You are required to write a report about the following topic.

The report must also include the following:

- The reasons behind torsional buckling of beams

- Examples of constructions where the torsional buckling of beams has occurred

## Properties of Beams and Mechanical Failure

Beams are horizontal structural members which primarily carry vertical loads. When a load is applied on a beam, the upper section of the beam experiences compression while the bottom section experiences tension. Concrete is weak in tension but good in compression. For this reason, concrete beams have more steel reinforcement at the bottom than at the top. Torsion is defined as the twisting of a structural member due to the applied torque. Torsion is represented in N/m^{2}. Unlike the axial loads, torque creates a non-uniform stress distribution in structural members. For circular sections, twisting occurs along the axis and the cross section remains circular. However, when torque is applied on non-circular members, the twisting is accompanied by warping hence the members do not remain plane.

Buckling is defined failure of structural members due to an applied load which is less than the design load. Failure of members is not members is in no way related to the strength of the members (Bansal, 2010). Buckling occurs in members under compression. When a load is applied on a member along its axis, it tends to be displaced along its axis. There is a sudden sideways displacement of a member from the axis. As loading is increased, the displacement increases and it ultimately leads to buckling. The member may collapse completely or remain in the same position and continue supporting loads. Buckling differs from bending which occurs when a load is applied along the length of the beam axis. Bending normally occurs in beams and slabs while buckling is common in struts and columns.

From the definitions above, torsional buckling can be defined as the failure of structural members due to longitudinal twisting of the member (Gupta, 2014). In beams, this type of buckling usually occurs in I-section beams. Doubly symmetrical sections such as W, and H sections are not prone to this type of buckling. Torsional buckling in such kind of sections can only occur if the torsional unbraced lengths are more than the flexural unbraced lengths (Gupta, 2014).

Buckling is defined failure of structural members due to an applied load which is less than the design load. Failure of members is not members is in no way related to the strength of the members. Buckling occurs in members under compression. When a load is applied on a member along its axis, it tends to be displaced along its axis. There is a sudden sideways displacement of a member from the axis. As loading is increased, the displacement increases and it ultimately leads to buckling. The member may collapse completely or remain in the same position and continue supporting loads.

Slenderness ratio is used to determine if a member is susceptible to buckling. Slenderness ratio is defined as the ratio between the effective length of the structural member and the radius of gyration of the member’s cross section.

Where, S = Slenderness ratio

= effective length

r = radius of gyration

I = Moment of Inertia

A = Cross sectional Area

A member with a high slenderness ratio is prone to buckling while that with small slenderness ratio is less susceptible for buckling.

## Torsional Buckling

Steel columns are considered to be linear members because the cross-sectional dimensions are smaller compared to their lengths. This implies that columns are more susceptible to fail due to buckling rather than yielding. There are two forms of failure for members under compression.

- Local instability- This occurs where parts of the members are relatively thin resulting in localized buckling. The sections can be determined to be slender or not based on the formulas provided above.
- Overall Instability- This type of phenomenon occurs when the whole member experiences buckling throughout its length leading to bending of the section from its axis. This type of buckling can occur in various ways:

- Flexural buckling- deformation occurs through bending about the weaker axis. Flexural buckling in steel can be either elastic or inelastic depending on the stress levels experienced before buckling compared with the steel stress limit(Kindmann & Kraus, 2012).
- Torsional buckling- In this type of buckling, the member twists along its axis. Common in I-section beams. Doubly symmetrical sections such as W, and H sections are not prone to this type of buckling. Torsional buckling in such kind of sections can only occur if the torsional unbraced lengths are more than the flexural unbraced lengths(Gupta, 2014).
- Flexural-torsional buckling- This occurs when there is simultaneous occurrence of both flexural and torsional buckling.

The orientation of a beam is reported according to the coordinate system i.e. x-axis, y-axis and z-axis. In most structures, the span usually runs along the x-axis while the cross section is in the x-axis and z-axis. A beam can deflect about either the y or z axes. The y-axis is referred to as the major axis while z-axis is the minor axis. If a beam subjected to a load so that it bends about the y axis and deflects in the z axis, it is referred to as the normal bending. However, if a beam is loaded in such a way that it bends about the z axis hence deflecting in the y axis, it is referred to as lateral bending (Carrera, Giunta, & Petrolo, 2011).

