# Chapter10-Beams On Elastic Foundations¶

## Ex1-pg366¶

In [1]:
##calculate the maximum deflection and maximum bending moment and maximum flexural stress in the rail for single wheel load and maximum bending moment
## initialization of variables
import math
##part(a)
E=200. ##GPa
d=184. ##mm
c=99.1 ##mm
Ix=36.9e+06##mm^4
k=14.0 ##N/mm^2
P=170. ##kN
##calculations
E=E*10**3
P=P*10**3
Beta=(k/(4.*E*Ix))**(1/4.)
y_max=P*Beta/(2.*k)
M_max=P/(4.*Beta)
S_max=M_max*c/Ix
print('part (a)')
print"%s %.2f %s"%('\n y_max = ',y_max,' mm')
print"%s %.2f %s"%('\n M_max = ',M_max/10**6,' kN.m')
print"%s %.2f %s"%('\n S_max = ',S_max,' MPa')
## part (b)
z1=1.7##m
z1=z1*10**3 ##mm
z2=2*z1
## A_bz=exp(-Beta*z)*(sin(Beta*z)+cos(Beta*z))
## C_bz=exp(-Beta*z)*(-sin(Beta*z)+cos(Beta*z))
A_bzo=1.
C_bzo=1.
A_bz1=math.exp(-Beta*z1)*(math.sin(Beta*z1)+math.cos(Beta*z1))
A_bz2=math.exp(-Beta*z2)*(math.sin(Beta*z2)+math.cos(Beta*z2))
C_bz1=math.exp(-Beta*z1)*(-math.sin(Beta*z1)+math.cos(Beta*z1))
C_bz2=math.exp(-Beta*z2)*(-math.sin(Beta*z2)+math.cos(Beta*z2))
y_end=P*Beta/(2.*k)*(A_bzo+A_bz1+A_bz2)
M_end=P/(4.*Beta)*(C_bzo+C_bz1+C_bz2)
y_center=P*Beta/(2.*k)*(A_bzo+2.*A_bz1)
M_center=P/(4.*Beta)*(C_bzo+2.*C_bz1)
y_max=max(y_end,y_center)
M_max=max(M_end,M_center)
S_max=M_max*c/Ix
print('\n part(b)')
print"%s %.2f %s"%('\n y_max = ',y_max,' mm')
print"%s %.2f %s"%('\n M_max = ',M_max/10**6,' kN.m')
print"%s %.2f %s"%('\n S_max =',S_max,' MPa')

part (a)

y_max =  5.04  mm

M_max =  51.21  kN.m

S_max =  137.54  MPa

part(b)

y_max =  7.86  mm

M_max =  37.02  kN.m

S_max = 99.42  MPa


## Ex2-pg367¶

In [2]:
import math
## initialization of variables
##calculate the load carried by each spring and deflection of beam
d=100. ##mm
Ix=2.45e+06 ##mm^4
E=72. ##GPa
L=6.8 ##m
K=110. ##N/mm
l=1.1 ##m
P=12. ##kN
##calculations
E=E*10**3
P=P*10**3
l=l*10**3
k=K/l
L1=7.*l
Beta=(k/(4.*E*Ix))**(1/4.)
if(l<math.pi/(4.*Beta)):
y_max=P*Beta/(2.*k)

if(L1>3*math.pi/(2.*Beta)):
M_max=P/(4.*Beta)
S_max=M_max*d/(2.*Ix)

print"%s %.2f %s"%('y_max = ',y_max,' mm')
print"%s %.2f %s"%('\n M_max = ',M_max/10**6,' kN.m')
print"%s %.2f %s"%('\n S_max = ',S_max,' MPa')
A_bl=math.exp(-Beta*l)*(math.sin(Beta*l)+math.cos(Beta*l))
A_2bl=math.exp(-Beta*2*l)*(math.sin(Beta*2*l)+math.cos(Beta*2*l))
A_3bl=math.exp(-Beta*3*l)*(math.sin(Beta*3*l)+math.cos(Beta*3*l))
y_C=P*Beta/(2.*k)*A_bl
y_B=P*Beta/(2.*k)*A_2bl
y_A=P*Beta/(2.*k)*A_3bl
print"%s %.2f %s"%('\n y_C = ',y_C,' mm')
print"%s %.2f %s"%('\n y_B = ',y_B,' mm')
print"%s %.2f %s"%('\n y_A = ',y_A,' mm' )

