ASME Section VIII – Division 1 - Appendix 13

rectangle_vessel_stresses_tmin.R

# ASME VIII 2017 
# 13-4 DESIGN OF VESSELS OF NONCIRCULAR CROSS SECTION

# Figure 13-2(a) Vessels of Rectangular Cross Section
# Selection  = Sketch (1)

#--- Begin single script - validate calculations against example problem

# ASME PTB-4-2013   - Example Problem - Header Box Data 
# Material = SA516, Grade 70 (S)
S = 20000 
# Design Conditions = 900 psig@350  F(P)
P = 400
# Inside Length (Short Side) = 10.625 in (H)
H = 7.125
# Inside Length (Long Side) = 126.0  in (h)
h = 9.25
# Overall Vessel Length = 346.38 + 8 in (Lv)
Lv = 40.0
# Thickness (Short Side) = 1.625 in (t1)
t1 = 1.0
# Thickness (Long Side) = 1.625 in (t2)
t2 = 1.0
# Thickness (End Plate) = 1.625 in (t5)
t5 = .75
# Corrosion Allowance = 0.125 in (CA)
CA = .125
# Weld Joint Efficiency (Corner Joint) = (E = 1.0) # 100% UT + Spot RT
E = 1.0
#  Tube Outside Diameter = 1.0000 in
do = 1.0
# Tube Pitch = 2.3910 in
p = 2.3910 


# Paragraph 13-5 – Calculate the equation constants.
h = h + 2*CA
H = H +2*(CA)
t1 = t1 - CA
t2 = t2 - CA
t5 = t5 - CA
t = t1 # Plates same t

# The design equations in this paragraph
# are based on vessels in which the ratio of the length of the vessel to the long side or short side
# lengths (aspect ratio) is greater than four. These equations are conservatively applicable to vessels
# of aspect ratio less than four. Vessels with aspect ratios less than four may be designed in
# accordance with the provisions of U-2(g).

if (Lv/h > 4) {
  print("Satisfied")
} else if (Lv/h < 4) {
  print("Not Satisifed")
}
  
# Paragraph 13-5 – Calculate the equation constants.
b = 1.0 
c = t / 2 # The sign of ci is positive. The sign of co is negative
I1 = b*(t1^3) / 12
I2 = b*(t2^3) / 12
a = H/h
K = (I2/I1)*a

# Paragraph 13-7(a) – Calculate the membrane and membrane plus bending stresses.

# The membrane stress on the short side plate, Equation (1):
Sm_eq1 = (P * h) / (2*t1)

# The membrane stress on the long side plate, Equation (2):
# 13-6 LIGAMENT EFFICIENCY OF MULTIDIAMETER HOLES IN PLATES
# In calculations made according to this Appendix for the
# case of a plate with uniform diameter holes, the ligament
# efficiency factors em and eb for membrane and bending
# stresses, respectively, are considered to be the same.
em = (p - do) / p
eb = em
Sm_eq2 = (P * H) / (2 * t2 * em) 

# The bending stress at Location N, short side plate, Equation (3):
SbN_eq3 = ((P*+c) / (12*I1)) * (-1.5 * (H)^2 + (h)^2*((1 + (a)^2 * K) / (1+K)))
SbN_eq3i = -SbN_eq3 # Inside is negative
SbN_eq3o = SbN_eq3 # Outside is positive

# The bending stress at Location Q, short side plate, Equation (4):
SbQ_eq4 = ((P * h^2 * c) / (12 * I1)) * ((1 + a^2 * K) / (1 + K))
SbQ_eq4i = SbQ_eq4 # Inside is positive
SbQ_eq4o = -SbQ_eq4 # Outside is negative 

# The bending stress at Location M, long side plate, Equation (5):
SbM_eq5 = ((P * h^2 * c) / (12 * I2 * eb)) * (-1.5+(1+a^2*K) / (1 + K))
SbM_eq5i = SbM_eq5 # Inside is negative
SbM_eq5o = abs(SbM_eq5) # Outside is positive

# The bending stress at Location Q, long side plate, Equation (6):
SbQ_eq6 = (((P * h^2 * c) / (12 * I2)) * ((1 + a^2 * K) / (1 + K)))
SbQ_eq6i = SbQ_eq6 # Inside is positive
SbQ_eq6o = -SbQ_eq6 # Outside is negative 

# Paragraphs 13-4(b), 13-4(c), and 13-7, Equations (7) through (10) – Acceptance Criteria:
if (Sm_eq1 <= (S * E)) { 
  cat("Short Side Membrane Stress",Sm_eq1,"psig is below Allowable Stress (S * E)", S*E,"psig")
}

