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Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

YOUR COMPANY LOGO Rectangular Open Top Tank Design Per AISC 360 Rev #

Rev Description

Rev By

Rev Date

1 2 3 4 Notes 1 2 3 4 5

www.mathcadcalcs.com

Page 1 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design This program is used to design large rectangular open top tanks per AISC 360. The tanks consist of plate sides and bottoms, a horizontal stiffener at the top (Wide Flange or Channel) and vertical stiffeners at some spacing on the sides (Wide Flange or Channel.)

A. Geometry Length of tank

Width of tank

Height of tank

Ltank := 38.4167 ⋅ ft

Btank := 12 ⋅ ft

Htank := 12 ⋅ ft

Thickness of tank walls

Location of horizontal stiffener above bottom

ts := .3125 ⋅ in

Hst := 11 ⋅ ft

Design liquid level

DLL := 12 ⋅ ft

Number of cross members

Ncross := 2

Spacing of tank stiffeners

Sv := 3.167 ⋅ ft

Vertical Stiffener Geometry Beam Selection (W or C shapes) Unbraced Length of Soil Side Flange

Stiffener Length

Unbraced Length of Product Side Flange

UBLB1 := .1 ⋅ ft

UBLT1 := Htank

LB1 := Htank

Horizontal Stiffener Geometry (Long Side) Beam Selection (W or C shapes)

Unbraced Length of Soil Side Flange

Unbraced Length of Product Side Flange

Ltank UBLB2 := Ncross + 1

Ltank UBLT2 := Ncross + 1

Stiffener Length

LB2 :=

Ltank Ncross + 1

Horizontal Stiffener Geometry (Short Side) Beam Selection (W or C shapes) Unbraced Length of Soil Side Flange

Unbraced Length of Product Side Flange

UBLB3 := Btank

UBLT3 := Btank

Stiffener Length

LB3 := Btank

Cross Member Geometry Column Selection (W shapes)

Column Length

Unbraced Length for Strong Axis

Unbraced Length for Weak Axis

LC := Btank

UBLs := Btank

UBLw := Btank

Eff. length factor for strong axis

Eff. length factor for weak axis

KxC := 1.0

KyC := 1.0

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Page 2 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design B. Material Properties Modulus of elasticity

Yield strength of platel

Es := 29000 ⋅ ksi

Fyp := 36 ⋅ ksi

Density of soil

Active pressure coefficient

γe := 110 ⋅ pcf

Ka := .35

Yield strength of stiffeners

Yield Strength or cross member

Safety factor for plate bending

FyB := 50 ⋅ ksi

FyC := 50 ⋅ ksi

Ωp := 1.67

Height to groundwater above bottom

Hgw := 5 ⋅ ft

Surcharge loading

Qsur := 400 ⋅ psf

Specific gravity

SG := 1.0

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Page 3 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design C. Loading Crteria Product pressure at bottom of tank

Pp := DLL ⋅ SG ⋅ γw = 5.20 psi

Peb := Hgw ⋅ γw + Hgw ⋅ γe − γw ⋅ Ka + Htank − Hgw ⋅ γe ⋅ Ka + Qsur ⋅ Ka ⋅ BURY  

(

)

(

)

Peb = 5.59 psi

Earth pressure at bottom of tank

Pet := Qsur ⋅ Ka ⋅ BURY = 0.97 psi

Earth pressure at top of tank

Pegw :=

( Htank − Hgw) ⋅ γe ⋅ Ka + Qsur ⋅ Ka ⋅ BURY = 2.84 psi

Earth pressure at ground water

D. Check Plate Thickness bvf1 :=

CBF( BEAM1) ⋅ in if BEAM1 ≤ 31

Flange width of vertical stiffener

= 5.25 ⋅ in

WBFBEAM1−31 ⋅ in otherwise

Mp1 :=

)2

(

P p ⋅ 1 ⋅ in ⋅ S v − bvf1

Plate moment due to fluid pressure at bottom of tank

Mp1 = 38.75 ⋅ ft ⋅ lbs

12 2

  DLL − Hst     DLL − Hst   1 1 Mp2 := max     ⋅ γw ⋅ SG ⋅ 1 ⋅ in ⋅ ⋅ ⋅ max    2 3   0 ⋅ ft     0 ⋅ ft   Plate moment due to fluid pressure in cantilvered plate above top stiffener

Mp2 = 0.87 ⋅ ft ⋅ lbs

Me1 :=

(

)2

Peb ⋅ 1 ⋅ in ⋅ Sv − bvf1 12

Plate moment due to earth pressure at bottom of tank

Me1 = 41.64 ⋅ ft ⋅ lbs

2  P ⋅ ( H  et tank − Hst ) 1 1 2 Me2 :=  + ⋅ ( Htank − Hst ) ⋅ γe ⋅ Ka ⋅ ⋅ ( Htank − Hst ) ⋅ 1 ⋅ in 2 2 3  

Plate moment due to earth pressure in cantilvered plate above top stiffener

Me2 = 6.37 ⋅ ft ⋅ lbs

  Mp1       Mp2   Msmax := max      Me1    M    e2  

Msmax = 41.64 ⋅ ft ⋅ lbs

Maximum plate moment over a 1" width

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Page 4 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design D. Check Plate Thickness 2

