The document summarizes the most common simply supported beam cases: 1) A beam with a central point load - the maximum moment occurs at the center and the maximum deflection also occurs at the center. 2) A beam with a point load at any point - the maximum moment and deflection occur at the point of load application. 3) A beam with two equal point loads symmetrically placed - the maximum moment is constant between the loads and the maximum deflection occurs at the center of the span. 4) A beam with a uniform distributed load - the maximum moment and deflection occur at the center of the span.

1. FORMULARIO PARA VIGAS
ISOSTÁTICAS MÁS COMUNES
TEMA:
MATERIA: CIENCIA Y RESISTENCIA DE
MATERIALES

2. 2
𝑪𝒂𝒔𝒐 𝟏: 𝑽𝒊𝒈𝒂 𝒔𝒊𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒆 𝒂𝒑𝒐𝒚𝒂𝒅𝒂 𝒄𝒐𝒏 𝒄𝒂𝒓𝒈𝒂 𝒑𝒖𝒏𝒕𝒖𝒂𝒍 𝒂𝒍 𝒄𝒆𝒏𝒕𝒓𝒐
P
L/2 L/2
L
A B
𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠:
𝑅𝐴 = 𝑅𝐵 =
𝑃
2
𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑚á𝑥𝑖𝑚𝑜:
𝑀𝑀𝐴𝑋 =
𝑃𝐿
4
𝑈𝑏𝑖𝑐𝑎𝑑𝑜 𝑒𝑛 𝑥 =
𝐿
2
𝐸𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛:
𝑥 =
𝑃𝑥
48𝐸𝐼
(3𝐿2
− 4𝑥2
) 𝑝𝑎𝑟𝑎 0 < 𝑥 <
𝐿
2
𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑚á𝑥𝑖𝑚𝑎:
𝑀𝐴𝑋 =
𝑃𝐿3
48𝐸𝐼
𝑈𝑏𝑖𝑐𝑎𝑑𝑎 𝑒𝑛 𝑥 =
𝐿
2
𝐸 = 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑 𝑑𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
𝐼 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑒𝑛 𝑒𝑙 𝑠𝑒𝑛𝑡𝑖𝑑𝑜 𝑑𝑒 𝑙𝑎 𝑓𝑙𝑒𝑥𝑖ó𝑛

3. 3
𝑪𝒂𝒔𝒐 𝟐: 𝑽𝒊𝒈𝒂 𝒔𝒊𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒆 𝒂𝒑𝒐𝒚𝒂𝒅𝒂 𝒄𝒐𝒏 𝒄𝒂𝒓𝒈𝒂 𝒑𝒖𝒏𝒕𝒖𝒂𝒍 𝒆𝒏 𝒄𝒖𝒂𝒍𝒒𝒖𝒊𝒆𝒓 𝒑𝒖𝒏𝒕𝒐
𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠:
𝑅𝐴 =
𝑃𝑏
𝐿
𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑚á𝑥𝑖𝑚𝑜:
𝑀𝑀𝐴𝑋 =
𝑃𝑎𝑏
𝐿
𝑈𝑏𝑖𝑐𝑎𝑑𝑜 𝑒𝑛 𝑒𝑙 𝑝𝑢𝑛𝑡𝑜 𝑑𝑒
𝐸𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛:
𝑥 =
𝑃𝑏𝑥
6𝐿𝐸𝐼
(𝐿2
− 𝑥2
− 𝑏2
)
𝑝𝑎𝑟𝑎 0 < 𝑥 < 𝑎
𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑚á𝑥𝑖𝑚𝑎:
𝑀𝐴𝑋 =
𝑃𝑎𝑏(𝑎 + 2𝑏) 3𝑎(𝑎 + 2𝑏)
27𝐿𝐸𝐼
b
L
A B
P
a
𝑅𝐵 =
𝑃𝑎
𝐿
𝑎𝑝𝑙𝑖𝑐𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑙𝑎 𝑐𝑎𝑟𝑔𝑎
𝑥 =
𝑃𝑏
6𝐿𝐸𝐼
𝐿
6
𝑥 − 𝑎 3
+ 𝐿2
− 𝑏2
𝑥 − 𝑥3
𝑝𝑎𝑟𝑎 𝑎 < 𝑥 < 𝐿
𝑈𝑏𝑖𝑐𝑎𝑑𝑎 𝑒𝑛 𝑒𝑙 𝑝𝑢𝑛𝑡𝑜 𝑑𝑒
𝑎𝑝𝑙𝑖𝑐𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑙𝑎 𝑐𝑎𝑟𝑔𝑎
𝐸 = 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑 𝑑𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
𝐼 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑒𝑛 𝑒𝑙 𝑠𝑒𝑛𝑡𝑖𝑑𝑜 𝑑𝑒 𝑙𝑎 𝑓𝑙𝑒𝑥𝑖ó𝑛

