Chapter 5 Beams on Elastic Foundation no load applied
Winkler foundation
elastic foundation
F
F
x
z, w
Winkler Foundation
reaction pressure pW
=
k0 w
3
k 0 [ N/m ] foundation modulus
floating on liquid (buoyancy) k0
= γ liquid =
g ρliquid
reaction force intensity (reaction force over unit length) qW
= − k0 b w = − k w
b width of the beam 2
k [ N/m ] foundation constant
Basic Equations q(x) M(x)
M(x+dx) x
V(x)
V(x+dx) k w(x) dx
z,w
dV dx
= − ( qW + q) = dM dx 2
d M dx
2
=
=
kw−q
2
d w dx
= − E I
V
4
E I
2
3
d w dx
d w
3
+kw =
q
+kw =
0
+ 4 β4 w =
0
dx
4
−q
V
= − E I
M
kw
homogeneous equation 4
E I
d w dx
4
d w dx
4
4
β4 =
k
4 E I
general solution of the homogeneous differential equation
w w ''''
− α4 w =
0
= eα x
⇔
w ''''
+ 4 β4 w =
0
α4 = − 4 β 4 , α2 = ± 2 i β 2 , α = ± ±2 i β = ± (1 ± i ) β = e±β x e± i β x
w
w
= eβ x (C1 sin βx + C2 cos βx) + e−βx (C3 sin βx + C4 cos βx)
example w w'
= β eβ x (sin β x + cos β x ) w ''
w ''' w ''''
= eβ x sin β x
= 2 β2 eβ x cos βx
= 2 β3 eβ x (cos β x − sin β x )
= − 4 β4 eβ x sin β x or
w ''''
+ 4 β4 w =
0
short-hand notation Aβ x
= e−β x (cos β x + sin β x) ,
Bβ x
= e−β x sin βx
Cβ x
= e−β x (cos β x − sin β x) ,
Dβ x
= e−β x cos β x
differentials
=
Aβ x
1 dAβ x − 2β dx
=
Bβ x
C β x
Dβ x
1 dDβ x − β dx
=
=
1 dBβ x
β
dx
=
=
= −
1 dCβ x − 2β dx
2 1 d Cβ x
2β
2
dx
2 1 d Dβx
2β
2
dx
2
2 1 d Aβx
2β
= −
2
dx
=
2
2
3 1 d Bβ x
2β3 dx 3
= −
=
2 1 d Bβx
2β 2 dx 2
3 1 d Cβ x
4β3 dx 3 3 1 d Dβ x
2β3
=
dx
3
3 1 d Aβ x
4β3 dx 3
distributions Bβ x
= e−β x sin βx
40
Dβ x
= e −β x cos β x 1 D(ßx)
) 20 x ß ( D , 0 ) x ß ( B -20
B(ßx) D(ßx)
-40
) x ß ( D , 0 ) x ß ( B
B(ßx)
-1 -5 -4 -3 -2 -1 0 1 2 3 4 5 x
0
1
2
3 ßx
4
5
Semi-Infinite Beams with Concentrated Loads
Po M o x
x
wo
θo z,w
z,w
general solution of the homogeneous differential equation w
= eβ x (C1 sin βx + C2 cos βx) + e−βx (C3 sin β x + C4 cos βx)
boundedness w
= e −β x (C3 sin β x + C4 cos β x ) =
C3 Bβ x
+
C4 Dβ x
boundary conditions
=
M o
M ( x = 0)
− Po =
w '''
M o
Po
=
dx
=
2E I
3
= 2β
x = 0
3
d w −E I 3 dx x = 0
= 2 β3 C3 Aβ x + 2 β3 C4 Cβ x
0 and Aβ x ( x = 0)
=
2