y_max =  36.81  mm

M_max =  4.89  kN.m

S_max =  99.78  MPa

y_C =  26.35  mm

y_B =  11.41  mm

y_A =  2.24  mm


## Ex4-pg372¶

In [3]:
import math
## initialization of variables
##calculate the maximum deflection maximum flexural stress and maximum pressure between the beam and foundation
E=10. ##GPa
h=200. ##mm
b=100. ##mm
ko=0.04 ##N/mm^3
w=35. ##N/mm
L1=3.61 ##m
##calculations
E=E*10**3
L1=L1*10**3
k=b*ko
Ix=b*h**3/12.
Beta=(k/(4.*E*Ix))**(1/4.)
ba=2.00 ## ba = Beta*a  based on the discussion
##D_bz=exp(-Beta*z)*sin(Beta*z)
D_ba=math.exp(-ba)*math.cos(ba)
y_max=w/k*(1-D_ba)
ba=0.777 ##Beta*a
bb=4.777 ##Beta*b
B_ba=math.exp(-ba)*math.sin(ba)
B_bb=math.exp(-bb)*math.sin(bb)
M_max=abs(-w*(B_ba-B_bb)/(4.*Beta**2))
c=h/2.
S_max=M_max*c/Ix
## calculation of M_H
ba=math.pi/4. ##Beta*a
bb=4-math.pi/4. ##Beta*b
B_ba=math.exp(-ba)*math.sin(ba)
B_bb=math.exp(-bb)*math.sin(bb)
M_H=w/(4.*Beta**2)*(B_ba+B_bb)
print"%s %.2f %s"%('y_max = ',y_max,' mm')
print"%s %.2f %s"%('\n M_max = ',M_max/10**6,' kN.m')
print"%s %.2f %s"%('\n S_max = ',S_max,' MPa')
print"%s %.2f %s"%('\n M_H = ',M_H/10**6,' kN.m')

y_max =  9.24  mm

M_max =  2.36  kN.m

S_max =  3.54  MPa

M_H =  2.28  kN.m


## Ex5-pg375¶

In [4]:
import math
## initialization of variables
##calculate the maximum deflection and maximum flexural stress in the beam and locations of each
E=200. ##GPa
h=102. ##mm
b=68. ##mm
Ix=2.53e+06 ##mm^4
L1=4. ##m
ko=0.35 ##N/mm^3
P=30.0 ##kN
##calculations
E=E*10**3
P=P*10**3
L1=L1*10**3
k=b*ko
Beta=(k/(4.*E*Ix))**(1/4.)
if(L1>3*math.pi/(2.*Beta)):
y_max=2.*P*Beta/k
M_max=-0.3224*P/Beta
S_max=abs(M_max*h/(2.*Ix))

z=math.pi/(4.*Beta)
print"%s %.2f %s"%('y_max = ',y_max,' mm')
print"%s %.2f %s"%('\n M_max = ',M_max,' kN.m')
print"%s %.2f %s"%('\n S_max = ',S_max,' MPa')
print"%s %.2f %s"%('\n Location of Sigma_max is z = ',z,' mm')


