Standard Test Methods for Elastic and Shear Properties

*Ahmed Mohamed*

### **Abstract**

A fundamental requirement for efficient use of timber-based composite structures like glulam beams is an accurate knowledge of their mechanical behavior and the material properties characterizing that behavior. Determining the elastic properties, such as the modulus of elasticity and shear modulus, for glulam beams requires careful experimentation and can be challengeable due to the anisotropic nature of wood. Determining these properties is not as simple and straightforward as in isotropic materials. Shear tests, such as torsion and shear field and compression loading tests, are commonly employed to determine the shear modulus of glulam beams. To determine the modulus of elasticity, experimental methods such as bending and compression tests are commonly used. In this chapter, we will discuss the experimental methods commonly used to determine the modulus of elasticity and shear modulus, for timber-based composite structures. These properties are crucial for understanding the structural behavior and design of these materials. This chapter describes the commonly used methods, bending tests, torsion tests, and compression loading tests, in determining their values. To obtain accurate and reliable results, it is essential to conduct these experimental methods following established standards and carefully controlling the test conditions, specimen preparation, loading configurations, and measurement techniques.

**Keywords:** modulus of elasticity, shear modulus, timer-based composite, glulam beams, experimental methods

## **1. Introduction**

The mechanical properties of wood composites rely on different parameters, including wood species, the type of adhesives used to glue the wood pieces, and the geometry of the wood pieces [1]. Mechanical properties play a vital role in evaluating the characteristics of wood-based composites. The elastic and strength properties provide important information for material selection, design, and establishing product specifications [1]. Elastic properties relate the resistance of a material to deformation under an applied load and the ability of the material to recover its original dimensions when the load is removed [1]. Known as elastic constants, the elastic properties include the basic and fundamental properties such as shear modulus (G) and modulus of elasticity (MOE).

The shear modulus (G), known as modulus of rigidity, is a material constant which relates shear stress to shear strain and indicates the resistance to deformation caused by shear stresses [1]. It is a fundamental mechanical property of wood that is used in the design of timber and engineered wood products. The shear modulus is critical when designing for lateral torsional buckling of the timber beams [2]. G is also significant in designing serviceability of wood-joist floors [3] and is an important input for setting up analytical and finite element models [4]. The modulus of elasticity (MOE), also known as the elastic modulus or Young's modulus, quantifies how much a material deforms under an applied stress within the elastic range. MOE can be calculated from the slope of the linear portion of the stress-strain graph obtained during a tensile or compressive test.

The European standard CEN, and the American Standard of Testing Methods (ASTM) have developed several test methods for the determination of shear and elastic properties of timber. These methods provide guidelines for testing timber in small clear wood specimens as well as in structural sizes under flexural and torsion loadings. The European standard CEN [5] provides test methods in laboratory for determining some physical and mechanical properties of in structural sizes timber samples. This standard specifically provides testing guidelines to determine the characteristic values of mechanical properties, including shear modulus and modulus of elasticity, of structural timber and glued laminated timber. The procedures of the test methods recommended by [5] are shown in the sections below.

### **2. Testing methods for shear modulus**

The EN408 standard has proposed the torsion and shear field test methods to determine shear modulus of timber in structural sizes.

#### **2.1 Torsion test method**

The torsion test method is a mechanical testing technique used to evaluate shear modulus (G) of materials, including timber. The proposed version of CEN [5] standards provide guidelines for conducting the torsion test on timber. This method involves subjecting a test specimen to a torque along its longitudinal axis, which is achieved by inducing a relative rotation between the supports where the specimen is clamped. The test specimen is clamped at the supports, that are positioned at a distance more than 16 times the cross-sectional depth of the specimen. To measure the response of the specimen, both torque and relative rotational displacements are measured at two specific sections, labeled as Section 1 and Section 2 in **Figure 1**. These sections are located within the free testing length, denoted as *l*1. The distance between the supports and these cross-sections, denoted as *l*2, should be two to three times the depth of the test specimen. The centers of the supports are aligned in a straight line. This alignment ensures that clamping the test specimen does not cause any deformation that could influence the torsion results. The torque required for the torsional loading can be applied by rotating one or both supports.

