1. C
CÁLCULO DE UNIONES ATORNILLADAS PARA UNA DISTRIBUCIÓN
RECTANGULAR SIMÉTRICA
Pernos en el lado H Pernos en el lado L
≔
nH 3 ≔
nL 3
Distancias entre
pernos lado H
Distancias entre
pernos lado L
≔
dH 50 mm ≔
dL 50 mm
Total de pernos
≔
n =
-
2 (
( +
nL nH)
) 4 8
≔
H =
⋅
(
( -
nH 1)
) dH 100 mm ≔
L =
⋅
(
( -
nL 1)
) dL 100 mm
a) Fuerzas y momentos aplicados al centroide del conjunto de pernos
1. Fuerzas axiales
≔
Fx 1000 kgf
2. Fuerzas de corte
≔
Fy 4000 kgf ≔
Fz -3000 kgf
Radio de giro y momento de inercia
3. Momentos
≔
RRy =
⋅
⋅
⎛
⎜
⎝
―
H
2
⎞
⎟
⎠
2
nL 2 ⎛
⎝ ⋅
1.5 104 ⎞
⎠ mm2
≔
My ⋅
5000 kgf mm ≔
Mz ⋅
5000 kgf mm ≔
Iy ―――
→
As RRy
,
float 3
⋅
⋅
⋅
0.15 105
As mm2
4. Torsion ≔
RRz =
⋅
⋅
⎛
⎜
⎝
―
L
2
⎞
⎟
⎠
2
nH 2 ⎛
⎝ ⋅
1.5 104 ⎞
⎠ mm2
≔
Mx ⋅
-1500 kgf mm ≔
Iz ―――
→
As RRz
,
float 3
⋅
⋅
⋅
0.15 105
As mm2
b) Esfuerzos en el perno crítico ≔
J0 ―――
→
+
Iy Iz
,
float 3
⋅
⋅
⋅
0.3 105
As mm2
1. Esfuerzo de corte directo
≔
στ1 ―――
→
――
0
Fy
Fz
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
⋅
As n
,
float 3
0.0
――――
⋅
500.0 kgf
As
―――――
⋅
-375.0 kgf
As
⎡
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎦
spreadsheet in symgear.blogspot.com

2. 2. Esfuerzo axial
≔
σN (
( ,
,
y z As)
) ―――
→
+
-
⋅
z ――
My
Iy
⋅
y ――
Mz
Iz
――
Fx
⋅
n As
,
float 3
+
―――――
⋅
⋅
0.333 kgf z
⋅
As mm
⎛
⎜
⎝
-
――――
⋅
125.0 kgf
As
⋅
1.0 ―――――
⋅
⋅
0.333 kgf y
⋅
As mm
⎞
⎟
⎠
≔
z (
(x)
) |
|
|
|
|
|
|
|
|
|
if
else
＝
My 0
‖
‖
‖
‖
-
⋅
⋅
x (
(1 m)
) ――
Mz
RRz
――
Fx
n
‖
‖
‖
‖
⎛
⎜
⎝
-
⋅
⋅
x (
(1 m)
) ――
Mz
RRz
――
Fx
n
⎞
⎟
⎠
――
RRy
My
Eje neutro del conjunto de pernos
≔
p11 =
――
-L
2
―
H
2
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
-50
50
⎡
⎢
⎣
⎤
⎥
⎦
mm ≔
p12
――
-L
4
―
H
2
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
≔
p13
0
―
H
2
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
≔
p14
―
L
4
―
H
2
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
≔
p15 =
―
L
2
―
H
2
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
50
50
⎡
⎢
⎣
⎤
⎥
⎦
mm
-8
-6
-4
-2
0
2
4
6
8
-12
-10
10
-6 -4 -2 0 2 4 6 8
-10 -8 10
z (
(y)
) (
(m)
)
y
≔
p21
――
-L
2
―
H
4
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
≔
p25
―
L
2
―
H
4
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
≔
p31
――
-L
2
0
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
≔
p35
―
L
2
0
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
≔
p41
――
-L
2
――
-H
4
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
≔
p45
―
L
2
――
-H
4
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
≔
p51 =
――
-L
2
――
-H
2
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
-50
-50
⎡
⎢
⎣
⎤
⎥
⎦
mm ≔
p52
-―
L
4
―
H
4
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
≔
p53
0
――
-H
2
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
≔
p54
―
L
4
―
H
4
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
≔
p55 =
―
L
2
――
-H
2
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
50
-50
⎡
⎢
⎣
⎤
⎥
⎦
mm
El perno crítico seleccionado es :
≔
pc p11
≔
yc
zc
⎡
⎢
⎣
⎤
⎥
⎦
pc
――
→
σN ⎛
⎝ ,
,
yc zc As⎞
⎠
float
――――
⋅
158.3 kgf
As
spreadsheet in symgear.blogspot.com

3. 3. Esfuerzo cortarte por torsión
≔
στ2 ―――
→
―――――
⨯
Mx
0
0
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
0
yc
zc
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
J0
,
float 3
0.0
―――
⋅
2.5 kgf
As
―――
⋅
2.5 kgf
As
⎡
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎦
c) Análisis de falla y factor de seguridad
Area de esfuerzo del perno Esfuerzo de fluencia del material del perno
≔
As 40 mm2
≔
σY 50 ――
kgf
mm2
Tensor de esfuerzos
≔
στ (
(As)
) ―――
→
+
στ1 στ2
,
float 3
0.0
――――
⋅
503.0 kgf
As
―――――
⋅
-373.0 kgf
As
⎡
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎥
⎦
=
στ (
(As)
)
0
12.575
-9.325
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
――
kgf
mm2
≔
τzx
τzy
τyz
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
στ (
(As)
)
=
σN ⎛
⎝ ,
,
yc zc As⎞
⎠ 3.958 ――
kgf
mm2
≔
σxx σN ⎛
⎝ ,
,
yc zc As⎞
⎠
≔
Π =
σxx 0 0
0 0 τyz
0 τzy 0
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
3.958 0 0
0 0 -9.325
0 12.575 0
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
――
kgf
mm2
Aplicamos el criterio de Von Mises
≔
σadm =
‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾
2
――――――――
+
2 σxx
2
6 ⎛
⎝ +
τyz
2
τzy
2 ⎞
⎠
2
27.403 ――
kgf
mm2
Finalmente el factor de seguridad
≔
F.S =
――
σY
σadm
1.825
spreadsheet in symgear.blogspot.com