Traditional methods focus on the ultimate bending moment of glulam beams and the fracture failure of materials with defects, which usually depends on empirical parameters. There is no systematic theoretical method to predict the stiffness and shear distribution of glulam beams in elastic-plastic stage, and consequently, the failure of such glulam beams cannot be predicted effectively. To address these issues, an analytical method considering material nonlinearity was proposed for glulam beams, and the calculating equations of deflection and shear stress distribution for different failure modes were established. The proposed method was verified by experiments and numerical models under the corresponding conditions. Results showed that the theoretical calculations were in good agreement with experimental and numerical results, indicating that the equations proposed in this paper were reliable and accurate for such glulam beams with wood material in the elastic-plastic stage ignoring the influence of mechanic properties in radial and tangential directions of wood. Furthermore, the experimental results reported by the previous studies indicated that the method was applicable and could be used as a theoretical reference for predicting the failure of glulam beams.

Wood is a lightweight but high-strength material with excellent acoustic and thermal insulation performance. The popularity of wood, especially engineered timber, has opened new potential for full utilization of wood and resources conservation for its sustainable characteristic. Compared with solid wood, glulam comes in various sizes and displays high stability and excellent mechanical performance [

Scholars have adopted varying methods to predict the mechanical performance of glulam beams [

Since the glulam beams in elastic-plastic stage have undergone a certain degree of plastic deformation, it is inept for calculating the deflection and shear stress based on elastic theory [

The following assumptions are made for the theoretical analysis:

The strain distribution over the beam section satisfied the plane-section assumption [

The beam specimens under bending have equal bending elastic modulus as compressive and tensile elastic modulus [

The glulam beam is in elastic stage under initial loading. As the load increases, the beam reaches the maximum elastic bending moment, after which the beam entered the elastic-plastic stage. At this moment, the compressive zone is considered as an ideal elastic-plastic body and the tensile zone as a perfectly elastic body [

The bonding between the laminates is intact without relative slip [

As shown in

where: _{c} and _{t} are the strains in compressive and tensile zones of the glulam beam, respectively; _{ce} and _{cu} are the maximum elastic compressive strain and the ultimate compressive strain in compressive zone, respectively; _{te} is the maximum elastic tensile strain in the compressive zone of the beam; _{c} and _{t} are the stresses in compressive and tensile zones, respectively; _{ce} and _{te} are the stresses of the glulam beam after reaching the ultimate elastic compressive strain (_{ce}) and tensile strain (_{te}), respectively.

The stresses in compressive and tensile zones of the beam are assumed equal, that is _{c} = _{t}. In accordance with the mechanical property and the phenomenon of tests [_{te} = _{0}_{ce}.

where: _{c} and _{t} are the respective resultant forces of compressive and tensile stresses before the beam enters elastic-plastic stage; _{ce} and _{cp} are the individual forces of compressive stresses in elastic and plastic zones after the beam enters elastic-plastic stage; _{te} is the resulting force of the tensile stresses in elastic zone after the beam enters elastic-plastic stage; _{1} and _{2} are the heights of plastic and elastic zones under compression, respectively; _{3} is the height of the elastic zone under tension; _{1} is the height of any position on the section with the beam base as the benchmark; _{0} the proportional relationship between ultimate elastic tensile stress and ultimate elastic compressive stress.

According to the static equilibrium equation of the section,

From

where: _{u} is the maximum bending moment of the glulam beam.

_{e} and _{p} are bending moments of the beam upon reaching the ultimate compressive elastic strain and ultimate compressive strain respectively, that is the maximum elastic bending moment and the maximum plastic bending moment. _{0} is the dividing point where the beam achieved the bending moment _{e} when entering elastic-plastic stage. Since symmetry exists about the axis defined by _{0}/2, the discussion of force analysis for the case _{0}/2 ≤ _{0} is omitted. Thus, under the action of the mid-span concentrated load, the force analysis of the integral beam shows that:

The bending moment at any position along the length

The proportional relationship in ultimate state _{0} is extended to any state when the compressive zone enters plasticity, as shown in

where: _{3} and the height of elastic zone under compression _{2}.

According to the theory of mechanics of materials [

The shear stress in the element body is

The shear stress calculation is in two parts by the section height _{1}

When _{2} + _{3} ≤ _{1} ≤ _{N} = _{ce}(_{1}). In this case, the shear stress is

When 0 ≤ _{1} ≤ _{2} + _{3},

When

Three-point bending mode

_{0} is the dividing point at which the beam reaches _{e}

The load-deflection relationship is in the following two situations depending on the load

When

According to the diagram multiplication method, it can be known

When

Taken together,

Four-point bending mode

When

According to the diagram multiplication method, it can be known that

When

The section height-strain curve is divided into two stages, namely elastic stage and elastic-plastic stage. At the elastic stage, the height of the neutral axis remains constant. However, the neutral axis drops gradually at the elastic-plastic stage and the extent of the decrease is related to the yield strength _{te} in tensile zone.

