 |
Section Properties
Section properties are used in various design calculations. For convenience, the following are formulas to calculate the section properties of rectangular beam cross sections.
Definitions
Neutral axis, in the cross section of a beam, is the line in which there is neither tension nor compression stress.
Moment of Inertia (I) of the cross section of beam is the sum of the products of each of its elementary areas multiplied by the square of their distance from the neutral axis of the section.
Section Modulus (S) is the moment of inertia divided by the distance from the neutral axis to the extreme fiber of the section.
Cross Section is a section taken through the member perpendicular to its longitudinal axis.
 Formulas
The following symbols and formulas apply to
rectangular beam cross sections:
X-X= neutral axis for edgewise bending (load applied to narrow face)
Y-Y= Neutral axis for flatwise bending (load applied to narrow face)
b= breadth of rectangular bending member(in.)
d= depth of rectangular bending member (in.)
A= bd=area of cross section (in.2)
c= distance from neutral axis to extreme fiber of cross section (in.)
Ixx= bd3/12 = moment of inertia about the X-X axis (in.4)
Iyy= db3/12 = moment of inertia about the Y-Y axis (in.4)
rxx= Square root of (Ixx/A) = d/Square root of 12 = radius of gyration about the X-X axis (in.)
ryy= Square root of (Iyy/A) = b/Square root of 12 = radius of gyration about the Y-Y axis (in.)
sxx= Ixx /c = bd2/6 = section modulus about the X-X axis (in.3)
syy= Iyy /c = db2/6 = section modulus about the Y-Y axis (in.3)
Sizes of rough and dressed Western Red Cedar are shown in Tables 5 and 6.
Base Design Values (United States Only)
Since different sizes of visually-graded lumber have different values, the design values shown in Table 8 are tabulated in a base value approach. Base values are provided for a base size that depends on the grade. For Select Structural, No.1, No.2 and No.3 grades, the base strength values are published on a 2x12 basis. For Construction Standard and Utility grades, the base strength values are published on a 2x4 basis (the size factor is always 1.0). For Stud grade, the base strength values are published on a 2x6 basis. These values are for use in the United States only.
To determine the value for a given size, the designer selects a base value for a given grade then multiplies the base value by a size factor from Table 9.
The base design values apply to Western Red Cedar manufactured by members of the Western Red Cedar Export Association and graded to National Lumber Grading Authority Rules (NLGA). Grades and sizes of Canadian dimension lumber are identical to those in use throughout the United States and conform to the requirements of applicable American Standards. Tabulated values are from The U.S. Span Book for Canadian Lumber published by the Canadian Wood Council (1-800-463-5091).
Span Tables
Spans for Western Red Cedar dimension lumber used as joists and rafters in residential and commercial structures are available from the Western Red Cedar Lumber Association, the Canadian Wood Council and the National Association of Home Builders. Please request publication The U.S Span Book for Canadian Lumber. Cost $10.
Table 1. Base Design Values For Use In The U.S.A. For Western Red Cedar - 2-4" Thick 2" and Wider
Base values in pounds per square inch (psi) - Use with Adjustment Factors (see Table 9)
Grade
Fb |
Extreme Fiber Stress in Bending
Ft |
Tension Parallel Parallel to Grain
Fv |
Horizontal Shear Fv |
Compression |
Modulus of Elasticity (million psi)
E |
| Perpendicular To Grain Fc (perp) |
Parallel To Grain Fc |
Select Structural
No.1/No.2
No.3
|
950
575
350
|
450
275
150
|
65
65
65
|
350
350
350
|
1,100
825
475
|
1.1
1.1
1.0 |
Construction
Standard
Utility
|
675
375
175
|
300
175
75
|
65
65
65
|
350
350
350
|
1,050
850
550
|
1.0
0.9
0.9 |
| Stud |
450 |
200 |
65 |
350 |
525 |
1.0 |
Notes:
- No.1/No.2 applies to either No.1 or No.2 grades.
- Values for Utility grade apply only to 2" and 4" lumber.
- For studs wider than 6" bearing the "Stud" grademark, use the property values and size factors for No.3 grade.
|
Table 2. Size Factors (CF) For Tabulated Design Values
| Grades |
Nominal Width (depth)(in) |
Fb less than 4 in. thick |
Fb
4 in. thick nominal |
Ft |
Fc |
Other Properties |
Select Structural
No.1
No.2
& No.3
|
4 & less
5
6
8
10
12
14 & wider
|
1.5
1.4
1.3
1.2
1.1
1.0
0.9
|
1.5
1.4
1.3
1.3
1.2
1.1
0.9
|
1.5
1.4
1.3
1.2
1.1
1.0
0.9
|
1.15
1.1
1.1
1.05
1.0
1.0
0.9
|
1.0
1.0
1.0
1.0
1.0
1.0
1.0 |
| Construction & Standard |
4 & less |
1.0 |
1.0 |
1.0 |
1.0 |
1.0 |
| Utility |
4 |
1.0 |
1.0 |
1.0 |
1.0 |
1.0 |
| Stud* |
4 & less
5 & 6 |
1.1
1.0 |
1.1
1.0 |
1.1
1.0 |
1.05
1.0 |
1.0
1.0 |
| MSR and plank decking All grades & sizes |
|
1.0 |
1.0 |
1.0 |
1.0 |
1.0 |
Note: Factors are for Stud grade widths 6" and less. For studs wider than 6", use the design values and size factors for No.3 grade.
