Chapter 14 : Combined Loading

Example 14.1 Page No : 349

In [1]:
			
# Variables
h = 6. 	    		#in
x1 = 7. 			#in
x2 = 1. 			#in
x3 = 2. 			#in
P = 600. 			#lb
			
# Calculations
By = P*(x1+x2+x3)/(x1+x2)
Bx = By*(x1+x2)/h
Fx = Bx
V = By-P
M = -P*(x2+x3)+By*x2
S1 = -Fx/(x3*h)
I = x3*h**3/12
S2 = -M*12*(h/2)/I
Scmax = S1-S2
Stmax = S1+S2
			
# Results
print  'Maximum tensile stress at  = %.1f psi'%(Scmax)
print  'Maximum compressive stress at  = %.1f psi'%(Stmax)

Maximum tensile stress at  = -1133.3 psi
Maximum compressive stress at  = 966.7 psi

Example 14.2 Page No : 350

In [3]:
			
# Variables
P = 10000. 			#lb
A = 11.77 			#in**2
Z = 51.9 			#in**3
x = 5. 			#ft
y = 12. 			#ft
			
# Calculations
S1 = round(-P/A)
S2 = round(P*x*y/Z,-2)
Sc = S1-S2
St = S1+S2

			
# Results
print  'Axial stress at c = %.1f psi'%(Sc)
print  'Axial stress at t = %.1f psi'%(St)

Axial stress at c = -12450.0 psi
Axial stress at t = 10750.0 psi

Example 14.3 Page No : 353

In [1]:
			
# Variables
b = 6. 		    	#in
h = 12. 			#in
l = 20. 			#ft
P = 100000. 		#lb
			
# Calculations
S = -P/(b*h)
S1 = l**2*6*12/(8*b*h**2)
w = -S/S1
			
# Results
print  'Safe distributed load = %.0f lb per ft'%(w)

Safe distributed load = 333 lb per ft

Example 14.4 Page No : 355

In [4]:
			
# Variables
b = 4. 			#in
h = 9. 			#in
l = 6. 			#in
Mx = 600. 			#lb
My = 100. 			#lb
			
# Calculations
Zx = b*h**3/(12*h/2)
Zy = b**3*h/(12*b/2)
S1 = Mx*l*12/Zx
S2 = My*b*12/Zy
Sb = S1+S2
Sd = -S1-S2
			
# Results
print  'Maximum stress = %.1f psi tension)'%(Sb)
print  '  Maximum stress =%.1f psi compression)'%(Sd) 

Maximum stress = 1000.0 psi tension)
  Maximum stress =-1000.0 psi compression)

Example 14.5 Page No : 356

In [4]:
import math 
			
# Variables
d = 2. 			#in
Px = -600. 			#lb
Py = 1200. 			#lb
x1 = 2. 			#in
x2 = 2. 			#in
x3 = 2. 			#in
Ray = -400. 			#lb
Rax = 400. 			#lb
Rbx = 200. 			#lb
Rby = -800. 			#lb
			
# Calculations
Mb = math.sqrt((Rax*x1)**2+(Ray*x1)**2)
Mc =math.sqrt((Rbx*x3)**2+(Rby*x3)**2)
if (Mb<Mc) :
    M =Mc
else: 
    M = Mb    

Smax = M*12*64*(d/2)/(math.pi*d**4)
			
# Results
print  'Maximum normal stress = %.1f psi'%(round(Smax,-2)) 

Maximum normal stress = 25200.0 psi

Example 14.6 Page No : 359

In [5]:
			
# Variables
P = 100. 			#kips
M = 400. 			#kip in
A = 14.7 			#in**2
Z = 80.7 			#in**3
			
# Calculations
Smax = -(P*10**3)/A-(M*10**3)/Z
Smin = -(P*10**3)/A+(M*10**3)/Z
			
# Results
print  'Maximum stress = %.1f psi'%(round(Smax,-2))
print  'Minimum stress =%.1f psi'%(round(Smin,-1))

Maximum stress = -11800.0 psi
Minimum stress =-1850.0 psi

Example 14.7 Page No : 360

In [7]:
			
# Variables
As = 1. 			#in**2
Zs = 0.167 			#in**3
Ah = 1. 			#in**2
Zh = 0.984 			#in**3
es = 0.5 			#in
eh = 0.5 			#in
			
# Calculations
phbyps = (1/As+es/Zs)/(1/Ah+eh/Zh)
			
# Results
print  'ratio = %.1f'%(phbyps)

ratio = 2.6

Example 14.8 Page No : 365

In [7]:
import math 
			
# Variables
Sx = 1800. 			#psi
Sy = 1000. 			#psi
angle = 30. 			#degrees
t = 0.25 			#in
t1 = 3. 			#in
t2 = 5. 			#in
			
# Calculations
Sx1 = Sx/(t1*t)
Sy1 = Sy/(t2*t)
S = ((Sx+Sy)/2+((Sx-Sy)/2)*math.cos(math.radians(2*angle)))+(Sx-Sy)*math.cos(math.radians(2*angle))
T = (Sx-Sy)*math.sin(math.radians(2*angle))
			
# Results
print  'S = %.1f psi'%(S)
print  'T =%.f psi'%(T)

S = 2000.0 psi
T =693 psi

Example 14.9 Page No : 365

In [8]:
			
