Problem 6.27

 

Given

 

roof rafters at 24 in. OC
D = 15 psf
S = 50 psf
No. 1 DF-L
fully braced
C_M = 1
C_t = 1

 

 

Loads

 

horizontal span = 14 ft
height = 4/12 x 14 ft = 4.667 ft  (similar triangles)
diagonal length = sqrt(14² + 4.667²) = 14.76 ft
D' = 14.76D/14 = 15.81 psf
w_D = (15.81 lb/ft²)(1 ft/12 in.)²(24 in.) = 2.636 lb/in.

w_S = (50 lb/ft²)(1 ft/12 in.)²(24 in.) = 8.333 lb/in.

 

 

Trial Size

 

F_b = 900 psi

vertical ground reactions:  wL/2
V_max = +/- wL/2
M_max = wL²/8

S ≥ M_max/F_b = (10.97 lb/in.) x (14 ft x 12 in./ft)² / (8 x 900 lb/in.²) = 43.00 in.³

S_2x12 = 31.64 in.³  ✘  (NDS Supplement table 1B)
S_2x14 = 43.89 in.³  ✓  (NDS Supplement table 1B)

Let's try to get a better estimate by using F'_b instead of F_b.

 

 

Adjusted ASD Values

 

NDS Supplement table 4A Cr = 1.15 CM = 1 Cfu = 1 CF 2x12 = 1 for Fb, 1 for Ft, 1 for Fc CF 2x14 = 0.9 for Fb, 0.9 for Ft, 0.9 for Fc NDS Supplement section 2.3 CD = 1.15 (S) Ct = 1

NDS Supplement section 3.3 CL = 1 (fully braced) NDS Supplement section 3.7 CP = 1 (not a column) NDS Supplement section 3.10 Cb = 1 (not enough info given)

NDS Supplement section 4.3 Ci = 1 NDS Supplement section 4.4 CT = 1 (not a truss)

 

Property		Reference Design Values (psi) (Table 4A)	Adjustment Factors (Table 4.3.1)											Adjusted Design Values (psi)
			CD	CM	Ct	CL	CF	Cfu	Cr	CP	Ci	CT	Cb
bending stress	Fb	1,000	1.15	1	1	1	1 0.9	1	1.15		1			1,323 for 2x12 1,190 for 2x14
tension stress parallel to grain	Ft	675	1.15	1	1		1 0.9				1			-- --
shear stress parallel to grain	Fv	180	1.15	1	1						1			207
compression stress perpendicular to grain	Fc⊥	625		1	1						1		1	--
compression stress parallel to grain	Fc	1,500	1.15	1	1		1 0.9			1	1			-- --
modulus of elasticity (or MOE)	E	1,700,000		1	1						1			--
modulus of elasticity for stability calculations	Emin	620,000		1	1						1	1		--

 

 

Trial Size

 

F'_{b 2x12} = 1,323 psi
F'_{b 2x14} = 1,190 psi

vertical ground reactions:  wL/2
V_max = +/- wL/2
M_max = wL²/8

S_2x12 ≥ M_max/F'_{b 2x8} = (10.97 lb/in.) x (14 ft x 12 in./ft)² / (8 x 1,323 lb/in.²) = 29.25 in.³
S_2x14 ≥ M_max/F'_{b 2x10} = (10.97 lb/in.) x (14 ft x 12 in./ft)² / (8 x 1,190 lb/in.²) = 32.52 in.³

S_2x12 = 31.64 in.³  ✓  (NDS Supplement table 1B)
S_2x14 = 43.89 in.³  ✓  (NDS Supplement table 1B)

∴ 2x12 or larger should be okay for bending.

 

Size Category  (NDS Supplement tables 1A and 1B)

 

dimension lumber
dressed size: 1.5" x 11.25"
A = 16.88 in.²
S_xx = 31.64 in.³
I_xx = 178.0 in.⁴

 

 

Actual Stresses (ASD) and Deflection

 

vertical ground reactions:  wL/2
V_max = +/- wL/2
M_max = wL²/8

f_b = M/S = (10.97 lb/in.) x (14 ft x 12 in./ft)² / (8 x 31.64 in.³) = 1,223 psi

f_v = 1.5V/A = 1.5 (10.97 lb/in. x 14 ft x 12 in./ft) / (2 x 16.88 in.²) = 81.89 psi

f_{v reduced} = 1.5 (10.97 lb/in. x (14 ft x 12 in./ft - 2 x 11.25 in.)) / (2 x 16.88 in.²) = 70.92 psi  (NDS section 3.4.3)

 

 

Is Member Adequate? (ASD)

 

bending:  1,323 psi > 1,223 psi  ✓

shear:  207 psi > 89.89 psi  ✓  (ok, so no need to consider reduced shear)

Notch dimensions not given in image, so they cannot be checked.

 

 

Nominal LRFD Values Fbn, Fvn, E Adjusted LRFD Values F'bn, F'vn, E' Adjusted LRFD Moment and Shear Resistances M'n, V'n Factored Moment and Shear (LRFD) and Actual Deflection Mu, Vn, Δ Is Member Adequate? (LRFD)