Design of flexural members requires consideration primarily of bending and shear strength, deflection, and end bearing.
Strength of Flexural Members
The stress induced in a beam (or other flexural member) when subjected to design loads should not exceed the strength of the member. The maximum bending stress ƒb at any section of a beam is given by the flexural formula
ƒb = M/S
where M is the bending moment and S the section modulus. For a rectangular beam, the section modulus is bd2 /6 and Eq. (10.14) transforms into
ƒ = 6M/bd^2
where b is the beam width and d the depth. At every section of the beam, ƒb should be equal to or less than the design value for bending Fb adjusted for all end-use modification factors (Art. 10.5).
Shear stress induced by design loads in a member should not exceed the allowable design value for shear FV. For wood beams, the shear parallel to the grain, that is, the horizontal shear, controls the design for shear. Checking the shear stress perpendicular to the grain is not necessary inasmuch as the vertical shear will never be a primary failure mode.
The maximum horizontal shear stress ƒV in a rectangular wood beam is given by
ƒ = 3V/2bd
where V is the vertical shear. In calculation of V for a beam, all loads occurring within a distance d from the supports may be ignored. This is based on the assumption that loads causing the shear will be transmitted at a 45 angle through the beam to the supports.
(K. F. Faherty and T. G. Williamson, ‘‘Wood Engineering and Construction Handbook.’’ McGraw-Hill Publishing Company, New York.)
Beams may require lateral support to prevent lateral buckling. Need for such bracing depends on the unsupported length and cross-sectional dimensions of the members.
When buckling occurs, a member deflects in the direction of its least dimension b.
In a beam, b usually is taken as the width. If bracing precludes buckling in that direction, deflection can still occur, but in the direction of the strong dimension.
Thus, the unsupported length L, width b, and depth d are key variables in formulas for lateral support and for reduction of design values for buckling.
Design for lateral stability of flexural members is based on a function of Ld/b2.
The beam stability factor, CL, for lumber beams of rectangular cross section having maximum depth-width ratios based on nominal dimensions, as summarized in Table 10.13, can be taken as unity.
No lateral support is required when the depth does not exceed the width of a beam. In that case also, the design value for bending does not have to be adjusted for lateral stability. Similarly, if continuous support prevents lateral movement of the compression flange, and the ends at points of bearing are braced to prevent rotation, then lateral buckling cannot occur and the design of value Fb need not be reduced.
When the beam depth exceeds the width, lateral support should be provided at end bearings. This support should be so placed as to prevent rotation of the beam about the longitudinal axis. Unless the compression flange is braced at sufficiently close intervals between supports, the design value should be adjusted for lateral buckling.
The slenderness ratio RB for beams is defined by
The effective length Le for Eq. (10.17) is given in Table 10.14 in terms of the unsupported length of beam. Unsupported length is the distance between supports or the length of a cantilever when the beam is laterally braced at the supports to prevent rotation and adequate bracing is not installed elsewhere in the span. When both rotational and lateral displacement are also prevented at intermediate points, the unsupported length may be taken as the distance between points of lateral support. If the compression edge is supported throughout the length of the beam and adequate bracing is installed at the supports, the unsupported length is zero.
Deflection of Wood Beams
The design of many structural systems, particularly those with long span, may be governed by deflection. Verifying structural adequacy based on allowable stresses alone may not be sufficient to prevent excessive deflection. Limitations on deflection may increase member stiffness.
Deflection of wood beams is calculated by conventional elastic theory. For example, for a uniformly loaded, simple-span beam, the maximum deflection is computed from
Deflection should not exceed limitations specified in the local building code nor industry-recommended limitations. (See, for example, K. F. Faherty and T. G. Williamson, ‘‘Wood Engineering and Construction Handbook,’’ McGraw-Hill Publishing Company, New York.) Deflections also should be evaluated with respect to other considerations, such as possibility of binding of doors or cracking of partitions or glass.
Table 10.15 gives recommended deflection limits, as a fraction of the beam span, for timber beams. The limitation applies to live load or total load, whichever governs.
