90363835 Rafter Design

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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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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

Number of lateral braces

Number of rafters in bay

Ro:= 50 ft⋅ _Ri:= 4 ft⋅ _Nrb:= 50 _Nbt:= 4

Roof slope Thickness of roof Effective Span of rafter

RS .75 in ft

⋅

:= _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

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D. Rafter Loadings

RT

Sr

2 π⋅ ⋅_Ro

Nrb

:= _{Sr 6.28ft}= Spacing of rafters at outer end

X1:= 0 ft⋅ =0.00 TL max LLr SLb

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tr γs ⋅ + +_DLmisc +_{0.4 Pext}⋅ := TL=32.24 psf⋅ Total load

DL:= _{tr γs}⋅ =7.66 psf⋅ Roof plate only load

qX1 LD( ) 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 + :=

qi TL( ) =32.20 plf⋅ Uniform load at inside of rafter

q x LD( , ) x X1⋅qX1 LD( ) if x<X1 qX1 LD( ) (qX1 LD( )−qi LD( )) (x−X1) LB 5 in− ⋅

(

)

−X1 ⋅ − otherwise := R2 LD( ) 0 ft⋅ LB x q x LD( , ) x⋅ ⌠  ⌡ d LB

:= R2 TL( )=2143.19 lbs⋅ Inside rafter reaction

R1 LD( ) 0 ft⋅ LB x q x LD( , ) ⌠  ⌡ d −R2 LD( )

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Rafter Design per API 650

D. Rafter Loadings

M x1 LD( , ) R1 LD( ) x1⋅ 0 ft⋅ x1 x q x LD( , ) x1⋅( −x) ⌠  ⌡ d − := Moment as a function of x MARRAY LD( ) m_i M LB 100⋅i, LD

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← i∈1 100.. for 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

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E. Member Properties

IB 103.00 in= ⋅ 4 _{ZxB 20.10 in}= ⋅ 3

Moment of inertia of rafter Plastic section modulus

CbB 1.00= _{ryB 0.77 in}= ⋅

Bending diagram factor Weak axis radius of gyration

CwB 96.90 in= ⋅ 6 _{IyB 2.82 in}= ⋅ 4

Torsional constant Weak axis moment of inertia

SxB 17.10 in= ⋅ 3 _{rtsB 0.98 in}= ⋅

Strong axis section modulus Torsional radius of gyration

dB 12.00 in= ⋅ _{tfB 0.27 in}= ⋅

Rafter depth Rafter flange thickness

twB 0.22 in= ⋅ _{bfB 3.99 in}= ⋅

Rafter web thickness Rafter flange width

cB 1.00= Factor used for LTB capacity hoB 11.73 in= ⋅ Center to center of flanges

UBLDL LB

Nbt 1+ =9.20 ft

:= Unbraced length of compression flange for roof weight

only - see API 650, Section 5.10.4.3

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

5.10.4.3 UBLTL 0.1 ft⋅ (INT=1_{) dB 15 in}⋅

(

≤ ⋅

)

RS 2 in⋅ ft ≤

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⋅ if LB Nbt 1+ otherwise 0.10 ft = :=

Yield strength reduction factor for rafter design per

API 650, Appendix M RFys RY1 Td °F

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if FyB 45 ksi≤ ⋅ RY3 Td °F

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if FyB 55 ksi> ⋅ RY2 Td °F

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otherwise 0.78 = :=

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Rafter Design per API 650

F. Bending Strength

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

LpB 1.76 ryB⋅ Es

FyB

⋅ :=

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

LrB 1.95 rtsB⋅ Es 0.7 FyB⋅ ⋅ JB cB ⋅ SxB hoB⋅ ⋅ 1 1 6.76 0.7 FyB ⋅ Es SxB hoB⋅ JB cB⋅ ⋅

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2 ⋅ + + ⋅ := FcrB UBL( ) CbB π 2 ⋅ ⋅_Es UBL rtsB

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2 1 0.078 JB cB⋅ SxB hoB⋅ ⋅ UBL rtsB

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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} MpB 0.7 FyB⋅ ⋅_SxB

(

)

− + ...

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UBL−_LpB LrB LpB−

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⋅

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− + ...