4. 4
𝑪𝒂𝒔𝒐 𝟑: 𝑽𝒊𝒈𝒂 𝒔𝒊𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒆 𝒂𝒑𝒐𝒚𝒂𝒅𝒂 𝒄𝒐𝒏 𝒅𝒐𝒔 𝒄𝒂𝒓𝒈𝒂𝒔 𝒑𝒖𝒏𝒕𝒖𝒂𝒍𝒆𝒔 𝒊𝒈𝒖𝒂𝒍𝒆𝒔
𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠:
𝑅𝐴 = 𝑅𝐵 = 𝑃
𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑚á𝑥𝑖𝑚𝑜:
𝑀𝑀𝐴𝑋 = 𝑃𝑎 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑒 𝑒𝑛𝑡𝑟𝑒 𝑐𝑎𝑟𝑔𝑎𝑠
𝐸𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛:
𝑥 =
𝑃𝑥
6𝐸𝐼
(3𝐿𝑎 − 3𝑎2 − 𝑥2) 𝑝𝑎𝑟𝑎 0 < 𝑥 < 𝑎
𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑚á𝑥𝑖𝑚𝑎:
𝑀𝐴𝑋 =
𝑃𝑎
24𝐸𝐼
(3𝐿2
− 4𝑎2
)
𝑝𝑢𝑛𝑡𝑢𝑎𝑙𝑒𝑠
𝑈𝑏𝑖𝑐𝑎𝑑𝑎 𝑎𝑙 𝑐𝑒𝑛𝑡𝑟𝑜 𝑑𝑒𝑙 𝑐𝑙𝑎𝑟𝑜
𝒄𝒐𝒍𝒐𝒄𝒂𝒅𝒂𝒔 𝒔𝒊𝒎é𝒕𝒓𝒊𝒄𝒂𝒎𝒆𝒏𝒕𝒆
P
a
L
A B
P
a
𝐸 = 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑 𝑑𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
𝐼 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑒𝑛 𝑒𝑙 𝑠𝑒𝑛𝑡𝑖𝑑𝑜 𝑑𝑒 𝑙𝑎 𝑓𝑙𝑒𝑥𝑖ó𝑛

5. 5
𝑪𝒂𝒔𝒐 𝟒: 𝑽𝒊𝒈𝒂 𝒔𝒊𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒆 𝒂𝒑𝒐𝒚𝒂𝒅𝒂 𝒄𝒐𝒏 𝒄𝒂𝒓𝒈𝒂 𝒖𝒏𝒊𝒇𝒐𝒓𝒎𝒆 𝒅𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒊𝒅𝒂
𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠:
𝑅𝐴 = 𝑅𝐵 =
𝑊𝐿
2
𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑚á𝑥𝑖𝑚𝑜:
𝑀𝑀𝐴𝑋 =
𝑊𝐿2
8
𝑈𝑏𝑖𝑐𝑎𝑑𝑜 𝑎𝑙 𝑐𝑒𝑛𝑡𝑟𝑜 𝑑𝑒𝑙
𝐸𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛:
𝑥 =
𝑊𝑥
24𝐸𝐼
(𝐿3 − 2𝐿𝑥2 − 𝑥3) 𝑝𝑎𝑟𝑎 0 < 𝑥 < 𝐿
𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑚á𝑥𝑖𝑚𝑎:
𝑀𝐴𝑋 =
5𝑊𝐿4
384𝐸𝐼
𝑐𝑙𝑎𝑟𝑜
𝑈𝑏𝑖𝑐𝑎𝑑𝑎 𝑎𝑙 𝑐𝑒𝑛𝑡𝑟𝑜 𝑑𝑒𝑙 𝑐𝑙𝑎𝑟𝑜
L
A B
W
𝐸 = 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑 𝑑𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
𝐼 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑒𝑛 𝑒𝑙 𝑠𝑒𝑛𝑡𝑖𝑑𝑜 𝑑𝑒 𝑙𝑎 𝑓𝑙𝑒𝑥𝑖ó𝑛

