90363835 Rafter Design

August 21, 2017 | Author: setak | Category: Structural Steel, Roof, Classical Mechanics, Materials, Chemical Product Engineering
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Client: Project Location: Project Desc:

Tank Desc: Job Number: Rev Number:

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YOUR COMPANY LOGO Steel Rafter Design Per AISC / API 650 Rev #

Rev Description

Rev By

Rev Date

1 2 3 4 Notes 1 2 3 4 5

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Page 1 of 12

Client: Project Location: Project Desc:

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Rafter Design per API 650 A. Introduction API 650 requires that the structural rafters be designed per AISC or other approved standard. These rafters are designed using the latest edition of AISC with temperature modification factors per API 650, Appendix M. API 650 requires that rafters not use roof plate for lateral support when considering the roof plate loads only. When considering the total load with live load and other dead loads included, the roof plate may be considered as effective in bracing the compression flange of the rafter (per API 650).

B. Geometry Beam Selection (W or C shapes)

Radius to outside rafter connection

Radius to inside rafter connection

Ro := 50 ⋅ ft Roof slope

RS := .75 ⋅

in ft

Ri := 4 ⋅ ft

Number of rafters in bay

Number of lateral braces

Nrb := 50

Nbt := 4

Thickness of roof

Effective Span of rafter

tr := .1875 ⋅ in

LB := Ro − Ri = 46.00 ft

C. Material Properties Yield Strength

Safety factor required per AISC 360

FyB := 50 ⋅ ksi

Ωb := 1.67

Rafter Design (AISC 360-05) D. Rafter Loadings Ground snow load

Balanced snow load on roof

SLg := 25 ⋅ psf

SLb := 0.84 ⋅ SLg = 21.00 ⋅ psf

Roof live load

Additional roof dead load

LLr := 20 ⋅ psf

DLmisc := 1.5 ⋅ psf

External pressure

Design temperature

Pext := 5.2 ⋅ psf

Td := 350 ⋅ °F

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Page 2 of 12

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Rafter Design per API 650 D. Rafter Loadings RT

Sr :=

2 ⋅ π ⋅ Ro Nrb

Sr = 6.28 ft

Spacing of rafters at outer end

X1 := 0 ⋅ ft = 0.00

TL := max

  LLr      + t ⋅ γ + DL misc + 0.4 ⋅ Pext   SL   r s  b 

( )

2 Ro ⋅ π Nrb

⋅ LD + wB Uniform load at outside of rafter

qX1 ( TL) = 218.55 ⋅ plf qi ( LD) := wB +

(Ri) ⋅ π ⋅ 2 Nrb

⋅ LD Uniform load at inside of rafter

qi ( TL) = 32.20 ⋅ plf

q ( x , LD) :=

x X1

Total load

Roof plate only load

DL := tr ⋅ γs = 7.66 ⋅ psf qX1 ( LD) :=

TL = 32.24 ⋅ psf

⋅ qX1 ( LD) if x < X1

qX1 ( LD) − ( qX1 ( LD) − qi ( LD) ) ⋅

( x − X1)

(LB − 5 ⋅ in) − X1

otherwise

L ⌠ B  q ( x , LD) ⋅ x dx ⌡

R2 ( LD) :=

0 ⋅ ft

LB

L ⌠ B R1 ( LD) :=  q ( x , LD) dx − R2 ( LD) ⌡

R2 ( TL) = 2143.19 ⋅ lbs

Inside rafter reaction

R1 ( TL) = 3584.88 lbf

Outside rafter reaction

0 ⋅ ft

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Page 3 of 12

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Rafter Design per API 650 D. Rafter Loadings ⌠ M ( x1 , LD) := R1 ( LD) ⋅ x1 −  ⌡

x1

q ( x , LD) ⋅ ( x1 − x) dx

Moment as a function of x

0 ⋅ ft

MARRAY ( LD) :=

for i ∈ 1 .. 100

 LB  ⋅ i , LD  100 

mi ← M  m

MmaxTL := max ( MARRAY ( TL) )

Maximum moment for total load

MmaxTL = 33442.58 ⋅ ft ⋅ lbs MmaxDL := max ( MARRAY ( DL) )

Maximum moment for dead load only case

MmaxDL = 11135.21 ⋅ ft ⋅ lbs

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Rafter Design per API 650 E. Member Properties IB = 103.00 ⋅ in

4

3

Moment of inertia of rafter

ZxB = 20.10 ⋅ in

Bending diagram factor

ryB = 0.77 ⋅ in

Torsional constant

IyB = 2.82 ⋅ in

Strong axis section modulus

rtsB = 0.98 ⋅ in

Torsional radius of gyration

dB = 12.00 ⋅ in

Rafter depth

tfB = 0.27 ⋅ in

Rafter flange thickness

twB = 0.22 ⋅ in

Rafter web thickness

bfB = 3.99 ⋅ in

Rafter flange width

cB = 1.00

Factor used for LTB capacity hoB = 11.73 ⋅ in

CbB = 1.00 CwB = 96.90 ⋅ in SxB = 17.10 ⋅ in

6

3

LB UBLDL := = 9.20 ft Nbt + 1 UBLTL :=

(

Nbt + 1

RFys :=

)

4

Weak axis moment of inertia

Center to center of flanges

= 0.10 ft

Unbraced length of compression flange for total load - see API 650, Section 5.10.4.3

otherwise

 Td   if FyB ≤ 45 ⋅ ksi = 0.78  °F   Td   if FyB > 55 ⋅ ksi RY3   °F   Td   otherwise RY2   °F 

RY1 

Weak axis radius of gyration

Unbraced length of compression flange for roof weight only - see API 650, Section 5.10.4.3

2 ⋅ in   0.1 ⋅ ft if ( INT = 1) ⋅ dB ≤ 15 ⋅ in ⋅  RS ≤  ft   LB

Plastic section modulus

Yield strength reduction factor for rafter design per API 650, Appendix M

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Page 5 of 12

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Rafter Design per API 650 F. 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

⋅ 1 + 0.078 ⋅

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

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

MnYB := MpB

MnLTB ( UBL) := CbB ⋅ MpB ...

