DERIVED FORMULAS FOR DIFFERENT TYPES OF LOADINGS AND SUPPORTS :>
TYPES OF LOADINGS & SUPPORTS
A. SIMPLY-SUPPORTED BEAMS
REACTION (R)
MOMENT (M)
𝑤𝐿
2
1 2
𝑤𝐿
8
3
𝑅𝐴 = 𝑅𝐵 = 𝑤𝐿
8
5
𝑅𝑚𝑖𝑑 = 𝑤𝐿
4
𝑀𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 =
SLOPE (θ)
DEFLECTION (δ)
1 𝑤𝐿3
24 𝐸𝐼
5 𝑤𝐿4
384 𝐸𝐼
7 𝑤𝐿3
360 𝐸𝐼
8 𝑤𝐿3
θ𝐵 =
360 𝐸𝐼
2.5 𝑤𝐿4
384 𝐸𝐼
θ𝑚𝑖𝑑 =
9
𝑤𝐿2
128
1
𝑀𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 = − 𝑤𝐿2
8
θ𝐴 =
1
𝑤𝐿2
12
𝑃
2
Location of X
θ𝐴 = θ𝐵 =
5 𝑤𝐿3
192 𝐸𝐼
1 𝑤𝐿4
120 𝐸𝐼
1 𝑃𝐿2
16 𝐸𝐼
1
𝑃𝐿
4
1 𝑃𝐿3
48 𝐸𝐼
3
𝑃𝑎𝑏
𝐿
𝜃𝐴 =
𝜃𝐵 =
𝑃𝑎𝑏
(𝑎2 + 3𝑎𝑏 + 2𝑏 2 )
6𝐿2 𝐸𝐼
𝑃𝑎𝑏
(𝑏 2 + 3𝑎𝑏 + 2𝑎2 )
6𝐿2 𝐸𝐼
𝑥𝑚𝑎𝑥 = 0.481𝐿
𝛿𝑚𝑎𝑥 =
𝑃𝑏(𝐿2 − 𝑏 2 )2
9√3𝐸𝐼𝐿
𝑜𝑟 𝛿𝑚𝑎𝑥 =
𝑃𝑎2 𝑏 2
3𝐿𝐸𝐼
If a > b
𝑃𝑏
𝛿𝑚𝑖𝑑 =
(3𝑙 2 − 4𝐵2 )⬚
48𝐸𝐼
If a < b
𝑃𝑎
(3𝐿2 − 4𝑎2 )⬚
𝛿𝑚𝑖𝑑 =
48𝐸𝐼
𝑥𝑚𝑎𝑥 =
√𝐿2 − 𝑏 2
√3
EVAN RAM VECINA MANCOL
1
θ𝐴 =
𝑀
1 𝑀𝐿
6 𝐸𝐼
𝑀𝐿2
𝛿𝑚𝑎𝑥 =
1 𝑀𝐿
θ𝐵 =
3 𝐸𝐼
9√3𝐸𝐼
𝑥𝑚𝑎𝑥 = 0.577𝐿
2
𝛿𝑚𝑖𝑑 =
1 𝑀𝐿
16 𝐸𝐼
B. CANTILEVER BEAMS
𝑃
−𝑃𝑎
θ𝐵 =
1 𝑃𝑎2
2 𝐸𝐼
𝑃
−𝑃𝐿
θ𝐵 =
1 𝑃𝐿2
2 𝐸𝐼
𝛿𝑚𝑎𝑥 =
1 𝑃𝐿3
3 𝐸𝐼
𝑃
−
𝑃𝐿
2
θ𝐵 =
1 𝑃𝐿2
8 𝐸𝐼
𝛿𝑚𝑎𝑥 =
5 𝑃𝐿
48 𝐸𝐼
𝑤𝐿
1
− 𝑤𝑙 2
2
θ𝐵 =
1 𝑤𝐿3
6 𝐸𝐼
𝛿𝑚𝑎𝑥 =
1 𝑤𝐿4
8 𝐸𝐼
1
− 𝑤𝑙 2
6
θ𝐵 =
1 𝑤𝐿3
24 𝐸𝐼
𝛿𝑚𝑎𝑥 =
1 𝑤𝐿4
30 𝐸𝐼
𝑀𝑙
𝐸𝐼
𝛿𝑚𝑎𝑥 =
1 𝑀𝐿2
2 𝐸𝐼
−𝑀
θ𝐵 =
𝛿𝑚𝑎𝑥 =
𝑃𝑎2
𝑎
𝑃𝑎2
(3𝐿 − 𝑎)
(𝐿 − ) =
2𝐸𝐼
3
6𝐸𝐼
EVAN RAM VECINA MANCOL
2
C. FIXED/FULLY RESTRAINED BEAMS
1
𝑀𝐴 = − 𝑃𝐿
8
𝛿𝑚𝑎𝑥 =
1
𝑀𝐵 = 𝑃𝐿
8
𝑀𝐴 = −
𝑀𝐵 =
𝑃𝑎𝑏 2
𝐿2
𝑃𝑎𝑏 2
𝐿2
1 𝑃𝐿3
192 𝐸𝐼
a>b
1 𝑃𝑏 2 (3𝐿 − 4𝑏)
𝛿𝑚𝑖𝑑 =
48
𝐸𝐼
1
𝑤𝐿2
12
1
𝑀𝐵 =
𝑤𝐿2
12
𝑀𝐴 = −
𝑤𝐿
2
Plastic Analysis
1
𝑀𝑃 =
𝑤𝐿2
16
𝑀𝐴 = −
𝑀𝐵 =
1 𝑤𝐿4
384 𝐸𝐼
1
𝑤𝐿2
30
1
𝑤𝐿2
20
𝑀𝐴 = −
𝑀𝐵 =
𝛿𝑚𝑎𝑥 =
5
𝑤𝐿2
96
5
𝑤𝐿2
96
EVAN RAM VECINA MANCOL
3
𝑀𝐴 = −
𝑀𝐴 =
𝑀𝐴 = −
𝑀𝐵 =
5
𝑤𝐿2
192
11
𝑤𝐿2
192
𝑀𝑏 3𝑎
( − 1)
𝐿 𝐿
𝑀𝑎 3𝑏
( − 1)
𝐿 𝐿
D. PROPPED BEAMS
𝑅𝐵 =
𝑃𝑎2
(3𝐿 − 𝑎)
2𝐿3
𝑅𝐴 =
11
𝑃
16
M=
𝑅𝐵 =
3
𝑃𝐿
16
𝛿𝑚𝑎𝑥 =
𝑃𝐿3
48√5 𝐸𝐼
1
5
𝑃
16
5
𝑅𝐴 = 𝑤𝐿
8
3
𝑅𝐵 = 𝑤𝐿
8
By Elastic Analysis
1
M = 𝑤𝐿2
8
By Plastic Analysis
MP = 0.0858𝑤𝐿2
1 𝑤𝐿4
𝛿𝑚𝑎𝑥 =
185.2 𝐸𝐼
EVAN RAM VECINA MANCOL
4
𝑅𝐵 =
1
𝑤𝐿
10
M=
𝑅𝐵 =
11
𝑤𝐿
40
M=
𝑅𝐵 =
11
𝑤𝐿
64
1
𝑤𝐿2
15
7
𝑤𝐿2
120
M=
5
𝑤𝐿2
64
4
𝑅𝐴 = 𝑃
3
2
𝑅𝐵 = 𝑃
3
EVAN RAM VECINA MANCOL
5