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Calculation of an elastic hyperstatic plane gantry
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V3.90 booklet: Theoretical references of tests in linear statics
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Organization (S):
EDF/IMA/MN
Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
Document: V3.90.001
Calculation of an elastic hyperstatic plane gantry
Summary:
The goal of this note is to expose the method of calculation used to determine the reference solution of
case-test SSL 14, entitled: “Plane Gantry articulated in foot”.
One uses the method of the forces (hyperstaticity 1), by taking account only of the energy of bending: assumption
slim beams.
One considers four loading cases separately.
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Calculation of an elastic hyperstatic plane gantry
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HI-75/96/037 - Ind A
Contents
1 isostatic Stresses under real load distributed
p
on
C C
1
....................................................... 4
1.1 Isostatic reactions of supports ....................................................................................................... 4
1.2 Stresses ..................................................................................................................................... 5
1.3 Diagrams ..................................................................................................................................... 6
2 Stresses under concentrated force
F
1
(downwards) ............................................................................. 7
2.1 Reactions of support ............................................................................................................................. 7
2.2 Stresses ..................................................................................................................................... 7
2.3 Diagrams (
F
1
downwards) .......................................................................................................... 8
3 Stresses under the concentrated force
F
2
(towards the left) ................................................................... 8
3.1 Reactions of support ............................................................................................................................. 8
3.2 Stresses ..................................................................................................................................... 9
3.3 Diagrams ..................................................................................................................................... 9
4 Stresses under the concentrated couple
(positive) ............................................................................... 10
4.1 Reactions of support ........................................................................................................................... 10
4.2 Stresses ................................................................................................................................... 10
4.3 Diagrams (
positive) ................................................................................................................. 11
5 Stresses under the moment
X
hyperstatic ................................................................................... 12
5.1 Reactions of support ........................................................................................................................... 12
5.2 Stresses ................................................................................................................................... 12
5.3 Diagrams ................................................................................................................................... 13
6 Stresses under specific dummy loads in
C
.......................................................................... 13
6.1 Reactions of support ........................................................................................................................... 13
6.2 Stresses ................................................................................................................................... 14
6.3 Diagrams ................................................................................................................................... 14
7 Determination of the moment
X
hyperstatic ........................................................................................ 15
7.1 Charge distributed
p
on
C C
1
......................................................................................................... 16
7.2 Concentrated loading
F
1
in
C
.......................................................................................................... 17
7.3 Concentrated loading
F
2
in
C
1
......................................................................................................... 18
7.4 Specific couple
in
C
1
............................................................................................................. 19
7.5 Summary ................................................................................................................................... 20
8 Calculation of displacement in
C
............................................................................................................... 20
8.1 Charge distributed
p
on
C C
1
......................................................................................................... 20
8.2 Concentrated loading
F
1
in
C
.......................................................................................................... 21
8.3 Concentrated loading
F
2
in
C
1
......................................................................................................... 21
8.4 Specific couple
in
C
1
............................................................................................................. 22
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8.5 Calculation of
D
m M
I.E.(internal excitation)
=
1
................................................................................................................ 22
8.6 Summary of displacements
U
C
and
v
C
.................................................................................... 23
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Calculation of an elastic hyperstatic plane gantry
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“
y
p
C
C
1
C
2
F
1
has
H
X
B
With
F
2
One considers the gantry opposite,
subjected to various loads.
Hyperstaticity of degree 1.
Hyperstatic unknown factor:
X
:
moment in
C
.
Top-load distributed
p
on
C
1
C
.
Two forces
F
1
, F
2
, and a couple
in
C
1
.
C
B
With
V
With
H
With
H
B
V
B
tg
=
2a
“
=
0, 4
cos
(
)
-
1
=
1,16
=
1, 077033
(
)
tg
=
“
2 A
+
H
(
)
=
1
1,2
B
=
“
2cos
; sin
=
B has
1
Isostatic stresses under real load distributed
p
on
C C
1
1.1
Isostatic reactions of supports
H
H
V
V
p
V
p
With
B
With
B
B
+
=
+
=
=
0
2
8
2
;
cos
;
cos
“
“
“
The part
CB
is articulated and charged only at its ends:
H
V
H
V
B
B
B
B
=
= -
BC
0
tg
From where isostatic reactions:
H
p
V
p
H
p
V
p
With
With
B
B
=
=
= -
=
“
“
“
“
8
3
8
8
8
cos
tg
;
cos
;
cos
tg
;
cos
Note:
(
)
“
“
tg
cos
8
8
=
+
B
has
H
.