 +  

MnLTBB ( 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

MpB if UBL ≤ LpB MnLTB ( UBL) if

(UBL > LpB) ⋅ (UBL ≤ LrB)

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 6 of 12

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Rafter Design per API 650 F. Bending Strength λB :=

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

CBFbyTF( BEAM) if BEAM ≤ 31 WBFby2TFBEAM− 31 otherwise

HbyTW :=

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

CHbyTW( BEAM) if BEAM ≤ 31 WHbyTWBEAM− 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

0.35 if

4

(AISC 360-05 F3-2)

< 0.35

HbyTW 0.76 if

4

> 0.76

HbyTW 4

otherwise

HbyTW

MnFLB := MpB ...

 +  

MnFLBB :=



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

−MpB ...

    

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

MpB if λB ≤ λpfB MnFLB if

(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 7 of 12

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Rafter Design per API 650 F. Bending Strength MnYB     φMnB ( UBL) := ⋅ min   MnLTBB ( UBL)     Ωb   MnFLBB   RFys

MnYB = 83750.00 ⋅ ft ⋅ lbs

Nominal moment strength of rafter

Beam Capacity as a Function of Unbraced Length

Moment Capacity (ft-kips)

30

20

10

0

5

10

Unbraced Length (ft) Nominal Moment Strength Positive Moment at Unbraced Length Negative Moment at Unbraced Length

MmaxTL

(

φMnB UBLTL

)

= 85.49 ⋅ %

MmaxDL

(

)

φMnB UBLDL

= 60.37 ⋅ %

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

Page 8 of 12

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Rafter Design per API 650 G. Shear Strength VnB := RFys ⋅ dB ⋅ twB ⋅ 0.6 ⋅ FyB

Nominal shear strength for rafter

VnB = 61.78 ⋅ kip R1 ( TL) VnB

= 5.80 ⋅ %

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

H. Web Compactness λpwB := 3.76 ⋅

Es FyB

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

λpwB = 90.55 HbyTW = 49.40 HbyTW λpwB

= 54.55 ⋅ %

Slenderness ratio for rafter

Ratio must be less than or equal to 100%

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Page 9 of 12

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Rafter Design per API 650 I. Check Rafter Spacing CArp := 0 ⋅ in

Corrosion allowance on roof plate

Fyrp := 36 ⋅ ksi

Yield strength of roof plate

 1.5 ⋅ Fyrp ⋅ RFys   t − CA ⋅  rp) Srmax := min ( r  TL   84 ⋅ in   Srmax = 6.78 ft

Maximum permissible spacing of rafters per API 650, Section 5.10.4.4

Sr = 6.28 ft

Actual rafter spacing

Sr Srmax

= 92.71 %

Ratio must be less than or equal to 100%

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Page 10 of 12

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Rafter Design per API 650 J. Brace Force Required Factor to determine brace force

Cd := 1 Fillet weld size

Length of fillet weld

Tensile strength of fillet weld

Lw := 2 ⋅ in

Fuw := 60 ⋅ ksi

tw := .25 ⋅ in

1.2 Ct := 1 + = 1.30 Nbt

Pbr :=

Ωw := 2

Factor to determine brace force

0.01 ⋅ MmaxTL ⋅ Ct ⋅ Cd hoB

Pbr = 444.76 ⋅ lbs Pw :=

Safety factor for weld per AISC 360-05

0.6 ⋅ 0.7071 ⋅ tw ⋅ Lw ⋅ Fuw

Brace force required per AISC 360-05

Allowable force on fillet weld

Ωw

Pw = 6.36 ⋅ kip Pbr Pw

= 6.99 ⋅ %

Must be less than 100%

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Page 11 of 12

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Rafter Design per API 650 K. Deflection of Beam ∆ ( q1 , q2 , L , E , I) :=

0.00652 ⋅ ( q2 − q1) ⋅ L

LB ∆allow := 180

4

+

E ⋅I

5 ⋅ q1 ⋅ L

4

384 ⋅ E ⋅ I

Beam deflection

∆max := ∆ qi LLr , qX1 LLr , LB , Es , IB

There are no live load deflection limits required - a good rule of thumb would be L/180. The roof plate can take a lot of deflection, so a limit is not actually required.

∆max = 2.83 ⋅ in

Maximum live load deflection

( ( )

∆max ∆allow

∆allow = 3.07 ⋅ in

( )

)

= 92.32 ⋅ %

LB ∆allowTL := 120

( (

There are no dead+live load deflection limits required a good rule of thumb would be L/120. The roof plate can take a lot of deflection, so a limit is not actually required.

∆allowTL = 4.60 ⋅ in

)

(

)

)

∆maxTL := ∆ qi LLr + tr ⋅ γs + DLmisc , qX1 LLr + tr ⋅ γs + DLmisc , LB , Es , IB ∆maxTL = 3.88 ⋅ in ∆maxTL ∆allowTL

Maximum dead plus live load deflection

= 84.35 ⋅ %

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Page 12 of 12

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