Elastic stage

The slope of the curve is

At this moment, the neutral axis is located at

b) Elastic-plastic stage

The slope of the curve is

As the load increases, the neutral axis moves downwards from the

To verify the accuracy of the theoretical derivation, test of the beams was performed under the four-point bending test.

Larix olgensis Henry lumber was adopted as the raw material for glulam beam with a density of 0.67 g/cm^{3} and a moisture content of 11.23%. Lumber grade was determined as II_{a}, in accordance with the visually stress-graded method in the Chinese Standard for Design of Timber Structures (GB 50005-2017) [

Category | Average value |
---|---|

Compressive strength parallel to grain (MPa) | 45 |

Ultimate compressive strain parallel to grain (με) | 5073 |

Ultimate tensile strain parallel to grain (με) | 5119 |

Bending elastic modulus (MPa) | 12791 |

Three glulam beams were designed as rectangular sections, which constituted one specimen group named EXP. The outer sections of all specimens were of the exact dimensions: beam length _{0}

Specimens were manufactured by Anhui Golden Pastoral Wood Manufacturing Co., Ltd. at a controlled indoor temperature of 25°C–27°C and humidity of 53%–57%. The raw materials were firstly dried to a moisture content below 12% and then sawn, shaped, glued, and subjected to cold-press adhesion and maintenance. The glue used was one-component polyurethane, with glue spread of 200 g/m^{2}. The glue line thickness was less than 0.2 mm and the continuous glue line was left exposed for only less than 20 min. Then the glued lumbers were pressed under 1–1.5 MPa. After gluing and pressing, the glulam lumbers were stacked and stabilized for at least 7 days before cutting to the tested beams.

The simply supported glulam beam was tested as shown in

Loading procedure was designed in accordance with Standard Test Methods of Static Tests of Lumber in Structural Sizes (ASTM D198-15) [

All the specimens underwent brittle failure as shown in

Since the dispersion of the test results is small, the three specimens are sufficient to respond to the relevant laws. _{u} is obtained from _{u} is obtained from

Specimen group | _{u} (kN) |
_{u} (mm) |
_{e} (kN/m) |
|||
---|---|---|---|---|---|---|

Result | Difference (%) | Result | Difference (%) | Result | Difference (%) | |

Theory | 58.50 | — | 66.93 | — | 1030.11 | — |

EXP-A | 60.20 | 2.91 | 66 | −1.39 | 1030.59 | 0.05 |

EXP-B | 59.00 | 0.85 | 66 | −1.39 | 1021.75 | −0.81 |

EXP-C | 58.40 | −0.17 | 70.81 | 5.89 | 1051.10 | 2.04 |

Mean | 58.44 | −0.10 | 67.33 | 0.60 | 1021.51 | 0.83 |

EXP specimens have similar load-deflection curve shapes: At the initial loading stage, the curves show linear changes, and the slopes of the linear portions are identical. Then the curves show a nonlinear increase as the load increases, with the deflection rising progressively at an accelerated rate. On reaching the ultimate load, the specimen fails, and the deflection decreases suddenly. In accordance with

The comparison of the mid-span section height-strain curves at the five measuring points and the corresponding curves obtained from calculation using

Neutral axis offset height (mm) | The slope of the curve (mm/με) | |||||||
---|---|---|---|---|---|---|---|---|

EXP-A | EXP-B | EXP-C | Theory | EXP-A | EXP-B | EXP-C | Theory | |

1.8 | 0 | 0 | 0 | 0 | −1.82 | −2.23 | −2.11 | −2.27 |

10 | 0.46 | 0.34 | 0.32 | 0 | −11.90 | −12.65 | −12.59 | −12.61 |

18 | 0.54 | 0.45 | 0.41 | 0 | −22.11 | −22.71 | −22.59 | −22.70 |

26 | 0.62 | 0.57 | 0.49 | 0 | −31.79 | −32.73 | −32.69 | −32.78 |

34 | 0.86 | 0.89 | 0.82 | 0 | −41.59 | −43.20 | −43.00 | −42.87 |

42 | 1.28 | 1.11 | 1.00 | 0.27 | −53.07 | −54.93 | −54.90 | −53.51 |

50 | 3.65 | 3.02 | 2.94 | 2.08 | −64.02 | −70.83 | −71.09 | −68.09 |

58 | 6.87 | 6.24 | 6.11 | 5.60 | −77.40 | −90.68 | −91.25 | −89.54 |

In order to further verify the accuracy of the theoretical equations, the experimental results reported by previous studies [

