γγγγ
γγγγ
sincos00
00sincos
sincos
sincos*
*
*
*
⋅=
ba
ba
ab
ab
ab
ba
ab
Z
X
Z
X
X
XT
*
*
=
γγγγ
sincos00
00sincosabT
T1abab TT =−
ab
ab
ab
ab
l
xx
l
zz
−=
−=
γ
γ
cos
sin
x
z
Xab
Zab
Xba
Zba
γ γ γ γ … … … … angle of angle of angle of angle of
transformation transformation transformation transformation
a
b
orthogonal matrix
−−
⋅=
−
−=
⋅=
⋅
−
−=
=
−
11
11*
*
**
*
*
*
*
ab
abab
ababab
b
aab
b
a
abab
abab
ba
ab
ba
ab
l
EA
l
EA
l
EAl
EA
l
EA
u
u
u
u
l
EA
l
EAl
EA
l
EA
X
X
N
N
k
k
Direct Stiffness Method, Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Member local stiffness matrix kab*
Direct Stiffness Method, Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Member global stiffness matrix kab*
−−−−
−−−−
=
⋅
−
−⋅
=⋅⋅=
⋅=
⋅⋅⋅=
⋅⋅=
⋅=
2
22
22
22
22
*T
*T
*
**T
*
*T
sscssc
sccscc
sscssc
sccscc
sincos00
00sincos
sin0
cos0
0sin
0cos
abab
abab
abababababab
b
b
a
a
ab
b
b
a
a
ababab
b
aabab
ba
abab
ba
ba
ab
ab
l
EA
l
EA
l
EAl
EA
l
EA
w
u
w
u
w
u
w
u
u
u
X
X
Z
X
Z
X
k
TkTk
kTkTkTT
γγγγ
γγ
γγ
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
F2 = 20 kN
F1 = 10 kN
a b
c
EA = const.
3 3
4
Member local stiffness matrix:
−−
⋅=
−−
⋅=
−−
⋅=
−−
⋅=
−−
⋅=
−−
⋅=
11
11
511
11
11
11
511
11
11
11
611
11
*
*
*
EA
l
EA
EA
l
EA
EA
l
EA
cbcb
acac
abab
k
k
k
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Transformation matrix:
6,05
36cos
8,05
04sin
6,05
03cos
8,05
40sin
16
06cos
06
44sin
=−=−=
=−=−=
=−=−=
−=−=−=
=−=−=
=−=−=
cb
cbcb
cb
cbcb
ac
acac
ac
acac
ab
abab
ab
abab
l
xx
l
zz
l
xx
l
zz
l
xx
l
zz
γ
γ
γ
γ
γ
γ
a b
c
EA = const.
3 3
4
x
z
[0 0] [3 0]
[0 4][6 4]
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Transformation matrix:
=
=
−−
=
=
=
=
8,06,000
008,06,0
sincos00
00sincos
8,06,000
008,06,0
sincos00
00sincos
0100
0001
sincos00
00sincos
cbcb
cbcbcb
acac
acacac
abab
ababab
γγγγ
γγγγ
γγγγ
T
T
T
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Member local stiffness matrix:
⋅
−−
⋅⋅
=⋅⋅=
−−
⋅
−−
⋅⋅
−
−=⋅⋅=
⋅
−−
⋅⋅
=⋅⋅=
8,06,000
008,06,0
11
11
5
8,00
6,00
08,0
06,0
8,06,000
008,06,0
11
11
5
8,00
6,00
08,0
06,0
0100
0001
11
11
6
00
10
00
01
*T
*T
*T
EA
EA
EA
bcbcbcbc
acacacac
abababab
TkTk
TkTk
TkTk
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Member local stiffness matrix:
−−−−
−−−−
⋅=
−−−−−−
−−
⋅=
−
−
⋅=
64,048,064,048,0
48,036,048,036,0
64,048,064,048,0
48,036,048,036,0
5
64,048,064,048,0
48,036,048,036,0
64,048,064,048,0
48,036,048,036,0
5
0000
0101
0000
0101
6
EA
EA
EA
bc
ac
ab
k
