Chapter 8
Structural Analysis Equations
Equations for deformation and stress, which are the basis for tension members and beam and column design, are discussed in this chapter. The first two sections cover tapered members, straight members, and special considerations such as notches, slits, and size effect. A third section presents stability criteria for members subject to buckling and for members subject to special conditions. The equations are based on mechanics principles and are not given in the design code format found in Allowable Stress Design or Load and Resistance Factor Design specifications.
Deformation Equations
Equations for deformation of wood members are presented as functions of applied loads, moduli of elasticity and rigidity, and member dimensions. They may be solved to determine minimum required cross-sectional dimensions to meet deformation limitations imposed in design. Average moduli of elasticity and rigidity are given in Chapter 4. Consideration must be given to variability in material properties and uncertainties in applied loads to control reliability of the design.
Axial Load
The deformation of an axially loaded member is not usually an important design consideration. More important considerations will be presented in later sections dealing with combined loads or stability. Axial load produces a change of length given by
where δ is change of length, L length, A cross-sectional area, E modulus of elasticity (EL when grain runs parallel to member axis), and P axial force parallel to grain.
Bending
Straight Beam Deflection
The deflection of straight beams that are elastically stressed and have a constant cross section throughout their length is given by
where δ is deflection, W total beam load acting perpendicular to beam neutral axis, L beam span, kb and ks constants dependent upon beam loading, support conditions, and location of point whose deflection is to be calculated, I beam moment of inertia, A′ modified beam area, E beam modulus of elasticity (for beams having grain direction parallel to their axis, E = EL), and G beam shear modulus (for beams with flat-grained vertical faces, G = GLT, and for beams with edge-grained vertical faces, G = GLR). Elastic property values are given in Tables 4–1 and 4–2 (Ch. 4).
The first term on the right side of Equation (8–2) gives the bending deflection and the second term the shear deflection. Values of kb and ks for several cases of loading and support are given in Table 8–1.
The moment of inertia I of the beams is given by
(8–3)
where b is beam width, h beam depth, and d beam diameter. The modified area A′ is given by
(8–4)
If the beam has initial deformations such as bow (lateral bend) or twist, these deformations will be increased by the bending loads. It may be necessary to provide lateral or torsional restraints to hold such members in line. (See Interaction of Buckling Modes section.)
Tapered Beam Deflection
Figures 8–1 and 8–2 are useful in the design of tapered beams. The ordinates are based on design criteria such as span, loading, difference in beam height (hc − h0) as required by roof slope or architectural effect, and maximum allowable deflection, together with material properties. From this, the value of the abscissa can be determined and the smallest beam depth h0 can be calculated for comparison with that given by the design criteria. Conversely, the deflection of a beam can be calculated if the value of the abscissa is known. Tapered beams deflect as a result of shear deflection in addition to bending deflections (Figs. 8–1 and 8–2), and this shear deflection ∆s can be closely approximated by
The final beam design should consider the total deflection as the sum of the shear and bending deflection, and it may be necessary to iterate to arrive at final beam dimensions. Equations (8–5) are applicable to either single-tapered or doubletapered beams. As with straight beams, lateral or torsional restraint may be necessary.
Effect of Notches and Holes
The deflection of beams is increased if reductions in cross-section dimensions occur, such as by holes or notches. The deflection of such beams can be determined by considering them of variable cross section along their length and appropriately solving the general differential equations of the elastic curves, EI(d2 y/dx2) = M, to obtain deflection expressions or by the application of Castigliano’s theorem. (These procedures are given in most texts on strength of materials.)
Table 8–1. Values of kb and ks for several beam loadings
Figure 8–1. Graph for determining tapered beam size based on deflection under uniformly distributed load.
Effect of Time: Creep Deflections
In addition to the elastic deflections previously discussed, wood beams usually sag in time; that is, the deflection increases beyond what it was immediately after the load was first applied. (See the discussion of creep in Time Under Load in Ch. 4.)
