Amongst the most prominent advantages of glulam is the possibility of designing curved structural members. During manufacture, the individual laminations are bent to the desired form before the glue has hardened. To avoid damaging of the laminations as they are bent, the curvature must be limited. Thicker laminations cannot be bent as sharply as thinner ones. Curvature must also be limited so that high residual bending stresses are not present in the finished member. As an example, let us try to estimate the bending stress in a lamination of a curved glulam beam during manufacturing, when it is bent to a given radius of curvature. Realistic values could be:
• lamination thickness: t = 33 mm • radius of curvature rin = 8 m
• modulus of elasticity E = 13 000 MPa.
The theoretical bending stress which occurs in the lamination due to bending is:
which is close to the bending strength of the lamination. However, due to the creep characteristics of timber, the bending stresses (σ) are significantly relaxed during gluing, which is carried out with added heat and moisture. These initial stresses can therefore often be ignored in design. However, when the ratio of radius of curvature (rin) to lamination thickness (t) is too small, the bending strength of the beam should be reduced. According to Eurocode 5, such a reduction factor must be applied when rin ⁄ t < 240.
When bending moment is applied to a beam that is initially curved in the plane of bending, radial stresses (as well as bending stresses) occur. These radial stresses may be either tensile or compressive, see Figure 3.31.
Eq. 3.45
Eq. 3.46
C T T C M T90 M h r σm σm dϑ d 2 dϑ M M M M Possible crack Tensile radial stresses Compressive radial stresses
Figure 3.31: Left: bending moment tending to increase the curvature of the glulam member. Right: bending moment tending to straighten the glulam member.
Figure 3.32: Simplified model for determining stresses perpendicular to grain at the apex of a curved beam subjected to pure bending moment.
When the applied moment tends to increase the curvature of the glulam member, the lamina- tions are pressed more firmly together, see to the left in Figure 3.31. This means that compres- sive radial stresses occur between laminations. On the other hand, when the applied bending moment tends to straighten the glulam member, the laminations try to move apart, see to the right in Figure 3.31. This means that tensile radial stresses occur between laminations. Tensile radial stresses should be kept as low as possible since they may cause cracking of the element. This effect can easily be demonstrated by the reader by holding together firmly several sheets of paper in an in initially curved shape. Bending the bundle one way separates the pages while bending in the other way compresses them together.
Figure 3.32 shows the apex zone of a curved beam subjected to constant bending moment. Assuming, for simplification, a linear stress distribution, it can easily been shown that the resulting tensile and compressive forces, T and C respectively, give rise to a force T90 in the radial direction:
Resultant forces T and C are equal to each other. For a beam cross section with breadth b and
By geometric considerations:
Substituting Eq. 3.49 and Equation 3.50 into Equation 3.48:
Knowing the tension force perpendicular to the grain, the corresponding tensile stress is:
Eq. 3.52 shows that the tension stress perpendicular to the grain σt,90 at the apex of a curved beam can be calculated approximately by modifying the bending stress parallel to the grain (σm = M ⁄ W) with a shape factor kp = h ⁄ (4 ∙ r). It is important to observe that increasing the beam depth h and/or decreasing the radius of curvature r will increase the magnitude of the stress perpendicular to the grain.
Various studies have shown that tension strength perpendicular to the grain ft,90 is highly dependent on the stressed volume of the timber. The basic design value of tensile strength perpendicular to the grain must therefore be modified, for example by multiplying it by a modification factor kvol and kdis:
where :
V0 reference volume. In EC 5, V0 = 0,01 m3
V stressed/curved volume determined with regard to the geometry of
the member
kdis modification factor with regard to stress distribution in the beam. Values of kdis and V for beams loaded by uniformly distributed load can be taken from Table 3.4. V needs, however, not be taken as more than 2 ⁄ 3Vb, where Vb is the total volume of the beam.
Eq. 3.50
Eq. 3.51
Eq. 3.52
Table 3.4 Values of kdis and V according to Eurocode 5 for typical beam types.
Beam type kdis V
Double tapered beam
αap
0,5hap 0,5hap
hap
(1)
1,4 Volume of the stressed part (1)
Curved beam αap = 0 rin r = rin + 0,5hap h = hap (1) t β
1,4 Volume of the curved part (1)
Pitched cambered beam αap rin t hap (1) r = rin + 0,5hap
1,7 Volume of the curved part (1)
Angles α and β in degrees.
In situations where the design tensile strength for stresses perpendicular to the grain is exceeded, mechanical fastenings such as glued-in rods or full-threaded screws may be used as reinforcement, see Figure 3.33.
