Analytical Approach to Predict Pre–Camber Deflection of the Pre-stressed Glulam Beams

: A new approach to reinforce glulam timber beams using compressed wood (CW) has been developed by the Author. The compressed wood blocks are inserted into pre-cut holes on the top of glulam beams to produce pre-camber and to generate initial tensile and compressive stresses on the top and on the bottom extreme fiber of the glulam beam, respectively. A simple analytical approach has been developed to predict pre-camber deflections of the beams reinforced with three CW blocks. Assuming that moisture-dependent expansion of the CW block creates an interactive linear pressure between the glulam beam and the CW block, this would generate a pair of eccentric force away from the neutral axis of the beam. It is shown that the pre-camber deflection predicted at midspan of the short beams with various reinforcing arrangements give good agreement with the measured pre-camber from previous experimental results.


Introduction
Recently, many studies on strengthening timber or glulam beam using FRP (fiber reinforced plastic) have been done, which indicated significant enhancement of bending capacity of timber or glulam beam [1][2][3][4]. Anshari et al. [5] introduced a novel way to enhance the load carrying capacity or bending capacity of glulam beam, by introducing a very small amount of compressed wood (CW) to pre-stress the beam. Compressed wood is made of a lower grade wood which is densified in radial direction under a high pressure and moderate temperature (100-180 0 C) condition, and becomes a wood product with high density and high strength. Figure 1 illustrates the initial thickness of softwood (t0=50 mm) and final thickness of CW (t1=15 mm), which is produced with compression rasio (CR) of 70%.  The technology developed is to make use of the moisture-dependent swelling nature of the compressed wood. By inserting the CW blocks on the top part of the glulam beam with the desired initial moisture content (MC) of 6%, a pre-camber deflection and the related initial stresses will be building up due to the swelling. Anshari, et al. [6] developed finite element modeling of upward deflection of glulam beams reinforced by compressed wood. Anshari and Guan [7] further worked to develop FE model of initial stress and pre-camber deflection of glulam beams. Figure 2 illustrates the mechanism of the prestressing process of glulam beam by using CW blocks. If the CW blocks absorb moisture from ambient it would expand in R direction. By expansion of the CW block, it will generate pre-camber at the mid span of the beam (see Figure 2c) and also generate initial compressive stress and tensile stress at the bottom and at the top extreme fibers of the beam, respectively. Once the CW blocks reach the equilibrium moisture content with ambient, the maximum pre-camber and initial stress will be achieved.

Mechanism of Pre-stressing
This study aimed to develop an analytical approach to determine pre-camber deflection of pre-stressed glulam beams using CW blocks.

Analytical Solution Procedure Basic Assumption
A simple analytical approach has been developed to predict pre-camber deflections of the beams reinforced with three CW blocks. The mechanisms of the pre-camber deflection generated in the pre-stressed beam are shown in Figure 3. Assuming that moisture-dependent expansion of the CW block creates an interactive linear pressure between the glulam beam and the CW block, this would generate a pair of eccentric force away from the neutral axis of the beam. As the result, a moment couple of M0 is generated, which produces a pre-camber deflection of the strengthened beam. It is assumed that the distribution of the swelling pressure is linearly increased from zero on the top of the beam to a certain pressure p at the bottom of the CW block. The CW block would expand if it absorbs moisture from ambient, because CW block is preconditioned with MC of 6%. The dimensions of the CW block and the location of the CW block are shown in Figure 3.
The pressure (p) acting on the glulam beam due to expansion of a CW block may be expressed, assuming isotropic elasticity applied, as: where Eglulam and ECW are moduli of glulam and CW, and εglulam and εCW are the maximum strains on glulam and CW, respectively. The resultant force generated by the above pressure therefore is; where e and f are depth and width of CW block, respectively, and p is the maximum pressure which is mutually acting on both the CW block and the glulam as shown in Figure 1. Assuming the resultant force F acting at a distance "d' away from the neutral axis of the beam, subsequently a moment couple is created as follow; Figure 4 shows sketches to represent linear moment distribution along the beam span, which are generated by expansion of one CW block on the mid span, L1, and L2 mm away from the left support, respectively. By Macaulay methods (double integration), the pre-camber deflections corresponding to the M0 generated at various locations of the beam can be expressed in the following equations. Here, for simplification, the same moment couple M0 is assumed to be generated, independent of the location of the CW block.  Although the bending moment generated by CW block swelling is localized initially with building-up the pre-camber, the influence of the moment is spread to the whole beam.