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Calculation of an elastic hyperstatic plane gantry
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1.2 Stresses
Beam
WITH C
1
NR
M
V
V
With
H
With
y
0
+
NR
p
V
p
M
p
y
Iso
Iso
Iso
= -
=
= -
3
8
8
8
“
“
“
cos
cos
tg
cos
tg
Beam
C B
2
NR
M
V
+
H
B
V
B
y
0
NR
p
V
p
M
p
y
Iso
Iso
Iso
= -
= -
= -
“
“
“
8
8
8
cos
cos
tg
cos
tg
Beam
C C
1
V
With
H
With
X
y
V
M
NR
+
p
(
)
(
)
NR
H
V
px
p
X
V
H
V
px
p
X
M
px
V X
H y
p
X
X
y
C
Iso
With
With
Iso
With
With
Iso
With
With
= -
-
+
= -
+
-
=
-
+
=
- +
= -
+
-
=
-
+
-
=
cos
sin
cos
sin
tg
tg
tg
sin
cos
cos
cos
tg
tg
/
cos
cos
tg
!
“
“
“
“
“
“
8
3
8
8
3 8
2
2
3
8
8
0
2
2
in
(
)
(
)
[]
M
p
has
H
S
has
H
B
S.A.
H
bh
S
X
B
Iso
= -
+
+
-
+
+
=
“
8
2
2
3
0
2
with
cos
,
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Calculation of an elastic hyperstatic plane gantry
Date:
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Key:
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Beam
C C
2
+
M
NR
V
V
B
H
B
X
y
“
(
)
(
)
(
)
(
)
(
)
(
)
NR
H
V
p
V
H
V
p
M
H y
V
X
p
y
X
C
Iso
B
B
Iso
B
B
Iso
B
B
=
-
= -
+
=
+
= -
-
=
+
-
= -
- -
=
cos
sin
tg
tg
sin
cos
tg
. tg
cos
tg
!
“
“
“
“
“
8
8
1
8
0
in
1.3 Diagrams
B
=
“
2 cos
+
-
3
8 cos
-
8cos
tg
8cos
-
tg
8cos
NR
Iso
V
Iso
M
Iso
-
tg
8cos
H
-
tg
8cos
= -
pbh
8
H
(a+h)
“
“
“
p
“
p
p
p
“
p
“
p
“
+
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2
Stresses under concentrated force
F
1
(downwards)
2.1 Reactions
of support
H
H
V
V
F
H
V
H
V
With
B
With
B
With
With
B
B
+
=
+
=
=
=
0
1
;
;
AC
0
BC
C
B
With
V
With
H
With
H
B
V
B
F
1
C
1
C
2
From where:
H
F
V
F
H
F
V
F
With
With
B
B
=
=
= -
=
1
2
1
2
1
2
1
2
1
1
1
1
tg
;
;
tg
;
2.2 Stresses
Beam
WITH C
1
:
tg
tg
NR
F
V
F
M
F y
Iso
Iso
Iso
= -
=
= -
1
2
1
2
1
2
1
1
1
Beam
C B
2
:
tg
tg
NR
F
V
F
M
F y
Iso
Iso
Iso
= -
= -
= -
1
2
1
2
1
2
1
1
1
Beam
C C
1
:
(
)
(
)
(
)
tg
cos
sin
tg
sin
cos
tg
NR
F
V
F
M
F y
X
Iso
Iso
Iso
= -
+
=
-
= -
-
1
2
1
2
1
2
1
1
1
Beam
C C
2
:
(
)
(
)
(
)
(
)
tg
cos
sin
tg
sin
cos
tg
NR
F
V
F
M
F y
X
Iso
Iso
Iso
= -
+
= -
-
= -
- -
1
2
1
2
1
2
1
1
1
“
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Calculation of an elastic hyperstatic plane gantry
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2.3 Diagrams
(
F
1
downwards)
NR
Iso
V
Iso
M
Iso
-
F
1
-
F
1
-
F
1
2
tg
H
F
1
2
tg
H
= -
F
1
lh
4 (a+h)
/2
/2
-
3
Stresses under the concentrated force
F
2
(towards the left)
3.1 Reactions
of support
·
+
=
+
=
+
=
·
=
H
H
F
V
V
V
H F
H
V
With
B
With
B
B
B
B
2
2
0
0
;
;
“
BC
0
C
B
With
V
With
H
With
H
B
V
B
C
1
C
2
F
2
From where:
H
F
H
V
F H
H
F H
V
F H
With
With
B
B
=
-
=
=
= -
1
2
1
2
1
“
“
“
“
tg
;
;
tg
;
Note:
(
)
(
)
H