There exists several degrees of freedom at the ends, some of which must be restrained with an aim of achieving equilibrium. In most cases, for a beam with two supports, there is a usually a restraint for translation in the y and z axes. Rotation about the major axis is prevented so as to prevent the beam from rotating about its axis. Depending on the end support condition, a beam with only end supports may rotate about the y and z axes and deflect on either axis depending on the direction the force is applied.

The shape of the cross section of the member is an important factor in buckling. I-Beams are susceptible to lateral torsional buckling than other shapes such a T-beams.

The center of gravity of a body is defined as that point where the whole weight of the body is concentrated (Rajput, 2015). This is the point at which the neutral axis passes through. Stress along the neutral axis is considered to be zero. However, any movement above or below the neutral axis implies an increase in either tensile or compressive stress. It is also assumed that the neutral axis remains plain even after bending has occurred. The center of gravity of sections with double symmetry such as I-sections is easy to determine geometrically as shown below.

Shear center is defined as the point at which the shear force acts so as to only cause in-plane bending (Megson, 2013). The shear center of sections with double symmetry such as I-sections is easy to determine geometrically as shown below. For this type of sections, the COG coincides with the shear center.

The COT of a body is the point along which the twisting axis will pass through. Twisting will happen about this point (Barretta, 2012). Usually, though with different definitions, both the shear center and center of twist always coincide. Therefore, for a section with double symmetry, the center of twist, center of gravity and shear center will always coincide. In our case, this will apply since we shall be considering an I-section with double symmetry i.e. symmetrical in both the x and y axis.

## Slenderness Ratio and Buckling

This is the point at which the load acts on a cross section. The point at which the load is applied on the beam determines its capacity to resist the lateral torsional buckling. Of importance to note is that the point of application of load doesn’t necessarily coincide with the center of the shear.

When torque is applied to a beam with a circular cross section, elements within the body of the member will move to a new point but still remain in the same position. However, this is not the case for no-circular symmetrical sections. Elements these types of members do not remain in the same plane due to the twisting moments (Atsuta & Chen, 2008). The twisting moments force the member to bend hence changing the positions of the sectional elements. This phenomenon is referred to as warping. The bending experienced in the beam flanges is similar to the behaviour of beams experiencing bending.

When a beam is subjected to compression forces, some flanges experiences compression while others are under tension. The compression flange pulls away from the axis while the tension flange ties to push the member back to the axis. However, due to the compression force being greater than the tensile force, the steel beam deflects away from the axis (Andrade, Camotim, & Providencia, 2006). The tensile forces determine the degree of deflection to be experienced by a member as they act as the restraints to deflection

Apart from deflection, the forces within the flanges of the beam cause the member to twist about the horizontal axis. The steel section offers resistance to the twisting. This resistance is called

torsional stiffness and is normally offered by the flanges. The degree of the resistance is determined by the flange thickness. For sections of same depth, the thicker the flange the more the bending strength and vice versa.

This type of buckling may occur in unrestrained beams. A beam is said to be unrestrained if the flange under compression is free to move about its axis and rotate. When a compression force is applied on a beam resulting in both lateral bending and twisting, lateral torsional buckling occurs (Brettle, 2006).

Consider a beam whose length runs parallel to the x axis. An increasing load is applied on the member. The beam will buckle at a load less than the capacity of the section if the beam under consideration is slender. This buckling involves two deformations, that is lateral deformation and twisting (Horacek, Melcher, Pesek, & Brodniansky, 2016). The result of both occurring in a member is lateral torsional buckling. Longer beams are more susceptible to lateral torsional buckling as compared to shorter ones. Furthermore, for beam of the same length, the buckling loads will differ depending on the size of the cross section. Slender sections of beams of similar lengths buckle at lesser loads compared to others of a larger cross sectional area.

The distance between the point of applied force and the shear center determine the extent to which a compression member will be prone to torsional buckling. A structural compression member is more prone to lateral torsional buckling if the point of application of the force is above the shear center. The susceptibility to this phenomenon is reduced in cases where the point of load application is through the center of the shear. A further reduction in the probability of the occurrence of the lateral torsional buckling is experienced when the point of load application is below the shear center of the section.