Green timbers, in particular, will sag if allowed to dry under load, although partially dried material will also sag to some extent. In thoroughly dried beams, small changes in deflection occur with changes in moisture content but with little permanent increase in deflection. If deflection under longtime load with initially green timber is to be limited, it has been customary to design for an initial deflection of about half the value permitted for longtime deflection. If deflection under longtime load with initially dry timber is to be limited, it has been customary to design for an initial deflection of about two-thirds the value permitted for longtime deflection.
Figure 8–2. Graph for determining tapered beam size on deflection under concentrated midspan load.
Water Ponding
Ponding of water on roofs already deflected by other loads can cause large increases in deflection. The total deflection ∆ due to design load plus ponded water can be closely estimated by
where ∆0 is deflection due to design load alone, S beam spacing, and Scr critical beam spacing (Eq. (8–31)).
Combined Bending and Axial Load
Concentric Load
Addition of a concentric axial load to a beam under loads acting perpendicular to the beam neutral axis causes increase in bending deflection for added axial compression and decrease in bending deflection for added axial tension.
The deflection under combined loading at midspan for pin-ended members can be estimated closely by
where the plus sign is chosen if the axial load is tension and the minus sign if the axial load is compression, ∆ is midspan deflection under combined loading, ∆0 beam midspan deflection without axial load, P axial load, and Pcr a constant equal to the buckling load of the beam under axial compressive load only (see Axial Compression in Stability Equations section.) based on flexural rigidity about the neutral axis perpendicular to the direction of bending loads. This constant appears regardless of whether P is tension or compression. If P is compression, it must be less than Pcr to avoid collapse. When the axial load is tension, it is conservative to ignore the P/Pcr term. (If the beam is not supported against lateral deflection, its buckling load should be checked using Eq. (8–35).)
Eccentric Load
If an axial load is eccentrically applied to a pin-ended member, it will induce bending deflections and change in length given by Equation (8–1). Equation (8–7) can be applied to find the bending deflection by writing the equation in the form
(8–8)
where δb is the induced bending deflection at midspan and ε0 the eccentricity of P from the centroid of the cross section.
Torsion
The angle of twist of wood members about the longitudinal axis can be computed by
where θ is angle of twist in radians, T applied torque, L member length, G shear modulus (use 
, or approximate G by EL/16 if measured G is not available), and K a cross-section shape factor. For a circular cross section, K is the polar moment of inertia:
(8–10)
where D is diameter. For a rectangular cross section,
(8–11)
where h is larger cross-section dimension, b is smaller cross-section dimension, and φ is given in Figure 8–3.
Figure 8–3. Coefficient φ for determining torsional rigidity of rectangular member (Eq. (8 -11)).
Stress Equations
The equations presented here are limited by the assumption that stress and strain are directly proportional (Hooke’s law) and by the fact that local stresses in the vicinity of points of support or points of load application are correct only to the extent of being statically equivalent to the true stress distribution (St. Venant’s principle). Local stress concentrations must be separately accounted for if they are to be limited in design.
Axial Load
Tensile Stress
Concentric axial load (along the line joining the centroids of the cross sections) produces a uniform stress:
where ft is tensile stress, P axial load, and A cross-sectional area.
Short-Block Compressive Stress
Equation (8–12) can also be used in compression if the member is short enough to fail by simple crushing without deflecting laterally. Such fiber crushing produces a local “wrinkle” caused by microstructural instability. The member as a whole remains structurally stable and able to bear load.
Bending
The strength of beams is determined by flexural stresses caused by bending moment, shear stresses caused by shear load, and compression across the grain at the end bearings and load points.
Straight Beam Stresses
The stress due to bending moment for a simply supported pin-ended beam is a maximum at the top and bottom edges. The concave edge is compressed, and the convex edge is under tension. The maximum stress is given by
where fb is bending stress, M bending moment, and Z beam section modulus (for a rectangular cross section, Z = bh2/6; for a circular cross section, Z = πD3/32).