# Short side plate at Location N, Membrane + Bending Stress:
N_totali = Sm_eq1 + SbN_eq3i
N_totalo = Sm_eq1 + SbN_eq3o
if (N_totali <= 1.5 * (S * E)) { 
  cat("Short Side Membrane + Bending Stress at location N (inside)",N_totali,"psig is below Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
}
if (N_totalo <= 1.5 * (S * E)) { 
  cat("Short Side Membrane + Bending Stress at location N (outside)",N_totalo,"psig is below Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
}

# Short side plate at Location Q, Membrane + Bending Stress:
Q_totali = Sm_eq1 + SbQ_eq4i
Q_totalo = Sm_eq1 + SbQ_eq4o
if (Q_totali <= 1.5 * (S * E)) { 
  cat("Short Side Membrane + Bending Stress at location Q (inside)",Q_totali,"psig is below Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
}
if (Q_totalo <= 1.5 * (S * E)) { 
  cat("Short Side Membrane + Bending Stress at location Q (outside)",Q_totalo,"psig is below Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
}

# Long side plate, Membrane Stress:
if (Sm_eq2 <= (S * E)) { 
  cat("Short Side Membrane Stress",Sm_eq2,"psig is below Allowable Stress (S * E)", S*E,"psig")
}

# Long side plate at Location M, Membrane + Bending Stress:
M_totali = Sm_eq2 + SbM_eq5i
M_totalo = Sm_eq2 + SbM_eq5o
if (M_totali <= 1.5 * (S * E)) { 
  cat("True - Long Side Membrane + Bending Stress at location M (inside)",M_totali,"psig is below Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
} else if (M_totali > 1.5 * (S * E)) {
  cat("Flase - Long Side Membrane + Bending Stress at location M (inside)",M_totali,"psig is above Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
}
if (M_totalo <= 1.5 * (S * E)) { 
  cat("True - Long Side Membrane + Bending Stress at location M (outside)",M_totalo,"psig is below Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
} else if (M_totalo > 1.5 * (S * E)) {
  cat("False - Long Side Membrane + Bending Stress at location M (outside)",M_totalo,"psig is above Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
}

# Long side plate at Location Q, Membrane + Bending Stress:
Q_totali = Sm_eq2 + SbQ_eq6i
Q_totalo = Sm_eq2 + SbQ_eq6o
if (Q_totali <= 1.5 * (S * E)) { 
  cat("True - Long Side Membrane + Bending Stress at location Q (inside)",Q_totali,"psig is below Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
} else if (Q_totali > 1.5 * (S * E)) {
  cat("Flase - Long Side Membrane + Bending Stress at location Q (inside)",Q_totali,"psig is above Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
}
if (Q_totalo <= 1.5 * (S * E)) { 
  cat("True - Long Side Membrane + Bending Stress at location Q (outside)",Q_totalo,"psig is below Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
} else if (Q_totalo > 1.5 * (S * E)) {
  cat("False - Long Side Membrane + Bending Stress at location Q (outside)",Q_totalo,"psig is above Allowable Stress 1.5*(S * E)", 1.5*(S*E),"psig")
} 

#--- End single script 

#--- Begin iterative script 

# Iterate to find the failure thickness's 
S = 20000 
P = 400
H = 7.125
h = 9.25
Lv = 40.0
t1 = 1.0
t2 = 1.0
t5 = .75
CA = .125
E = 1.0
do = 1.0
p = 2.3910 

# Substitute t1 and t2 for a range of thickness
t1 = seq(from=0.01,to=3.05,by=0.05)
t2 = seq(from=0.01,to=3.05,by=0.05)

# Establish this appendix can be used
if (Lv/h > 4) {
  print("Satisfied")
} else if (Lv/h < 4) {
  print("Not Satisifed")
}

# Iterate through varying t1 and t2 thickness to find failure points 
# Initialize output 
output = data.frame(short_side_membrane_stress = numeric(0), short_side_membrane_plus_bending_stress_at_location_N_inside = numeric(0), short_side_membrane_plus_bending_stress_at_location_N_outside = numeric(0),
                    short_side_membrane_plus_bending_stress_at_location_Q_inside = numeric(0), short_side_membrane_plus_bending_stress_at_location_Q_outside = numeric(0),
                    long_side_membrane_stress = numeric(0),long_side_membrane_plus_bending_stress_at_location_M_inside = numeric(0),long_side_membrane_plus_bending_stress_at_location_M_outside = numeric(0),
                    long_side_membrane_plus_bending_stress_at_location_Q_inside = numeric(0), long_side_membrane_plus_bending_stress_at_location_Q_outside = numeric(0),t1_thickness = numeric(0),t2_thickness = numeric(0), all_pass = numeric(0))

for (i in 1:length(t1)) { 
  # Paragraph 13-5 – Calculate the equation constants.
  b = 1.0 
  c = t1[i] / 2 # The sign of ci is positive. The sign of co is negative
  I1 = b*(t1[i]^3) / 12
  I2 = b*(t2[i]^3) / 12
  a = H/h
  K = (I2/I1)*a
  