## Cross-Sectional Shape and Buckling

The end supports of a compression affect the susceptibility to buckling. Several end support methods are used in structural engineering and include; both ends fixed, two ends will roller supper, pin support at both ends. These end conditions determine the extent of buckling and twisting. In instances where end support offers a high restraint, the member experiences high buckling moment and hence becomes more susceptible to buckling and twisting (Zhang, Liu, & Xi, 2018). An example of this type of support that offers high restraint at the ends is the fixed-fixed end condition. In contrast, members whose ends have a low restraint at the ends are less prone to buckling and twisting because the buckling moment is low. This is the case for simply supported beams with both ends having roller supports.

The shape of the resulting bending moment after a load is applied on the compression member determines if it is prone to torsional buckling. If the bending moment distribution is uniform along the length of the member, a greater buckling resistance is developed. However, if the same member is subjected to a non-uniform bending moment distribution, a lower buckling resistance is developed resulting into the section being more prone to torsional buckling.

If the flanges under compressive stress are restrained from bending as shown below, then torsional buckling will be prevented (McCann, Gardner, & Wadee, 2013). For steel members, steel restraints can be installed across the major axis of the beam. The space separation between the consecutive lateral restraints should be small enough to ensure that buckling is prevented.

Steel beam sections are manufactured in different cross sectional shapes such as circular, rectangular, H, T and I sections. Lateral torsional buckling only occurs when the resultant bending is about the major axis. If the stiffness in the weaker axis is high enough, buckling may be prevented. I-Cross sectional beams are more susceptible to lateral torsional buckling that other cross sectional shapes.

Conclusion

Lateral torsional buckling only occurs in I section steel beams. It is mainly caused by insufficient bracing of the member. When an unbraced I-beam is subjected to a sufficient loading, it undergoes bending while the compression flanges behave like a column resulting into buckling. The flanges experiencing tensile stress tends to maintain is original location while the flange under compression tends to move laterally due to the absence of bracing. The result of this deformations and movements is torsional buckling. Lateral torsional buckling is common in slender beams. These are beams which have considerable length compared to the cross sectional area.

References

Andrade, A., Camotim, D., & Providencia, e. P. (2006). On the evaluation of elastic critical moments in doubly and singly symmetric I-section cantilevers. Journal of Constructional Steel Research, 894-908.

Atsuta, T., & Chen, W. F. (2008). Theory of Beam-Columns - Volume 2 Space,Behaviour and Design. Fort Launderdale: J. Ross Publishing.

Bansal, R. K. (2010). A Textbook Strength of Materials. New Delhi: Laxmi Publications (P) Ltd.

Barretta, R. (2012). On the relative position of twist and shear centres in the orthotropic and fiberwise homogeneous Saint–Venant beam theory. International Journal of Solids and Structures, 3038-3046.

Brettle, M. (2006). Lateral torsional buckling and slenderness. New Steel Conmstruction,30-34.

Carrera, E., Giunta, G., & Petrolo, M. (2011). Beam Structures: Classical and Advanced Theories. Chichester: John Wiley & Sons, Ltd.

Gupta, R. S. (2014). Principles of Structural Design: Wood, Steel, and Concrete. (2nd ed.). Boca Ranton: CRC Press.

Horacek, M., Melcher, J., Pesek, O., & Brodniansky, J. (2016). Focusing on Problem of Lateral Torsional Buckling of Beams with Web Holes. Procedia Engineering, 549-555.

Kindmann, R., & Kraus, M. (2012). Steel Structures: Design using FEM. Berlin: Ernst & Sohn: A Wiley Company.

McCann, F., Gardner, L., & Wadee, M. A. (2013). Design of steel beams with discrete lateral restraints. Journal of Constructional Steel Research, 82-90.

Megson, T. G. (2013). Aircraft Structures for Engineering Students. Kidlington: Elsevier Ltd.

Rajput, R. K. (2015). A Textbook of Applied Mechanics (3rd ed.). New Delhi: Laxmi Publications (P) LTD.

Zhang, W., Liu, F., & Xi, F. (2018). Lateral Torsional Buckling of Steel Beams under Transverse Impact Loading. Shock and Vibration.

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