Beams, Girders, and Beam-Columns

Introduction and Context

Beams are structural members whose primary function is to resist transverse loads through bending and shear. In buildings, bridges, industrial platforms, cranes, and temporary works, beams transfer floor, roof, equipment, wind, and construction loads to columns, walls, or foundations. A girder is generally a major beam that collects loads from secondary beams or other framing members. A beam-column carries axial force together with bending moment and shear.

The design of these members is governed not only by material strength but also by stability, serviceability, connection behavior, load paths, fabrication, erection, inspection, and durability. A steel beam may have sufficient plastic moment resistance and still fail by lateral-torsional buckling, web crippling, shear buckling, excessive deflection, or vibration. Sound design therefore requires resistance checks and serviceability checks based on a recognized standard, such as AISC 360, Eurocode 3, or the applicable national steel code. The numerical coefficients and resistance factors differ among standards; the mechanics explained here are general, while final design must follow one consistent code.

Learning Objectives

By the end of this chapter, a student should be able to:

1. Explain bending, shear, axial force, torsion, and their interaction in steel members.
2. Distinguish laterally supported beams from beams susceptible to lateral-torsional buckling.
3. Determine elastic and plastic section properties and use them in preliminary design.
4. Check beam strength for yielding, local buckling, lateral-torsional buckling, shear, web crippling, and web buckling.
5. Explain the purpose and design logic of transverse and longitudinal stiffeners in plate girders.
6. Evaluate deflection and vibration as serviceability requirements.
7. Analyze beam-columns using second-order effects and interaction equations.
8. Select practical sections and communicate assumptions, limitations, and construction requirements.

1. Structural Action of Beams

For a simply supported beam under a uniformly distributed load w over span L, the maximum shear and bending moment occur at the supports and midspan, respectively:

Here w is the total design line load in force per unit length and L is the span. These expressions assume a prismatic, simply supported beam with a uniform load over the full span and small deformations. Other support conditions and load arrangements require the appropriate shear-force and bending-moment diagrams.

The flexure relationship for an elastic, homogeneous beam is:

where σ is the longitudinal bending stress, M is bending moment, y is the distance from the neutral axis, and I is the second moment of area about the bending axis. At the extreme fiber, y = c, and the elastic section modulus is S = I/c, giving:

The formula assumes plane sections remain plane, the material is elastic, and the member is not significantly affected by local or lateral instability. The corresponding elastic moment resistance is approximately M_y = F_y S, where F_y is yield stress. For a compact section with adequate lateral restraint, yielding may spread through the section and the plastic resistance is:

Z is the plastic section modulus. Since Z is generally greater than S, M_p exceeds M_y. Plastic resistance is applicable only when code requirements for compactness, bracing, ductility, and load history are satisfied.

Shear stress in a thin-web approximation is often estimated by:

V is shear force, Q is the first moment of the area outside the level considered about the neutral axis, I is the section second moment of area, and t_w is web thickness. For an I-section, most vertical shear is carried by the web. Exact resistance expressions depend on the adopted design code and account for web slenderness, yielding, and post-buckling action.

2. Section Selection and Classification

A rolled I-section is efficient in major-axis bending because its flanges place material far from the neutral axis, while its web provides depth and carries shear. Important geometric quantities include area A, depth d, flange width b_f, flange thickness t_f, web thickness t_w, I, S, Z, radius of gyration r, and torsional constant J.

Section classification limits determine whether plate elements can develop yield, plastic, or only elastic resistance. The flange outstand and web depth-to-thickness ratio are compared with code limits involving the material parameter:

when f_y is expressed in MPa and 235 MPa is the reference value used in Eurocode-style classification. Other standards use an equivalent parameter, commonly involving √(E/F_y). Classification must never be transferred from one code to another without checking definitions and limits.

A preliminary design commonly follows this sequence:

1. Establish factored and service load combinations.
2. Determine reactions, shear, and moment envelopes.
3. Select a trial section using required section modulus.
4. Check compactness or plate-element classification.
5. Check lateral-torsional buckling and unbraced length.
6. Check shear, web bearing, web crippling, and local buckling.
7. Check deflection, vibration, connections, fire, corrosion, and constructability.
8. Revise the member or restraint system if any requirement is not satisfied.

For preliminary elastic sizing, a required section modulus may be estimated as:

for an allowable-stress approach, or as M_u/(appropriate design stress or resistance factor) in a limit-state approach. This is only a sizing step; it is not a complete code design.