  # Paragraph 13-7(a) – Calculate the membrane and membrane plus bending stresses.
  
  # The membrane stress on the short side plate, Equation (1):
  Sm_eq1 = (P * h) / (2*t1[i])
  
  # The membrane stress on the long side plate, Equation (2):
  em = (p - do) / p
  eb = em
  Sm_eq2 = (P * H) / (2 * t2[i] * em) 
  
  # The bending stress at Location N, short side plate, Equation (3):
  SbN_eq3 = ((P*+c) / (12*I1)) * (-1.5 * (H)^2 + (h)^2*((1 + (a)^2 * K) / (1+K)))
  SbN_eq3i = -SbN_eq3 # Inside is negative - compression
  SbN_eq3o = SbN_eq3 # Outside is positive + positive
  
  # The bending stress at Location Q, short side plate, Equation (4):
  SbQ_eq4 = ((P * h^2 * c) / (12 * I1)) * ((1 + a^2 * K) / (1 + K))
  SbQ_eq4i = SbQ_eq4 # Inside is positive + positive
  SbQ_eq4o = -SbQ_eq4 # Outside is negative - compression
  
  # The bending stress at Location M, long side plate, Equation (5):
  SbM_eq5 = ((P * h^2 * c) / (12 * I2 * eb)) * (-1.5+(1+a^2*K) / (1 + K))
  SbM_eq5i = SbM_eq5 # Inside is negative - compression
  SbM_eq5o = abs(SbM_eq5) # Outside is positive + positive
  
  # The bending stress at Location Q, long side plate, Equation (6):
  SbQ_eq6 = (((P * h^2 * c) / (12 * I2)) * ((1 + a^2 * K) / (1 + K)))
  SbQ_eq6i = SbQ_eq6 # Inside is positive + positive
  SbQ_eq6o = -SbQ_eq6 # Outside is negative - compression
  
  # Paragraphs 13-4(b), 13-4(c), and 13-7, Equations (7) through (10) – Acceptance Criteria:
  # 1 = true, 0 = false
  if (Sm_eq1 <= (S * E)) { 
    short_side_membrane_stress = 1 
  } else if (Sm_eq1 > (S * E)) {
    short_side_membrane_stress = 0
  }
  
  # Short side plate at Location N, Membrane + Bending Stress:
  N_totali = Sm_eq1 + SbN_eq3i
  N_totalo = Sm_eq1 + SbN_eq3o
  if (N_totali <= 1.5 * (S * E)) { 
  short_side_membrane_plus_bending_stress_at_location_N_inside = 1
      } else if (N_totali > 1.5 * (S * E)) {
        short_side_membrane_plus_bending_stress_at_location_N_inside = 0
      }
  if (N_totalo <= 1.5 * (S * E)) { 
    short_side_membrane_plus_bending_stress_at_location_N_outside = 1
  } else if (N_totalo > 1.5 * (S * E)) {
    short_side_membrane_plus_bending_stress_at_location_N_outside = 0
  }
  
  # Short side plate at Location Q, Membrane + Bending Stress:
  Q_totali = Sm_eq1 + SbQ_eq4i
  Q_totalo = Sm_eq1 + SbQ_eq4o
  if (Q_totali <= 1.5 * (S * E)) { 
    short_side_membrane_plus_bending_stress_at_location_Q_inside = 1
  } else if (Q_totali > 1.5 * (S * E)) {
    short_side_membrane_plus_bending_stress_at_location_Q_inside = 0
  }
  if (Q_totalo <= 1.5 * (S * E)) { 
    short_side_membrane_plus_bending_stress_at_location_Q_outside = 1
  } else if (Q_totalo > 1.5 * (S * E)) {
    short_side_membrane_plus_bending_stress_at_location_Q_outside = 0
  }
  
  # Long side plate, Membrane Stress:
  if (Sm_eq2 <= (S * E)) { 
    long_side_membrane_stress = 1 
  } else if (Sm_eq2 > (S * E)) {
    long_side_membrane_stress = 0
  }
  