The following equation is used to calculate shear modulus (*GTor*)

$$\mathbf{G}\_{Tor} = \frac{K\_{Tor} l\_1}{\eta \ h \, b^3} \tag{1}$$

*Standard Test Methods for Elastic and Shear Properties DOI: http://dx.doi.org/10.5772/intechopen.112375*

**Figure 1.**

*Example of test setup with requirements of specific gauge locations [5].*

**Figure 2.**

*Torque versus relative rotation [5].*


#### **Table 1.**

*Shape factor values for torsion test [5].*

The torque stiffness, *KTor*, can be determined using a linear regression analysis as shown in **Figure 2**. Linear regression analysis can be conducted on the linear elastic portion of the graph of torque and relative twist of a specimen within its proportional limits. Where:

*l*<sup>1</sup> is the gauge length, *h* is the depth of the specimen, *b* is the width of the specimen,

*η* is the shape factor that is dependent on the depth to width ratio of the specimen and can be obtained from **Table 1**.

#### **2.2 Shear filed test method**

The shear field test method is a recommended approach for determining the shear modulus of structural-size timber beams. It involves conducting standardized four-point bending tests on the timber beams according to the CEN [5] standard. The objective of this test is to measure the shear distortion within the areas of constant transverse force. Precise measuring instruments are placed on both sides of the constant transverse force, opposite each side of the beam. These instruments accurately measure length displacements across the shear fields of the sample during testing.

The setup of the shear field test is shown in **Figure 3**. According to the test method, the test specimen should be symmetrically loaded in bending at two points over a span of 18 times its cross-sectionals' depth. However, if these requirements cannot be met exactly, the distance between the load points and the supports can be changed by a maximum of 1.5 times the specimen depth. The span and length of the test specimen can also be changed by a maximum of three times its depth while maintaining the symmetry of the test. To minimize local indentation, small steel plates are recommended to be inserted between the specimen and the loading heads or supports. The length of these plates should not exceed half the depth of the specimen.

The applied load during the test should be applied at a constant rate. In the middle of the area under constant shear stress, a square is marked on both side faces of the specimen. These squares are placed symmetrically with respect to the height of the test piece. A device that measures the change in the diagonals of the square is fixed to the test specimen at the square corners, as shown in **Figure 4**. The shear deformation

**Figure 3.** *Test SETUP for shear field test [5].*

**Figure 4.** *Example of the shear field test apparatus fixed on one of both sides [5].*

*Standard Test Methods for Elastic and Shear Properties DOI: http://dx.doi.org/10.5772/intechopen.112375*

**Figure 5.** *Deformation of the square with diagonals [5].*

can be determined from these measurements by calculating the mean value of the summation of the absolute readings of both diagonals at each side face of the crosssection, as shown in **Figure 5**. This measurement provides the necessary data to calculate the shear modulus of the timber beam using the shear field test method.

The shear modulus *Gtor*,*<sup>s</sup>* is given by the equation:

$$G\_{tor, \varepsilon} = a \frac{h\_0}{bh} \frac{(V\_{\varepsilon, 2} - V\_{\varepsilon, 1})}{(w\_2 - w\_1)} \tag{2}$$

Where:

$$\alpha = \frac{3}{2} - \frac{h\_0^2}{4h^2} \tag{3}$$

$$w\_i = \frac{(|w\_{i,1}| + |w\_{i,2}|)}{2} \\ \\xi = 1,2 \tag{4}$$

*wi* is the mean deformation of both diagonals on opposite side faces of the beam for a given shear load *Vs*,*<sup>i</sup>*, in millimeters.

*Vs*,2 � *Vs*,1 represents the shear load increment, which is the difference between the shear load values, in Newton.

## **3. Test methods for determining the modulus of elasticity (MOE)**

There are static and dynamic test methods that can be used to determine the MOE of materials. The static methods are used more commonly and are based on the relationship between the load and the deformation of the sample, while the dynamic methods are based on the relationship between the frequency of vibration of a sample and its MOE.

The European standard CEN [5] provides various test methods to obtain the modulus of elasticity values of structural timber and glued laminated timber. These methods are used to measure the modulus of elasticity in different conditions such as bending, tension parallel and perpendicular to the grain, and compression parallel and perpendicular to the grain. The following sections provides review of some of these test methods.

#### **3.1 Four-point bending test method**

The CEN edition [6] introduced a global bending modulus of elasticity and renamed the existing test as the local MOE. The four-point bending test method can be used to determine both the local and global MOE [5]. In the four-point bending test, a full-size specimen is loaded symmetrically in bending at two points over a span that is 18 times the depth of the section as provided in **Figure 6**. If it is not possible to meet this exact span requirement, the distance between the load points and the supports can be changed by up to 1.5 times the specimen depth. Additionally, the span and length of the test specimen can be changed by up to three times its depth, as long as the symmetry of the test is maintained. To minimize local indentation, small steel plates can be inserted between the specimen and the loading heads or supports. The length of these plates should not exceed one-half of the depth of the specimen. The test method also suggests that the test specimen should be supported simply, and lateral restraint should be provided to prevent lateral torsional buckling. This allows the specimen to deflect without significant frictional resistance. The applied load during the test should be at a constant rate, as recommended by the standard. The test setup for determining the local MOE in bending is provided in **Figure 6**.