Fyp 1 ⋅ in ⋅ ts Mnp := ⋅ Ωp 4

Bending strength of plate

Mnp = 43.86 ⋅ ft ⋅ lbs Msmax Mnp

max ∆p :=

Plate bending strength check - Must be less than or equal to 100%

= 94.95 ⋅ %

  Pp   4 ⋅ 1 ⋅ in ⋅ ( Sv − bvf1)  P    eb   3

384 ⋅ E s ⋅

1 ⋅ in ⋅ ts 12

∆p = 0.23 ⋅ in

Deflection of plate

∆p

Plate deflection check - Must be less than or equal to 100%

ts

= 72.69 ⋅ %

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Page 5 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design E. Loading in Vertical Stiffener Uniform fluid load in stiffener as a function of height above the bottom

qp ( x) := Pp ⋅ S v − γw ⋅ SG ⋅ x ⋅ Sv

(

)

qp Htank = 0.00 ⋅ plf qp ( 0 ⋅ ft) = 2372.21 ⋅ plf

 (Peb − Pegw) ⋅ x Peb − ⋅ S if x < Hgw   v Hgw

qe ( x) :=

Uniform soil load in stiffener as a function of height above the bottom

 (Pegw − Pet) ⋅ (x − Hgw) ⋅ S otherwise Pegw −   v Htank − Hgw x := 0 ⋅ ft ,

Htank 50

.. Htank

Uniform Soil Load

Uniform Product Load 12

Height Above Bottom

Height Above Bottom

12

6

0

0

1275

2549

6

0

0

Uniform Load

1186

2372

Uniform Load

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Page 6 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design E. Loading in Vertical Stiffener H ⌠ tank  qe ( x) ⋅ x dx  ⌡

R2e :=

0 ⋅ ft

R2e = 5809.49 ⋅ lbs

Htank

H ⌠ tank  R1e := qe ( x) dx − R2e  ⌡

R1e = 9895.76 ⋅ lbs

Reaction at bottom of vertical stiffener due to soil loading Reaction at top of vertical stiffener due to soil loading

0 ⋅ ft

⌠ Me ( x1) := R1e ⋅ x1 −  ⌡

x1

Moment as a function of x due to soil loading

qe ( x) ⋅ ( x1 − x) dx

0 ⋅ ft

MARRAYe :=

for i ∈ 1 .. 100

 Htank  ⋅ i  100 

m i ← Me  m

Maximum moment for soil load

Mmaxe := max ( MARRAYe ) Mmaxe = 22861.83 ⋅ ft ⋅ lbs H ⌠ tank  qp ( x) ⋅ x dx  ⌡

R2p :=

0 ⋅ ft

R2p = 4744.42 ⋅ lbs

Htank

H ⌠ tank  R1p := qp ( x) dx − R2p  ⌡

R1e = 9895.76 ⋅ lbs

Reaction at bottom of vertical stiffener due to product loading Reaction at top of vertical stiffener due to product loading

0 ⋅ ft

⌠ Mp ( x1) := R1p ⋅ x1 −  ⌡

x1

qp ( x) ⋅ ( x1 − x) dx

0 ⋅ ft

MARRAYp :=

Moment as a function of x due to product loading

for i ∈ 1 .. 100

 Htank  ⋅ i  100 

m i ← Mp  m Mmaxp := max ( MARRAYp)

Maximum moment for product load

Mmaxp = 21912.84 ⋅ ft ⋅ lbs

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Page 7 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design E. Loading in Vertical Stiffener Moment in Vertical Stiffener from Soil

Moment

22862

11431

0

0

6

12

Height Above Bottom

Moment in Vertical Stiffener from Product

Moment

21913

10956

0

0

6

12

Height Above Bottom

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Page 8 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design F. Vertical Stiffener Design 1. Beam Loadings Ultimate Positive Bending Moment

Mup := 1.6 ⋅ Mmaxp Ultimate Negative Bending Moment

Mun := 1.6 ⋅ Mmaxe Ultimate shear in beam

  R2e      R2p  

VB := 1.6 ⋅ max  

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Page 9 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design F. Vertical Stiffener Design RT

2. Member Properties UBLT1 = 12.00 ft

Unbraced length of top flange

UBLB1 = 0.10 ft

Unbraced length of bottom flange

Mup = 35.06 ⋅ ft ⋅ kip

Ultimate positive moment

Mun = 36.58 ⋅ ft ⋅ kip

Ultimate negative moment

Moment of inertia of beam

ZxB = 17.00 ⋅ in

Bending diagram factor

ryB = 1.23 ⋅ in

Torsional constant

IyB = 7.97 ⋅ in

Strong axis section modulus

rtsB = 1.43 ⋅ in

Torsional radius of gyration

dB = 8.14 ⋅ in

Beam depth

tfB = 0.33 ⋅ in

Beam flange thickness

twB = 0.23 ⋅ in

Beam web thickness

bfB = 5.25 ⋅ in

Beam flange width

cB = 1.00

Factor used for LTB capacity

hoB = 7.81 ⋅ in

Center to center of flanges

IB = 61.90 ⋅ in

4

CbB = 1.00 CwB = 122.00 ⋅ in SxB = 15.20 ⋅ in

6

3

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3

Plastic section modulus Weak axis radius of gyration

4

Weak axis moment of inertia

Page 10 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design F. Vertical Stiffener Design 3. Bending Strength LpB := 1.76 ⋅ ryB ⋅

Es

Critical unbraced flange length for which inelastic bukling applies (AISC 360-05, F2-5)