This equation is also used beyond the limits of Hooke’s law with M as the ultimate moment at failure. The resulting pseudo-stress is called the “modulus of rupture,” values of which are tabulated in Chapter 4. The modulus of rupture has been found to decrease with increasing size of member. (See Size Effect section.)
The shear stress due to bending is a maximum at the centroidal axis of the beam, where the bending stress happens to be zero. (This statement is not true if the beam is tapered—see following section.) In wood beams this shear stress may produce a failure crack near mid-depth running along the axis of the member. Unless the beam is sufficiently short and deep, it will fail in bending before shear failure can develop; but wood beams are relatively weak in shear, and shear strength can sometimes govern a design. The maximum shear stress is
(8–14)
where fs is shear stress, V vertical shear force on cross section, A cross-sectional area, and k = 3/2 for a rectangular cross section or k = 4/3 for a circular cross section.
Tapered Beam Stresses
For beams of constant width that taper in depth at a slope less than 25°, the bending stress can be obtained from Equation (8–13) with an error of less than 5%. The shear stress, however, differs markedly from that found in uniform beams. It can be determined from the basic theory presented by Maki and Kuenzi (1965). The shear stress at the tapered edge can reach a maximum value as great as that at the neutral axis at a reaction.
Consider the example shown in Figure 8–4, in which concentrated loads farther to the right have produced a support reaction V at the left end. In this case the maximum stresses occur at the cross section that is double the depth of the beam at the reaction. For other loadings, the location of the cross section with maximum shear stress at the tapered edge will be different.
For the beam depicted in Figure 8–4, the bending stress is also a maximum at the same cross section where the shear stress is maximum at the tapered edge. This stress situation also causes a stress in the direction perpendicular to the neutral axis that is maximum at the tapered edge. The effect of combined stresses at a point can be approximately accounted for by an interaction equation based on the Henky- von Mises theory of energy due to the change of shape. This theory applied by Norris (1950) to wood results in
where fx is bending stress, fy stress perpendicular to the neutral axis, and fxy shear stress. Values of Fx, Fy, and Fxy are corresponding stresses chosen at design values or maximum values in accordance with allowable or maximum values being determined for the tapered beam. Maximum stresses in
Figure 8–4. Shear stress distribution for a tapered beam.
the beam depicted in Figure 8–4 are given by
Substitution of these equations into the interaction Equation (8–15) will result in an expression for the moment capacity M of the beam. If the taper is on the beam tension edge, the values of fx and fy are tensile stresses.
Example: Determine the moment capacity (newton-meters) of a tapered beam of width b = 100 mm, depth h0 = 200 mm, and taper tan θ = 1/10. Substituting these dimensions into Equation (8–16) (with stresses in pascals) results in
Substituting these into Equation (8–15) and solving for M results in
where appropriate allowable or maximum values of the F stresses (pascals) are chosen.
Size Effect
The modulus of rupture (maximum bending stress) of wood beams depends on beam size and method of loading, and the strength of clear, straight-grained beams decreases as size increases. These effects were found to be describable by statistical strength theory involving “weakest link” hypotheses and can be summarized as follows: For two beams under two equal concentrated loads applied symmetrical to the midspan points, the ratio of the modulus of rupture of beam 1 to the modulus of rupture of beam 2 is given by
where subscripts 1 and 2 refer to beam 1 and beam 2, R is modulus of rupture, h beam depth, L beam span, a distance between loads placed a/2 each side of midspan, and m a constant. For clear, straight-grained Douglas-fir beams, m = 18. If Equation (8–17) is used for beam 2 size (Ch. 4) loaded at midspan, then h2 = 5.08 mm (2 in.), L2 = 71.112 mm (28 in.), and a2 = 0 and Equation (8–17) becomes
(8–18a)
(8–18b)
Example: Determine modulus of rupture for a beam 10 in. deep, spanning 18 ft, and loaded at one-third span points compared with a beam 2 in. deep, spanning 28 in., and loaded at midspan that had a modulus of rupture of 10,000 lb/in2. Assume m = 18. Substituting the dimensions into Equation (8–18) produces
Application of the statistical strength theory to beams under uniformly distributed load resulted in the following relationship between modulus of rupture of beams under uniformly distributed load and modulus of rupture of beams under concentrated loads:
(8–19)
where subscripts u and c refer to beams under uniformly distributed and concentrated loads, respectively, and other terms are as previously defined.