Glulam beams may be cambered to offset the effects of deflections due to design loads. These members are cambered during fabrication by creation of curvature opposite in direction to that of deflections under load. Camber, however, does not increase stiffness. Table 10.16 lists recommended minimum cambers for glulam beams.
Minimum Roof Slopes. Flat roofs have collapsed during rainstorms even though they were adequately designed for allowable stresses and definite deflection limitations.
The failures were caused by ponding of water as increasing deflections permitted more and more water to collect.
Roof beams should have a continuous upward slope equivalent to 1⁄4 in / ft between a drain and the high point of a roof, in addition to minimum recommended camber (Table 10.16), to avoid ponding. As a general guideline, when flat roofs have insufficient slope for drainage (less than 1⁄4 in / ft), the stiffness of supporting members should be such that a 5-lb / ft2 load will cause no more than 1⁄2-in deflection.
Because of ponding, snow loads or water trapped by gravel stops, parapet walls, or ice dams magnify stresses and deflections from existing roof loads by
where Cp factor for multiplying stresses and deflections under existing loads to determine stresses and deflections under existing loads plus ponding
W weight of 1 in of water on roof area supported by beam, lb
L span of beam, in
E modulus of elasticity of beam material, psi
I moment of inertia of beam, in4
(Kuenzi and Bohannan, ‘‘Increases in Deflection and Stresses Caused by Ponding of Water on Roofs,’’ Forest Products Laboratory, Madison, Wis.)
Bearing Stresses in Beams
Bearing stresses, or compression stresses perpendicular to the grain, in a beam occur at the supports or at places where other framing members are supported on the beam. The compressive stress in the beam ƒc is given by
where P = load transmitted to or from the beam and A bearing area. This stress should be less than the design value for compression perpendicular to the grain Fc multiplied by applicable adjustment factors (Art. 10.5). (The duration-of-load factor does not apply to Fc for either solid sawn lumber of glulam timber.)
Limitations on compressive stress perpendicular to the grain are set to keep deformations within an acceptable range. An expected failure mode is excessive localized deformation rather than a catastrophic type of failure.
Design values for Fc are averages based on a maximum deformation of 0.04 in in tests conforming with ASTM D143. Design values Fc for glulam beams are generally lower than for solid sawn lumber with the same deformation limit. This is due partly to use of larger-size sections for glulam beams, length of bearing and partly to the method used to derive the design values.
Where deformations are critical, the deformation limit may be decreased, with resulting reduction in Fc. For example, for a deformation maximum of 0.02 in.
the ‘‘National Design Specification for Wood Construction,’’ American Forest & Paper Association, recommends that Fc, psi, be reduced to 0.73 Fc.
Cantilever systems may be composed of any of the various types and combinations of beam illustrated in Fig. 10.1. Cantilever systems permit longer spans or larger loads for a given size member than do simple-span systems, if member size is not controlled by compression perpendicular to grain at the supports or by horizontal shear. Substantial design economies can be effected by decreasing the depths of the members in the suspended portions of a cantilever system.
For economy, the negative bending moment at the supports of a cantilevered beam should be equal in magnitude to the positive moment.
Consideration must be given to deflection and camber in cantilevered multiple spans. When possible, roofs should be sloped the equivalent of 1⁄4 in per foot of horizontal distance between the level of drains and the high point of the roof to eliminate water pockets, or provisions should be made to ensure that accumulation of water does not produce greater deflection and live loads than anticipated. Unbalanced loading conditions should be investigated for maximum bending moment, deflection, and stability.
(For further information on the design of cantilevered beam systems, see K. F. Faherty and T. G. Williamson, ‘‘Wood Engineering and Construction Handbook,’’ 2d ed., McGraw-Hill Publishing Company, New York; D. E. Breyer, ‘‘Design of Wood Structures,’’ 3d ed., McGraw-Hill Publishing Company, New York; ‘‘Wood Structural Design Data,’’ American Forest and Paper Association, Washington, D.C.)