3. Laterally Supported Beams

A beam is laterally supported when the compression flange is restrained against sufficient lateral translation and the cross-section is restrained against the twist associated with lateral-torsional buckling. Bracing may be provided by floor slabs, metal decking, secondary beams, diaphragms, bridging, or explicit braces. A slab is not automatically adequate: the connection, continuity, stiffness, load transfer, construction stage, and possible separation must be demonstrated.

For a laterally supported compact beam, the governing major-axis resistance may approach the plastic moment M_p, subject to shear interaction and code limitations. For noncompact or slender flange and web elements, local buckling occurs before full plastic capacity. Local buckling is a plate phenomenon caused by compression in an individual flange or web element. It is controlled by width-to-thickness ratios, edge restraint, stress distribution, and material properties.

Lateral-Torsional Buckling

In an unbraced beam, compression-flange instability causes the member to move laterally and twist. The critical unbraced length L_b is the distance between points that provide effective lateral restraint to the compression flange, not simply the beam span. Restraint conditions at supports and load application points are significant.

A simplified elastic critical moment for a doubly symmetric prismatic member under idealized conditions can be represented conceptually by:

E is Young’s modulus, G is shear modulus, I_y is the weak-axis second moment of area, J is the torsional constant, and C_w is the warping constant. The expression is sensitive to end restraint, moment gradient, load height, and imperfections; therefore, design standards use modified equations or tables. It should not be used as a code resistance formula without confirming the assumptions.

The general behavior is:

• Short unbraced lengths: yielding or plastic resistance may govern.
• Intermediate unbraced lengths: inelastic lateral-torsional buckling reduces resistance.
• Long unbraced lengths: elastic lateral-torsional buckling governs.

The moment-gradient factor accounts for the shape of the bending-moment diagram. A beam with a high moment only in a short region may have greater resistance than one with constant high moment over the same unbraced length. Bracing should be designed for both strength and stiffness; a brace that cannot prevent the required lateral movement is not effective.

4. Plate Girders and Web Behavior

A plate girder is fabricated from steel plates, usually with two flange plates and a web plate joined by welding or bolting. Plate girders are economical for long spans or heavy loads when a rolled section would be unavailable, excessively heavy, or inefficient. Their depth and flange area can be proportioned to match bending and shear demands.

The idealized bending resistance is primarily provided by the flanges, while the web supplies most shear resistance and maintains flange separation. A preliminary bending design may estimate the required flange force as:

where F_f is the force in one flange and d_0 is the distance between flange centroids. This is an approximation that neglects the web’s bending contribution and must be followed by a complete section and stability design.

Web Slenderness and Shear Buckling

A slender web can buckle diagonally under shear before reaching shear yield. The elastic shear-buckling stress for an idealized simply supported rectangular plate is:

where k_v is the shear buckling coefficient, ν is Poisson’s ratio, h is the clear web depth, and t_w is web thickness. The coefficient depends on aspect ratio and boundary restraint. The formula assumes a flat, elastic, initially perfect plate; real webs contain residual stress, fabrication imperfections, and varying restraint.

When a web is slender, a girder may develop post-buckling tension-field action. Diagonal tensile forces, anchored by flanges and stiffeners, can provide additional shear resistance. This mechanism is valid only when the code’s anchorage, stiffener, panel, and detailing requirements are satisfied. It should not be assumed merely because a web is visibly thin.

Web Stiffeners

Transverse stiffeners divide a web into panels and improve resistance to shear buckling, concentrated loads, and bearing. Bearing stiffeners are placed near supports or load introduction points where concentrated compression may cripple or yield the web. Intermediate transverse stiffeners are placed between supports to reduce the effective web panel width. Longitudinal stiffeners are used in very deep slender webs to increase shear-buckling resistance by subdividing the web plate.

Stiffeners must themselves be checked for axial force, bending, local buckling, weld strength, fatigue where applicable, and load transfer. Detailing must allow drainage, inspection, corrosion protection, and weld access. A stiffener that terminates improperly or is not connected to the force path may provide little structural benefit.

Concentrated Forces and Web Crippling

A point load or reaction introduces local compression into the web. Potential limit states include web yielding, web crippling, web buckling, flange bending, and local flange instability. The resistance depends on load length, bearing length, web thickness, flange restraint, distance from the beam end, and stiffener arrangement. Concentrated reactions should therefore be checked separately from global shear and moment.