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⋅ :=

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

MnLTBB UBL( ) _MpB if UBL≤_LpB

MnLTB UBL( ) if

(

UBL>_LpB

)

⋅

(

UBL≤_LrB

)

FcrB UBL( _{) SxB}⋅ otherwise

:=

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

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F. Bending Strength

λB CBFbyTF₍_BEAM₎ if BEAM≤31 WBFby2TF_{BEAM 31}₋ otherwise

:= Flange slenderness ratio for local buckling

(AISC 360-05 F3-1)

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

HbyTW CHbyTW₍_BEAM₎ if BEAM≤31 WHbyTW_{BEAM 31}₋ otherwise

:=

Limiting slenderness for compact flange (Table B4.1)

λpfB 0.38 Es

FyB

⋅ :=

Limiting slenderness for non-compact flange (Table B4.1) λrfB 1.0 Es FyB ⋅ := kcB 0.35 4 HbyTW 0.35 < if 0.76 4 HbyTW 0.76 > if 4 HbyTW otherwise := (AISC 360-05 F3-2) MnFLB MpB MpB 0.7 FyB⋅ ⋅_SxB

(

)

− + ...

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λB λpfB− λrfB λpfB−

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⋅

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− + ...

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:=

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

MnFLBB MpB if λB λpfB≤ MnFLB if

(

λB λpfB>

)

⋅

(

_{λB λrfB}≤

)

0.9 Es⋅ ⋅_kcB⋅_SxB λB

( )

2 otherwise :=

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

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Rafter Design per API 650

F. Bending Strength

MnYB 83750.00 ft lbs= ⋅ ⋅ Nominal moment strength of rafter φ_{MnB UBL}( ) RFys Ωb min MnYB MnLTBB UBL( ) MnFLBB

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⋅ := 0 5 10 10 20 30

Nominal Moment Strength

Positive Moment at Unbraced Length Negative Moment at Unbraced Length

Beam Capacity as a Function of Unbraced Length

Unbraced Length (ft) M o m en t C ap ac it y ( ft -k ip s)

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

MmaxTL

φ_{MnB UBLTL}

(

)

=85.49 %⋅

MmaxDL

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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

Limiting slenderness ratio for web compactness (AISC 360-05, Table B4.1) λpwB 3.76 Es FyB ⋅ := λpwB 90.55=

Slenderness ratio for rafter

HbyTW=49.40

HbyTW

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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

Srmax min

(

tr CArp−

)

1.5 Fyrp⋅ ⋅_RFys TL ⋅ 84 in⋅

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:=

Srmax 6.78ft= Maximum permissible spacing of rafters per API 650,

Section 5.10.4.4 Sr 6.28ft= Actual rafter spacing

Sr

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J. Brace Force Required

Factor to determine brace force Cd:= 1 Tensile strength of fillet weld

Safety factor for weld per AISC 360-05 Fillet weld size Length of fillet weld

tw:= .25 in⋅ _Lw:= 2 in⋅ _Fuw:= 60 ksi⋅ _Ωw:= 2

Factor to determine brace force

Ct 1 1.2 Nbt + =1.30 := Pbr 0.01 MmaxTL ⋅ ⋅_Ct⋅_Cd hoB :=

Brace force required per AISC 360-05

Pbr 444.76 lbs= ⋅

Allowable force on fillet weld

Pw

0.6 0.7071⋅ ⋅_tw⋅_Lw⋅_Fuw Ωw

:=

Pw 6.36 kip= ⋅

Must be less than 100%

Pbr

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Rafter Design per API 650

K. Deflection of Beam

∆(q1 q2, , L, E, I) 0.00652 q2⋅( −q1) L 4 ⋅ E I⋅ 5 q1⋅ ⋅L4 384 E⋅ ⋅I + := Beam deflection

∆allow:= ₁₈₀LB _{∆allow 3.07 in}= ⋅ 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:= ∆

(

_{qi LLr}

(

)

, _{qX1 LLr}

(

)

, _LB, _Es, _IB

)

∆max 2.83 in= ⋅

Maximum live load deflection ∆max

∆allow =92.32 %⋅

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

∆allowTL:= ₁₂₀LB _{∆allowTL 4.60 in}= ⋅

∆maxTL:= ∆

(

_{qi LLr tr γs}

(

+ ⋅ +_DLmisc

)

, _{qX1 LLr tr γs}

(

+ ⋅ + _DLmisc

)

, _LB, _Es, _IB

)

∆maxTL 3.88 in= ⋅ Maximum dead plus live load deflection

∆maxTL