# Shear Deformations of Framework Structures in Timber Construction

### Technical Article

In current literature, the formulas to determine internal forces and deformations manually are usually specified without considering the shear deformation. Especially in timber construction, the deformations resulting from shear force are often underestimated.

Symbols used:

h ... cross-section height

L ... span

E ... modulus of elasticity

G ... shear modulus

κ ... shear correction factor

A ... cross-section area

w ... deformation

The low shear modulus respectively the low ratio of G/E are governing here. This is given as 1/16 for softwood due to the anisotropy according to [1]. Isotropic materials provide a much larger ratio. For steel, for example, the result is a G/E ratio of 1/2.6.

#### Beam Theory

Whereas in the classical Bernoulli beam theory it is assumed that the cross-section of a member remains perpendicular to the member axis when deformed, the shear sliding is considered for the Timoshenko beam theory (flexible beam). As a result, the cross-section of a member no longer remains perpendicular to the member axis when deformed (see Figure 01). By assuming that the cross-section remains planar, it results in a uniform shear stress distribution along the beam height. However, since the distribution is parabolic, a shear correction factor is taken into account for the determination of shear areas. This is 5/6 for a rectangular cross-section. Thus, the shear stiffness of a rectangular member results in:

${\mathrm{GA}}^{*}=\mathrm{\kappa}\xb7\mathrm{G}\xb7\mathrm{A}=\frac{5}{6}\xb7\mathrm{G}\xb7\mathrm{A}$

Image 01 - Comparison of the deformations of a Bernoulli beam and a Timoshenko beam

#### Standardization

The standard gives no indication as to whether or from which criterion shear deformations for members have to be considered. Threfore, the structural engineer has to take the decisions.

#### Example:

A simple example will demonstrate the influence of shear deformations. We consider a pinned single-span beam designed as a downstand beam. The details are shown in Figure 02.

Image 02 - Example of a single -span beam

First, we only want to determine the deformation from the moment's curvature. For the shown system, the characteristic deformation is:

${\mathrm{w}}_{\mathrm{M}}=\frac{{\mathrm{q}}_{\mathrm{z}}\xb7{\mathrm{L}}^{4}}{384\xb7\mathrm{E}\xb7{\mathrm{I}}_{\mathrm{y}}}=3.09\mathrm{mm}$

The component of the shear deformation can be derived, for example, with the working set or simplified from the investigations from [2] or [3]. For a pinned single-span beam, this results in:

${\mathrm{w}}_{\mathrm{V}}=\frac{{\mathrm{q}}_{\mathrm{z}}\xb7{\mathrm{L}}^{2}}{8\xb7\mathrm{\kappa}\xb7\mathrm{G}\xb7\mathrm{A}}=1.03\mathrm{mm}$

The total deformation is thus:

${\mathrm{w}}_{\mathrm{tot}}={\mathrm{w}}_{\mathrm{M}}{\mathrm{w}}_{\mathrm{V}}=3.091.03=4.12\mathrm{mm}$

In this example, the shear deformation ratio is already 25% of the total deformation. Figure 03 graphically shows the individual deformation components.

Image 03 - Superposition of deformation components

#### Slenderness

Decisive for the shear deformation component is the slenderness of a member. While shear deformations are negligible for slender members with a large L/h ratio, they have a significant influence on compact members with a small L/h ratio.

Figure 04 shows the influence of the shear deformation on the total deformation in a diagram. For hinged single-span beams with a rectangular cross-section, the shear deformation is predominant up to an L/h ratio of 4. Only then does the ratio predominate from the moment curvature. From an L/h ratio of 12, the influence of the shear deformation is only 10% of the total deformation.

#### Statically Indeterminate Systems

In statically indeterminate systems, shear deformation has a greater influence than in statically determined systems. In this case, the deformations due to shear force have an influence on the bending moment and thus also on the bending deformations. This redistribution may, for example, have a positive effect on supporting moments (see Figure 05).

Image 05 - Moment redistribution by shear deformation

#### Shear Deformations in RFEM and RSTAB

In RFEM and RSTAB, the shear deformations for members are automatically taken into account. For control calculations, however, they can also be neglected with the function shown in Figure 06. If the check box is selected, the shear deformations are considered. If it is deactivated, only the deformation components from the bending moment are considered.

Image 06 - Consideration of shear deformation in RFEM and RSTAB

#### Summary

In many practical situations, shear deformations can be neglected because they do not contribute significantly to the total deformation. For compact members, the shear deformation must not be neglected any more. For RFEM and RSTAB, the shear deformation is always considered by default; for manual calculations, it is necessary to use tools (see [2] or [3]).

#### Author

#### Dipl.-Ing. (FH) Gerhard Rehm

Product Engineering & Customer Support

Mr. Rehm is responsible for the development of products for timber structures, and provides technical support for customers.

#### Keywords

Shear deformation Shear area Shear correction factor Bernoulli Timoshenko

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