H
has
H
H
has
H
has
H
“
“
tg
;
tg
=
+
-
=
+
+
2
1
2
2
(
)
(
)
(
)
tg
sin
cos
;
tg
cos
sin
-
= -
+
-
=
-
+
+
H
B has
H
has
ah
B has
H
“
“
2
4
4
2
2
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3.2 Stresses
Beam
WITH C
1
:
/
tg
tg
NR
F H
V
F
H
M
F y
H
Iso
Iso
Iso
= -
=
-
= -
-
2
2
2
1
1
“
“
“
Beam
C B
2
:
/
tg
tg
NR
F H
V
F H
M
F H y
Iso
Iso
Iso
=
=
= -
2
2
2
“
“
“
Beam
C C
1
:
tg
cos
sin
tg
sin
cos
tg
NR
F
H
H
V
F
H
H
M
F
H X
H
y
Iso
Iso
Iso
= -
-
+
=
-
-
=
- -
2
2
2
1
1
1
“
“
“
“
“
“
Beam
C C
2
:
(
)
(
)
(
)
(
)
tg
cos
sin
tg
sin
cos
tg
NR
F H
V
F H
M
F H y
X
Iso
Iso
Iso
=
+
=
-
=
- -
2
2
2
“
“
“
“
3.3 Diagrams
NR
Iso
V
Iso
+
M
Iso
-
F
2
H
-
F
2
H2 has
+
H
(
)
2 A
+
H
(
)
F
2
H
2
2 A
+
H
(
)
“
-
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4
Stresses under the concentrated couple
(positive)
4.1 Reactions
of support
·
+
=
+
=
+
=
·
=
H
H
V
V
V
H
V
With
B
With
B
B
B
B
0
0
0
;
;
“
BC
0
C
B
With
V
With
H
With
H
B
V
B
C
2
From where:
H
V
H
V
With
With
B
B
= -
=
=
= -
tg
;
tg
;
“
“
“
“
Note:
(
)
tg
“
=
+
1
2 A
H
4.2 Stresses
Beam
WITH C
1
:
NR
V
M
y
Iso
Iso
Iso
= -
= -
=
“
“
“
tg
tg
Beam
C B
2
:
NR
V
M
y
Iso
Iso
Iso
=
=
=
“
“
“
tg
tg
Beam
C C
1
:
(
)
(
)
(
)
NR
V
M
X
y
Iso
Iso
Iso
=
-
= -
+
=
+
-
“
“
“
“
tg
cos
sin
tg
sin
cos
tg
Beam
C C
2
:
(
)
(
)
(
)
(
)
NR
V
M
y
X
Iso
Iso
Iso
=
+
=
-
=
- -
“
“
“
“
tg
cos
sin
tg
sin
cos
tg
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Calculation of an elastic hyperstatic plane gantry
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4.3 Diagrams
(
positive)
NR
Iso
V
Iso
M
Iso
-
/
/
-
tg
-
H
+
2 A
(
)
2 A
+
H
(
)
H
2 A
+
H
(
)
H
2 A
+
H
(
)
“
“
“
-
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5
Stresses under the moment
X
hyperstatic
5.1 Reactions
of support
(
)
H
H
V
V
V
H
has
H
X
With
B
With
B
B
B
+
=
+
=
=
+ -
=
0
0
0
0
;
;
;
“
C
B
With
V
With
H
With
H
B
V
B
C
1
C
2
+
X
X
From where reactions:
H
X
has
H
V
H
X
has
H
V
With
With
B
B
= - +
=
=
+
=
;
;
;
0
0
5.2 Stresses
Beam
WITH C
1
:
NR
V
X
has
H
M
X
has
H y
X
X
X
=
= - +
=
+
0
Beam
C B
2
:
NR
V
X
has
H
M
X
has
H y
X
X
X
=
=
+
=
+
0
Beam
C C
1
:
(
)
cos
sin
tg
NR
X
has
H
V
X
has
H
M
X
has
H y
X
has
H H
X
X
X
X
=
+
= - +
=
+
=
+
+
Beam
C C
2
:
cos
sin
NR
X
has
H
V
X
has
H
M
X
has
H y
X
X
X
=
+
=
+
=
+
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5.3 Diagrams
+
X
X
X H
has
+
H
X H
has
+
H
X
has
+
H
X sin
has
+
H
X cos
has
+
H
NR
X
V
X
M
X
O
O
6
Stresses under specific dummy loads in
C
In order to calculate displacement in
C
, using the Principle of Virtual work (cf.
paragraph [§8]), it is necessary to establish the diagrams of stresses under the action of two
“fictitious” forces
F
and
G
applied in
C
.
