90363835 Rafter Design
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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
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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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Page 4 of 12
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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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