y_max =  4.67  mm

M_max =  -5223055.92  kN.m

S_max =  105.29  MPa

Location of Sigma_max is z =  424.13  mm


## Ex6-pg377¶

In [5]:
import math
## initialization of variables
##calculate the maximum deflection and maximum flexural stress
P=30.0 ##kN
a=500. ##mm
h=102. ##mm
b=68. ##mm
k=23.8 ##N/mm**2
Beta=0.001852
Ix=2.53e+06 ##mm**4
##calculations
P=P*10**3.
C_ba=math.exp(-Beta*a)*(-math.sin(Beta*a)+math.cos(Beta*a))
D_ba=math.exp(-Beta*a)*math.cos(Beta*a)
## y = P*Beta/(2*k)*(A_bz+2*D_ba*D_baz+C_ba*C_baz))
## Mx = P/(4*Beta)*(C_bz-2*D_ba*B_baz-C_ba*A_baz)
A_ba=math.exp(-Beta*a)*(math.sin(Beta*a)+math.cos(Beta*a))
B_ba=math.exp(-Beta*a)*math.sin(Beta*a)
C_ba=math.exp(-Beta*a)*(-math.sin(Beta*a)+math.cos(Beta*a))
D_ba=math.exp(-Beta*a)*math.cos(Beta*a)
z1=424 ##mm
z=z1-a
A_bz=math.exp(-Beta*z)*(math.sin(Beta*z)+math.cos(Beta*z))
B_bz=math.exp(-Beta*z)*math.sin(Beta*z)
C_bz=math.exp(-Beta*z)*(-math.sin(Beta*z)+math.cos(Beta*z))
D_bz=math.exp(-Beta*z)*math.cos(Beta*z)
## to find out X_baz
z=a+z
A_baz=math.exp(-Beta*z)*(math.sin(Beta*z)+math.cos(Beta*z))
B_baz=math.exp(-Beta*z)*math.sin(Beta*z)
C_baz=math.exp(-Beta*z)*(-math.sin(Beta*z)+math.cos(Beta*z))
D_baz=math.exp(-Beta*z)*math.cos(Beta*z)
y_max = P*Beta/(2*k)*(A_bz+2*D_ba*D_baz+C_ba*C_baz)
print"%s %.2f %s"%('y_max = ',y_max,' mm')
## For M_max
z1=500. ##mm
z=z1-a
A_bz=math.exp(-Beta*z)*(math.sin(Beta*z)+math.cos(Beta*z))
B_bz=math.exp(-Beta*z)*math.sin(Beta*z)
C_bz=math.exp(-Beta*z)*(-math.sin(Beta*z)+math.cos(Beta*z))
D_bz=math.exp(-Beta*z)*math.cos(Beta*z)
## to find out X_baz
z=a+z
A_baz=math.exp(-Beta*z)*(math.sin(Beta*z)+math.cos(Beta*z))
B_baz=math.exp(-Beta*z)*math.sin(Beta*z)
C_baz=math.exp(-Beta*z)*(-math.sin(Beta*z)+math.cos(Beta*z))
D_baz=math.exp(-Beta*z)*math.cos(Beta*z)
M_max = P/(4.*Beta)*(C_bz-2.*D_ba*B_baz-C_ba*A_baz)
print"%s %.2f %s"%('\n M_max = ',M_max,' N.mm')
S_max=M_max*h/(2.*Ix)
print"%s %.2f %s"%('\n Sigma_max = ',S_max,' MPa')

y_max =  1.32  mm

M_max =  3615504.81  N.mm

Sigma_max =  72.88  MPa


## Ex7-pg381¶

In [6]:
import math
## initialization of variables
##calculate the maximum shear stress and
D=30. ##m
t=10. ##m
h=20. ##mm
E=200. ##GPa
v=0.29
rho=900. ##kg/m**3
##calculations
##part (a)
E=E*10**3.
a=D/2*10**3.
p=t*10**3*9.807*rho*10**-9
S_th=p*a/h
tau_max=S_th/2.
print('part (a)')
print"%s %.2f %s"%('\n Maximum shear stress= ',tau_max,' MPa')
## part (b)
k=E*h/(a**2)
Beta=(3*(1.-v**2)/(h**2*a**2))**(1/4.)
L1=3.*math.pi/(4.*Beta) ##L1=L/2
u=S_th*a/E
w=2.*k*u/(Beta)
M_max=w/(4.*Beta)
Szz_max=M_max*(h/2.)/(h**3/12.)
Sth_max=v*Szz_max
tau_max=Szz_max/2.
u_b=w*(1-v)*a/(2*E*h)
print('\n part (b)')
print"%s %.2f %s"%('\n Maximum shear stress= ',tau_max,' MPa')
print"%s %.2f %s"%('\n u_bottom = ',u_b,' mm')
## part (c)
w=u*k/(2.*Beta)
z=math.pi/(4.*Beta)
B_bz=math.exp(-Beta*z)*math.sin(Beta*z)
M_max=-w*B_bz/Beta
c=6.
I=h**2
Szz_max=(M_max*c/I)
S_th1=v*(Szz_max)
k=0.3224
S_th2=(1-k)*S_th
Sigma_th=S_th1+S_th2
tau_max=(Sigma_th-Szz_max)/2.
print('\n part (c)')
print"%s %.2f %s"%('\n Maximum shear stress= ',tau_max,' MPa')

part (a)

Maximum shear stress=  33.10  MPa

part (b)

Maximum shear stress=  59.90  MPa

u_bottom =  0.10  mm

part (c)

Maximum shear stress=  36.14  MPa