5. Deflection, Vibration, and Serviceability

Strength design prevents collapse; serviceability design controls performance in normal use. Excessive deflection can damage partitions, finishes, glazing, drainage systems, ceilings, and machinery. A common elastic deflection for a simply supported beam carrying a full-span uniform load is:

For a central point load P on the same support condition:

These equations assume small elastic deflection, constant E and I, ideal supports, and the stated load arrangement. Real design must use the service load combination and the deflection limits specified by the project or code. Long-term effects, composite action, connection slip, camber, construction sequence, and creep of attached materials may alter actual response.

Vibration is governed by mass, stiffness, damping, forcing frequency, and occupancy sensitivity. A simplified estimate of the fundamental frequency of a single-degree-of-freedom system is:

where f_n is frequency in hertz, k is effective stiffness in force per unit displacement, and m is effective mass. This is a conceptual model, not a replacement for a floor-system vibration analysis. Walking, rhythmic activity, rotating equipment, and impact loads may cause discomfort or operational problems even when deflection limits are satisfied.

6. Beam-Columns and Combined Forces

Beam-columns occur in columns supporting eccentric floor loads, portal frames, crane columns, braced-frame members, and members exposed to wind or seismic drift. They carry axial compression or tension, major- and minor-axis moments, and often shear and torsion.

The first-order moment is produced by applied loads using the undeformed geometry. Under compression, lateral displacement creates an additional moment P∆, where P is axial force and ∆ is lateral displacement. This second-order effect increases deflection and moment. At member level, curvature and end moments also produce a P–δ effect.

For an ideal pin-ended prismatic member under concentric compression, Euler’s elastic critical load is:

E is Young’s modulus, I is the relevant-axis second moment of area, L is member length, and K is the effective-length factor. The formula assumes a straight, elastic, centrally loaded member with ideal end conditions. Real design includes residual stress, initial crookedness, connection stiffness, load eccentricity, and inelastic behavior.

The nondimensional slenderness concept compares applied compression with buckling resistance. Codes commonly require interaction equations of the general form:

where P_u and M_u are factored axial force and moment, while P_c and M_c are design resistances. Actual code equations may include separate major- and minor-axis terms, moment-gradient factors, amplification factors, and nonlinear interaction. The expression above is a conceptual linear interaction, not a universal design equation.

An elastic moment amplification may be estimated for a simplified member as:

where M_1 is a first-order moment and M_2 is an amplified moment. This approximation is valid only for a simplified elastic model and becomes unreliable near buckling, where imperfections and inelasticity dominate. In professional design, second-order analysis or the code-prescribed moment-amplification procedure is required.

For tension beam-columns, axial tension may reduce lateral buckling risk but can increase total stress on one side of the section. For compression beam-columns, both member buckling and cross-section local buckling must be considered. Connections must transfer the combined force resultants and may govern before the member itself.

7. Worked Example: Preliminary Beam Design

Consider an illustrative simply supported steel beam spanning 6.0 m. The service uniform load, including self-weight for this preliminary calculation, is 20 kN/m. Assume the beam is laterally supported continuously by a verified floor system. Take E = 200 GPa and an allowable service-stress screening limit of 0.66F_y, with F_y = 345 MPa. A final design would use the governing national standard and factored load combinations.

The maximum service moment is:

The maximum shear is:

The screening bending stress is 0.66 × 345 = 227.7 MPa. Thus:

or approximately 395 cm³. A trial section must have at least this elastic section modulus about the major axis, subject to classification and all other checks.

For a serviceability screening limit of L/360:

Rearranging the uniform-load deflection equation gives:

Using w = 20 N/mm, L = 6000 mm, E = 200,000 N/mm², and δ = 16.7 mm gives approximately I_req = 1.07 × 10^8 mm^4. A selected section must satisfy both S and I requirements and then be checked for shear, local buckling, connections, vibration, fire, and construction-stage stability. This example demonstrates preliminary screening, not approval for construction.

8. Applied Case: Illustrative Plate-Girder Floor Transfer

Background

A six-storey building requires a transfer girder to support several secondary beams over a 15 m column-free opening. The architectural depth limit is 1.2 m, and the girder receives concentrated reactions from the secondary framing.

Problem

A rolled section would require excessive weight and would not provide adequate shear resistance within the depth limit. The design team must select a welded plate girder while controlling web instability, concentrated-load effects, deflection, vibration, and fabrication risk.

Analysis and Approach

The engineer first establishes factored and service reaction envelopes, including construction-stage loads. Flange plates are proportioned primarily for bending, and the web is selected for shear and required depth. The web slenderness is checked against the chosen code. Because the web is slender, transverse stiffeners are placed at supports and under major concentrated reactions; intermediate stiffeners divide shear panels. The compression flange is connected to the floor system only after verifying that the slab, studs, diaphragm, and construction sequence provide reliable lateral restraint.