FyB

Critical unbraced flange length for which elastic bukling applies (AISC 360-05, F2-6) Es

LrB := 1.95 ⋅ rtsB ⋅ ⋅ 0.7 ⋅ FyB

JB ⋅ cB SxB⋅ hoB

⋅ 1+

2

FcrB ( UBL) :=

CbB ⋅ π ⋅ Es

 UBL  r   tsB 

2

 0.7 ⋅ FyB SxB⋅ hoB  1 + 6.76 ⋅  ⋅  Es JB ⋅ cB  

⋅ 1 + 0.078 ⋅

JB ⋅ cB

 UBL   SxB ⋅ hoB rtsB   ⋅

2

Critical stress based on LTB (AISC 360-05, F2-4)

Plastic moment strength (AISC 360-05, F2-1)

MpB := FyB ⋅ ZxB

Nominal moment strength based on yielding

φMnYB := 0.9 ⋅ MpB

φMnLTB ( UBL) := 0.9 ⋅ CbB ⋅ MpB ...

 +  

φMnLTBB ( UBL) :=

2

  UBL − LpB   −MpB ...  ⋅   LrB − LpB  + − 0.7 ⋅ F ⋅ S yB xB)   ( 

    

Nominal moment strength based on LTB (AISC 360-05, F2-2 and F2-3

0.9 ⋅ MpB if UBL ≤ LpB φMnLTB ( UBL) if

(UBL > LpB) ⋅ (UBL ≤ LrB)

0.9 ⋅ FcrB ( UBL) ⋅ SxB otherwise

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Nominal moment strength based on LTB with limits (AISC 360-05, F2-2 and F2-3)

Page 11 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design F. Vertical Stiffener Design 3. Bending Strength λB :=

Flange slenderness ratio for local buckling (AISC 360-05 F3-1)

CBFbyTF( BEAM1) if BEAM1 ≤ 31 WBFby2TFBEAM1−31 otherwise

HbyTW :=

Web slenderness ratio (AISC 360-05 F3-2)

CHbyTW( BEAM1) if BEAM1 ≤ 31 WHbyTWBEAM1− 31 otherwise Es

λpfB := 0.38 ⋅

FyB

Limiting slenderness for non-compact flange (Table B4.1)

Es

λrfB := 1.0 ⋅

kcB :=

Limiting slenderness for compact flange (Table B4.1)

FyB 4

0.35 if

(AISC 360-05 F3-2)

< 0.35

HbyTW 4

0.76 if

> 0.76

HbyTW 4

otherwise

HbyTW

φMnFLB := 0.9 ⋅ MpB ...

 +  

φMnFLBB :=



 λB − λpfB    ⋅  λrfB − λpfB     −( 0.7 ⋅ FyB ⋅ SxB)  

−MpB ...

+ 

    

Moment strength based on flange local buckling (AISC 360-05 F3-1)

0.9 ⋅ MpB if λB ≤ λpfB φMnFLB if 0.9 ⋅

(λB > λpfB) ⋅ (λB ≤ λrfB)

0.9 ⋅ Es ⋅ kcB⋅ SxB

( ) λB

2

otherwise

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Moment strength based on flange local buckling with limits (AISC 360-05 F3-1)

Page 12 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design F. Vertical Stiffener Design 3. Bending Strength φMnYB     φMnB ( UBL) := min   φMnLTBB ( UBL)       φMnFLBB  

Nominal moment strength of beam

Beam Capacity as a Function of Unbraced Length 70

Moment Capacity (ft-kips)

60

50

40

30

0

5

10 Unbraced Length (ft)

Nominal Moment Strength Positive Moment at Unbraced Length Negative Moment at Unbraced Length

Mup

(

)

φMnB UBLT1

= 80.14 ⋅ %

Mun

(

)

φMnB UBLB1

= 57.38 ⋅ %

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All ratios must be at 100% or less try another beam shape if over 100%

Page 13 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design F. Vertical Stiffener Design 4. Shear Strength φVnB := 1.0 ⋅ dB ⋅ twB ⋅ 0.6 ⋅ FyB

Nominal shear strength for beam

φVnB = 56.17 ⋅ kip VB φVnB

= 16.55 ⋅ %

Ratio must be less than or equal to 100% - try another beam shape if over 100%

5. Web Compactness λpwB := 3.76 ⋅

Es FyB

Limiting slenderness ratio for web compactness (AISC 360-05, Table B4.1)

λpwB = 90.55 HbyTW = 29.90 HbyTW λpwB

= 33.02 ⋅ %

Slenderness ratio for beam

Ratio must be less than or equal to 100%

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Page 14 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design G. Loading in Horizontal Stiffener R1e whe := = 3124.65 ⋅ plf Sv

Horizontal unit load due to soil pressure

R1p whp := = 2996.16 ⋅ plf Sv

Horizontal unit load due to product pressure

MheL :=

 Ltank whe ⋅   Ncross + 

 1 

2

if Ltank = Btank

12

 Ltank  whe ⋅   ( Ncross + 1) 8

Moment in long side due to soil pressure

2

otherwise

MheL = 64.05 ⋅ ft ⋅ kip

MheB :=

whe ⋅ Btank

2

if Ltank = Btank

12 whe ⋅ Btank

Moment in short side due to soil pressure

2

8

otherwise

MheB = 56.24 ⋅ ft ⋅ kip

VheL :=

Ltank whe ⋅ Ncross + 1

Shear in long direction due to soil pressure

2

VheL = 20006.44 ⋅ lbs VheB :=

whe ⋅ Btank

Shear in short direction due to soil pressure

2

VheB = 18747.88 ⋅ lbs

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Page 15 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

heB

Rectangular Tank Design G. Loading in Horizontal Stiffener

MhpL :=

 Ltank  whp ⋅   ( Ncross + 1) 

