1. CÁLCULO DE LA DEFLEXIÓN DE UNA VIGA APLICANDO ECUACIONES DIFERENCIALES
1. Cálculo de las reacciones (N)
≔
a 0
≔
b 4 ≔
Ra Rb
[
[ ]
] ――――
→
＝
+
Ra Rb +
⋅
350 4 400
＝
-
+
⋅
⋅
-350 4 2 ⋅
Rb 4 ⋅
400 5 0
⎡
⎢
⎣
⎤
⎥
⎦
,
,
solve Ra Rb
600 1200
[
[ ]
]
≔
c 5
Rigidez a flexión ( ) :
⋅
N m2
≔
EI 1270000
2. Diagrama de carga (N/m)
≔
W1
(
(x)
) -350 ≔
W2
(
(x)
) 0
≔
W(
(x)
) |
|
|
|
|
|
|
if
else if
<
≤
a x b
‖
‖W1
(
(x)
)
≤
b x
‖
‖W2
(
(x)
)
-280
-245
-210
-175
-140
-105
-70
-35
-350
-315
0
-6 -4 -2 0 2 4 6 8
-10 -8 10
x
W(
(x)
)
worksheet in symgear.blogspot.com

2. 3. Diagrama de fuerzas cortantes (N)
≔
N1
(
(x)
) ――――
→
＝
――
d
dx
N1
(
(x)
) W1
(
(x)
)
＝
N1
(
(a)
) Ra
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
,
solve N1
(
(x)
)
+
-(
( ⋅
350 x)
) 600
≔
N2'
(
(x)
) ――――
→
＝
――
d
dx
N2'
(
(x)
) W2
(
(x)
)
＝
N2'
(
(a)
) Rb
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
,
solve N2'
(
(x)
)
1200
≔
N2
(
(x)
) →
+
N2'
(
(x)
) N1
(
(b)
) 400
≔
N2
(
(x)
) N2
(
( -
x b)
)
≔
x , ‥
0 0.01 5
≔
N(
(x)
) |
|
|
|
|
|
|
if
else if
<
≤
0 x b
‖
‖N1
(
(x)
)
≤
b x
‖
‖N2
(
(x)
)
-600
-450
-300
-150
0
150
300
450
-900
-750
600
1 1.5 2 2.5 3 3.5 4 4.5
0 0.5 5
x
N(
(x)
)
clear (
(x)
)
worksheet in symgear.blogspot.com

3. 4. Diagrama de momento flector (N.m)
≔
M1
(
(x)
) ――――
→
＝
――
d
dx
M1
(
(x)
) N1
(
(x)
)
＝
M1
(
(a)
) 0
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
,
solve M1
(
(x)
)
+
-⎛
⎝ ⋅
175 x2 ⎞
⎠ ⋅
600 x
≔
M2'
(
(x)
) ――――
→
＝
――
d
dx
M2'
(
(x)
) N2
(
(x)
)
＝
M2'
(
(a)
) M1
(
(b)
)
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
,
solve M2'
(
(x)
)
-
⋅
400 x 400
≔
M2
(
(x)
) M2'
(
( -
x b)
)
≔
x , ‥
0 0.01 5
≔
M(
(x)
) |
|
|
|
|
|
|
if
else if
<
≤
0 x b
‖
‖M1
(
(x)
)
≤
b x
‖
‖M2
(
(x)
)
-200
-100
0
100
200
300
400
500
-400
-300
600
1 1.5 2 2.5 3 3.5 4 4.5
0 0.5 5
x
M(
(x)
)
clear (
(x)
)
worksheet in symgear.blogspot.com

4. 3. Diagrama de deflexiones (mm)
≔
y1
(
(x)
) ――――
→
＝
――
d
d
2
x2
y1
(
(x)
) M1
(
(x)
)
＝
y1
(
(a)
) 0
＝
y1
(
(b)
) 0
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
,
solve y1
(
(x)
)
+
-――――――
+
⋅
175 x4
⋅
8000 x
12
⋅
100 x3
≔
y2'
(
(x)
) ――――
→
＝
――
d
d
2
x2
y2'
(
(x)
) M2
(
(x)
)
＝
y2'
(
(a)
) 0
＝
′
y2'
(
(a)
) ′
y1
(
(b)
)
⎡
⎢
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎥
⎦
,
solve y2'
(
(x)
)
+
―――
⋅
200 x3
3
⎛
⎝ -
⋅
400 x ⋅
1000 x2 ⎞
⎠
≔
y1
(
(x)
) ⋅
――
y1
(
(x)
)
EI
1000
≔
y2
(
(x)
) ⋅
―――
y2'
(
( -
x b)
)
EI
1000
≔
t , ‥
0 0.01 5
≔
y(
(t)
) |
|
|
|
|
|
|
if
else if
<
≤
0 t b
‖
‖y1
(
(t)
)
≤
≤
b t c
‖
‖y2
(
(t)
)
-0.52
-0.455
-0.39
-0.325
-0.26
-0.195
-0.13
-0.065
0
-0.65
-0.585
0.065
1 1.5 2 2.5 3 3.5 4 4.5
0 0.5 5
t
y(
(t)
)
worksheet in symgear.blogspot.com

5. 4. Cálculo de la deflexión máxima
≔
x1 ―――――――――――
→
＝
――
d
dx
⎛
⎝y1
(
(x)
)⎞
⎠ 0
,
,
,
,
,
solve x float 3 assume ≥
>
b x a
1.87
≔
x2 ―――――――――――
→
＝
――
d
dx
⎛
⎝y2
(
(x)
)⎞
⎠ 0
,
,
,
,
,
solve x float 3 assume ≥
≥
c x b
4.2
≔
yMAX =
max⎛
⎝ ,
|
|y1
⎛
⎝x1
⎞
⎠|
| |
|y2
⎛
⎝x2
⎞
⎠|
|⎞
⎠ 0.607
Respuesta : 0.607 mm