6. 6
𝑪𝒂𝒔𝒐 𝟓: 𝑽𝒊𝒈𝒂 𝒔𝒊𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒆 𝒂𝒑𝒐𝒚𝒂𝒅𝒂 𝒄𝒐𝒏 𝒄𝒂𝒓𝒈𝒂 𝒅𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒊𝒅𝒂 𝒄𝒐𝒏 𝒗𝒂𝒓𝒊𝒂𝒄𝒊ó𝒏
𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠:
𝑅𝐴 =
𝑊𝐿
6
𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑚á𝑥𝑖𝑚𝑜:
𝑀𝑀𝐴𝑋 =
321𝑊𝐿2
5000
𝑈𝑏𝑖𝑐𝑎𝑑𝑜 𝑎
577𝐿
1000
𝐸𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛:
𝑥 =
𝑊𝑥
360𝐿𝐸𝐼
(4𝐿4 − 10𝐿2𝑥2 + 3𝑥4) 𝑝𝑎𝑟𝑎 0 < 𝑥 < 𝐿
𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑚á𝑥𝑖𝑚𝑎:
𝑀𝐴𝑋 =
163𝑊𝐿4
25000𝐸𝐼
𝑈𝑏𝑖𝑐𝑎𝑑𝑎 𝑎
519𝐿
1000
𝒍𝒊𝒏𝒆𝒂𝒍
L
A B
W
𝑅𝐵 =
𝑊𝐿
3
𝐸 = 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑 𝑑𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
𝐼 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑒𝑛 𝑒𝑙 𝑠𝑒𝑛𝑡𝑖𝑑𝑜 𝑑𝑒 𝑙𝑎 𝑓𝑙𝑒𝑥𝑖ó𝑛

7. 7
𝑪𝒂𝒔𝒐 𝟔: 𝑽𝒊𝒈𝒂 𝒔𝒊𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕𝒆 𝒂𝒑𝒐𝒚𝒂𝒅𝒂 𝒄𝒐𝒏 𝒎𝒐𝒎𝒆𝒏𝒕𝒐 𝒆𝒏 𝒂𝒍𝒈𝒖𝒏𝒐 𝒅𝒆 𝒍𝒐𝒔 𝒂𝒑𝒐𝒚𝒐𝒔
𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠:
𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑚á𝑥𝑖𝑚𝑜:
𝑀𝑀𝐴𝑋 = 𝑀 𝑈𝑏𝑖𝑐𝑎𝑑𝑜 𝑒𝑛 𝑒𝑙 𝑒𝑥𝑡𝑟𝑒𝑚𝑜 𝑑𝑜𝑛𝑑𝑒
𝐸𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛:
𝑥 =
𝑀𝑥
6𝐿𝐸𝐼
(2𝐿2 − 3𝐿𝑥 + 𝑥2) 𝑝𝑎𝑟𝑎 0 < 𝑥 < 𝐿
𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑚á𝑥𝑖𝑚𝑎:
𝑀𝐴𝑋 =
8𝑀𝐿2
125𝐸𝐼
𝑈𝑏𝑖𝑐𝑎𝑑𝑎 𝑎
423𝐿
1000
𝑑𝑒𝑠𝑑𝑒 𝑒𝑙 𝑎𝑝𝑜𝑦𝑜 𝑑𝑜𝑛𝑑𝑒 𝑠𝑒 𝑐𝑜𝑙𝑜𝑞𝑢𝑒 𝑀
L
A B
M
𝑅𝐴 = 𝑅𝐵 =
𝑀
𝐿
𝑠𝑒 𝑐𝑜𝑙𝑜𝑞𝑢𝑒 𝑀
𝐸 = 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑 𝑑𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
𝐼 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑒𝑛 𝑒𝑙 𝑠𝑒𝑛𝑡𝑖𝑑𝑜 𝑑𝑒 𝑙𝑎 𝑓𝑙𝑒𝑥𝑖ó𝑛

