CE-REF.COM
Maximum bending stress, fb must not exceed allowable stress parallel to grain,
F’_{b} = F_{b}*C_{D}*C_{M}*C_{t}*C_{F}*C_{V}*C_{fu}*C_{r}*C_{c}*C_{f}
Where
F_{b} is allowable bending stress in NDS supplement.
C_{D} is load duration factor, (see NDS Table 2.3.2 reproduced below)
C_{M} is wet service factor, (use when moisture of timber is higher than 19%)
C_{t} is temperature factor, (when timber is used in temperature higher than 150°F)
C_{L} is beam stability factor, (See below)
C_{F} is size factor, (apply only to visually graded sawn lumber members, and to round timber bending members, not apply simultaneously with Cv for glued laminated timber)
C_{V} is volume factor, (apply only to glued laminated timber bending member)
C_{fu} is flat use factor, (when 2”-4” timber is loaded at wide face)
C_{r} is repetitive member factor, (apply to dimension bending member 2”-4” thick)
C_{c} is curvature factor (apply to curved glued laminated bending member)
C_{f} is form factor. (for round or diamond section)
Deflection should not exceed allowable limit. The elastic modulus shall be calculated as E’=E*C_{M}*C_{t}, Where E is modulus of elasticity in NDS supplement
Maximum shear stress, fv shall not exceed allowable shear stress,
F’_{v} = F_{v}*C_{D}*C_{M}*C_{t}*C_{H}
Where F_{v} is allowable shear stress in NDS supplement and,
C_{H} is shear stress factor depends on length of split and shake. Value of C_{H} varies from 2 for no split to 1 with 1-1/2 split.
Load duration |
C_{D} |
Design load |
Permanent |
0.9 |
Dead load |
Ten years |
1.0 |
Occupancy live load |
Two months |
1.15 |
Snow load |
Seven days |
1.25 |
Construction load |
Ten minutes |
1.6 |
Wind/Earthquake load |
Impact |
2.0 |
Impact load |
C_{L} = 1 for the following condition for member, with nominal depth, B and width, D.
D/B £ 2
2 < D/B £ 4 – solid blocking is provided at both ends of member.
D/B = 5, one edge (tension or compression) is fully supported.
D/B = 6, bridge, full depth blocking, cross bracing at 8 ft maximum, and both edges are fully supported or compressive edge is fully supported to prevent lateral displacement, and the ends at the point of bearing are laterally supported to prevent rotation;
D/B = 7, both edge fully supported.
When the conditions were not met, C_{L} is calculated based on a complicated equation in NDS section 3.3.3.7. Normally, it is easier to meet the requirement then to go through the complicated equation.
The size, C_{F}, for timber species other than southern pine are listed in Table 4-A. For southern pine 2” to 4” thick, size factor needs not be applied. For southern pine 4” thick, 8” and wider, C_{F} = 1.1. For dimension lumber, wider than 12”, C_{F} = 0.9 except Dense structural 86. 72, and 65. in which, C_{F} =0.9. When the depth of Dense structural 86, 72, and 65, dimension lumber exceeds 12”, C_{F} =(12/d)1/9.
C_{r} applies to dimension lumber 2” to 4” thick that subjected to bending. C_{r} =1.15 when members are used as joist, truss chords, rafters, etc and spacing is not exceed 24” and not less than 3”.
When the moisture of dimension lumber exceeds 19%, the design value Fb shall be multiplied by C_{M} = 0.85 except that when F_{b} * C_{M} £ 1500 psi, C_{M} =1.
Calculate design load and moment
Select timber species and cross section. Determine maximum bending stress, f”_{b}=M/S, where M is design moment with load duration factor, S is section modulus.
Determine allowable bending stress, with the rest of multiplication factors
F”_{b} = F_{b}*C_{D}*C_{M}*C_{t}*C_{F}*C_{V}*C_{fu}*C_{r}*C_{c}*C_{f}
Calculate elastic modulus
E’=E*C_{M}*C_{t}
Calculate deflection of beam with load without load duration factor.
Calculate shear stress, f”_{v} = VQ/Ib or for rectangular member, f”_{v} = 3V/2b
Where, V is shear force with load duration factor, Q is first moment of inertia, I is second moment of inertia, b is width of the member, d is depth of the member.
Calculate allowable shear stress with the rest of multiplication factors
F”_{v} = F_{v}*C_{D}*C_{M}*C_{t}*C_{H}
Design data:
Length of floor joist: L = 16 ft
Spacing of floor joist: s = 16 in.
Top of joist supported by plywood sheathing.
Design load:
Floor live load: W_{L} =
40 psf
Floor dead load: W_{D} =
10 psf
Superimposed dead load including mechanical and electric load,
W_{SD} = 8 psf
Timber: Southern pine, moisture less than 19%, used in normal
room temperature.
Solution:
Calculate Design load: W = [W_{D} +
W_{SD}+ W_{L}]*s = 77.3 lb/f
Design moment: M = W*L^{2}/8 = 2475 lb-ft
Try 2x10 joist
Nominal dimension, B = 2 in, D = 10 in
Actual dimension, b = 1.5 in, d = 9.25 in
Section modulus: S = 21.39 in^{3}, Modulus of inertia,
I = 98.93 in^{4}.