2

if Ltank = Btank

12

 Ltank  whp ⋅   ( Ncross + 1)    8

Moment in long side due to product pressure

2

otherwise

MhpL = 61.41 ⋅ ft ⋅ kip

MhpB :=

whp ⋅ Btank

2

if Ltank = Btank

12 whp ⋅ Btank 8

Moment in short side due to product pressure

2

otherwise

MhpB = 53.93 ⋅ ft ⋅ kip

VhpL :=

Ltank whp ⋅ Ncross + 1

(

)

Shear in long direction due to product pressure

2

VhpL = 19183.76 ⋅ lbs VhpB :=

whp ⋅ Btank

Shear in short direction due to product pressure

2

VhpB = 17976.96 ⋅ lbs

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Page 16 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design H. Horizontal Stiffener Design (Long Side) 1. Beam Loadings Ultimate Positive Bending Moment

Mup := 1.6 ⋅ MhpL Ultimate Negative Bending Moment

Mun := 1.6 ⋅ MheL Ultimate shear in beam

VB := 1.6 ⋅ max

  VheL     VhpL    

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Page 17 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design H. Horizontal Stiffener Design (Long Side) RT

2. Member Properties UBLT1 = 12.81 ft

Unbraced length of top flange

UBLB1 = 12.81 ft

Unbraced length of bottom flange

Mup = 98.26 ⋅ ft ⋅ kip

Ultimate positive moment

Mun = 102.48 ⋅ ft ⋅ kip

Ultimate negative moment

Moment of inertia of beam

ZxB = 47.30 ⋅ in

Bending diagram factor

ryB = 1.49 ⋅ in

Torsional constant

IyB = 19.60 ⋅ in

Strong axis section modulus

rtsB = 1.77 ⋅ in

Torsional radius of gyration

dB = 13.80 ⋅ in

Beam depth

tfB = 0.39 ⋅ in

Beam flange thickness

twB = 0.27 ⋅ in

Beam web thickness

bfB = 6.73 ⋅ in

Beam flange width

cB = 1.00

Factor used for LTB capacity

hoB = 13.41 ⋅ in

Center to center of flanges

IB = 291.00 ⋅ in

4

CbB = 1.00 CwB = 887.00 ⋅ in SxB = 42.00 ⋅ in

6

3

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3

Plastic section modulus Weak axis radius of gyration

4

Weak axis moment of inertia

Page 18 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design H. Horizontal Stiffener Design (Long Side) 3. Bending Strength LpB := 1.76 ⋅ ryB ⋅

Es

Critical unbraced flange length for which inelastic bukling applies (AISC 360-05, F2-5)

FyB

Critical unbraced flange length for which elastic bukling applies (AISC 360-05, F2-6) Es

LrB := 1.95 ⋅ rtsB ⋅ ⋅ 0.7 ⋅ FyB

JB ⋅ cB SxB⋅ hoB

⋅ 1+

2

FcrB ( UBL) :=

CbB ⋅ π ⋅ Es

 UBL  r   tsB 

2

 0.7 ⋅ FyB SxB⋅ hoB  1 + 6.76 ⋅  ⋅  Es JB ⋅ cB  

⋅ 1 + 0.078 ⋅

JB ⋅ cB

 UBL   SxB ⋅ hoB rtsB  

2

⋅

Critical stress based on LTB (AISC 360-05, F2-4)

Plastic moment strength (AISC 360-05, F2-1)

MpB := FyB ⋅ ZxB

Nominal moment strength based on yielding

φMnYB := 0.9 ⋅ MpB

φMnLTB ( UBL) := 0.9 ⋅ CbB ⋅ MpB ...

 +  

φMnLTBB ( UBL) :=

2

  UBL − LpB   −MpB ...  ⋅   LrB − LpB  + − 0.7 ⋅ F ⋅ S yB xB)   ( 

    

Nominal moment strength based on LTB (AISC 360-05, F2-2 and F2-3

0.9 ⋅ MpB if UBL ≤ LpB φMnLTB ( UBL) if

(UBL > LpB) ⋅ (UBL ≤ LrB)

0.9 ⋅ FcrB ( UBL) ⋅ SxB otherwise

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Nominal moment strength based on LTB with limits (AISC 360-05, F2-2 and F2-3)

Page 19 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design H. Horizontal Stiffener Design (Long Side) 3. Bending Strength λB :=

Flange slenderness ratio for local buckling (AISC 360-05 F3-1)

CBFbyTF( BEAM2) if BEAM2 ≤ 31 WBFby2TFBEAM2−31 otherwise

HbyTW :=

Web slenderness ratio (AISC 360-05 F3-2)

CHbyTW( BEAM2) if BEAM2 ≤ 31 WHbyTWBEAM2− 31 otherwise Es

λpfB := 0.38 ⋅

FyB

Limiting slenderness for non-compact flange (Table B4.1)

Es

λrfB := 1.0 ⋅

kcB :=

Limiting slenderness for compact flange (Table B4.1)

FyB 4

0.35 if

(AISC 360-05 F3-2)

< 0.35

HbyTW 4

0.76 if

> 0.76

HbyTW 4

otherwise

HbyTW

φMnFLB := 0.9 ⋅ MpB ...