α x = x0 x = /2 at x = x0 at x = /2 b b h0 hap σm,0 σm,0 σt,90 σm,α 3.3.3 Design procedures 3.3.3.1 Tapered beams
For slopes α ≤ 10°, the design bending stresses σm,α,d and σm,0,d (see Figure 3.34) may be taken as:
Eq. 3.54
Figure 3.34: Single tapered beam showing critical cross sections for bending stresses.
At the outermost fibre of the tapered edge, the stresses should satisfy the following expression:
where:
• σm,α,d and σm,0,d are the design bending stresses at an angle to grain and at the straight edge, respectively
• Md is the design bending moment in the section x = x0
• x = x0 is the position of maximum bending stress (x0 = (h0 ∙ l) ⁄ (2 ∙ hap) for simply supported beams with uniformly distributed load)
• Wx0 is the section modulus at the cross section x = x0 • fm,d is the design bending strength
• km,α is a reduction factor that takes into account the simultaneous action of bending stress, shear stress and compression/tension stress at the tapered edge.
The values of km,α for different slopes of the tapered edge are shown in Figure 3.35. The values are derived for the glulam class GL30c.
km,α 0,2 0,1 0,4 0,3 0,6 0,5 0,8 0,9 1 1,1 0,7 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 α [º] α α
Figure 3.35: Values of km,α according to Eurocode 5 for different slopes of the tapered edge,
glulam class GL30c.
The bending stress at the apex must also be checked, even though it seldom governs the design:
with:
• kl is a factor determined by finite element analysis that takes into account the tapering
of laminations. Values of kl for glulam GL30c are given in Figure 3.37 • Map,d is the design moment at the apex
• Wap is the section modulus of the beam at the apex.
The design tensile stress perpendicular to the grain due to bending moment can be calculated as follows:
where:
• kp is a factor determined by finite element analysis, defined as the ratio between perpendicular to grain stress and bending stress at the apex. Values of kp for glulam GL30c are given in Figure 3.38.
Eq. 3.56
Stresses at the apex Stresses at the apex
b b hap hap σm,0 σ m,0 σt,90 σt,90 σm,α h = hap hap
The design tensile strength perpendicular to the grain must then be reduced to take into account the volume effect. According to Eurocode 5, the following inequality must be satisfied:
with:
• kdis, see Table 3.4 • V, see Table 3.4
• ft,90,d is the tensile strength perpendicular to the grain.
3.3.3.2 Curved beams and pitched cambered beams
The most critical section of curved beams and pitched cambered beams is normally the one at the apex, see Figure 3.36.
Eq. 3.58
Figure 3.36: Bending stresses and tension stresses perpendicular to the grain for: curved beam (left) and pitched cambered beam (right).
The bending stress at the apex can be calculated as follows:
where:
• kl is a factor determined by finite element analysis that takes into account the geometry
of the beam. Values of kl for glulam GL30c are given in Figure 3.37.
0,04 0 0,08 0,12 0,16 0,2 0,24 0,28 0,32 0,36 0,4 1,7 1,6 1,5 1,4 1,3 1,2 1,1 1 α =14° α =12° α =10° α =8° α =6° α =4° α =2° α =0° k hap/rin
Figure 3.37: Factor kl according to Eurocode 5 for different radii of curvature, glulam class GL30c.
The bending strength of curved laminations should be reduced for taking into account the eigenstresses that occur when the laminations are bent during the manufacturing of the structural element. This can be done by multiplying the basic value of bending strength
fm,d by a reduction factor kr:
The value of kr decreases with decreasing ratio rin ⁄ t (see table below)
Eq. 3.60
Table 3.5 Reduction factor for bending strength kr according to Eurocode 5 as a function of
the ratio rin/t, where rin = inner radius of curvature and t = thickness of the lamination.
rin/t kr
≥ 240 1
0,04 0 0,08 0,12 0,16 0,2 0,24 0,28 0,32 0,36 0,4 0,12 0,02 0,04 0,06 0 α =14° α =12° α =10° α =8° α =6° α =4° α =2° α =0° kp 0,08 0,1 hap/rin
The design tensile stress perpendicular to the grain due to bending moment can be calculated as follows:
where:
• kp is a factor determined by finite element analysis, defined as the ratio between perpendicular to grain stress and bending stress at the apex. Values of kl for glulam GL30c are given in Figure 3.38.
Eq. 3.61
Figure 3.38: Factor kp according to Eurocode 5 for different radii of curvature, glulam class GL30c.
The design tensile strength perpendicular to the grain shall be reduced then in the same manner as for tapered beams, see Equation 3.58.