k
k
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Member vector of unknown forces:
⋅
−−−−
−−−−
⋅=
⋅=
⋅
−−−−−−
−−
⋅=
⋅=
=
⋅
−
−
⋅=
⋅=
0
0
64,048,064,048,0
48,036,048,036,0
64,048,064,048,0
48,036,048,036,0
5
0
0
64,048,064,048,0
48,036,048,036,0
64,048,064,048,0
48,036,048,036,0
5
0
0
0
0
0
0
0
0
0000
0101
0000
0101
6
c
c
b
b
c
c
cb
bc
bc
cb
cb
c
c
c
c
a
a
ac
ca
ca
ac
ac
b
b
a
a
ab
ba
ba
ab
ab
w
u
EA
w
u
w
u
Z
X
Z
X
w
u
EA
w
u
w
u
Z
X
Z
X
EA
w
u
w
u
Z
X
Z
X
k
k
k
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
The load displacement equation for the truss:
S … the vector of joint loads acting on the truss
r … the vector of joint displacements
K … the global stiffness matrixglobal stiffness matrixglobal stiffness matrixglobal stiffness matrix
∑
∑
=+=
=+=
2
1
:0
:0
FZZF
FXXF
cbcazc
cbcaxcF2 = 20 kN
F1 = 10 kN
c
SrK =⋅
=
⋅
++−+−+
⋅
=
⋅
⋅+
⋅
−−
⋅
20
10
64,064,048,048,0
48,048,036,036,0
5
20
10
64,048,0
48,036,0
564,048,0
48,036,0
5
c
c
c
c
c
c
w
uEA
w
uEA
w
uEA
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
The load displacement equation for the truss:
=
⋅=
⋅
−
=⋅=
=
⋅=
++−+−+
⋅
−
=
c
c
w
u
EAEA
EA
EAEA
125,78
444,691
20
10
1
144,00
0144,0
20
10
144,00
0144,0
64,064,048,048,0
48,048,036,036,0
5
SKr
S
K
1
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Member vector of joint displacements
=
=
=
=
=
=
=
⋅=
0
0
125,78
444,69
0
0
125,78
444,69
0
0
0
0
125,78
444,691
EA
EA
w
u
w
u
EA
EA
w
u
w
u
w
u
w
u
w
u
EA
b
b
c
c
cb
c
c
a
a
ac
b
b
a
a
ab
c
c
rr
r
r
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Member vector of knownknownknownknown globalglobalglobalglobal forces:
−−
=
⋅
−−−−
−−−−
⋅=⋅=
−−
=
⋅
−−−−−−
−−
⋅=⋅=
=
⋅
−
−
⋅=⋅=
67,16
5,12
67,16
5,12
0
0
125,78
444,69
64,048,064,048,0
48,036,048,036,0
64,048,064,048,0
48,036,048,036,0
5
33,3
5,2
33,3
5,2
0
0
125,78
444,69
64,048,064,048,0
48,036,048,036,0
64,048,064,048,0
48,036,048,036,0
5
0
0
0
0
0
0
0
0
0000
0101
0000
0101
6
EA
EAEA
Z
X
Z
X
EA
EAEA
Z
X
Z
X
EA
Z
X
Z
X
cbcb
bc
bc
cb
cb
acac
ca
ca
ac
ac
abab
ba
ba
ab
ab
rk
rk
rk
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Member vector of knownknownknownknown locallocallocallocal forces:
−=
−−
⋅
=
⋅=
−=
−−
⋅
−−
=
⋅=
=
⋅
=
⋅=
833,20
833,20
67,16
5,12
67,16
5,12
8,06,000
008,06,0
167,4
167,4
33,3
5,2
33,3
5,2
8,06,000
008,06,0
0
0
0
0
0
0
0100
0001
*
*
*
*
*
*
bc
bc
cb
cb
cb
bc
cb
ca
ca
ac
ac
ac
bc
ac
ba
ba
ab
ab
ab
ba
ab
Z
X
Z
X
X
X
Z
X
Z
X
X
X
Z
X
Z
X
X
X
T
T
T