To determine the local modulus of elasticity (MOE) in bending, the deformation is measured at the center of a central gauge length, which is defined as 5 times the depth of the section. The deformation and load are observed at two different times during the test, denoted as (*w*1, *F*1) and (*w*2, *F*2) in **Figure 7**. It is crucial to ensure that these two measurements are made within the proportionality limit of the beam. The recommended approach is to take the deformation as the average of measurements on both side faces at the neutral axis.

The equation to calculate the local modulus of elasticity, *Em*,*<sup>l</sup>* of structural size specimen in the four-point bending test is

$$E\_{m,l} = \frac{al\_1^2(F\_2 - F\_1)}{16I(w\_2 - w\_1)}\tag{5}$$

Where:

*Em,l*: Local modulus of elasticity in the four-point bending test, in Newtons per square millimeter *<sup>N</sup>=mm*<sup>2</sup> ð Þ.

**Figure 6.** *Typical test configuration for measuring local modulus of elasticity in bending [5].* *Standard Test Methods for Elastic and Shear Properties DOI: http://dx.doi.org/10.5772/intechopen.112375*

#### **Figure 7.**

*Load-deformation graph within the elastic range of deformation [5].*

*F*<sup>1</sup> and *F*2: Loads observed at two different times during the test, measured in Newton (N),

*w*<sup>1</sup> and *w*2: Average deformations measured on both side faces at the neutral axis at the corresponding times, measured in millimeters (mm).

*I*: Moment of inertia of the beam's cross-sectional shape, in (mm<sup>4</sup> ).

*l*1: Gauge length for the determination of modulus of elasticity, measured in millimeters (mm).

a: Distance between a loading position and the nearest support, in millimeters (mm).

The global MOE in bending is evaluated using the same test configuration as the one used for measuring the local MOE. The test arrangement is illustrated in **Figure 8**.

**Figure 8.**

*Typical test configuration for measuring the global MOE in bending [5].*

To determine the global MOE, the deformation is measured at two locations:

1.At the center of the span.

2.From the center of the tension or compression edge.

Additionally, when the deformation is measured at the neutral axis, it is recommended to take the average of measurements on both side faces of the test specimen.

The equation to calculate the global modulus of elasticity, *Em*,*<sup>g</sup>* of structural size specimen in the four-point bending test is.

$$E\_{m, \text{g}} = \frac{3al^2 - 4a^3}{2bh^3 \left(2\frac{w\_2 - w\_1}{F\_2 - F\_1} - \frac{6a}{5Gbh}\right)}\tag{6}$$

Where:

*l*: The span in bending, measured in millimeters (mm).

*b*: The width or the smaller dimension of the cross-section, also measured in millimeters (mm).

*h*: The depth or the larger dimension of the cross-section, measured in millimeters (mm).

*G*: The shear modulus, which is taken as *G* = 650 N*=*mm2.

#### **3.2 Compression test methods**

The compression test is one of the most used methods for determining the modulus of elasticity and other elastic and strength properties of materials. The specific characteristics of the compression tests for measuring MOE parallel to grain and MOE perpendicular to grain in structural and glued-laminated timber, as outlined in the CEN [5] standard.

1.MOE Parallel to Grain Test:

This test is widely used for determining the modulus of elasticity of timber parallel to the grain direction. The deformation, or strain, of the specimen is measured over a central gauge length.

2.MOE Perpendicular to Grain Test:

The perpendicular to grain test is specifically conducted to assess the modulus of elasticity of timber when compressed perpendicular to the grain. In this test, the MOE is reported at the proportional limit.

## *3.2.1 Test specimen*

According to CEN [5] standard, the test specimen requirements for the parallel to grain compression tests are as follows:

The test specimen should have a full cross-section.

The length of the specimen should be six times the smaller cross-sectional dimension.

The two contact surfaces of the specimen must be accurately prepared to ensure they are parallel to each other and aligned perpendicular to the longitudinal axis of the specimen.

**Table 2** provides the dimensions of the test specimen for structural and glued laminated timber samples in compression perpendicular to grain. These samples are shown in **Figures 9** and **10**.