 +  

φMnFLBB :=



 λB − λpfB    ⋅  λrfB − λpfB     −( 0.7 ⋅ FyB ⋅ SxB)  

−MpB ...

+ 

    

Moment strength based on flange local buckling (AISC 360-05 F3-1)

0.9 ⋅ MpB if λB ≤ λpfB φMnFLB if 0.9 ⋅

(λB > λpfB) ⋅ (λB ≤ λrfB)

0.9 ⋅ Es ⋅ kcB⋅ SxB

( ) λB

2

otherwise

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Moment strength based on flange local buckling with limits (AISC 360-05 F3-1)

Page 20 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design H. Horizontal Stiffener Design (Long Side) 3. Bending Strength φMnYB     φMnB ( UBL) := min   φMnLTBB ( UBL)       φMnFLBB    

Nominal moment strength of beam

Beam Capacity as a Function of Unbraced Length 180

Moment Capacity (ft-kips)

160

140

120

100

80

0

5

10 Unbraced Length (ft)

Nominal Moment Strength Positive Moment at Unbraced Length Negative Moment at Unbraced Length

Mup

(

)

φMnB UBLT2

= 78.82 ⋅ %

Mun

(

)

φMnB UBLB2

= 82.20 ⋅ %

www.mathcadcalcs.com

All ratios must be at 100% or less try another beam shape if over 100%

Page 21 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design H. Horizontal Stiffener Design (Long Side) 4. Shear Strength φVnB := 1.0 ⋅ dB ⋅ twB ⋅ 0.6 ⋅ FyB

Nominal shear strength for beam

φVnB = 111.78 ⋅ kip VB φVnB

= 28.64 ⋅ %

Ratio must be less than or equal to 100% - try another beam shape if over 100%

5. Web Compactness λpwB := 3.76 ⋅

Es

Limiting slenderness ratio for web compactness (AISC 360-05, Table B4.1)

FyB

λpwB = 90.55 Slenderness ratio for beam

HbyTW = 45.40 HbyTW λpwB

Ratio must be less than or equal to 100%

= 50.14 ⋅ %

6. Beam Deflection

∆L := 5 ⋅

  whe    Ltank max ⋅   whp    Ncross +    384 ⋅ Es ⋅ IB

∆maxL := ∆L ∆maxL

Ltank 180

= 8.75 ⋅ %

= 2.56 ⋅ in

 1 

4

= 0.22 ⋅ in

Beam deflection at top of tank

Allowable beam deflection at top of tank

Ratio must be less than or equal to 100%

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Page 22 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design I. Horizontal Stiffener Design (Short Side) 1. Beam Loadings Ultimate Positive Bending Moment

Mup := 1.6 ⋅ MhpB Ultimate Negative Bending Moment

Mun := 1.6 ⋅ MheB Ultimate shear in beam

VB := 1.6 ⋅ max

  VheB     VhpB    

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Page 23 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

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Rectangular Tank Design I. Horizontal Stiffener Design (Short Side) RT

2. Member Properties UBLT1 = 12.00 ft

Unbraced length of top flange

UBLB1 = 12.00 ft

Unbraced length of bottom flange

Mup = 86.29 ⋅ ft ⋅ kip

Ultimate positive moment

Mun = 89.99 ⋅ ft ⋅ kip

Ultimate negative moment

Moment of inertia of beam

ZxB = 47.30 ⋅ in

Bending diagram factor

ryB = 1.49 ⋅ in

Torsional constant

IyB = 19.60 ⋅ in

Strong axis section modulus

rtsB = 1.77 ⋅ in

Torsional radius of gyration

dB = 13.80 ⋅ in

Beam depth

tfB = 0.39 ⋅ in

Beam flange thickness

twB = 0.27 ⋅ in

Beam web thickness

bfB = 6.73 ⋅ in

Beam flange width

cB = 1.00

Factor used for LTB capacity

hoB = 13.41 ⋅ in

Center to center of flanges

IB = 291.00 ⋅ in

4

CbB = 1.00 CwB = 887.00 ⋅ in SxB = 42.00 ⋅ in

6

3

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3

Plastic section modulus Weak axis radius of gyration

4

Weak axis moment of inertia

Page 24 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design I. Horizontal Stiffener Design (Short Side) 3. Bending Strength LpB := 1.76 ⋅ ryB ⋅

Es

Critical unbraced flange length for which inelastic bukling applies (AISC 360-05, F2-5)

FyB

Critical unbraced flange length for which elastic bukling applies (AISC 360-05, F2-6) Es

LrB := 1.95 ⋅ rtsB ⋅ ⋅ 0.7 ⋅ FyB

JB ⋅ cB SxB⋅ hoB

⋅ 1+

2

FcrB ( UBL) :=

CbB ⋅ π ⋅ Es

 UBL  r   tsB 

2

 0.7 ⋅ FyB SxB⋅ hoB  1 + 6.76 ⋅  ⋅  Es JB ⋅ cB  

⋅ 1 + 0.078 ⋅

JB ⋅ cB

 UBL   SxB ⋅ hoB rtsB  

2

⋅

Critical stress based on LTB (AISC 360-05, F2-4)

Plastic moment strength (AISC 360-05, F2-1)

MpB := FyB ⋅ ZxB

Nominal moment strength based on yielding

φMnYB := 0.9 ⋅ MpB

φMnLTB ( UBL) := 0.9 ⋅ CbB ⋅ MpB ...