= − 2 β2C3 Dβ x + 2 β2C4 Bβ x
w ''
Bβ x ( x = 0)
V ( x = 0)
= − E I
2
d w
β
2
C 3 or C 3
E I (C3
+ C4 )
= Cβx ( x = 0) =
=
M o
2 E I β2
or C 4
=
=
2 β Po k
Dβ x ( x = 0)
2 β2 M o k
−
2 β2 M o k
=
1
full solution
w
=
2 β2 M o k
Bβ x
Cβ x
w
θ =
M
V
=
w'
2 β Po k
= −
+( =
2 β 2 Po
−
k
Dβx
Dβ x
k
2 β Po
−
Aβ x
= − E I w '' = −
2 β2 M o k
) Dβ x
− Bβ x 2 β 2 M o k
+
C βx
4 β3 M o
Po Bβx
β
k
+
Dβ x
M o Aβ x
= − E I w ''' = − Po Cβx − 2 β M o Bβx
end force load Po
end moment load M o x
x
z
z
π 2β x
x
π 4β
w w
π 4β
3π 4β x
x M
M
Infinite Beams with Concentrated Loads Po
x
z,w
symmetric deflection
=
w( x )
w( − x )
= − w '( − x)
w '( x )
=
w '(0) Po /2
w '(0)
= −
2 β2 ( Po / 2) k
Aβ x (0)
M o
w
=
k
Dβ x
−
θ =
w'
= −
k
Aβ x
=
+
βPo 2 k
Cβ x
k
=
k
β2 Po k
=
β Po 2 k
(2 Dβx
− Cβx )
Aβ x
4 β3 ( Po / 4 β )
θ = −
k
+
4 β3 M o
4β
k
=
= −
β 2 Po
Po
2 β 2 ( Po / 4 β )
w
2 β 2 ( Po / 2)
Dβ x (0)
k M o
2 β( Po / 2)
Po /2
M o
4 β3 M o
+
0
Bβ x
Dβ x
= −
β 2 Po k
( Aβ x
−
Dβ x )
0
M
= − E I w '' = −
( Po / 2) Bβx
β
=
M
V
+ ( Po / 4 β) Aβx = − Po
4β
Po
4β
= −
Po
Po
2
( Cβ x
Dβ x
2
Po
x
3π 4β
z
x
π β
w
x
θ x
π 4β M Po
π 2β x
V
− Aβx )
C β x
= − E I w ''' = − ( Po / 2) Cβx − 2 β ( Po / 4β) Bβx = − V
(2 Bβx
+ Bβ x )
M o x
z,w
symmetric deflection w( x )
= − w( − x) =
w(0) w '( x )
=
0
w '( − x)
M o /2
Po
w(0)
=
2 β Po k
Dβ x (0)
−
M o /2
2 β2 ( M o / 2) k Po
w
=
2 β(β M o / 2) k
Dβ x
−
θ =
w'
= −
k
Aβ x
=
C βx (0)
+
β2 M o k
Cβx
k
=
0
k
β3 M o k
k
( Dβ x
− Cβ x )
Bβ x
4 β3 ( M o / 2)
θ =
=
β 2 M o
k
−
β2 M o
2
k
=
=
2 β Po
β M o
2 β2 ( M o / 2)
w
2 β 2 (β M o / 2)
Po
C β x
Dβ x
= −
β3M o k
( Aβ x
−
2 Dβ x )
M
= − E I w '' = −
(β M o / 2) Bβ x
β M
V
+ ( M o / 2) Aβx = − M o
=
= −
2
( Bβx
β M o 2
β M o 2
( Cβ x
Aβ x
M o x
π β
z
x
w x
π 4β θ
π 2β
M o x
M
3π 4β x
V
− Aβ x )
Dβ x
2
= − E I w ''' = − (β M o / 2) Cβx − 2 β ( Mo / 2) Bβ x = − V
M o
+ 2 Bβ x )
Continuous running load on an infinite beam q( x) a
b x
z,w q ( x)
q( x+dx) x
z,w dP = q( x ) dx
x
z,w
dw
=
β dP 2 k
Aβx ,
dθ
= −
β2 dP k
Bβ x ,
dM
=
dP
4β
C βx ,
dV
= −
for running load between x = a and b:
w
=
β
b
∫ Aβ x q( x) dx
2 k a
etc.