6.1 Reactions
of support
H
H
F
V
V
G
H
V
H
V
With
B
With
B
With
With
B
B
+
= -
+
= -
=
=
;
;
AC
0
BC
C
B
With
V
With
H
With
H
B
V
B
C
2
C
1
G
F
From where:
(
)
(
)
(
)
(
)
H
F
G tg
V
G
F
H
F
G tg
V
G
F
With
With
B
B
= -
+
= -
+
= -
-
= -
-
1
2
1
2
1
2
1
2
;
cot
;
cot
G
G
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6.2 Stresses
Beam
WITH C
1
:
(
)
(
)
(
)
cot
tg
tg
N
G
F
v
F
G
m
F
G
y
=
+
= -
+
=
+
1
2
1
2
1
2
G
Beam
C B
2
:
(
)
(
)
(
)
cot
tg
tg
N
G
F
v
F
G
m
F
G
y
=
-
= -
-
= -
-
1
2
1
2
1
2
G
Beam
C C
1
:
(
)
(
)
(
)
(
)
(
)
(
)
tg
cos
cot
sin
tg
sin
cot
cos
tg
cot
N
F
G
G
F
v
F
G
G
F
m
F
G
y
G
F
X
=
+
+
+
= -
+
+
+
= +
+
-
+
1
2
1
2
1
2
1
2
1
2
1
2
G
G
G
Beam
C C
2
:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
tg
cos
cot
sin
tg
sin
cot
cos
tg
cot
N
F
G
G
F
v
F
G
G
F
m
F
G
y
G
F
X
= -
-
+
-
= -
-
-
-
= -
-
-
-
-
1
2
1
2
1
2
1
2
1
2
1
2
G
G
G
“
6.3 Diagrams
Here diagrams of stresses under the action of the two “fictitious” forces
F
and
G
. One considers
here:
F
G
F
0, cotg
.
+
G
F
-
fh/2
+
4 A
+
H
(
)
fh
2
+
gh
4 A
+
H
(
)
m
v
N
“
“
gh
Code_Aster
®
Version
4.0
Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
01/09/99
Author (S):
F. VOLDOIRE
Key:
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Page:
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Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
7
Determination of the moment
X
hyperstatic
One places oneself in elasticity; one considers only the energy of bending, the beams being slim. The state
naturalness is supposed to be virgin (not prestressings nor of displacement of support).
The complementary potential is then:
()
(
)
(
)
F
* X
M
M
X
I.E.(internal excitation)
M
M
X
I.E.(internal excitation)
Iso
posts
Iso
frames
=
+
+
+
1
2
1
1
2
2
It is stationary with balance, from where:
=
+
= -
-
=
X
M
I.E.(internal excitation)
M
I.E.(internal excitation)
X
MR. M
I.E.(internal excitation)
MR. M
I.E.(internal excitation)
S
pot
charp
Iso
pot
Iso
charp
1
2
1
1
2
2
1
1
1
2
.
.
.
.
The coefficient of flexibility
is the sum of:
(
)
M
I.E.(internal excitation)
H
I.E.(internal excitation)
H
has
H
M
I.E.(internal excitation)
B
I.E.(internal excitation)
H
has
H
has
has
H
ah
has
H
posts
frames
1
2
1
1
2
1
2
2
2
2
2
2
2
3
2
1
3
=
+
=
+
+
+
+ +
that is to say:
(
)
(
)
E
has
H
H
I
B H
has
ah
I
=
+
+
+
+
2
3
3
3
3
2
3
1
2
2
2
Numerical application:
In the example considered:
I
I
m
H
has
m
m
B
1
2
4
4
2
5 0 10
2
8
20
2 1 16
=
=
=
=
=
=
-
,
;
;
,
“
“
From where:
(
)
=
+
+
2
3
19
2
2353 45347
2
1
2
3
E has
H
I
H
H
B
m
.
One studies one after the other the various loadings to calculate the second members
S
.