Finite-element analysis is used as a supplementary tool to investigate local stress concentrations around openings and stiffeners. It does not replace code checks, load combinations, or engineering judgment. Weld access, inspection, corrosion protection, fire protection, and transportation limits are incorporated before fabrication drawings are released.

Outcome

The final arrangement uses a deeper, lighter web with carefully spaced stiffeners rather than simply increasing web thickness. Bearing stiffeners transfer secondary-beam reactions into the girder supports. Service checks confirm acceptable deflection and vibration, while erection bracing maintains stability before the floor diaphragm becomes effective.

Learning Points

The governing decision was not based on moment capacity alone. Efficient plate-girder design integrates global strength, web-panel behavior, concentrated forces, lateral restraint, serviceability, fabrication, and erection. The case also illustrates why a computer model cannot compensate for an incomplete load path or inadequate detailing.

9. Professional Practice, Safety, and Contemporary Tools

Designers must identify the governing standard, material grade, weld and bolt specifications, load combinations, fire rating, corrosion environment, and inspection category. Drawings should show bracing locations, stiffener continuity, weld sizes, camber, erection sequence, temporary supports, and connection assumptions. Unclear responsibility for lateral restraint is a frequent source of construction risk.

Digital analysis, parametric section selection, building information modelling, laser scanning, and sensor-based monitoring can improve coordination and quality control. Artificial intelligence may assist with preliminary optimization, anomaly detection, or document checking, but it cannot independently validate load paths, code applicability, material certificates, hidden defects, or unusual failure modes. The licensed engineer remains responsible for assumptions, verification, professional judgment, and public safety.

Sustainability considerations include minimizing steel tonnage without compromising robustness, designing bolted or accessible details for disassembly, specifying recycled steel where appropriate, reducing repainting requirements, and considering whole-life carbon. Weight reduction must not create excessive vibration, fragile weld details, or difficult erection conditions.

10. Common Design Errors and Best Practices

Common errors include treating the full span as the unbraced length without checking actual restraints; counting a slab as lateral bracing without verifying its connection; ignoring construction-stage instability; using plastic resistance for a slender section; overlooking concentrated-load web failure; applying a shear-buckling equation outside its boundary assumptions; and checking deflection using factored rather than service loads.

Best practice is to maintain a clear design basis, draw the load path, separate strength and serviceability combinations, identify every potential instability mode, and coordinate member design with connections and erection. Section selection should be iterative: a heavier rolled section may be more economical than a fabricated girder once welding, inspection, fire protection, and transport are included. Every analytical model should be compared with a simple hand calculation to detect unit errors and unrealistic results.

Chapter Summary

Beams resist transverse loading primarily through bending and shear; girders are major load-collecting beams; and beam-columns resist combined axial force and bending. Elastic bending depends on M, I, and S, while plastic resistance depends on F_y and Z and is available only for suitably compact and restrained sections. Lateral-torsional buckling is controlled by compression-flange restraint, unbraced length, section torsional properties, warping stiffness, and moment gradient. Plate girders require coordinated design of flanges, web panels, stiffeners, concentrated-load regions, welds, and erection bracing. Slender webs may rely on code-recognized post-buckling tension-field action, but only with suitable anchorage and detailing. Deflection and vibration are serviceability requirements distinct from strength. Beam-columns require consideration of member buckling, local buckling, P–∆ and P–δ effects, and interaction between axial force and moments. Professional design combines mechanics, code procedures, constructability, inspection, durability, digital tools, and engineering judgment.

Key Terms

Beam: A member primarily resisting transverse loads through bending and shear.

Girder: A major beam that supports secondary members or collects substantial loads.

Beam-column: A member subjected to axial force and bending simultaneously.

Elastic section modulus: S = I/c, used to calculate extreme-fiber elastic stress.

Plastic section modulus: Z, the first moment of yielded areas about the plastic neutral axis.

Lateral-torsional buckling: Instability involving lateral movement and twisting of a beam.

Unbraced length: Distance between points providing effective lateral restraint to the compression flange.

Local buckling: Instability of an individual flange or web plate element.

Plate girder: A fabricated girder assembled from plates, commonly by welding.

Web slenderness: A measure commonly expressed as web depth divided by web thickness.

Shear buckling: Buckling of a web caused by diagonal shear stresses.

Tension-field action: Post-buckling shear resistance carried through diagonal web tension.