Constant running load intensity on an infinite beam
w
=
β q0
b
∫ Aβ x dx
2 k a
dP
2
Dβx
Uniformly distributed load on a semi-infinite beam q0 x
x
z
z 4
E I
d w dx
4
w
+ kw =
q0
q0
=
k q0 x
R q0
z
k x
w w
= −
2 β R k
Dβ x
+
q0 k
so that
R
w
=
q0 k
(1 − Dβ x ) ,
θ =
β q0 k
=
Aβ x ,
w(0)
= −
=
q0
2 β R k
+
q0 k
=
0
=
q0
q0
2β M
2β
2
Bβ x ,
V
2β
C βx
q0 x M w
R q0
z
k x
w
w
= −
2 β R
2 β2 R
θ =
w(0)
= −
2 β R k
+
2
2 β M w k
k q0
+
+
Dβ x
k
2 β2 M w
Aβ x
k
= 0
k
4 β3 M w
−
k
w
q0
=
k
V
w
=
q0 k
(1 − Aβ x ) ,
θ =
k
=
M
(1 − 2 Dβ x
2 β q0
θ =
=
q0
2β
2
q0
2 β q0 k
R
β
=
q0
2 β 2 R k
4 β3 M w
−
=
k
= 0
q0
2β
2
+ C βx )
−
(2 Bβ x
− Aβ x )
Bβ x ,
k
M w
,
β
( Aβ x
(Cβ x
q0
Dβ x
θ(0) =
and
⎡ q0 ⎤ ⎡ 2 β −2 β2 ⎤ ⎡ R ⎤ ⎥⎢ ⎥ , ⎢0⎥ = ⎢ 2 3 M ⎢ ⎥ ⎣ ⎦ ⎣2 β − 4 β ⎦ ⎣ w ⎦
+
C β x
Dβx )
+ Bβx ) M
= −
q0
2β
2
C β x ,
V
=
q0
D β β x
Uniformly distributed load on an infinite beam A
q0 x Q a
b z q0 d ξ d ξ q0 x Q
ξ
O
z dwQ
=
β q0 d ξ 2 k
wQ
wQ
=
Aβ x ( −ξ) if
b
∫
ξ = −a
dwQ
ξ≤0
=
and dwQ
=
β q0 d ξ 2 k
Aβx (ξ) if
b ⎤ β q0 ⎡ 0 ( ) ( ) A −ξ d ξ + A ξ d ξ ∫ βx ⎢ ∫ βx ⎥ 2 k ⎣ − a 0 ⎦
ξ>0
a b⎤ b ⎤ β q0 ⎡⎛ 1 β q0 ⎡ a ⎞ ⎛ ⎞ 1 ⎢⎜ − Dβx ⎟ + ⎜ − Dβ x ⎟ ⎥ = ⎢ ∫ Aβx (ξ) d ξ + ∫ Aβx (ξ) d ξ⎥ = 2 k ⎣0 2 k ⎢⎝ β ⎠0 ⎝ β ⎠0 ⎥⎦ 0 ⎦ ⎣
wQ
θQ =
=
∂wQ = ∂ x
q0
⎡ 2 − Dβa − Dβb ⎤⎦
2 k ⎣
⎡ ∂Dβa ∂a ∂Dβb ∂b ⎤ ⎢ − ∂a ∂ x − ∂ b ∂ x ⎥ 2k ⎣ ⎦ q0
∂a = ∂ x
1,
∂b = −1 ∂ x
θQ =
β q0 ⎡ Aβ a − Aβ b ⎤⎦ , 2 k ⎣
q0
⎡⎣ Bβa + Bβ b ⎤⎦ , 4 β2
=
M Q
VQ
=
q0
maximum bending moment VQ
short loading span,
=
q0
⎡⎣Cβa − C β b ⎤⎦ = 4β
=
M max
long loading span,
0
βA ≤ π only one zero at a q0
2β
2
= b = A/2 BβA / 2
βA > 2 π dominant zero close to left end VQ
Cβa
q0
≈
4β
=
C β a
= e −βa (cos β a − sin β a ) = M max
≈
q0
4β
2
Bπ / 4
≈
βb > 2 π
0
0
βa =
or
0.0806
q
β2
⎡Cβa − C β b ⎤⎦
4β ⎣
π 4
Beams of finite length A
Po x a
b z
general deflection solution
= eβ x (C1A sin βx + C2A cos βx ) + e−βx ( C3A sin β x + C4 A cos β x)
wA ( x )
if
= eβ x (C1r sin βx + C2 r cos βx ) + e−βx ( C3r sin β x + C4r cos β x)
wr ( x )
−a ≤
x
≤0
≤
x
≤b
wr '''(0)
−
Po
if 0
boundary conditions wA ''( −a )
wA (0)
=
wr (0) ,
=
wA '''( −a )
0,
wA '(0)
=
wr '(0) ,
=
wr ''(b)
0,
wA ''(0)
=
=
0,
wr ''(0) ,
wr '''(b) wA '''(0)
=
=
0
E I
symmetric loading Po x A /2
A /2
z
w( x )
=
w( − x)
w( x )
= eβ x (C1 sin βx + C2 cos βx) + e−βx ( C3 sin βx + C4 cos βx)
if 0
w( x )
=
D4 sinh βxcos βx
D1 cosh βx sin β x
+
D2 cosh βx cos β x
+
D3 sinh β x sin β x
+
≤ x ≤ A / 2
boundary conditions w ''(A / 2)
=
0,
w0
w '''( A / 2)
=
M 0
w(0)
=
lim w0 βA → ∞ lim w0 βA → 0
=
=
=
M (0)
=
β Po
0,
β Po
=
=
=
0,
w '''(0)
=
+ cosh βA + cos βA sinh βA + sin βA
Po cosh βA
− cos βA 4 β sinh βA + sin βA and
2 k
2 k 2β A
2
2 k
β Po 4
w '(0)
Po k A
lim M 0 βA → ∞ and
=
Po
4β
lim M 0 = 0 βA → 0
Po
2 E I