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Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
01/09/99
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Key:
V3.90.001-A
Page:
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Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
7.1 Charge
distributed
p
on
C C
1
The second member
S
had with
p
is:
(
)
(
)
(
)
(
)
(
)
-
=
+
+
=
+
-
=
+
+
+
+
=
+
+
-
=
MR. M
I.E.(internal excitation)
H
I.E.(internal excitation)
H
has
H
Pb H
has
H
E has
H
I
pH B
MR. M
I.E.(internal excitation)
Pb H
has
H I.E.(internal excitation)
H
has
H
has
has
H
E has
H
I
phb
H
has
MR. M
I.E.(internal excitation)
I.E.(internal excitation)
p
Iso
posts
Iso
C C
Iso
C C
1
1
1
2
1
3
1
2
2
2
2
2
2
1
2
2
2
3
8
2
24
8
1
2
1
6
1
3
48
1
2
1
“
“
“
“
“
(
)
(
)
(
)
(
)
8
2
2
3
1
48
2
2
2
0
2
2
2
2
2
has
H
S
has
H
B
S.A.
H
bh
H
S.A.
B ds
E has
H
I
p B
H
ah
has
B
+
+
-
+
+
+
=
+
-
-
“
From where:
(
)
(
)
(
)
S
E has
H
p B
H
I
hb H
has
I
B H
ah
has
I
=
+
+
+ +
-
-
2
96
4
3
2
2
3
1
2
2
2
2
“
Numerical application:
(
)
I
I
H
has
p
NR m
S
E has
H
I
p bh
H
B
NR m
1
2
2
1
2
4
2
2
3 000
2
96
4
13
2
43.946 02189
=
=
=
=
+
+
;
;
/
.
(downwards)
“
From where:
·
moment in
C
:
X
18.672.994 NR m
=
.
·
reaction in
With
:
(
)
(
)
H
p
B
has
H
X
has
H
Pb
X
has
H
H
V
Pb
V
With
With
With
With
=
+ - +
=
-
+
=
=
-
=
“
“
8
8
3
4
0
5.175 37 NR
24.233 24 NR
.
.
Code_Aster
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Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
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Key:
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Page:
17/24
Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
7.2 Charge
specific
F
1
in
C
The second member is obtained using:
(
)
(
)
(
)
(
)
(
)
-
=
+
+
=
+
-
=
+
+
+
+
=
+
+
MR. M
I.E.(internal excitation)
H
I.E.(internal excitation)
H
has
H
F H
has
H
E has
H
I
F H
MR. M
I.E.(internal excitation)
B
I.E.(internal excitation)
F
H
has
H
H
has
H
has
has
H
E has
H
I
F B H H
has
Iso
posts
Iso
frames
1
1
1
1
2
1
1
3
1
2
2
1
2
2
1
2
3
4
2
12
2
4
1
2
1
6
2
3
24
“
“
“
“
From where:
(
)
(
)
S
E has
H
F H
H
I
B H
has
I
=
+
+
+
2
24
2
3
2
1
2
1
2
“
Numerical application:
(
)
[
]
I
I
H
has
F
NR
S
E has
H
I
F H
H
B
NR m
1
2
1
2
1
1
2
4
2
2
20 000
2
24
2
7
97.485.127 76
=
=
=
=
+
+
;
;
,
(downwards)
“
From where
·
moment in
C
:
X
41.422.161 NR m
=
.
·
reaction in
With
:
(
)
H
F has H
X
has
H
F
X
has
H
H
V
F
V
With
With
With
With
=
+ - +
=
-
+
=
=
-
=
1
4
4
1
2
0
1
1
1
“
“
4.881 4866 NR
10.000 0 NR
.
.
Code_Aster
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Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
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Key:
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Page:
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Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
7.3
Concentrated loading
F
2
in
C
1
The second member is obtained using:
(
)
(
)
(
)
(
)
-
=
+
+
+
=
+
+
MR. M
I.E.(internal excitation)
H
I.E.(internal excitation)
H
has
H
F H has
H
has
H
E has
H
I
F H
has
H
Iso
WITH C
1
1
1
2
2
1
2
3
1
3
2
2
2
2
12
(
)
(
)
(
)
-
=
+
- +
=
+
-
MR. M
I.E.(internal excitation)
H
I.E.(internal excitation)
H
has
H
F H
has
H
E has
H
I
F H
Iso
C B
1
1
1
2
2
2
1
2
4
2
3
2
2
12
(
)
(
)
(
)
(
)
-
=
+
+
+
+
+
=
+
+
+
MR. M
I.E.(internal excitation)
B
I.E.(internal excitation)
F H has
H
has
H
H
has
H
has
has
H
E has
H
I
F bh H
ah
has
Iso
C C
1
2
2
2
2
2
2
2
2
1
2
2
1
2
1
6
2
3
7
2
24
(
)
(
)
(
)
-
=
-
+
+
+
+
=
+
-
+
MR. M
I.E.(internal excitation)
B
I.E.(internal excitation)
F H
has
H
H
has
H
has
has
H
E has
H
I
F bh
H
has
Iso
C C
1
2
2
2
2
2
2
2
2
2
2
1
2
1
6
2
3
24
From where:
(
)
(
)
S
E has
H
F ha
H
I
B
I
H
has
=
+
+
+
2
12
2
3
2
2
2
1
2
Numerical application:
I
I
H
has
F
NR
1
2
2
2
2
10 000
=
=
=
;
;
(towards the left)
(
)
(
)
S
E has
H
I
F H has
H
B
NR m
=
+
+
2
12
2
7
19.497.025 55
2
1
2
2
4
.