Shear strength for non-split, non-checked, solid-sawn, and glulam beams also decreases as beam size increases. A relationship between beam shear τ and ASTM shear block strength τASTM, including a stress concentration factor for the re-entrant corner of the shear block, Cf, and the shear area A, is
(8–20a)
(8–20b)
where τ is beam shear (MPa, lb/in2), Cf stress concentration factor, τASTM ASTM shear block strength (MPa, lb/in2), and A shear area (cm2, in2).
This relationship was determined by empirical fit to test data. The shear block re-entrant corner concentration factor is approximately 2; the shear area is defined as beam width multiplied by the length of beam subjected to shear force.
Effect of Notches, Slits, and Holes
In beams having notches, slits, or holes with sharp interior corners, large stress concentrations exist at the corners. The local stresses include shear parallel to grain and tension perpendicular to grain. As a result, even moderately low loads can cause a crack to initiate at the sharp corner and propagate along the grain. An estimate of the crack-initiation load can be obtained by the fracture mechanics analysis of Murphy (1979) for a beam with a slit, but it is generally more economical to avoid sharp notches entirely in wood beams, especially large wood beams, since there is a size effect: sharp notches cause greater reductions in strength for larger beams. A conservative criterion for crack initiation for a beam with a slit is
where h is beam depth, b beam width, M bending moment, and V vertical shear force, and coefficients A and B are presented in Figure 8–5 as functions of a/h, where a is slit depth. The value of A depends on whether the slit is on the tension edge or the compression edge. Therefore, use either At or Ac as appropriate. The values of A and B are dependent upon species; however, the values given in Figure 8–5 are conservative for most softwood species.
Figure 8–5. Coefficients A and B for crack-initiation criterion (Eq. (8–21)).
Effects of Time: Creep Rupture, Fatigue, and Aging
See Chapter 4 for a discussion of fatigue and aging. Creep rupture is accounted for by duration-of-load adjustment in the setting of allowable stresses, as discussed in Chapters 4 and 6.
Water Ponding
Ponding of water on roofs can cause increases in bending stresses that can be computed by the same amplification factor (Eq. (8–6)) used with deflection. (See Water Ponding in the Deformation Equations section.)
Combined Bending and Axial Load
Concentric Load
Equation (8–7) gives the effect on deflection of adding an end load to a simply supported pin-ended beam already bent by transverse loads. The bending stress in the member is modified by the same factor as the deflection:
(8–22)
where the plus sign is chosen if the axial load is tension and the minus sign is chosen if the axial load is compression, fb is net bending stress from combined bending and axial load, fb0 bending stress without axial load, P axial load, and Pcr the buckling load of the beam under axial compressive load only (see Axial Compression in the Stability Equations section), based on flexural rigidity about the neutral axis perpendicular to the direction of the bending loads. This Pcr is not necessarily the minimum buckling load of the member. If P is compressive, the possibility of buckling under combined loading must be checked. (See Interaction of Buckling Modes.)
The total stress under combined bending and axial load is obtained by superposition of the stresses given by Equations (8–12) and (8–22).
Example: Suppose transverse loads produce a bending stress fb0 tensile on the convex edge and compressive on the concave edge of the beam. Then the addition of a tensile axial force P at the centroids of the end sections will produce a maximum tensile stress on the convex edge of
and a maximum compressive stress on the concave edge of
where a negative result would indicate that the stress was in fact tensile.
Torsion
For a circular cross section, the shear stress induced by torsion is
(8–23)
where T is applied torque and D diameter. For a rectangular cross section,
(8–24)
where T is applied torque, h larger cross-section dimension, and b smaller cross-section dimension, and β is presented in Figure 8–6.