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Check: Check: Check: Check:
� forces equilibrium equation
a b
c
Xac
ZacXac
*
Xca*
Xca
ZcaXcb*
Xbc*
Zcb
Xcb
Zbc
Xbc
F2
F1
2067,1633,3
105,125,2
:0
:0
2
1
=+=+−
=+=
=+=
∑∑
FZZF
FXXF
cbcazc
cbcaxc
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
Normal forces:Normal forces:Normal forces:Normal forces:
−
=
−=
−
=
−=
−
=
=
bc
cb
bc
cb
ca
ac
bc
ac
ba
ab
ba
ab
N
N
X
X
N
N
X
X
N
N
X
X
833,20
833,20
167,4
167,4
0
0
*
*
*
*
*
*
a b
c
Xac
Zac-Nac = Xac
*
Nca = Xca*
Xca
Zca-Ncb = Xcb*
Nbc = Xbc*
Zcb
Xcb
Xbc
F2
F1
Zbc
kN833,20
kN167,4
−==
−==
cbbc
caac
NN
NN
Direct Stiffness Method: Plane Truss
Example 1Example 1Example 1Example 1, , , , Matrix NotationMatrix NotationMatrix NotationMatrix Notation
RRRReactionseactionseactionseactions::::
a b
c
Rax = Xac
-Raz = ZacXac
*
Xca*
( )( )( )( )↑=−=
←=−=
↑=−=
→==
kN67,16
kN5,12
kN33,3
kN5,2
bcbz
bcbx
acaz
acax
ZR
XR
ZR
XR
Xca
ZcaXcb*
Xbc*
Zcb
Xcb
-Rbx = Xbc
F2
F1
-Rbz = Zbc
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Degrees of freedom:
� Truss is kinematically
indeterminate to 7th
degree.
� Seven translations are
unknown – horizontal
ub, uc, ud, ue and
vertical wc, wd, we.
eF1 = 4kN
F2 = 8kN
a b
c d
1
4
5
6
2 2
3
323
7
E = 20 GPaA1,4,5,6 = 0,02 m2
A2,3,7= 0,01 m2
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Code number:
� Non-zero code number is assigned code number is assigned code number is assigned code number is assigned to each unknown
translation.
7
6
5
4
3
2
1
=
b
d
d
c
c
e
e
u
w
u
w
u
w
u
r
e
a b
c d
1
4
5
6
23
7
(1 2)
(3 4) (5 6)
(0 0) (7 0)
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Member parameters:
( ) m61,332 227,3,2 =+=l
Member E [kPa] A l cos sin
1 (ab) 20000000 0,02 4 1 0
2 (db) 20000000 0,01 3,606 0,555 0,832
3 (ad) 20000000 0,01 3,606 0,555 -0,83
4 (ca) 20000000 0,02 3 0 1
5 (cd) 20000000 0,02 2 1 0
6 (ec) 20000000 0,02 3 0 1
7 (ed) 20000000 0,01 3,606 0,555 0,832
ab
abab
ab
abab
l
xx
l
zz
−=
−=
γ
γ
cos
sin
eF1
F2
a b
c d
1
4
5
6
2 2
3
323
7
[0 0]
[0 3]
[0 6]
x
z
[2 3]
[4 6]
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Member local stiffness matrix:
−
−=
l
EA
l
EAl
EA
l
EA*k
k1* 100000,0 -100000,0
-100000,0 100000,0
k2* 55470,0 -55470,0
-55470,0 55470,0
k3* 55470,0 -55470,0
-55470,0 55470,0
k4* 133333,3 -133333,3
-133333,3 133333,3
k5* 200000,0 -200000,0
-200000,0 200000,0
k6* 133333,3 -133333,3
-133333,3 133333,3
k7* 55470,0 -55470,0
-55470,0 55470,0