 +  

φMnLTBB ( UBL) :=

2

 UBL − LpB 

  −MpB ...  ⋅  L − L   rB pB  + − 0.7 ⋅ F ⋅ S yB xB)   ( 

    

Nominal moment strength based on LTB (AISC 360-05, F2-2 and F2-3

0.9 ⋅ MpB if UBL ≤ LpB φMnLTB ( UBL) if

(UBL > LpB) ⋅ (UBL ≤ LrB)

0.9 ⋅ FcrB ( UBL) ⋅ SxB otherwise

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Nominal moment strength based on LTB with limits (AISC 360-05, F2-2 and F2-3)

Page 25 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design I. Horizontal Stiffener Design (Short Side) 3. Bending Strength λB :=

Flange slenderness ratio for local buckling (AISC 360-05 F3-1)

CBFbyTF( BEAM3) if BEAM3 ≤ 31 WBFby2TFBEAM3−31 otherwise

HbyTW :=

Web slenderness ratio (AISC 360-05 F3-2)

CHbyTW( BEAM3) if BEAM3 ≤ 31 WHbyTWBEAM3− 31 otherwise Es

λpfB := 0.38 ⋅

FyB

Limiting slenderness for non-compact flange (Table B4.1)

Es

λrfB := 1.0 ⋅

kcB :=

Limiting slenderness for compact flange (Table B4.1)

FyB 4

0.35 if

(AISC 360-05 F3-2)

< 0.35

HbyTW 4

0.76 if

> 0.76

HbyTW 4

otherwise

HbyTW

φMnFLB := 0.9 ⋅ MpB ...

 +  

φMnFLBB :=



    ⋅    λrfB − λpfB  + − 0.7 ⋅ F ⋅ S yB xB)   (  λB − λpfB

−MpB ...

    

Moment strength based on flange local buckling (AISC 360-05 F3-1)

0.9 ⋅ MpB if λB ≤ λpfB φMnFLB if 0.9 ⋅

(λB > λpfB) ⋅ (λB ≤ λrfB)

0.9 ⋅ Es ⋅ kcB⋅ SxB

(λB)

2

otherwise

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Moment strength based on flange local buckling with limits (AISC 360-05 F3-1)

Page 26 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design I. Horizontal Stiffener Design (Short Side) 3. Bending Strength φMnYB     φMnB ( UBL) := min   φMnLTBB ( UBL)       φMnFLBB    

Nominal moment strength of beam

Beam Capacity as a Function of Unbraced Length 180

Moment Capacity (ft-kips)

160

140

120

100

80

0

5

10 Unbraced Length (ft)

Nominal Moment Strength Positive Moment at Unbraced Length Negative Moment at Unbraced Length

Mup

(

)

φMnB UBLT3

= 66.23 ⋅ %

Mun

(

)

φMnB UBLB3

= 69.07 ⋅ %

www.mathcadcalcs.com

All ratios must be at 100% or less try another beam shape if over 100%

Page 27 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design I. Horizontal Stiffener Design (Short Side) 4. Shear Strength φVnB := 1.0 ⋅ dB ⋅ twB ⋅ 0.6 ⋅ FyB

Nominal shear strength for beam

φVnB = 111.78 ⋅ kip VB φVnB

= 26.84 ⋅ %

Ratio must be less than or equal to 100% - try another beam shape if over 100%

5. Web Compactness Es

λpwB := 3.76 ⋅

Limiting slenderness ratio for web compactness (AISC 360-05, Table B4.1)

FyB

λpwB = 90.55 Slenderness ratio for beam

HbyTW = 45.40 HbyTW λpwB

Ratio must be less than or equal to 100%

= 50.14 ⋅ %

6. Beam Deflection max ∆B := 5 ⋅

384 ⋅ Es ⋅ IB

∆maxB := ∆B ∆maxB

  whe   4 ⋅B   whp   tank  

Btank 180

= 0.80 ⋅ in

= 21.59 ⋅ %

= 0.17 ⋅ in

Deflection of beam at top of tank

Allowable deflection of beam

Ratio must be less than or equal to 100%

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Page 28 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design J. Cross Stiffener Design 1. Member Loadings Ultimate Axial Load

PC := 1.6 2 ⋅ VheL = 64020.61 ⋅ lbs Ultimate strong axis shear in column

VxC := 0 ⋅ kip Ultimate weak axis shear in column

VyC := 0 ⋅ kip Ultimate Strong Axis Bending Moment at Top

MusT := 0 ⋅ ft ⋅ kip Ultimate Strong Axis Bending Moment at Bottom

MusB := 0 ⋅ ft ⋅ kip Ultimate Weak Axis Bending Moment at Top

MuwT := 0 ⋅ ft ⋅ kip Ultimate Weak Axis Bending Moment at Bottom

MuwB := 0 ⋅ ft ⋅ kip Ultimate Tensile Load

TC := 1.6 2 ⋅ VhpL

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Page 29 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design J. Cross Stiffener Design RT