Stability Equations
Axial Compression
For slender members under axial compression, stability is the principal design criterion. The following equations are for concentrically loaded members. For eccentrically loaded columns, see Interaction of Buckling Modes section.
Long Columns
A column long enough to buckle before the compressive stress P/A exceeds the proportional limit stress is called a “long column.” The critical stress at buckling is calculated by Euler’s formula:
where EL is elastic modulus parallel to the axis of the member, L unbraced length, and r least radius of gyration (for a rectangular cross section with b as its least dimension, r = 
, and for a circular cross section, r = d/4). Equation (8–25) is based on a pinned-end condition but may be used conservatively for square ends as well.
Short Columns
Columns that buckle at a compressive stress P/A beyond the proportional limit stress are called “short columns.” Usually the short column range is explored empirically, and appropriate design equations are proposed. Material of this nature is presented in USDA Technical Bulletin 167 (Newlin and Gahagan 1930). The final equation is a fourth-power parabolic function that can be written as
where Fc is compressive strength and remaining terms are defined as in Equation (8–25). Figure 8–7 is a graphical representation of Equations (8–25) and (8–26).
Figure 8–6. Coefficient β for computing maximum shear stress in torsion of rectangular member (Eq. (8 -24)).
Figure 8–7. Graph for determining critical buckling stress of wood columns.
Short columns can be analyzed by fitting a nonlinear function to compressive stress-strain data and using it in place of Hooke’s law. One such nonlinear function proposed by Ylinen (1956) is
where ε is compressive strain, f compressive stress, c a constant between 0 and 1, and EL and Fc are as previously defined. Using the slope of Equation (8–27) in place of EL in Euler’s formula (Eq. (8–25)) leads to Ylinen’s buckling equation
where Fc is compressive strength and fe buckling stress given by Euler’s formula (Eq. (8–25)). Equation (8–28) can be made to agree closely with Figure 8–7 by choosing c = 0.957.
Comparing the fourth-power parabolic function Equation (8–26) to experimental data indicates the function is nonconservative for intermediate L/r range columns. Using Ylinen’s buckling equation with c = 0.8 results in a better approximation of the solid-sawn and glued-laminated data.
Built-Up and Spaced Columns
Built-up columns of nearly square cross section with the lumber nailed or bolted together will not support loads as great as if the lumber were glued together. The reason is that shear distortions can occur in the mechanical joints.
If built-up columns are adequately connected and the axial load is near the geometric center of the cross section, Equation (8–28) is reduced with a factor that depends on the type of mechanical connection. The built-up column capacity is
(8–29)
where Fc, fe, and c are as defined for Equation (8–28). Kf is the built-up stability factor, which accounts for the efficiency of the connection; for bolts, Kf = 0.75, and for nails, Kf = 0.6, provided bolt and nail spacing requirements meet design specification approval.
If the built-up column is of several spaced pieces, the spacer blocks should be placed close enough together, lengthwise in the column, so that the unsupported portion of the spaced member will not buckle at the same or lower stress than that of the complete member. “Spaced columns” are designed with previously presented column equations, considering each compression member as an unsupported simple column; the sum of column loads for all the members is taken as the column load for the spaced column.
Columns With Flanges
Columns with thin, outstanding flanges can fail by elastic instability of the outstanding flange, causing wrinkling of the flange and twisting of the column at stresses less than those for general column instability as given by Equations (8–25) and (8–26). For outstanding flanges of cross sections such as I, H, +, and L, the flange instability stress can be estimated by
where E is column modulus of elasticity, t thickness of the outstanding flange, and b width of the outstanding flange. If the joints between the column members are glued and reinforced with glued fillets, the instability stress increases to as much as 1.6 times that given by Equation (8–30).
Bending
Beams are subject to two kinds of instability: lateral- torsional buckling and progressive deflection under water ponding, both of which are determined by member stiffness.
