Problem 6.25

 

Given

 

roof rafters at 24 in. OC
D = 12 psf
L_r = 20 psf  (ASCE 7 table 4-1 or IBC table 1607.1)
No. 2 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 = 12.65 psf
w_D = (12.65 lb/ft²)(1 ft/12 in.)²(24 in.) = 2.108 lb/in.

A_T = 14 ft x (24 in. x 1 ft/12 in.) = 28 ft² < 200 ft²  (ASCE 7 section 4.9.1 or IBC section 1607.11)
R₁ = 1
R₂ = 1  (F ≤ 4)
L_r = 1 x 1 x 20 psf = 20 psf
w_Lr = (20 lb/ft²)(1 ft/12 in.)²(24 in.) = 3.333 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 = (5.441 lb/in.) x (14 ft x 12 in./ft)² / (8 x 900 lb/in.²) = 21.33 in.³

S_2x8 = 13.14 in.³  ✘  (NDS Supplement table 1B)
S_2x10 = 21.39 in.³  ✓  (NDS Supplement table 1B)

This doesn't agree with the solutions manual, so 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 2x8 = 1.2 for Fb, 1.2 for Ft, 1.05 for Fc CF 2x10 = 1.1 for Fb, 1.1 for Ft, 1 for Fc NDS Supplement section 2.3 CD = 1.25 (Lr) 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	900	1.25	1	1	1	1.2 1.1	1	1.15		1			1,553 for 2x8 1,423 for 2x10
tension stress parallel to grain	Ft	575	1.25	1	1		1.2 1.1				1			-- --
shear stress parallel to grain	Fv	180	1.25	1	1						1			225
compression stress perpendicular to grain	Fc⊥	625		1	1						1		1	--
compression stress parallel to grain	Fc	1,350	1.25	1	1		1.05 1			1	1			-- --
modulus of elasticity (or MOE)	E	1,600,000		1	1						1			--
modulus of elasticity for stability calculations	Emin	580,000		1	1						1	1		--

 

 

Trial Size

 

F'_{b 2x8} = 1,553 psi
F'_{b 2x10} = 1,423 psi

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

S_2x8 ≥ M_max/F'_{b 2x8} = (5.441 lb/in.) x (14 ft x 12 in./ft)² / (8 x 1,553 lb/in.²) = 12.36 in.³
S_2x10 ≥ M_max/F'_{b 2x10} = (5.441 lb/in.) x (14 ft x 12 in./ft)² / (8 x 1,423 lb/in.²) = 13.49 in.³

S_2x8 = 13.14 in.³  ✓  (NDS Supplement table 1B)
S_2x10 = 21.39 in.³  ✓  (NDS Supplement table 1B)

∴ 2x8 or larger should be okay for bending.

 

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

 

dimension lumber
dressed size: 1.5" x 7.25"
A = 10.88 in.²
S_xx = 13.14 in.³
I_xx = 47.63 in.⁴

 

 

Actual Stresses (ASD) and Deflection

 

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

f_b = M/S = (5.441 lb/in.) x (14 ft x 12 in./ft)² / (8 x 13.14 in.³) = 1,461 psi

f_v = 1.5V/A = 1.5 (5.441 lb/in. x 14 ft x 12 in./ft) / (2 x 10.88 in.²) = 63.01 psi

f_{v reduced} = 1.5 (5.441 lb/in. x (14 ft x 12 in./ft - 2 x 7.25 in.)) / (2 x 10.88 in.²) = 57.57 psi  (NDS section 3.4.3)

 

 

Is Member Adequate? (ASD)

 

bending:  1,553 psi > 1,461 psi  ✓

shear:  225 psi > 63.01 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)