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BEAM DEFLECTION FORMULAE BEAM TYPE SLOPE AT FREE END DEFLECTION AT ANY SECTION IN TERMS OF x 1. Cantilever Beam – Concentrated load P at the free end
Pl 2 2 EI
θ=
y=
MAXIMUM DEFLECTION
Px 2 ( 3l − x ) 6 EI
δ max =
Pl 3 3EI
2. Cantilever Beam – Concentrated load P at any point
Px 2 ( 3a − x ) for 0 < x < a 6 EI Pa 2 y= ( 3x − a ) for a < x < l 6 EI
y=
Pa 2 θ= 2 EI
δ max
Pa 2 = ( 3l − a ) 6 EI
3. Cantilever Beam – Uniformly distributed load ω (N/m)
ωl 3 6 EI
θ=
δ max =
ωl 4 8 EI
ωo x 2 (10l 3 − 10l 2 x + 5lx2 − x3 ) 120lEI
δ max =
ωo l 4 30 EI
Mx 2 2 EI
δ max =
Ml 2 2 EI
y=
ωx 2 x 2 + 6l 2 − 4lx ) ( 24 EI
4. Cantilever Beam – Uniformly varying load: Maximum intensity ωo (N/m)
θ=
ωol 3 24 EI
y=
5. Cantilever Beam – Couple moment M at the free end
θ=
Ml EI
y=
BEAM DEFLECTION FORMULAS BEAM TYPE
SLOPE AT ENDS
DEFLECTION AT ANY SECTION IN TERMS OF x
MAXIMUM AND CENTER DEFLECTION
6. Beam Simply Supported at Ends – Concentrated load P at the center
Pl 2 θ1 = θ2 = 16 EI
⎞ Px ⎛ 3l 2 l − x 2 ⎟ for 0 < x < y= ⎜ 12 EI ⎝ 4 2 ⎠
δ max =
Pl 3 48 EI
7. Beam Simply Supported at Ends – Concentrated load P at any point
Pb(l 2 − b 2 ) θ1 = 6lEI Pab(2l − b) θ2 = 6lEI
Pbx 2 l − x 2 − b 2 ) for 0 < x < a ( 6lEI Pb ⎡ l 3 y= ( x − a ) + ( l 2 − b2 ) x − x3 ⎤⎥ ⎢ 6lEI ⎣ b ⎦ for a < x < l y=
δ max = δ=
Pb ( l 2 − b 2 )
32
9 3 lEI
at x =
(l
2
− b2 ) 3
Pb ( 3l 2 − 4b2 ) at the center, if a > b 48 EI
8. Beam Simply Supported at Ends – Uniformly distributed load ω (N/m)
θ1 = θ2 =
ωl 3 24 EI
y=
ωx 3 l − 2lx 2 + x3 ) ( 24 EI
δmax =
5ωl 4 384 EI
9. Beam Simply Supported at Ends – Couple moment M at the right end
Ml θ1 = 6 EI Ml θ2 = 3EI
y=
Mlx ⎛ x 2 ⎞ ⎜1 − ⎟ 6 EI ⎝ l 2 ⎠
δmax = δ=
Ml 2 l at x = 9 3 EI 3
Ml 2 at the center 16 EI
10. Beam Simply Supported at Ends – Uniformly varying load: Maximum intensity ωo (N/m)
7ωol 3 360 EI ω l3 θ2 = o 45 EI
θ1 =
y=
ωo x ( 7l 4 − 10l 2 x 2 + 3x4 ) 360lEI
δ max = 0.00652 δ = 0.00651
ωo l 4 at x = 0.519 l EI
ωol 4 at the center EI
Table 61.1. Frequencies and Eigenfunctions for Uniform Beams
_..._._________________ ···---------·
b �
<,6(0) • lf,'(0) • 0 1/)(l) - -'(I} - 0
Cln1111,cd-clam1>Cd
l�
CJ1unpo
�
-
�
Clnmpod-,i:11ido
l Hineod-hinaod
Sr
""
equation
"111('1:)
__
...
ooe >. Coeh >. .,. I
(>.nz) J(>.11) (>.,,.r) J - ·- --11 l l II(>.,,)
...
41(0) • cJ,'(O) • 0 (/>(l) • lfi"(l) • 0
tnn >. .. 'l'n.nh >.
J
(""�) J(>.,,) (>.,,x) -
l
- ---·· JI
II(>.,,)
-·-
I
·--
+
"• ""
3.9266
>.t "" 7.0686 >.a "" 10. 2102
>-•• l3.Hl8
1/l(O) • cJ,'(0) • 0 ,t,"(l) • (/1111(1) • 0
UOII
·---
>. C(JHh >. ... - 1
----···
G(>..) (>.ft-:e) .r (>.1-l1%) --1! P(>. ) l ,,
-:,
-
>.a • 4'.7300 >., - 7.8S32 >., • 10. 99.S6 "• • 14. 1372 For n lnrge, l)1r/2 >.. ""(211
--·�---·------------ ----·�·---·---· ------------..- ---
,A. --···-···••"-·�·····-�--
CJo.mpo
equation
oonditioD8
Roota of frequenoy
El1enfunotlon
Frequenoy
Boundary
Typo
,p(O) • 41'(0) • 0
q,(0) • ,t>"(O) • 0 rp(Z) - c/1"(l) - 0
---.... -....·-·�-·····-··· · ---·'>" ·-��····-tBD >. • -1'nnh �
sin>. ... 0
·----------J
(""�) - - /•'(>.,.) --II J(>. n ) l
n,rx
sin l
(>,."x) l
I•'or n lnrgo, >." ,.. Hn
+- 1).-1•
>., - I. 8751 At • 4.69•1 >., - 7.8.S.8 >.1 • 10. 99.SS For n lnrge, >.,. ""' (2n - 1)11' /2 >.1 - 2.3650
>..2 - S.4978
>., - 8.6394 >.� • 11. 7810 I•'or n large, ).,. ,_ (.fn - l)T/4 >.,, .. nir