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Member transformation matrix:
=
γγγγ
sincos00
00sincosT
T1 1,00 0,00 0,00 0,00
0,00 0,00 1,00 0,00
T2 0,55 0,83 0,00 0,00
0,00 0,00 0,55 0,83
T3 0,55 -0,83 0,00 0,00
0,00 0,00 0,55 -0,83
T4 0,00 1,00 0,00 0,00
0,00 0,00 0,00 1,00
T5 1,00 0,00 0,00 0,00
0,00 0,00 1,00 0,00
T6 0,00 1,00 0,00 0,00
0,00 0,00 0,00 1,00
T7 0,55 0,83 0,00 0,00
0,00 0,00 0,55 0,83
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Member global stiffness matrix: TkTk ⋅⋅= *T
0 0 7 0 code number 5 6 7 0 code number 0 0 5 6 code number
k1 100000 0 -100000 0 0 k2 17067,7 25601,55 -17067,7 -25601,5 5 k3 17067,7 -25601,5 -17067,7 25601,55 0
0 0 0 0 0 25601,55 38402,32 -25601,5 -38402,3 6 -25601,5 38402,32 25601,55 -38402,3 0
-100000 0 100000 0 7 -17067,7 -25601,5 17067,7 25601,55 7 -17067,7 25601,55 17067,7 -25601,5 5
0 0 0 0 0 -25601,5 -38402,3 25601,55 38402,32 0 25601,55 -38402,3 -25601,5 38402,32 6
3 4 0 0 code number 3 4 5 6 code number 1 2 3 4 code number
k4 0 0 0 0 3 k5 200000 0 -200000 0 3 k6 0 0 0 0 1
0 133333,3 0 -133333 4 0 0 0 0 4 0 133333,3 0 -133333 2
0 0 0 0 0 -200000 0 200000 0 5 0 0 0 0 3
0 -133333 0 133333,3 0 0 0 0 0 6 0 -133333 0 133333,3 4
1 2 5 6 code number
k7 17067,7 25601,55 -17067,7 -25601,5 1
25601,55 38402,32 -25601,5 -38402,3 2
-17067,7 -25601,5 17067,7 25601,55 5
-25601,5 -38402,3 25601,55 38402,32 6
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Global stiffness matrix (partial calculation):
� “Localization” “Localization” “Localization” “Localization” according to the code numberMember 1 1 2 3 4 5 6 7 Member 5 1 2 3 4 5 6 7
1 12 23 3 200000 0 -200000 04 4 0 0 0 05 5 -200000 0 200000 06 6 0 0 0 07 100000 7
Member 2 1 2 3 4 5 6 7 Member 6 1 2 3 4 5 6 71 1 0 0 0 02 2 0 133333,3 0 -1333333 3 0 0 0 04 4 0 -133333 0 133333,35 17067,7 25601,55 -17067,7 56 25601,55 38402,32 -25601,5 67 -17067,7 -25601,5 17067,7 7
Member 3 1 2 3 4 5 6 7 Member 7 1 2 3 4 5 6 71 1 17067,7 25601,55 -17067,7 -25601,52 2 25601,55 38402,32 -25601,5 -38402,33 34 45 17067,7 -25601,5 5 -17067,7 -25601,5 17067,7 25601,556 -25601,5 38402,32 6 -25601,5 -38402,3 25601,55 38402,327 7
Member 4 1 2 3 4 5 6 7123 0 04 0 133333,3567
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Global stiffness matrix for truss (summation of partial calculations)
K 1 2 3 4 5 6 7
1 17067,7 25601,55 0 0 -17067,7 -25601,5 0
2 25601,55 171735,7 0 -133333 -25601,5 -38402,3 0
3 0 0 200000 0 -200000 0 0