Bending stress: f_{b} =M/S
= 1388 psi
Try Southern pine No. 2, F_{b} =
1500 psi
Load duration factor for dead load: C_{D} =
0.9
Load duration factors for live load: C_{D} =
1.0
The depth to width ratio based on nominal dimension, D/B = 5
Since compressive edge is fully supported by plywood floor, C_{L} =
1
Repetition factor for joist: C_{r} =
1.15
Wet service factor: C_{M} =
1
Temperature factor: C_{t} =
1
Other factors not applicable
Allowable stress, F’_{b} =
F_{b}*C_{D}* C_{L}*
C_{r}* C_{M}*
C_{t} = 1725
psi O.K.
Check deflection:
Elastic modulus: E = 1600000 psi*C_{M}*
C_{t} = 1600000
psi
Deflection: D =
5*W*L^{4}/(384*E*I) = 0.75 in <
L/240 O.K.
Check shear stress
Maximum shear force. V = W*L/2 = 640 lb
Shear stress, f_{v} =
V/bd = 46 psi
Conservatively assume shear stress factor, C_{H} =
1
Allowable shear stress, F_{v} = 90 psi * C_{D}* C_{M}* C_{t }*C_{H} = 90 psi O.K.
Design data:
Length of beam: L = 16 ft
Tributary width: s = 8 ft
Top of beam supported by floor joists at 16 in O.C.
Design load:
Floor live load: W_{L} =
40 psf
Floor dead load: W_{D} =
10 psf
Superimposed dead load including mechanical and electric load,
W_{SD} = 8 psf
Timber: Southern pine, moisture less than 19%, used in normal
room temperature.
Solution:
Calculate Design load: W = [(W_{D} +
W_{SD}+ W_{L}]*s = 464 lb/ft
Design moment: M = W*L^{2}/8 = 14850 lb-ft
Use 3-3x12 nailed together with 12d nails at 12 in O.C. from
both sides staggered.
Nominal dimension, B = 9 in, D = 12 in
Actual dimension, b = 7.5 in, d = 11.25 in
Section modulus: S = 158.2 in^{3}, Modulus of inertia,
I = 890 in^{4}.
Bending stress: f_{b} =M/S
= 1126 psi
Try Southern pine No. 2, F_{b} =
1500 psi
The depth to width ratio based on nominal dimension, D/B =
1.33
Since compressive edge is supported by floor joist at 16 in
O.C., C_{L} = 1
Wet service factor: C_{M} =
1
Temperature factor: C_{t} =
1
Load duration factor for dead load: C_{D} =
0.9
Load duration factors for live load: C_{D} =
1.0
Other factors not applicable
Allowable stress, F’_{b} =
F_{b}* C_{D}*C_{L}*
C_{M}* C_{t} =
1500 psi O.K.
Check deflection:
Elastic modulus: E = 1600000 psi*CM*
Ct = 1600000 psi
Deflection: D =
5*W*L^{4}/(384*E*I) = 0.48 in <
L/240 O.K.
Check shear stress
Maximum shear force. V = W*L/2 = 3840 lb
Shear stress, fv = V/bd = 46 psi
Conservatively assume shear stress factor, C_{H} =
1
Allowable shear stress, Fv = 90 psi * C_{D}*C_{M}* C_{t }*C_{H} = 90 psi O.K.
Design data:
Length of floor joist: L = 16 ft
Spacing of floor joist: s = 16 in.
Top of joist supported by plywood sheathing.
Design load:
Floor live load: W_{L} =
40 psf
Floor dead load: W_{D} =
10 psf
Superimposed dead load including mechanical and electric load,
W_{SD} = 8 psf
Timber: Southern pine, moisture less than 19%, used in normal
room temperature.
Solution:
Calculate Design load: W = [W_{D} +
W_{SD}+ W_{L}]*s = 77.3 lb/f
Design moment: M = W*L^{2}/8 = 2475 lb-ft
Try 2x12 joist
Nominal dimension, B = 2 in, D = 12 in
Actual dimension, b = 1.5 in, d = 11.25 in
Section modulus: S = 31.64 in^{3}, Modulus of inertia,
I = 178 in^{4}.
Bending stress: f_{b} =M/S
= 938.5 psi
Try Douglas
Fir-Larch No. 1, F_{b} =
1000 psi
The depth to width ratio based on nominal dimension, D/B = 6
Since compressive edge is fully supported by plywood floor,
Provide solid blocking at both ends, and cross bracing at
mid-span,
Maximum spacing = 8 ft, C_{L} =
1
Repetition factor for joist: C_{r} =
1.15
Wet service factor: C_{M} =
1
Temperature factor: C_{t} =
1
From NDS Table, size factor, C_{F} =
1
Load duration factor for dead load: C_{D} =
0.9
Load duration factors for live load: C_{D} =
1.0
Other factors not applicable
Allowable stress, F’_{b} =
F_{b}* C_{D}*C_{L}*
C_{r}* C_{M}*
C_{t}*C_{F} =
1150 psi O.K.
Check deflection:
Elastic modulus: E = 1700000 psi*C_{M}*
C_{t} = 1700000
psi
Deflection: D =
5*W*L^{4}/(384*E*I) = 0.38 in <
L/240 O.K.
Check shear stress
Maximum shear force. V = W*L/2 = 640 lb
Shear stress, fv = V/bd = 38 psi
Conservatively assume shear stress factor, C_{H} =
1
Allowable shear stress, Fv = 95 psi * C_{D}*C_{M}* C_{t} *C_{H} = 95 psi O.K.