From where:
·
moment in
C
:
X
8.284 4321 NR m
=
.
·
reaction in
With
:
(
)
(
)
(
)
H
F
has
H
has
H
X
has
H
F has
H
X
has
H
With
=
+
+ - +
=
+
-
+
2
2
2
2
2
/
H
With
=
5.976.297 NR
.
V
F H
With
=
2
/
“
V
With
=
4.000 00 NR
.
Code_Aster
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Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
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Key:
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Page:
19/24
Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
7.4 Couples
specific
in
C
1
The second member is obtained using:
(
)
(
)
-
= -
+
+
=
+
-
MR. M
I.E.(internal excitation)
H
I.E.(internal excitation)
H
has
H
H
has
H
E has
H
I
H
Iso
posts
1
1
1
2
1
3
2
3
2
2
6
(
)
(
)
(
)
(
) (
)
-
=
+
+
+
+
+
=
+
+
+
MR. M
I.E.(internal excitation)
B
I.E.(internal excitation)
H
has
has
H
H
has
H
has
has
H
E has
H
I
H
has
H
B has
Iso
C C
1
2
2
2
2
1
2
2
1
2
1
6
2
2
3
24
(
)
(
)
(
)
-
=
-
+
+
+
+
=
+
-
+
MR. M
I.E.(internal excitation)
B
I.E.(internal excitation)
H
has
H
H
has
H
has
has
H
E has
H
I
hb
H
has
Iso
C C
1
2
2
2
2
2
2
1
2
1
6
2
3
24
From where:
(
)
(
)
S
E has
H
H
I
ab H
has
I
=
+
-
-
+
2
12
2
3
2
3
1
2
Numerical application:
I
I
H
has
NR m
1
2
2
2
100 000
=
=
= -
;
;
(direction hands clock)
(
)
(
)
[
]
S
E has
H
I
H
ab H
has
NR m
=
+
-
-
+
2
6
3
11.571.281 93
2
1
3
4
.
From where
·
moment in
C
:
X
4.916 7243 NR m
=
.
·
reaction in
With
:
(
)
(
)
H
has
H
X
has
H
X
has
H
With
=
-
+ - +
= -
-
+
2
2
H
With
=
4.576.394 NR
.
V
With
=
“
V
With
=
5.000.000 NR
.
Code_Aster
®
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4.0
Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
01/09/99
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Key:
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Page:
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Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
7.5 Summary
CASE
Moment in
C
Reactions in
With
(NR)
(N.m)
H
With
V
With
p
on
C C
1
18672.994
5175.37
24233.240
F
1
in
C
41422.161
4881.487
10000.000
F
2
in
C
1
8284.432
5976.297
4000.000
in
C
1
4916.724
4576.394
5000.000
TOTAL
22033.31
43233.24
Notice
Recall: in the post
WITH C
1
: normal effort =
-
V
With
, shearing action =
H
With
.
8
Calculation of displacement in
C
One considers also only the elastic energy of bending (slim beams). By applying the Principle
virtual Work on the structure subjected to the fictitious forces of the paragraph [§6], working in
sought displacements, the numbers are calculated
W
and
D
depending linearly on
F
and
G
:
(
)
(
)
()
F U
G v
m M
X M
I.E.(internal excitation)
m M
X M
I.E.(internal excitation)
W
Xd
F G
C
C
Iso
pot
Iso
charp
+
=
+
+
+
=
+
1
1
1
2
.
.