Table 3. Wet Use Factors (CM) For Tabulated Design Values
The recommended design values are for applications where the moisture content of the wood does not exceed 19%. For use conditions where the moisture content of dimension lumber will exceed 19%, the Wet Use Adjustment Factors below are recommended:
| Property |
Adjustment Factor |
| Fb Extreme Fiber Stress in Bending |
0.85* |
| Ft Tension Parallel to Grain |
1.0 |
| Fc Compression Parellel to Grain |
0.8** |
| Fv Horizontal Shear |
0.97 |
| Fc(perp) Compresion Perpendicular to Grain |
0.67 |
| E Modulus of Elasticity |
0.9 |
Notes:
- Bending Wet Use Factor = 1.0 where Fb Cf (base value size factor) does not exceed 1,150 psi.
- Compression Parallel Wet Use Factor=1.0 where Fc Cf (base value size factor) does not exceed 750 psi.
|
Table 4. Flat Use Factors (Cfu)
Apply to Tabulated Design Values for Extreme Fiber Stress in Bending Where Lumber is used Flatwise Rather than on Edge.
| Nominal Width (inches) |
Nominal Thickness (inches) |
|
less than 4 |
4 |
less than 4 |
1.00 |
|
4 |
1.10 |
1.00 |
5 |
1.10 |
1.05 |
6 |
1.15 |
1.05 |
8 |
1.15 |
1.05 |
10 & Wider |
1.20 |
1.10 |
Note: These factors apply to all dimension lumber except tongue-and-grove decking grades. For T & G decking, the following adjustments may be used:
| Nominal thickness |
2" |
3" |
4" |
| Flat use factor |
1.10 |
1.04 |
1.00 |
Table 5. Repetitive Member Factor (Cr)
Applies to Tabulated Design Values for Extreme Fiber Stress in Bending when members are used as joists, truss chords, rafters, studs, planks, decking or similar members which are in contact or spaced not more than 24" on centers, are not less than 3 in number and are joined by floor, roof or other load distributing elements adequate to support the design load.
Table 6. Duration of Load Adjustment (CD) For Tabulated Design Values
| Load Duration |
Factor |
| Permanent |
0.9 |
| Ten Years (normal load) |
1.0 |
| Two Months (snow load) |
1.15 |
| Seven Days |
1.25 |
| Ten Minutes (wind, earthquake) |
1.6 |
| Impact |
2.0 |
Note: Confirm load requirements with local codes. Refer to Model Building Codes or the National Design Specification for high-temperature or fire-retardant treated adjustment factors.
Table 7. Horizontal Shear Adjustment For Tabulated Design Values
(CH) All horizontal shear base values are established as if a piece were split full length and as such the values are reduced from those permitted to be assigned in accordance with ASTM standards. This reduction is made to compensate for any degree of shake, check or split that might develop in a piece.
| 2 inches Thick (Nominal) Lumber |
3 inches Thicker (Nominal) Lumber |
| For convenience, the table below may be used to determine horizontal shear values for any grade of 2" thick lumber in any species when the length of split or check is known: |
Horizontal shear values for 3" and thicker lumber also are established as if a piece were split full length. When specific lengths of splits are known and any increase in them is not anticipated, the following adjustments may be applied: |
| When length of split on wide face does not exceed: |
Multiply tabulated FV value by: |
When length of split on wide face does not exceed |
Multiply tabulated FV value by: |
No split
1/2 wide face
3/4 wide face
1 x wide face
1-1/2 wide face or more
|
2.00
1.67
1.50
1.33
1.00 |
No split
1/2 x narrow face
1 x narrow face
1-1/2 x narrow face or more
|
2.00
1.67
1.33
1.00 |
Table 8. Adjustments for Compression Perpendicular To Grain To Deformation Basis of 0.02"
| Design values for compression perpendicular to grain are established in accordance with the procedures set forth in ASTM D 2555 and D 245. ASTM procedures consider deformation under bearing loads as a serviceability limit state comparable to bending deflection because bearing loads rarely cause structural failures. Therefore, ASTM procedures for determining compression perpendicular to grain values are based on a deformation of 0.04" and are considered adequate for most classes of structures. Where more stringent measures need be taken in design, the following permits the designer to adjust design values to a more conservative deformation basis of 0.02". |
| Y02=0.73Y04+5.60 |
Table 9. Design Values For Use In the U.S.A. For Visually Graded (NLGA) Western Red Cedar Timbers (5" 5" and Larger)
| Grade |
Size Class-ification |
Design Values in pounds per square inch (psi) |
Extreme Fiber Stress in Bending
Fb |
Tension Parallel to Grain
F1 |
Shear Parallel to Grain
FV |
Compression Perpendicular to Grain
Fc(perp) |
Compression Parallelto Grain
Fc |
Modulus of Elasticity
E |
Select
Strctl.
No.1
No.2
|
Beams and Stringers |
1,150
925
625
|
675
475
300
|
65
65
65
|
425
425
425
|
850
700
450
|
1,000,000
1,000,000
800,000 |
Select
Strctl.
No.1
No.2
|
Posts and Timber |
1,050
875
500
|
700
575
350
|
65
65
65
|
425
425
425
|
900
800
550
|
1,000,000
1,000,000
800,000 |
Notes:
- Allowable Extreme Fiber Stress in Bending applies only when Beams and Stringers are loaded on narrow face.
- Where applicable see Tables 9 through 13 for conditions of use and adjustment factors.
|
|