  # Long side plate at Location M, Membrane + Bending Stress:
  M_totali = Sm_eq2 + SbM_eq5i
  M_totalo = Sm_eq2 + SbM_eq5o
  if (M_totali <= 1.5 * (S * E)) { 
    long_side_membrane_plus_bending_stress_at_location_M_inside = 1
  } else if (M_totali > 1.5 * (S * E)) {
    long_side_membrane_plus_bending_stress_at_location_M_inside = 0
  }
  if (M_totalo <= 1.5 * (S * E)) { 
    long_side_membrane_plus_bending_stress_at_location_M_outside = 1
  } else if (M_totalo > 1.5 * (S * E)) {
    long_side_membrane_plus_bending_stress_at_location_M_outside = 0
  }
  
  # Long side plate at Location Q, Membrane + Bending Stress:
  Q_totali = Sm_eq2 + SbQ_eq6i
  Q_totalo = Sm_eq2 + SbQ_eq6o
  if (Q_totali <= 1.5 * (S * E)) { 
    long_side_membrane_plus_bending_stress_at_location_Q_inside = 1
  } else if (Q_totali > 1.5 * (S * E)) {
    long_side_membrane_plus_bending_stress_at_location_Q_inside = 0
  }
  if (Q_totalo <= 1.5 * (S * E)) { 
    long_side_membrane_plus_bending_stress_at_location_Q_outside = 1
  } else if (Q_totalo > 1.5 * (S * E)) {
    long_side_membrane_plus_bending_stress_at_location_Q_outside = 0
  }
  
  t1_thickness = t1[i]
  t2_thickness = t2[i]
  
  # For varying t1,t2 check if Stresses at N,M and Q (inside and outside) Pass
  test_all_pass = sum(short_side_membrane_stress,short_side_membrane_plus_bending_stress_at_location_N_inside,short_side_membrane_plus_bending_stress_at_location_N_outside,
                 short_side_membrane_plus_bending_stress_at_location_Q_inside,short_side_membrane_plus_bending_stress_at_location_Q_outside,long_side_membrane_stress,
                 long_side_membrane_plus_bending_stress_at_location_M_inside,long_side_membrane_plus_bending_stress_at_location_M_outside,long_side_membrane_plus_bending_stress_at_location_Q_inside,
                 long_side_membrane_plus_bending_stress_at_location_Q_outside)
  
  if(test_all_pass == 10) {
    all_pass = 1
  } else if (test_all_pass != 10) {
    all_pass = 0
}
  
  
  temp_row = data.frame(short_side_membrane_stress,short_side_membrane_plus_bending_stress_at_location_N_inside,short_side_membrane_plus_bending_stress_at_location_N_outside,
                        short_side_membrane_plus_bending_stress_at_location_Q_inside,short_side_membrane_plus_bending_stress_at_location_Q_outside,long_side_membrane_stress,
                        long_side_membrane_plus_bending_stress_at_location_M_inside,long_side_membrane_plus_bending_stress_at_location_M_outside,long_side_membrane_plus_bending_stress_at_location_Q_inside,
                        long_side_membrane_plus_bending_stress_at_location_Q_outside,t1_thickness,t2_thickness,all_pass)
  output = rbind(output,temp_row)
  
  
  } # end

# Find first passing thickness (tmin)
for (i in 2:nrow(output)) { 
  if(output$all_pass[i-1] == 0 & output$all_pass[i] == 1) {
    tmin = max(output$t1_thickness[i],output$t2_thickness[i])
    }
}

cat("Resulting Pressure Tmin = ",tmin)

# plot data
t = output$t1_thickness
index = 1:nrow(output)

plot(x = index,y = t, xlab = "No.", ylab = "t", main = "Thickness Needed For Stresses At Location N, M and Q To All Pass", cex.main=1.0,axes = FALSE,col = ifelse(t < tmin,'red','green'), pch = 19 )
axis(side = 1, at = 1:length(t),labels = 1:length(t), las =1,cex.axis=.7,tck = 0, lty = 2, col = "grey")
axis(side = 2, at = seq(from=0.01,to=3.05,by=0.05),labels = seq(from=0.01,to=3.05,by=0.05),  las =0,cex.axis=.7,tck = 0, lty = 2)
box()
abline(h=tmin)
text(13, 1.0, expression("Tmin = 0.86"), col = "black")
text(25, .7, expression("Thickness = Fail"), col = "red")
text(55, 2.0, expression("Thickness = Pass"), col = "green")

#--- End iterative script 

Hello flare9x. Do you have a user manual of it?. Looks very interesting !

Hello flare9x. Do you have a user manual of it?. Looks very interesting !
ASME Section VIII Div 1 Mandatory Appendix 13

how do i run this?

its written for R. There's a single script for a given plate thickness - what the stresses are, end line 157. There is another script that at a specified plate dimension, will iterate over a 'range' of plate thickness's and check that the stresses induced are within code allowable stress - that output is saved in a dataframe and the plot at the bottom will show this result as well. Run script in R, RStudio.