2. Member Properties Unbraced length of compression flange for flexure

ZyC = 4.75 ⋅ in

Moment of inertia of column

ZxC = 10.80 ⋅ in

Area of column

ryC = 1.45 ⋅ in

Weak axis radius of gyration

Torsional constant

rxC = 2.56 ⋅ in

Weak axis radius of gyration

Strong axis section modulus

IyC = 9.32 ⋅ in

Weak axis section modulus

rtsC = 1.65 ⋅ in

Torsional radius of gyration

dC = 5.99 ⋅ in

Beam depth

tfC = 0.26 ⋅ in

Beam flange thickness

twC = 0.23 ⋅ in

Beam web thickness

bfC = 5.99 ⋅ in

Beam flange width

cC = 1.00

Factor used for LTB capacity

hoC = 5.73 ⋅ in

Center to center of flanges

UBLF = 12.00 ft IC = 29.30 ⋅ in AC = 4.45 ⋅ in

4

2

CwC = 76.50 ⋅ in SxC = 9.77 ⋅ in SyC = 3.11 ⋅ in

6

3 3

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3

Plastic section modulus-weak axis 3

Plastic section modulus-strong axis

4

Weak axis moment of inertia

Page 30 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design J. Cross Stiffener Design 3. Strong Axis Bending Strength LpC := 1.76 ⋅ ryC ⋅

Es

Critical unbraced flange length for which inelastic bukling applies (AISC 360-05, F2-5)

FyC

Critical unbraced flange length for which elastic bukling applies (AISC 360-05, F2-6) Es

LrC := 1.95 ⋅ rtsC ⋅ ⋅ 0.7 ⋅ FyC

JC ⋅ cC SxC⋅ hoC

⋅ 1+

2

FcrC ( UBL) :=

CbC ⋅ π ⋅ Es

 UBL  r   tsC 

2

 0.7 ⋅ FyC SxC⋅ hoC  1 + 6.76 ⋅  ⋅  Es JC ⋅ cC  

⋅ 1 + 0.078 ⋅

JC ⋅ cC

 UBL   SxC ⋅ hoC rtsC  

2

⋅

Critical stress based on LTB (AISC 360-05, F2-4)

Plastic moment strength (AISC 360-05, F2-1)

MpC := FyC ⋅ ZxC

Nominal moment strength based on yielding

φMnYC := 0.9 ⋅ MpC

φMnLTB ( UBL) := 0.9 ⋅ CbC ⋅ MpC ...

 +  

φMnLTBC ( UBL) :=

2

 UBL − LpC 

  −MpC ...  ⋅  L − L   rC pC  + − 0.7 ⋅ F ⋅ S yC xC)   ( 

    

Nominal moment strength based on LTB (AISC 360-05, F2-2 and F2-3

0.9 ⋅ MpC if UBL ≤ LpC φMnLTB ( UBL) if

(UBL > LpC) ⋅ (UBL ≤ LrC)

0.9 ⋅ FcrC ( UBL) ⋅ SxC otherwise

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Nominal moment strength based on LTB with limits (AISC 360-05, F2-2 and F2-3)

Page 31 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design J. Cross Stiffener Design 3. Strong Axis Bending Strength Flange slenderness ratio for local buckling (AISC 360-05 F3-1)

λC := WBFby2TFCOL− 31

Web slenderness ratio (AISC 360-05 F3-2)

HbyTW := WHbyTWCOL−31

Es

λpfC := 0.38 ⋅

FyC

Limiting slenderness for non-compact flange (Table B4.1)

Es

λrfC := 1.0 ⋅

kcC :=

Limiting slenderness for compact flange (Table B4.1)

FyC 4

0.35 if

(AISC 360-05 F3-2)

< 0.35

HbyTW 4

0.76 if

> 0.76

HbyTW 4

otherwise

HbyTW

φMnFLB := 0.9 ⋅ MpC ...

 +  

φMnFLBC :=



 λC − λpfC    ⋅  λrfC − λpfC     −( 0.7 ⋅ FyC ⋅ SxC)  

−MpC ...

+ 

    

Moment strength based on flange local buckling (AISC 360-05 F3-1)

0.9 ⋅ MpC if λC ≤ λpfC φMnFLB if 0.9 ⋅

(λC > λpfC) ⋅ (λC ≤ λrfC)

0.9 ⋅ Es ⋅ kcC⋅ SxC

( ) λC

2

otherwise

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Moment strength based on flange local buckling with limits (AISC 360-05 F3-1)

Page 32 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design J. Cross Stiffener Design 3. Strong Axis Bending Strength φMnYC     φMnC ( UBL) := min   φMnLTBC ( UBL)       φMnFLBC    

Nominal moment strength of beam

( )

φMnC LC = 31.72 ⋅ ft ⋅ kip

MusC := max

  MusB    M    usT  

Maximum strong axis moment in column

MusC = 0.00 ⋅ ft ⋅ kip

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Page 33 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

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Rectangular Tank Design J. Cross Stiffener Design 4. Weak Axis Bending Strength MpwC := FyC ⋅ ZyC

Plastic moment strength (AISC 360-05, F6-1)

φMnwYC := 0.9 ⋅ MpwC

Nominal moment strength based on yielding

φMnwFLB := 0.9 ⋅ MpwC ...

 +  

φMnwFLBC :=

    ⋅   λrfC − λpfC  + − 0.7 ⋅ F ⋅ S yC yC)   ( 



−MpwC ...