# Variables
Sx = 1800. 			#lb
Sy = 1000. 			#lb
angle = 30. 			#degrees
			
# Calculations
Sa =-((Sx+Sy)/2+((Sx-Sy)/2)*math.cos(math.radians(2*angle)))*math.cos(math.radians(2*angle))-(Sx-Sy)
Ta = -((Sx+Sy)/2+((Sx-Sy)/2)*math.cos(math.radians(2*angle)))*math.sin(math.radians(2*angle))
Sb =((Sx+Sy)/2+((Sx-Sy)/2)*math.cos(math.radians(2*angle)))*math.cos(math.radians(2*angle))-(Sx-Sy)
Tb = ((Sx+Sy)/2+((Sx-Sy)/2)*math.cos(math.radians(2*angle)))*math.sin(math.radians(2*angle))
			
# Results
print  'Sa = %.1f psi'%(Sa)
print  'Sb =%.1f psi'%(Sb)
print  'Ta =%.f psi'%(round(Ta,-1))
print  'Tb =%.1f psi'%(round(Tb,-1))

Sa = -1600.0 psi
Sb =0.0 psi
Ta =-1390 psi
Tb =1390.0 psi

Example 14.10 Page No : 369

In [9]:
import math 

		
# Variables
angle = 15. 			#degrees
Tyx = -1000. 			#psi
Txy = 1000. 			#psi
			
# Calculations
Sx = Txy*math.sin(math.radians(2*angle))
Tx = Txy*math.cos(math.radians(2*angle))
Sy = Tyx*math.sin(math.radians(2*angle))
Ty = Tyx*math.cos(math.radians(2*angle))
Sx1 = Txy
Sy1 = Tyx
Txy = 0
			
# Results
print  'Sx = %.1f psi'%(Sx)
print  'Tx = %.1f psi'%(Tx)
print  'Sy = %.1f psi'%(Sy)
print  'Ty = %.1f psi'%(Ty)
print  'Sx1 = %.1f psi'%(Sx1)
print  'Sy1 = %.1f psi'%(Sy1)
print  'Txy = %.1f psi'%(Txy)

Sx = 500.0 psi
Tx = 866.0 psi
Sy = -500.0 psi
Ty = -866.0 psi
Sx1 = 1000.0 psi
Sy1 = -1000.0 psi
Txy = 0.0 psi

Example 14.11 Page No : 370

In [12]:
import math 
			
# Variables
d = 4. 			#in
n = 315. 			#rpm
Ss = 8000. 			#psi
Ns = 12000. 			#psi
			
# Calculations
T = Ss*d**4/(32*(d/2))
hp = T*math.pi*n/63000
			
# Results
print  'T = %.1f pi lb in'%(T)
print  'horsepower rating =%.1f hp'%(round(hp,-1))

T = 32000.0 pi lb in
horsepower rating =500.0 hp

Example 14.12 Page No : 372

In [16]:
import math 
			
# Variables
Sx = 9. 			#ksi
Sy = -5. 			#ksi
Txy = 4. 			#ksi
			
# Calculations
R = math.sqrt(((Sx-Sy)/2)**2+Txy**2)
Smax = ((Sx+Sy)/2)+R
Smin = ((Sx+Sy)/2)-R
ap1 = (1./2)*math.degrees(math.atan(2*Txy/(Sx-Sy)))
ap2 = 90+ap1
Sc = (Sx+Sy)/2
Tc = R
Sd = (Sx+Sy)/2
Td = -R
a1 = (90-2*ap1)/2
a2 = 90+a1
			
# Results
print  'Smax = %.2f ksi'%(Smax)
print  'Smin = %.2f ksi'%(Smin)
print  'R = %.2f psi'%(R)
print  'palne1 = %.2f degrees'%(ap1)
print  'plane 2 = %.2f degrees'%(ap2)
print  'Sc = %.2f ksi'%(Sc)
print  'Sd = %.2f ksi'%(Sd)
print  'Tc = %.2f ksi'%(Tc)
print  'Td = %.2f ksi'%(Td)
print  'palne1 = %.2f degrees'%(a1)
print  'plane 2 = %.2f degrees'%(a2)

# note : rounding off error.

Smax = 10.06 ksi
Smin = -6.06 ksi
R = 8.06 psi
palne1 = 14.87 degrees
plane 2 = 104.87 degrees
Sc = 2.00 ksi
Sd = 2.00 ksi
Tc = 8.06 ksi
Td = -8.06 ksi
palne1 = 30.13 degrees
plane 2 = 120.13 degrees

Example 14.13 Page No : 373

In [21]:
import math 
			
# Variables
d = 4. 		        	#in
T = 40000. 			    #lb in
Th = 20000. 			#lb in
			
# Calculations
t = round(T*(d/2)*32/(math.pi*d**4),-1)
S = round(Th/(math.pi*(d/2)**2),-1)
Smax = -(S/2)-math.sqrt(t**2+(S/2)**2)
Tmax = math.sqrt(t**2+(S/2)**2)

			
# Results
print  'Maximum normal stress = %.f psi'%(Smax)
print  'Maximum shearing stress =%.f psi'%(Tmax)

Maximum normal stress = -4073 psi
Maximum shearing stress =3278 psi