8. 8
𝑪𝒂𝒔𝒐 𝟕: 𝑽𝒊𝒈𝒂 𝒆𝒏 𝒗𝒐𝒍𝒂𝒅𝒊𝒛𝒐 𝒄𝒐𝒏 𝒄𝒂𝒓𝒈𝒂 𝒑𝒖𝒏𝒕𝒖𝒂𝒍 𝒆𝒏 𝒆𝒍 𝒆𝒙𝒕𝒓𝒆𝒎𝒐 𝒍𝒊𝒃𝒓𝒆
𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠:
𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑚á𝑥𝑖𝑚𝑜:
𝑀𝑀𝐴𝑋 = 𝑃𝐿 𝑈𝑏𝑖𝑐𝑎𝑑𝑜 𝑒𝑛 𝑒𝑙 𝑒𝑥𝑡𝑟𝑒𝑚𝑜 𝐴
𝐸𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛:
𝑥 =
𝑃𝑥2
6𝐸𝐼
(3𝐿 − 𝑥) 𝑝𝑎𝑟𝑎 0 < 𝑥 < 𝐿
𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑚á𝑥𝑖𝑚𝑎:
𝑀𝐴𝑋 =
𝑃𝐿3
3𝐸𝐼
𝑈𝑏𝑖𝑐𝑎𝑑𝑎 𝑒𝑛 𝑒𝑙 𝑒𝑥𝑡𝑟𝑒𝑚𝑜 𝑙𝑖𝑏𝑟𝑒
𝑅𝐴 = 𝑃
P
L
A
𝐸 = 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑 𝑑𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
𝐼 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑒𝑛 𝑒𝑙 𝑠𝑒𝑛𝑡𝑖𝑑𝑜 𝑑𝑒 𝑙𝑎 𝑓𝑙𝑒𝑥𝑖ó𝑛

9. 9
𝑪𝒂𝒔𝒐 𝟖: 𝑽𝒊𝒈𝒂 𝒆𝒏 𝒗𝒐𝒍𝒂𝒅𝒊𝒛𝒐 𝒄𝒐𝒏 𝒄𝒂𝒓𝒈𝒂 𝒑𝒖𝒏𝒕𝒖𝒂𝒍 𝒆𝒏 𝒄𝒖𝒂𝒍𝒒𝒖𝒊𝒆𝒓 𝒑𝒖𝒏𝒕𝒐
𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠:
𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑚á𝑥𝑖𝑚𝑜:
𝑀𝑀𝐴𝑋 = 𝑃𝑎 𝑈𝑏𝑖𝑐𝑎𝑑𝑜 𝑒𝑛 𝑒𝑙 𝑒𝑥𝑡𝑟𝑒𝑚𝑜 𝐴
𝐸𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛:
𝑥 =
𝑃𝑥2
6𝐸𝐼
(3𝑎 − 𝑥)
𝑝𝑎𝑟𝑎 0 < 𝑥 < 𝑎
𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑚á𝑥𝑖𝑚𝑎:
𝑀𝐴𝑋 =
𝑃𝑎2
6𝐸𝐼
(3𝐿 − 𝑎) 𝐸𝑛 𝑒𝑙 𝑒𝑥𝑡𝑟𝑒𝑚𝑜 𝑙𝑖𝑏𝑟𝑒
𝑅𝐴 = 𝑃
P
L
A
a b
𝑥 =
𝑃𝑎2
6𝐸𝐼
(3𝑥 − 𝑎)
𝑝𝑎𝑟𝑎 𝑎 < 𝑥 < 𝐿
𝐸 = 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑 𝑑𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
𝐼 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑒𝑛 𝑒𝑙 𝑠𝑒𝑛𝑡𝑖𝑑𝑜 𝑑𝑒 𝑙𝑎 𝑓𝑙𝑒𝑥𝑖ó𝑛