Linear Distribution of Moment along the Beam Span
Therefore, a linear distribution of the moment along the beam span is assumed, as shown in Figure 4(a). Deflection at mid span generated that was obtained can be expressed as: where "E' is modulus of elasticity of glulam and "I' is second moment inertia of the beam cross-section.
Using the same procedure as those applied to one CW block at mid span, if M0 at "L1" away from the left support, and x start from left support as shown in Figure 4 If M0 at "L2" away from the left support, and x start from left support as shown in Figure 4(c), the deflection was obtained can be expressed as: Due to the symmetrical positions of the CW blocks at the left and the right sides, it is only necessary to determine the deflection at L1 mm from the right support as that to the left support of the beam can be mirrored. Assuming the principle of superposition applies, the total deflection at a specific location along a pre-stressed beam with the prescribed locations of CW blocks may be expressed as;

Parabolic Distribution of Moment along the Beam Span
Similar to the linear distribution of moment, an assumption of parabolic moment distribution can also be made. The upward deflections due to one CW block at the mid-span and one CW block at either a1 or a2 from the left support can be determined based on the relationship between the pre-camber deflecttion and M0 sketched in Figure 5. The related precambers can be calculated using Equations. 8 to 11. If a beam is pre-stressed by five or seven CW blocks, the total pre-camber deflection can also be determined following the above procedure. Table 1 lists parameters used to calculate the predicted pre-camber deflections in this analytical approach, which is related to the CW block with a thickness (t) of 45 mm where the size of CW blocks is 45 mm in thickness, 35 mm in depth and 65 mm in width and the dimension of glulam is 105x105x1500 in mm. The iterative approach starts with assuming the stress on glulam which resulted from swelled CW. This relationship between pressure on glulam and moisture dependent swelling of CW was used to obtain an interactive strain on the glulam in the glulam-CW interface at the bottom of the CW block to produce reasonable prediction of the pre-camber deflection. The strain, shown in Table 1, was then used to work out the corresponding pressure on the glulam through Equation 1, which gives a pressure of 10 MPa for the 45 mm thick CW block. The interactive strains generated on the glulam corres-ponding to the thinner CW blocks were proportionnally reduced following the related measurements of the top extreme fiber strain no. 1 as shown in Figure  6. Therefore, the pressures generated on the glulam corresponding to CW blocks of 30 and 15 mm thick are 9.8 and 5.2 MPa, respectively [9].     Tables 2 and 3 for 3CW45. The notation 3CW45 stands for 3 CW blocks of 45 mm width.

Validation and Discussions
From Table 3 with the parabolic moment distribution assumption, the total deflection of 2.77 mm was predicted at mid span of 3CW45, which is 20.6% higher than that of 2.20 mm corresponding to the linear moment distribution assumption (Table 2). This value is slightly lower than the measured precamber of 2.96 mm [9] for a glulam beam reinforced with three 45 mm thick CW blocks, 3CW45. Figure 7 shows the comparison of the deflections from the analytical prediction and the measurement. The results indicated the analytical deflecttions related to the parabolic moment distribution correlate reasonably well to the measured ones, with less than 10% error. However, the theoretical predictions of the deflections corresponding to the linear moment distribution show relatively large discrepancies, i.e. up to 20% lower than the experimental measurement for the 45 mm thick CW reinforcement. This indicates that the linear moment distribution assumption may not be appropriate to describe the pre-stressed beams;

Conclusion
Based on the simple assumptions related to the parabolic moment distribution assumption in analytical approach, the pre-camber deflection predicted at midspan of the short beams with various reinforcing arrangements give a good agreement with the measured pre-camber from experimental results. This approach is not only applicable for glulam timber but also would be suitable for solid timber.