4 0 -133333 0 266666,7 0 0 0
5 -17067,7 -25601,5 -200000 0 251203,1 25601,55 -17067,7
6 -25601,5 -38402,3 0 0 25601,55 115207 -25601,5
7 0 0 0 0 -17067,7 -25601,5 117067,7
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Vector of joint loads acting on the truss:
eF1 = 4kN
F2 = 8kN
a b
c d
1
4
5
6
2 2
3
323
7
=
=
0
0
0
0
8
0
4
xb
zd
xd
zc
xc
ze
xe
F
F
F
F
F
F
F
S
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222The load displacement equation for the truss:
SKrSrK 1 ⋅=⇒=⋅ −
7
6
5
4
3
2
1
000080,0
000105,0
000392,0
000045,0
000432,0
000090,0
000918,0
−
−
=
=
b
d
d
c
c
e
e
u
w
u
w
u
w
u
r
code n. Member1 code n. Member2 code n. Member3 code n. Member4 code n. Member5 code n. Member6 code n. Member7
0 0,000000 5 0,000392 0 0,000000 3 0,000432 3 0,000432 1 0,000918 1 0,000918
0 0,000000 6 0,000105 0 0,000000 4 -0,000045 4 -0,000045 2 -0,000090 2 -0,000090
7 0,000080 7 0,000080 5 0,000392 0 0,000000 5 0,000392 3 0,000432 5 0,000392
0 0,000000 0 0,000000 6 0,000105 0 0,000000 6 0,000105 4 -0,000045 6 0,000105
Member vector of joint displacements:
� Creating according to the code number
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Member vector of known global forces:
Member vector of known local forces:
Member1 Member2 Member3 Member4 Member5 Member6 Member7
0 -8 5 8 0 -4 3 0 3 8 1 0 1 4
0 0 6 12 0 6 4 -6 4 0 2 -6 2 6
7 8 7 -8 5 4 0 0 5 -8 3 0 5 -4
0 0 0 -12 6 -6 0 6 6 0 4 6 6 -6
abab
ba
ba
ab
ab
Z
X
Z
X
rk ⋅=
Member1 Member2 Member3 Member4 Member5 Member6 Member7
-8,00 14,42 -7,21 -6,00 8,00 -6,00 7,21
8,00 -14,42 7,21 6,00 -8,00 6,00 -7,21
⋅=
ba
ba
ab
ab
ab
ba
ab
Z
X
Z
X
X
XT
*
*
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Normal forces:Member N [kN]
1 (ab) 8,00
2 (db) -14,42
3 (ad) 7,21
4 (ca) 6,00
5 (cd) -8,00
6 (ec) 6,00
7 (ed) -7,21
−
=
ba
ab
ba
ab
N
N
X
X*
*
Member1 Member2 Member3 Member4 Member5 Member6 Member7
-8,00 14,42 -7,21 -6,00 8,00 -6,00 7,21
8,00 -14,42 7,21 6,00 -8,00 6,00 -7,21
Member vector of known local forces:
Direct Stiffness Method: Plane Truss
Example Example Example Example 2222
Reactions:
( ) ( )
( ) kN12120
kN12660
kN12048
−=−+=
+=
+=++=
++=
−=+−+−=
++=
bz
bdbabz
az
acadabaz
ax
acadabax
R
ZZR
R
ZZZR
R
XXXR
eF1 = 4kN
F2 = 8kN
a b
c d
1
4
5
6
23
7
Rax
Raz Rbz
Member1 Member2 Member3 Member4
Xab -8 Xdb 8 Xad -4 Xca 0
Zab 0 Zdb 12 Zad 6 Zca -6
Xba 8 Xbd -8 Xda 4 Xac 0
Zba 0 Zbd -12 Zda -6 Zac 6
Member vector of known global forces