Group | _{0} (mm) |
Material | ||||
---|---|---|---|---|---|---|

Reference [ |
1620 | 56 | 88 | 1.57 | Larix olgensis Henry | |

Reference [ |
900 | 80 | 130 | 1.63 | Spruce S10/MS10 | |

Reference [ |
5000 | 80 | 304 | 3.80 | 20f-E grade Douglas fir | |

Reference [ |
[ |
2286 | 60 | 127 | 2.12 | Larix olgensis Henry |

[ |
2790 | 60 | 155 | 2.58 | Larix olgensis Henry | |

[ |
3185 | 60 | 177 | 2.95 | Larix olgensis Henry | |

[ |
3654 | 60 | 203 | 3.38 | Larix olgensis Henry |

The bending performance parameters of the above specimens are obtained from theoretical calculation. The ultimate load _{u} and its corresponding maximum deflection _{u} are compared as shown in

3D finite element models (FEM) of the specimens were established using the simulation software ABAQUS. Depending on the tensile and compressive stress-strain relationship and the loading mode, innovative nonlinear elastic-plastic FE models were established and named BS1-BS2, respectively. Specimen groups BS1-2 were configured conform to the theoretical assumption (3): A dividing plane was set up at

Specimen group | BS1 | BS2 |
---|---|---|

Loading pattern | Three-point bending | Four-point bending |

Length |
2900 | 2900 |

Span _{0} (mm) |
2700 | 2700 |

Width |
100 | 100 |

Height |
150 | 150 |

Elastic modulus (MPa) | Shear modulus (MPa) | Poisson’s ratio | Elastic stage (MPa) | Plastic stage (MPa) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

_{L} |
_{R} |
_{T} |
_{LR} |
_{LT} |
_{RT} |
_{LT}, _{LR} |
_{RT} |
Yield stress | Plastic strain | Yield stress | Plastic strain | |

Upper section | 12791 | 1279 | 640 | 959 | 767 | 230 | 0.3 | 0.5 | 45 | 0 | 45 | 0.005 |

Lower section | 12791 | 1279 | 640 | 959 | 767 | 230 | 0.3 | 0.5 | 45 | 0 | — | — |

_{L} is elastic modulus presented from the test and

The boundary condition at the end of the beam model was set as simply support and the loading mode is two-point loading, as shown in

The finite element analysis (FEA) results of two specimen groups are compared against the theoretical values, as shown in

_{u} and the elastic stiffness _{e} obtained from numerical simulation and theoretical calculation in the two groups of specimens. Based on theoretical analysis and the actual situation, the ultimate tensile strength of specimens is fixed as 80 MPa. In the simulation, the dividing plane is set up at

Group | _{u} (kN) |
_{u} (mm) |
_{e} (kN/m) |
||||||
---|---|---|---|---|---|---|---|---|---|

Theory | FEA | Difference (%) | Theory | FEA | Difference (%) | Theory | FEA | Difference (%) | |

BS1 | 48.00 | 48.33 | 0.69 | 69.10 | 75.26 | 8.91 | 877.19 | 816.88 | −6.88 |

BS2 | 58.50 | 58.35 | −0.26 | 66.93 | 71.06 | 6.17 | 1030.11 | 965.58 | −6.26 |

After the beams entered elastic-plastic stage, the shear stress can be calculated from

This paper presents theoretical and numerical study of stiffness and shear stress distribution of glulam beams in elastic-plastic stage, and the main conclusions are as follows:

Combined with a suitable stress-strain relationship model of wood, the bending failure modes of the glulam beams in elastic-plastic stage were divided into two situations. The deflection and shear stress calculation equations of different failure modes were established, and the movement of the neutral axis was estimated, which can predict the failure of glulam beams in elastic-plastic stage closely ignoring the influence of mechanic properties in radial and tangential directions.

Finite element models were established for each of the two failure modes, comparing the load-deflection relationship and shear stress distribution. Results show that the calculations obtained from the proposed equations have a good agreement with the numerical results.

Four-point bending tests were conducted investigating the stiffness and shear stress distribution of glulam beams in elastic-plastic stage under assumed conditions, and main parameters corresponding to the failure modes were calculated by applying the theoretical equations. Theoretical results are in good agreement with experimental results, indicating that the equations have high precision and conform to actual situations. Furthermore, the theoretical analysis of the corresponding conditions for different reference tests is carried out and compared with the experimental results. It can be concluded that the proposed method can accurately predict the failure of glulam beams in elastic-plastic stage, and are suitable for specimens with different section sizes and materials.