    

Moment strength based on flange local buckling (AISC 360-05 F6-2)

0.9 ⋅ MpwC if λC ≤ λpfC φMnwFLB if 0.9 ⋅

(λC > λpfC) ⋅ (λC ≤ λrfC)

0.69 ⋅ Es ⋅ S yC

(λC)

φMnwC := min

λC − λpfC

2

  φMnYC     φMnFLBC    

Moment strength based on flange local buckling with limits (AISC 360-05 F6)

otherwise

Nominal weak axis moment strength of column

φMnwC = 38.16 ⋅ ft ⋅ kip

MuwC := max

  MuwB     MuwT    

MuwC = 0.00 ⋅ ft ⋅ kip

Maximum weak axis bending moment

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Page 34 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design J. Cross Stiffener Design 5. Column Axial Strength λfxC :=

KxC⋅ UBLs

λfyC :=

rxC 2

FexC :=

(

)

ryC

Slenderness ratio for Y-axis and X-axis

2

π ⋅ Es λfxC

KyC ⋅ UBLw

FeyC :=

2

π ⋅ Es Elastic critical buckling stress for X-axis and Y-axis - AISC 360-05 E3-4

(λfyC)2

FyC

FcrxC :=

0.658

FexC

⋅ FyC if λfxC ≤ 4.71 ⋅

Es FyC

Critical compressive stress for X-axis without consideration to local buckling AISC 360-05 E3-2 and E3-3

0.877 ⋅ FexC otherwise

FyC

FcryC :=

0.658

FeyC

⋅ FyC if λfyC ≤ 4.71 ⋅

Es FyC

Critical compressive stress for Y-axis without consideration to local buckling AISC 360-05 E3-2 and E3-3

0.877 ⋅ FeyC otherwise

FcrC := min

QsC :=

  FcryC    F    crxC  

1 if λC ≤ 0.56 ⋅

1.415 − 0.74 ⋅ λC ⋅ 0.69 ⋅ Es

( )

FyC ⋅ λC

2

Controlling critical compressive stress

Es FyC Es FyC

if

 Es    ⋅  λ ≤ 1.03 ⋅ Es  λC > 0.56 ⋅  FyC   C FyC    

otherwise

QsC = 1.00

Local buckling factor for unstiffened elements, AISC 360-05 E7-4, E7-5, E7-6

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Page 35 of 37

Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design J. Cross Stiffener Design 5. Column Axial Strength    Es  Es  .34     max 1.92 ⋅ twC ⋅ ⋅ ⋅ 1− 0    FcrC  FcrC  HbyTW bewC := min          HbyTW ⋅ twC  

(

AeffC := AC − HbyTW ⋅ twC

QaC :=

AeffC

)2 + bewC ⋅ twC

AeffC = 4.45 ⋅ in

2

Local buckling factor for stiffened elements, AISC 360-05 E7-16

QaC = 1.00

AC

QC := QsC⋅ QaC

Effective width of web, AISC 360-05 E7-17

Combined factor for determining critical compressive stress including local buckling

QC = 1.00

QC ⋅ FyC

FcrxQC :=

FexC

QC ⋅ 0.658

⋅ FyC if λfxC ≤ 4.71 ⋅

Es

Critical compressive stress for X-axis with consideration to local buckling AISC 360-05 E3-2 and E3-3

QC ⋅ FyC

0.877 ⋅ FexC otherwise

QC ⋅ FyC

FcryQC :=

QC ⋅ 0.658

FeyC

⋅ FyC if λfyC ≤ 4.71 ⋅

Es

Critical compressive stress for Y-axis with consideration to local buckling AISC 360-05 E3-2 and E3-3

QC ⋅ FyC

0.877 ⋅ FeyC otherwise

FcrQC := min

  FcryC     FcrxC    

Controlling critical compressive stress with consideration to local buckling

FcrQC = 24310.31 psi φPnC := 0.9 ⋅ FcrQC ⋅ AC

Compressive strength of column section AISC 360-05 E7-1

φPnC = 97.36 ⋅ kip www.mathcadcalcs.com

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Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

Designed By: Checked By: Date:

Rectangular Tank Design J. Cross Stiffener Design 6. Check Interaction PC MuwC   MusC  if + ≥ 0.2 φPnC 9  φMnC ( LC) φMnwC  φPnC   PC MuwC   MusC  otherwise + + φMnwC  2 ⋅ φPnC  φMnC ( LC)   PC

INTC :=

+

8

⋅

Interaction of bending and compression per AISC 360-05 H1-1a and H1-1b. Moments and axial force must consider 2nd order effects.

INTC = 65.75 ⋅ %

7. Shear Strength φVnxC := 1.0 ⋅ dC ⋅ twC ⋅ 0.6 ⋅ FyC

Nominal shear strength for column

φVnxC = 41.33 ⋅ kip VxC φVnxC

= 0.00 ⋅ %

Ratio must be less than or equal to 100% - try another column shape if over 100%

φVnyC := 1.0 ⋅ 2 ⋅ bfC ⋅ tfC ⋅ 0.6 ⋅ FyC

Nominal shear strength for column

φVnyC = 93.44 ⋅ kip VyC φVnyC

= 0.00 ⋅ %

Ratio must be less than or equal to 100% - try another column shape if over 100%

8. Web Compactness λpwC := 3.76 ⋅

Es FyC

Limiting slenderness ratio for web compactness (AISC 360-05, Table B4.1)

λpwC = 90.55 HbyTW = 21.20 HbyTW λpwC

= 23.41 ⋅ %

Slenderness ratio for beam

Ratio must be less than or equal to 100%

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