,
,
8.1 Charge
distributed
p
on
C C
1
(
) (
)
(
)
m M
I.E.(internal excitation)
H
I.E.(internal excitation)
gh
has
H
pbh
has
H
E has
H
I
gpbh
Iso
posts
=
+
-
+
=
+
-
1
1
2
1
3 2
2
3
4
8
2
96
“
“
“
(
)
(
)
(
)
(
)
m M
I.E.(internal excitation)
E has
H
I
p hb
F has
H
G
H
has
Iso
C C
=
+
-
+ +
-
2
2
2
2
1
2
384
2
“
“
(
)
(
)
(
)
(
)
(
)
M
I.E.(internal excitation)
B
I.E.(internal excitation)
pbh
has
H
fh
G H
has
H
E has
H
I
Pb
H G
F has
H
Iso
C C
2
2
2
2
2
2
2
3
8
2
4
2
2
192
=
+
-
+
=
+
-
-
+
“
“
“
“
From where:
(
)
(
)
(
)
W
E has
H
pbh
G H
I
G B H
has
Bfr has
H
I
=
+
-
+
- -
+
2
384
4
3
2
2
2
1
2
2
“
“
“
Numerical application:
I
I
H
has
p
NR m
1
2
2
2
3 000
=
=
=
;
;
/
(downwards)
(
)
W
E has
H
I
G
H
B
fbh
p bh
NR m
=
+
+
-
-
-
2
2
5
2
9
2
192
215406 5922
2
1
2
3
“
“
.
Code_Aster
®
Version
4.0
Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
01/09/99
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Key:
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Page:
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Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
8.2 Charge
specific
F
1
in
C
(
) (
)
(
)
m M
I.E.(internal excitation)
H
I.E.(internal excitation)
gh
has
H
F H
has
H
E has
H
I
F gh
Iso
posts
=
+
-
+
=
+
-
1
1
1
2
1
1
3 2
2
3
4
4
2
48
“
“
“
(
) (
)
(
)
m M
I.E.(internal excitation)
B
I.E.(internal excitation)
gh
has
H
F H
has
H
E has
H
I
F gbh
Iso
charp
=
+
-
+
=
+
-
2
2
1
2
2
1
2 2
2
3
4
4
2
48
.
“
“
“
From where (it is noted that
W
does not depend on
F
for this loading):
(
)
W
E has
H
F gh
H
I
B
I
=
+
-
+
2
48
2
1
2 2
1
2
“
Numerical application:
I
I
H
has
F
NR
1
2
1
2
2
20 000
=
=
=
;
;
(downwards)
(
)
(
)
W
G
E has
H
I
F H
H
B
NR m
=
+
-
+
2
48
2
315.100.365 0
2
1
1
2 2
5
“
.
8.3 Charge
specific
F
2
in
C
1
(
)
(
)
(
)
(
)
(
)
(
)
(
)
m M
I.E.(internal excitation)
H
I.E.(internal excitation)
F H
has
H
has
H
fh
gh
has
H
H
fh
gh
has
H
E has
H
I
F H
Ag
F has
H
Iso
posts
=
+ -
+
+
+
+ -
+
+
=
+
-
+
+
1
1
2
2
1
2
3
2
3
2
2
2
4
2
4
2
24
2
“
“
“
(
)
(
)
(
)
m M
I.E.(internal excitation)
E has
H
I
F bh
Ag
F has
H
Iso
charp
=
+
-
+
+
2
2
2
2
2
2
2
24
2
.
“
From where:
(
)
(
)
(
)
W
E has
H
F H
Ag
F has
H
H
I
B
I
=
+
-
+
+
+
2
24
2
2
2
2
2
1
2
“
Numerical application:
I
I
H
has
F
NR
1
2
2
2
2
10 000
=
=
=
;
;
(towards the left)
(
)
(
)
(
)
W
E has
H
I
G
gh
F H H
B
NR m
=
+
+
-
+
-
2
9
2
48
3.151.003 65
2
1
2
3
4
“
.
Code_Aster
®
Version
4.0
Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
01/09/99
Author (S):
F. VOLDOIRE
Key:
V3.90.001-A
Page:
22/24
Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
8.4 Couples
specific
in
C
1
(
)
(
)
(
)
(
)
m M
I.E.(internal excitation)
H
I.E.(internal excitation)
H
has
H
fh
G H
has
H
fh
G H
has
H
E has
H
I
H G
Iso
posts
=
+
+
+
+ -
+
+
=
+
1
1
2
1
3
3
2
2
4
2
4
2
24
“
“
“
(
)
(
)
(
)
(
)
(
)
(
)
(
)
m M
I.E.(internal excitation)
B
I.E.(internal excitation)
has
H
has
H
fh
G H
has
H
H
fh
G H
has
H
E has
H
I
bh
Ag
F has
H
Iso
charp
=
+ -
+
+
+
+ -
+
+
=
-
+
+
+
2
2
2
2
2
3
2
2
2
4
2
4
2
24
2
.
“
“
“
Numerical application:
I
I
H
has
Nm
1
2
2
2
100 000
=
=
= -
;
;
(
)
(
)
(
)
W
E has
H
I
H
NR m
G H
B
fhb
=
+
-
- -
2
24
266.666.667
9
2
1
2
3
.