#### *3.2.2 Conditioning test specimen*

The conditioning of the test specimens for structural and glued-laminated timber should be carried out in a controlled environment with specific temperature and relative humidity conditions. The test specimens should be conditioned at a temperature of 20 2 degrees Celsius (°C). The relative humidity during conditioning should be set at 65 5%. The specimens should be conditioned until they reach a state of constant mass. This is achieved when the results of two successive weightings do not differ more than 0.1% of the mass of the test specimen.


*Where:*

b*: The width of the cross-section or the smaller dimension of the cross-section. Measure in mm.*

h*: The height of the test specimen perpendicular to the grain, measured in mm.*

l*: The length of the test piece between the testing machine grips in compression and tension, measured in mm.*

#### **Table 2.**

*Dimension of structural timber or glued laminated timber test pieces [5].*

**Figure 9.**

*A typical structural timber test specimen [5].*

#### **Figure 10.**

*A typical glued laminated timber test specimen [5].*

#### *3.2.3 Loading procedure*

The CEN (European Committee for Standardization) test configuration for measuring the modulus of elasticity (MOE) in compression is outlined in **Figure 11**. The test specimen needs to be positioned vertically between the platens of the testing machine. The test specimen needs to be centrally loaded to ensure that the compressive force is applied uniformly. This is achieved by using spherically seated loadingheads. These loading-heads allow the application of compressive loads without causing any bending or uneven loading. After pre-loading the test specimen, the spherical loading head should be securely fastened to prevent slipping. The test specimen should be loaded uniformly with a constant rate of loading throughout the test. The applied load during the test should be measured with an accuracy of 1%.

**Figure 11.** *Typical compression test configuration in accordance with [5].*

To minimize the effects of distortion during the MOE compression test, the use of displacement sensors, such as extensometers, is recommended. Two displacement sensors (extensometers) should be positioned symmetrically in the middle of the two opposite surfaces of the test sample. The readings from the two displacement sensors should be averaged to obtain a more reliable and representative result. The displacement sensors should be positioned at least at a distance from the contact surfaces of one third of the largest cross-sectional size of the test sample. This distance helps to negate the grip effect, which refers to the influence of the grips or clamps used to hold the specimen during testing.

In the CEN test configuration, the deformation in compression parallel to the grain is measured over a central gauge length. This gauge length should be four times the smaller cross-sectional dimension of the test specimen. The deformation in compression perpendicular to the grain is measured over a gauge length denoted as *h*0. The gauge length *h*<sup>0</sup> is approximately 0.6 times the height of the test specimen, h. The gauge length *h*<sup>0</sup> must be located centrally in the height of the test specimen. It should be positioned at a distance of at least b/3 from the loaded ends of the test specimen. Here, b represents the width or thickness of the test specimen, depending on its orientation.

#### *3.2.4 Calculation of MOE in compression parallel to the grain*

The equation for calculating the modulus of elasticity parallel to the grain in compression *Ec*,0 according to [5] standard, is as follows:

$$E\_{c,0} = \frac{l\_1 (F\_2 - F\_1)}{A(W\_2 - W\_1)}\tag{7}$$

Where:

*l*<sup>1</sup> refers to the span in bending, measured in ð Þ *mm* ,

*F*<sup>2</sup> – *F*<sup>1</sup> represents the increment of load on the straight-line portion of the load-deformation curve. (Measured in Newton),

*W*<sup>2</sup> – *W*<sup>1</sup> indicates the increment of deformation corresponding to *F*<sup>2</sup> – *F*<sup>1</sup> (measured in millimeters),

A is the initial crossed section of the sample.

#### *3.2.5 Calculation of MOE in compression perpendicular to the grain*

The modulus of elasticity perpendicular to the grain in compression is assessed with the same test configuration for measuring MOE parallel to the grain. According to EN 408 standard, *Ec*,90 may be determined using the iterative process as follows:


**Figure 12.** *Load-deformation graph for compression perpendicular to the grain according to [5].*


The equation for calculating the modulus of elasticity perpendicular to the grain in compression, *Ec*,90, according to the EN 408:2012 standard, is as follows:

$$E\_{c,90} = \frac{(F\_{40} - F\_{10})h\_0}{(w\_{40} - w\_{10})\,\text{bl}}\tag{8}$$

Where:

*F*40: Load at 0.4 of *Ec*,90, max (N). *F*10: Load at 0.1 of *Ec*,90, max (N). *w*40: Deformations at *F*<sup>40</sup> (mm). *w*10: Deformations at *F*<sup>10</sup> (mm). *h*0: Gauge length (mm). *b:* Width of the specimens (mm). *l*: Length of the specimens (mm).