Bearing stiffener: A stiffener placed to resist concentrated reaction or load effects.

Intermediate stiffener: A web stiffener used between supports or load points to improve panel behavior.

Web crippling: Local web failure under a concentrated force or reaction.

Serviceability: Performance requirements concerning deflection, vibration, appearance, and use.

Second-order effect: Additional response caused by loads acting on displaced geometry.

P–∆ effect: Global second-order moment from axial load acting through lateral displacement.

P–δ effect: Member-level second-order response caused by curvature and local displacement.

Moment interaction: Combined resistance behavior under axial force and bending moments.

Review Questions

5 Short Answer Questions

1. What is the difference between a beam, a girder, and a beam-column?
2. Why is the compression flange important in lateral-torsional buckling?
3. State two functions of transverse web stiffeners.
4. Distinguish strength design from serviceability design.
5. What do the P–∆ and P–δ effects represent?

5 Long Answer Questions

1. Explain the mechanics of lateral-torsional buckling and discuss how bracing, unbraced length, and moment gradient affect resistance.
2. Describe the complete design process for a welded plate girder, including web behavior, stiffeners, concentrated forces, and fabrication.
3. Derive or explain the elastic bending-stress relationship and discuss the conditions required for plastic moment resistance.
4. Discuss the causes of excessive beam deflection and vibration and identify practical methods of control.
5. Explain why beam-column design cannot be completed by checking axial compression and bending separately.

3 Analytical / Application Questions

1. A simply supported beam spans 8 m and carries a uniform service load of 25 kN/m. Calculate its maximum shear and moment, then identify the additional checks required before selecting a steel section.
2. A transfer girder has a slender web and several concentrated reactions. Develop a design-and-detailing strategy addressing shear buckling, bearing stiffeners, intermediate stiffeners, lateral restraint, deflection, and erection stability.
3. A frame member carries high compression and wind-induced bending. Explain how you would assess first-order response, second-order amplification, member buckling, cross-section classification, connection resistance, and serviceability before finalizing the design.

Multiple Choice Questions with Answers

Question 1

For a simply supported beam carrying a uniformly distributed load w over the full span L, the maximum bending moment is:

A. wL/2

B. wL²/8

C. wL²/2

D. wL/8

Correct Answer: B. wL²/8

Question 2

Which property is used with yield stress to estimate the elastic bending moment at first yield?

A. Plastic section modulus Z

B. Torsional constant J

C. Elastic section modulus S

D. Warping constant Cw

Correct Answer: C. Elastic section modulus S

Question 3

Lateral-torsional buckling involves:

A. Only vertical shear yielding

B. Lateral movement and twisting of the beam

C. Only local buckling of the web

D. Axial elongation without rotation

Correct Answer: B. Lateral movement and twisting of the beam

Question 4

A bearing stiffener is primarily provided to:

A. Increase paint adhesion

B. Resist concentrated reaction or load effects

C. Reduce steel yield stress

D. Replace all flange bracing

Correct Answer: B. Resist concentrated reaction or load effects

Question 5

For a simply supported beam with a central point load P, the elastic maximum deflection is:

A. PL³/(48EI)

B. 5PL⁴/(384EI)

C. PL²/(8EI)

D. P/(48EIL³)

Correct Answer: A. PL³/(48EI)

Question 6

The idealized Euler critical load for a pin-ended compression member is:

A. π²EI/(KL)²

B. π²EI/(KL)

C. KL/(π²EI)

D. F_yA/(KL)²

Correct Answer: A. π²EI/(KL)²

Question 7

Which effect is represented by an axial compression force acting through a global lateral displacement?

A. P–∆ effect

B. Plastic hinge effect

C. Shear-lag effect

D. Poisson effect

Correct Answer: A. P–∆ effect

Question 8

Most vertical shear in a conventional rolled I-beam is carried primarily by the:

A. Compression flange

B. Tension flange

C. Web

D. Lateral brace

Correct Answer: C. Web

Question 9

A preliminary beam design using service loads should primarily address which category?

A. Serviceability screening such as deflection

B. Only ultimate fracture

C. Only weld metallurgy

D. Only foundation settlement

Correct Answer: A. Serviceability screening such as deflection

Question 10

For a slender web, post-buckling tension-field action is acceptable only when:

A. The web has no stiffeners or anchorage

B. Code requirements for anchorage and detailing are satisfied

C. The beam is unloaded during service

D. The flange plates are removed

Correct Answer: B. Code requirements for anchorage and detailing are satisfied