Water Ponding
Roof beams that are insufficiently stiff or spaced too far apart for their given stiffness can fail by progressive deflection under the weight of water from steady rain or another continuous source. The critical beam spacing Scr is given by
where E is beam modulus of elasticity, I beam moment of inertia, ρ density of water (1,000 kg/m3, 0.0361 lb/in3), L beam length, and m = 1 for simple support or m = 16/3 for fixed-end condition. To prevent ponding, the beam spacing must be less than Scr.
Lateral-Torsional Buckling
Since beams are compressed on the concave edge when bent under load, they can buckle by a combination of lateral deflection and twist. Because most wood beams are rectangular in cross section, the equations presented here are for rectangular members only. Beams of I, H, or other built-up cross section exhibit a more complex resistance to twisting and are more stable than the following equations would predict.
Long Beams—Long slender beams that are restrained against axial rotation at their points of support but are otherwise free to twist and to deflect laterally will buckle when the maximum bending stress fb equals or exceeds the following critical value:
where α is the slenderness factor given by
where EI y is lateral flexural rigidity equal to EL hb3/12, h is beam depth, b beam width, GK torsional rigidity defined in Equation (8–9), and Le effective length determined by type of loading and support as given in Table 8–2. Equation (8–32) is valid for bending stresses below the proportional limit.
Short Beams—Short beams can buckle at stresses beyond the proportional limit. In view of the similarity of Equation (8–32) to Euler’s formula (Eq. (8–25)) for column buckling, it is recommended that short-beam buckling be analyzed by using the column buckling criterion in Figure 8–7 applied with α in place of L/r on the abscissa and fbcr/Fb in place of fcr/Fc on the ordinate. Here Fb is beam modulus of rupture.
Table 8–2. Effective length for checking lateral- torsional stability of beamsa
Effect of Deck Support—The most common form of support against lateral deflection is a deck continuously attached to the top edge of the beam. If this deck is rigid against shear in the plane of the deck and is attached to the compression edge of the beam, the beam cannot buckle. In regions where the deck is attached to the tension edge of the beam, as where a beam is continuous over a support, the deck cannot be counted on to prevent buckling and restraint against axial rotation should be provided at the support point.
If the deck is not very rigid against in-plane shear, as for example standard 38-mm (nominal 2-in.) wood decking, Equation (8–32) and Figure 8–7 can still be used to check stability except that now the effective length is modified by dividing by θ, as given in Figure 8–8. The abscissa of this figure is a deck shear stiffness parameter τ given by
(8–34)
where EIy is lateral flexural rigidity as in Equation (8–33), S beam spacing, GD in-plane shear rigidity of deck (ratio of shear force per unit length of edge to shear strain), and L actual beam length. This figure applies only to simply supported beams. Cantilevers with the deck on top have their tension edge supported and do not derive much support from the deck.
Figure 8–8. Increase in buckling stress resulting from attached deck; simply supported beams. To apply this graph, divide the effective length by θ.
Interaction of Buckling Modes
When two or more loads are acting and each of them has a critical value associated with a mode of buckling, the combination can produce buckling even though each load is less than its own critical value.
The general case of a beam of unbraced length le includes a primary (edgewise) moment M1, a lateral (flatwise) moment M2, and axial load P. The axial load creates a secondary moment on both edgewise and flatwise moments due to the deflection under combined loading given by Equation (8–7). In addition, the edgewise moment has an effect like the secondary moment effect on the flatwise moment.
The following equation contains two moment modification factors, one on the edgewise bending stress and one on the flatwise bending stress that includes the interaction of biaxial bending. The equation also contains a squared term for axial load to better predict experimental data:
where f is actual stress in compression, edgewise bending, or flatwise bending (subscripts c, b1, or b2, respectively), F buckling strength in compression or bending (a single prime denotes the strength is reduced for slenderness), e/d ratio of eccentricity of the axial compression to member depth ratio for edgewise or flatwise bending (subscripts 1 or 2, respectively), and θc moment magnification factors for edgewise and flatwise bending, given by
(8–36)
(8–37)
(8–38)
(8–39)
(8–40)
where le is effective length of member and S and Scr are previously defined ponding beam spacing.
References
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