Pl 2
2 EI
Zl 3
6 EI
y
y
y
Zx 2
x 2 6l 2 4lx
24 EI
Zol 3
24 EI
T
Ml
EI
5. Cantilever Beam – Couple moment M at the free end
T
y
for a x l
for 0 x a
Px 2
3l x
6 EI
Px 2
3a x
6 EI
Pa 2
3x a
6 EI
y
y
Mx 2
2 EI
Zo x 2
10l 3 10l 2 x 5lx 2 x 3
120lEI
4. Cantilever Beam – Uniformly varying load: Maximum intensity o (N/m)
T
3. Cantilever Beam – Uniformly distributed load (N/m)
T
Pa 2
2 EI
2. Cantilever Beam – Concentrated load P at any point
T
BEAM TYPE
SLOPE AT FREE END DEFLECTION AT ANY SECTION IN TERMS OF x
1. Cantilever Beam – Concentrated load P at the free end
BEAM DEFLECTION FORMULAE
Pl 3
3EI
G max
G max
Ml 2
2 EI
Zo l 4
30 EI
Zl 4
8 EI
Pa 2
3l a
6 EI
G max
G max
G max
MAXIMUM DEFLECTION
SLOPE AT ENDS
DEFLECTION AT ANY SECTION IN TERMS OF x
T2
Pl 2
16 EI
y
·
Px § 3l 2
l
x 2 ¸ for 0 x ¨
12 EI © 4
2
¹
y
y
Pbx 2
l x 2 b 2 for 0 x a
6lEI
Pb ª l
3
º
x a l 2 b 2 x x3 »
«
6lEI ¬ b
¼
for a x l
T2
Zl 3
24 EI
y
Zx 3
l 2lx 2 x3
24 EI
y
Mlx § x 2 ·
¨1 ¸
6 EI © l 2 ¹
T2
T1
7Zol 3
360 EI
Zo l 3
45 EI
y
Zo x
7l 4 10l 2 x 2 3x 4
360lEI
10. Beam Simply Supported at Ends – Uniformly varying load: Maximum intensity o (N/m)
T2
T1
Ml
6 EI
Ml
3EI
9. Beam Simply Supported at Ends – Couple moment M at the right end
T1
8. Beam Simply Supported at Ends – Uniformly distributed load (N/m)
T2
T1
Pb(l 2 b 2 )
6lEI
Pab(2l b)
6lEI
7. Beam Simply Supported at Ends – Concentrated load P at any point
T1
6. Beam Simply Supported at Ends – Concentrated load P at the center
BEAM TYPE
BEAM DEFLECTION FORMULAS
G
9 3 lEI
32
at x
Pl 3
48 EI
l 2 b2 3
Ml 2
at x
9 3 EI
5Zl 4
384 EI
Zo l 4
at x 0.519 l
EI
l
3
Zol 4
at the center
EI
0.00652
Ml 2
at the center
16 EI
G 0.00651
G max
G
Gmax
Gmax
Pb
3l 2 4b 2 at the center, if a ! b
48 EI
G max
Pb l 2 b 2
G max
MAXIMUM AND CENTER
DEFLECTION