“
8.5 Calculation
of
D
m M
I.E.(internal excitation)
=
1
(
)
(
)
m M
I.E.(internal excitation)
H
I.E.(internal excitation)
G H
has
H
H
has
H
E has
H
I
G H
posts
=
+ +
=
+
1
1
1
2
1
3
2
3
4
2
12
“
“
(
)
(
)
(
)
m M
I.E.(internal excitation)
B
I.E.(internal excitation)
H
has
H
has
has
H
G H
has
H
E has
H
I
H
has
frame
=
+
+
+
+
=
+
+
1
2
2
2
2
2
1
2
1
6
4
2
3
24
“
“
G bh
From where (it is noted that
D
does not depend on
F
):
(
)
(
)
D
E has
H
G H
H
I
B H
has
I
=
+
+
+
2
24
2
3
2
2
1
2
“
Numerical application:
I
I
H
has
1
2
2
2
=
=
;
(
)
(
)
D
E has
H
I
G
H
H
B
m
=
+
+
2
24 2
7
4.874 2564
2
1
2
4
“
.
Code_Aster
®
Version
4.0
Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
01/09/99
Author (S):
F. VOLDOIRE
Key:
V3.90.001-A
Page:
23/24
Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
8.6
Summary of displacements
U
C
and
v
C
I
m
E
MPa
1
4
4
5 0 10
210 000
=
=
-
,
CASE
X
X D
W
V
pressure on
C C
1
18672.994
91016960.3
184930109.4
F
1
in
C
41422.161
201902233.4
315100365.0
F
2
in
C
1
8284.432
40380445.6
63020073.0
in
C
1
4916.724
23965373.4
14775091.25
CASE
W
H
()
U
m
C
()
v
m
C
pressure on
C C
1
83519999.94
0.0110476
0.012422374
F
1
in
C
0.00
0.00
0.01497330
F
2
in
C
1
226872262.8
0.03000956
0.00299466
in
C
1
206790328.5
0.0273532
0.001215646
Note:
(
)
D
E has
H
I
G D
=
+
2
2
1
, with:
D
m
=
4.874 2564
4
.
(
)
(
)
W
E has
H
I
G W
F W
V
H
=
+
+
2
2
1
to see higher
(
)
(
)
(
)
U
E has
H
I
W
v
E has
H
I
W
Xd
C
H
C
V
=
+
=
+
+
2
2
2
1
2
1
;
(
)
2
1 32275132 10
2
1
10
1
4
E has
H
I
NR
m
+
=
-
-
-
.
Code_Aster
®
Version
4.0
Titrate:
Calculation of an elastic hyperstatic plane gantry
Date:
01/09/99
Author (S):
F. VOLDOIRE
Key:
V3.90.001-A
Page:
24/24
Manual of Validation
V3.90 booklet: Theoretical references of tests in linear statics
HI-75/96/037 - Ind A
Comparison
Aster
- analytical reference (R.)
CASE
Moment in
C
(N.m)
Reaction
H
With
(NR)
Reaction
V
With
(NR)
Displacement
U
C
(m)
Displacement
v
C
(m)
p
on
C C
1
R:
Aster
:
18672.994
18673.20
5175.37
5175.36
24233.24
24233.2
0.0110476
0.0110472
0.012422374
0.0124233
F
1
in
C
R:
Aster
:
41422.161
41422.40
4881.487
4881.47
10000.00
10000.0
0.00000
0.0000
0.01497330
0.0
F
2
in
C
1
R:
Aster
:
8284.432
8284.34
5976.297
5976.31
4000.00
4000.0
0.03000956
0.0300098
0.00299466
0.00299450
in
C
1
R:
Aster
:
4916.724
4916.62
4576.394
4576.38
5000.00
5000.0
0.0273532
0.0273536
0.001215646
0.00121583
Note:
Calculation
Aster
was realized by taking very slim elements, so that:
S
I
“
2
<<
. Thus,
the energy of bending is prevalent. Values of calculation
Aster
are resulting from case-test VPCS called SSLL14,
with the following data:
I
m
I
m
E
MPa
1
4
4
2
4
4
5 0 10
2 5 10
210 000
=
=
=
-
-
.
;
.
;
,
H
has
m
m
B
=
=
=
=
2
8
20
2 116
;
;
.
“
“
,
p
NR m
=
3 000
/
(downwards),
F
NR
1
20 000
=
(downwards),
F
NR
2
10 000
=
(towards the left),
= -
100.000 Nm
.