10. 10
𝑪𝒂𝒔𝒐 𝟗: 𝑽𝒊𝒈𝒂 𝒆𝒏 𝒗𝒐𝒍𝒂𝒅𝒊𝒛𝒐 𝒄𝒐𝒏 𝒄𝒂𝒓𝒈𝒂 𝒖𝒏𝒊𝒇𝒐𝒓𝒎𝒆𝒎𝒆𝒏𝒕𝒆 𝒅𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒊𝒅𝒂
𝑅𝑒𝑎𝑐𝑐𝑖𝑜𝑛𝑒𝑠:
𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑚á𝑥𝑖𝑚𝑜:
𝑀𝑀𝐴𝑋 =
𝑊𝐿2
2
𝑈𝑏𝑖𝑐𝑎𝑑𝑜 𝑒𝑛 𝑒𝑙 𝑒𝑥𝑡𝑟𝑒𝑚𝑜 𝐴
𝐸𝑐𝑢𝑎𝑐𝑖ó𝑛 𝑑𝑒 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛:
𝑥 =
𝑊𝑥2
24𝐸𝐼
(6𝐿2 − 4𝐿𝑥 + 𝑥2)
𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑐𝑖ó𝑛 𝑚á𝑥𝑖𝑚𝑎:
𝑀𝐴𝑋 =
𝑊𝐿4
8𝐸𝐼
𝐸𝑛 𝑒𝑙 𝑒𝑥𝑡𝑟𝑒𝑚𝑜 𝑙𝑖𝑏𝑟𝑒
𝑅𝐴 = 𝑊𝐿
𝑝𝑎𝑟𝑎 0 < 𝑥 < 𝐿
L
A
W
𝐸 = 𝑀ó𝑑𝑢𝑙𝑜 𝑑𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑑𝑎𝑑 𝑑𝑒𝑙 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
𝐼 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑜 𝑑𝑒 𝑖𝑛𝑒𝑟𝑐𝑖𝑎 𝑑𝑒 𝑙𝑎 𝑠𝑒𝑐𝑐𝑖ó𝑛 𝑒𝑛 𝑒𝑙 𝑠𝑒𝑛𝑡𝑖𝑑𝑜 𝑑𝑒 𝑙𝑎 𝑓𝑙𝑒𝑥𝑖ó𝑛

11. 11
𝑹𝑬𝑭𝑬𝑹𝑬𝑵𝑪𝑰𝑨 𝑩𝑰𝑩𝑳𝑰𝑶𝑮𝑹𝑨𝑭𝑰𝑪𝑨:
𝑅𝑜𝑏𝑒𝑟𝑡 𝑊. 𝐹𝑖𝑡𝑧𝑔𝑒𝑟𝑎𝑙𝑑 1996 . 𝑀𝑒𝑐á𝑛𝑖𝑐𝑎 𝑑𝑒 𝑀𝑎𝑡𝑒𝑟𝑖𝑎𝑙𝑒𝑠
𝐼𝑆𝐵𝑁 9701501543, 9789701501542
𝐴𝑝é𝑛𝑑𝑖𝑐𝑒 𝐷: 𝐷𝑖𝑎𝑔𝑟𝑎𝑚𝑎𝑠 𝑑𝑒 𝑣𝑖𝑔𝑎𝑠
𝐸𝑑𝑖𝑡𝑜𝑟𝑖𝑎𝑙 𝐴𝑙𝑓𝑎𝑜𝑚𝑒𝑔𝑎