Problem 7.8

 

Given

 

Member size	4x10 door header
Load	D = 120 lb/ft Lr = 120 lb/ft W = 8 kip (T)
Stress grade and species	No. 2 Hem-Fir
Fasteners	ignore
Adjustment factors	CM = 1 Ct = 1 Ci = 1

Lateral stability is not an issue.

 

 

Size (NDS Supplement table 1B)

 

A_g = 32.38 in.²
A_n = A_g - 1.5 in. x (XX in. + 1/16 in.) = XX in.²  (NDS section 11.1.2.2 and appendix L table L1)
S_xx = 49.91 in.³
I_xx = 230.8 in.⁴

 

 

Load Combinations

 

Load	Symbol	Given Value
dead load	D	120 lb/ft
weight of ice	Di	0
earthquake load	E	0
load due to fluids	F	0
flood loads	Fa	0
load due to sub-surface pressure	H	0
live load	L	0
roof live load	Lr	120 lb/ft
rain load	R	0
snow load	S	55
self-straining force	T	0
wind load	W	8,000 lb
wind-on-ice	Wi	0

 

ASCE 7 section 2.4

ASCE 7	IBC	ASD Load Combination	Possible Values
1	16-8	D + F	120 lb/ft
2	16-9	D + H + F + L + T	120 lb/ft
3	16-10	D + H + F + (Lr or S or R)	240 lb/ft
4	16-11	D + H + F + 0.75(L + T) + 0.75(Lr or S or R)	210 lb/ft
5	16-12	D + H + F + (W or 0.7E)	120 lb/ft, 8,000 lb
6	16-13	D + H + F + 0.75(W or 0.7E) + 0.75L + 0.75(Lr or S or R)	210 lb/ft, 6,000 lb
7	16-14	0.6D + W + H	--A
8	16-15	0.6D + 0.7E + H	--A

^A Combinations 7 and 8 are not applicable because S is acting in the same direction (downward) as D.  See highlighted paragraph in ASCE 7 section C2.4.

 

(D)_max = 120 lb/ft
(D+L_r)_max = 240 lb/ft
(D+W)_max = 120 lb/ft, 8,000 lb
(D+0.75W+0.75L_r)_max = 210 lb/ft, 6,000 lb

 

 

Adjusted Design Values

 

NDS Supplement table 4A Cr = 1  (does not repeat) CM = 1 (given) CF = 1.2 for Fb, 1.1 for Ft, 1.0 for Fc Cfu = 1  (headers are bent around strong axis) NDS Supplement section 2.3 CD = 0.9 (D), 1.25 (D+Lr), 1.6 (D+W), or 1.6 (D+0.75W+0.75Lr) depending on load combination Ct = 1 (given)

NDS Supplement section 3.3 CL = 1 (given) 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 (given) 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	850	0.9 1.25 1.6	1	1	1	1.2	1	1		1			918 1,275 1,632
tension stress parallel to grain	Ft	525	0.9 1.25 1.6	1	1		1.1				1			520 722 924
shear stress parallel to grain	Fv	150	0.9 1.25 1.6	1	1						1			-- -- --
compression stress perpendicular to grain	Fc⊥	405		1	1						1		1	--
compression stress parallel to grain	Fc	1,300	0.9 1.25 1.6	1	1		1			1	1			-- -- --
modulus of elasticity (or MOE)	E	1,300,000		1	1						1			--
modulus of elasticity for stability calculations	Emin	470,000		1	1						1	1		--

 

 

Actual Stresses for (D) Load Combination (ASD)

 

(D)_max = 120 lb/ft

Axial tension stress...

f_t = P/A_n = X lb / 32.38 in.² = X psi ≤ 520 psi

Bending stress...

w_D = 120 lb/ft
V_max = 120 lb/ft x 12 ft / 2 = 720 lb
M_max = 0.5 x 6 ft x 720 lb = 2,160 lb-ft x 12 in./ft = 25,920 lb-in.

f_b = M/S = 25,920 lb-in. / 49.91 in.³ = 519 psi ≤ 918 psi  ✓

Check formulas in NDS section 3.9.1 for combined axial tension and bending...

f_t/F'_t + f_b/F^*_b = X/520 + 519/918 = X ≤ 1.0
(f_b-f_t)/F^**_b = (519-X)/918 = X ≤ 1.0

Severity...

519 / 918 = 0.565

 

 

Actual Stresses for (D+L_r) Load Combination (ASD)

 

(D+L_r)_max = 240 lb/ft

Axial tension stress...

f_t = P/A_n = X lb / 32.38 in.² = X psi ≤ 722 psi

Bending stress...

w_D+Lr = 240 lb/ft
V_max = 240 lb/ft x 12 ft / 2 = 1,440 lb
M_max = 0.5 x 6 ft x 1,440 lb = 4,320 lb-ft x 12 in./ft = 51,840 lb-in.

f_b = M/S = 51,840 lb-in. / 49.91 in.³ = 1,039 psi ≤ 1,275 psi  ✓

Check formulas in NDS section 3.9.1 for combined axial tension and bending...

f_t/F'_t + f_b/F^*_b = X/722 + 1,039/1,275 = X ≤ 1.0
(f_b-f_t)/F^**_b = (1,039-X)/1,275 = X ≤ 1.0

Severity...

1,039 / 1,275 = 0.815

 

 

Actual Stresses for (D+W) Load Combination (ASD)

 

(D+W)_max = 120 lb/ft, 8000 lb

Axial tension stress...

f_t = P/A_n = 8,000 lb / 32.38 in.² = 247 psi ≤ 924 psi  ✓

Bending stress...

w_D = 1 20 lb/ft
V_max = 1 20 lb/ft x 12 ft / 2 = 720 lb
M_max = 0.5 x 6 ft x 720 lb = 2,160 lb-ft x 12 in./ft = 25,920 lb-in.

f_b = M/S = 25,920 lb-in. / 49.91 in.³ = 519 psi ≤ 1,632 psi  ✓

Check formulas in NDS section 3.9.1 for combined axial tension and bending...

f_t/F'_t + f_b/F^*_b = 247/924 + 519/1,632 = 0.585 ≤ 1.0  ✓
(f_b-f_t)/F^**_b = (519-247)/1,632 = 0.167 ≤ 1.0  ✓

 Severity...

247 / 924 = 0.267
519 / 1,632 = 0.318

 

 

Actual Stresses for (D+W) Load Combination (ASD)

 

(D+0.75W+0.75L_r)_max = 210 lb/ft, 6,000 lb

Axial tension stress...

f_t = P/A_n = 6,000 lb / 32.38 in.² = 186 psi ≤ 924 psi  ✓

Bending stress...

w_D = 210 lb/ft
V_max = 210 lb/ft x 12 ft / 2 = 1,260 lb
M_max = 0.5 x 6 ft x 1,260 lb = 3,780 lb-ft x 12 in./ft = 45,360 lb-in.

f_b = M/S = 45,360 lb-in. / 49.91 in.³ = 909 psi ≤ 1,632 psi  ✓

Check formulas in NDS section 3.9.1 for combined axial tension and bending...

f_t/F'_t + f_b/F^*_b = 186/924 + 909/1,632 = 0.758 ≤ 1.0  ✓
(f_b-f_t)/F^**_b = (909-186)/1,632 = 0.443 ≤ 1.0  ✓

 Severity...

186 / 924 = 0.298
909 / 1,632 = 0.557

 

 

∴ The 4x10 is okay for all load combinations.  The D+L_r combination is the most severe (0.815).