rafter design

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Steel Rafter Design Per AISC / API 650

Rev # Rev Description Rev By Rev Date

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Notes

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Date:

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 rafterconnection

Radius to inside rafterconnection

Number of lateralbracesNumber of rafters in bay

Ro 50 ft⋅:= Ri 4 ft⋅:= Nrb 50:= Nbt 4:=

Roof slope Thickness of roof Effective Span of rafter

RS .75in

ft⋅:= tr .1875 in⋅:= LB Ro Ri− 46.00ft=:=

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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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

SLb

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 rafterqX1 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

X1qX1 LD( )⋅ x X1<if

qX1 LD( ) qX1 LD( ) qi LD( )−( )x X1−( )

LB 5 in⋅−( ) X1−⋅− otherwise

:=

R2 LD( )0 ft⋅

LB

xq x LD, ( ) x⋅⌠⌡

d

LB

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

R1 LD( )

0 ft⋅

LB

xq x LD, ( )⌠⌡

d R2 LD( )−:= R1 TL( ) 3584.88 lbf= Outside rafter reaction





Date:


D. Rafter Loadings

M x1 LD, ( ) R1 LD( ) x1⋅

0 ft⋅

x1

xq x LD, ( ) x1 x−( )⋅⌠⌡

d−:= Moment as a function of x

MARRAY LD( )

mi MLB

100i⋅ LD,

←

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

MmaxDL 11135.21 ft lbs⋅⋅=





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E. Member Properties

IB 103.00 in4

⋅= ZxB 20.10 in3

⋅=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 in6

⋅= IyB 2.82 in4

⋅=Torsional constant Weak axis moment of inertia

SxB 17.10 in3

⋅= 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⋅≤( )⋅ RS2 in⋅

ft≤

⋅if

LB

Nbt 1+otherwise

0.10ft=:=

Yield strength reduction factor for rafter design per

API 650, Appendix MRFys RY1

Td

°F

FyB 45 ksi⋅≤if

RY3Td

°F

FyB 55 ksi⋅>if

RY2Td

°F

otherwise

0.78=:=





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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⋅⋅

2

⋅++⋅:=

FcrB UBL( )CbB π

2⋅ Es⋅

UBL

rtsB

21 0.078

JB cB⋅

SxB hoB⋅⋅

UBL

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 yieldingMnYB MpB:=

MnLTB UBL( ) CbB MpB

MpB

0.7 FyB⋅ SxB⋅( )−+

...

UBL LpB−

LrB LpB−

⋅

−+

...

⋅:=


based on LTB (AISC 360-05,

F2-2 and F2-3

MnLTBB UBL( ) MpB UBL LpB≤if

MnLTB UBL( ) UBL LpB>( ) UBL LrB≤( )⋅if

FcrB UBL( ) SxB⋅ otherwise

:=


based on LTB with limits

(AISC 360-05, F2-2 and F2-3)





Date:


F. Bending Strength

λB CBFbyTF BEAM( ) BEAM 31≤if

WBFby2TFBEAM 31− otherwise

:= Flange slenderness ratio for local buckling

(AISC 360-05 F3-1)

Web slenderness ratio (AISC 360-05 F3-2)HbyTW CHbyTW BEAM( ) BEAM 31≤if

WHbyTWBEAM 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 1.0Es

FyB

⋅:=

kcB 0.354

HbyTW0.35<if

0.764

HbyTW0.76>if

4

HbyTWotherwise

:= (AISC 360-05 F3-2)

MnFLB MpB

MpB

0.7 FyB⋅ SxB⋅( )−+

...

λB λpfB−

λrfB λpfB−

⋅

−+

...

:=

Moment strength based on flange

local buckling (AISC 360-05 F3-1)

MnFLBB MpB λB λpfB≤if

MnFLB λB λpfB>( ) λB λrfB≤( )⋅if

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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F. Bending Strength

MnYB 83750.00 ft lbs⋅⋅= Nominal moment strength of

rafterφMnB UBL( )

RFys

Ωb

min

MnYB

MnLTBB UBL( )

MnFLBB

⋅:=

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)

Mo

men

t C

apac

ity

(ft

-kip

s)

All ratios must be at 100% or less -

try another rafter shape if over 100%

MmaxTL

φMnB UBLTL( )85.49 %⋅=

MmaxDL

φMnB UBLDL( )60.37 %⋅=





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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 rafterHbyTW 49.40=

HbyTW

λpwB

54.55 %⋅= Ratio must be less than or equal to 100%





Date:


I. Check Rafter Spacing

CArp 0 in⋅:= Corrosion allowance on roof plate

Fyrp 36 ksi⋅:= Yield strength of roof plate

Srmax mintr CArp−( )

1.5 Fyrp⋅ RFys⋅

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%





Date:


J. Brace Force RequiredFactor to determinebrace force

Cd 1:=

Tensile strength offillet weld

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

tw .25 in⋅:= Lw 2 in⋅:= Fuw 60 ksi⋅:= Ωw 2:=

Factor to determine brace forceCt 1

1.2

Nbt

+ 1.30=:=

Pbr

0.01 MmaxTL⋅ Ct⋅ Cd⋅

hoB

:=

Brace force required per AISC 360-05Pbr 444.76 lbs⋅=

Allowable force on fillet weldPw

0.6 0.7071⋅ tw⋅ Lw⋅ Fuw⋅

Ωw

:=

Pw 6.36 kip⋅=

Must be less than 100% Pbr

Pw

6.99 %⋅=





Date:


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

LB

180:= ∆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 plate

can take a lot of deflection, so a limit is not actually

required.

∆allowTL

LB

120:= ∆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

∆allowTL

84.35 %⋅=


rafter design

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