Stresses in Beams

Introduction

Beams are structural members that primarily carry loads transverse to their longitudinal axis. Their internal actions are usually bending moment and shear force. These actions produce normal stresses, commonly called flexural stresses, and tangential stresses, commonly called transverse shear stresses. Safe and economical beam design requires both types of stress to remain within acceptable limits.

This chapter develops the stress relationships for prismatic beams, explains the role of section properties, and extends the analysis to composite, built-up, and unsymmetrical beams. It also introduces shear flow and stress concentrations, which are especially important in thin-walled members, connections, openings, and fabricated structures.

Learning Objectives

After studying this chapter, you should be able to:

1. Derive and apply the flexure formula for beams.
2. Locate the neutral axis and calculate centroidal and second-moment properties.
3. Determine transverse shear stress distributions for common sections.
4. Analyze composite and built-up beams using transformed sections and shear flow.
5. Apply the theory of unsymmetrical bending.
6. Explain stress concentrations and select appropriate design responses.
7. Recognize the assumptions, limitations, and practical significance of beam-stress equations.

1. Beam Action and Fundamental Assumptions

Consider a beam subjected to transverse loads. At any cut section, the external loads on one side can be replaced by a resultant internal shear force V and bending moment M. Axial force may also exist, but the present chapter concentrates on bending and transverse shear.

The classical theory of simple bending is based on the following assumptions:

• The beam is initially straight and prismatic, or its properties vary gradually.
• The material is homogeneous, isotropic, and linearly elastic within the range considered.
• Plane cross-sections remain plane after bending (Euler–Bernoulli assumption).
• Deflections and rotations are small.
• The radius of curvature is large compared with the beam depth.
• The bending moment acts about a principal centroidal axis for the simplest form of the flexure equation.

These assumptions are highly effective for slender beams. They become less reliable for deep beams, short members, highly curved beams, anisotropic materials, plastic deformation, or regions close to load introduction and supports.

2. Section Properties

2.1 Centroid and neutral axis

The centroid is the geometric center of an area. For an area divided into elements dA, its coordinates are

where A is the total cross-sectional area and x̄, ȳ are centroid coordinates. For composite areas, the integral is replaced by a summation:

Here Aᵢ is a component area and xᵢ, yᵢ locate its centroid. Areas removed from a section are entered as negative areas.

The neutral axis is the line in the cross-section where longitudinal bending strain and, in elastic pure bending, longitudinal stress are zero. For symmetric sections under bending about a symmetry axis, it passes through the centroid. In general, the neutral axis depends on the bending moment direction and the section’s principal axes.

2.2 Second moment of area

The second moment of area about the x-axis is

and about the y-axis is

The product of inertia is

The units of I are length⁴, such as mm⁴ or m⁴. A large second moment of area indicates that more material is distributed away from the relevant axis, increasing bending resistance.

For a rectangle of width b and depth h about its centroidal horizontal axis,

The parallel-axis theorem permits transfer from a centroidal axis to a parallel axis:

where I_c is the centroidal second moment, A is area, and d is the perpendicular distance between axes.

The section modulus is

where c is the distance from the neutral axis to the extreme fiber. Section modulus has units of length³ and is useful because maximum elastic bending stress is M/S.

3. Flexural Stress

3.1 Derivation of the flexure formula

Suppose a straight beam bends into an arc of radius ρ. A longitudinal fiber at distance y from the neutral axis has strain

The negative sign indicates that fibers on one side shorten while those on the opposite side lengthen. Under linear elasticity, Hooke’s law gives

where E is Young’s modulus. The resisting moment is

Substitution gives the elastic flexure relation

or, commonly,

The sign depends on the selected coordinate and moment convention. In magnitude, the stress at a distance y is |σₓ| = |M|y/I. The maximum stress occurs at the farthest fiber:

M is the internal bending moment, I is the second moment of area about the bending axis, y is the distance from the neutral axis, c is the extreme-fiber distance, and σₓ is longitudinal normal stress. Consistent units must be used; for example, N·mm, mm⁴, and mm produce N/mm² (MPa).

3.2 Stress distribution

For a homogeneous beam in elastic bending, flexural stress varies linearly through the depth. It is zero at the neutral axis, tensile on one side, and compressive on the other. A rectangular section therefore has a triangular stress diagram. The stress is not generally uniform across the width unless the width changes or the section is nonstandard.

At a free outer surface, transverse shear stress is normally zero, but flexural stress may be maximum. At the neutral axis, flexural stress is zero while transverse shear stress is often maximum. This distinction is central to beam design.

Worked Example 1: Flexural stress in a rectangular beam

A simply supported beam has a rectangular cross-section 80 mm wide and 160 mm deep. At a critical section, the bending moment is 4.8 kN·m. Determine the extreme-fiber stress.

Convert the moment:

M = 4.8 × 10⁶ N·mm. For the rectangle,

I = bh³/12 = (80)(160³)/12 = 27.31 × 10⁶ mm⁴.

The extreme-fiber distance is c = 160/2 = 80 mm. Therefore,

σ_max = Mc/I = (4.8 × 10⁶)(80)/(27.31 × 10⁶) = 14.1 MPa.

One face is in tension and the opposite face is in compression. The result assumes elastic, plane bending about the centroidal horizontal axis.

4. Transverse Shear Stress

4.1 General shear formula

A beam carrying shear force V develops a distribution of longitudinal shear stress over its cross-section. For a section of arbitrary shape, the elastic shear formula is

where τ is shear stress at the selected level, V is internal shear force, I is the second moment of area about the neutral axis, b is the width of material at that level, and Q is the first moment of the area above or below the selected level about the neutral axis:

A′ is the partial area between the selected level and the free surface, and ȳ′ is the distance from the neutral axis to the centroid of A′. The formula assumes a slender beam, elastic behavior, and a cross-section for which beam theory is appropriate.

The units are consistent: VQ has units force × length³, while Ib has units length⁵, giving force/area.

4.2 Rectangular section

For a rectangle of width b and depth h, the shear stress distribution is parabolic:

where A = bh and y is measured from the neutral axis. The maximum occurs at the neutral axis:

The average shear stress is V/A, so the maximum is 1.5 times the average.

4.3 Circular section

For a solid circular cross-section of area A, the maximum elastic shear stress is

It occurs at the neutral axis and is zero at the circumference. For I-sections, most of the shear is carried by the web, while the flanges primarily resist bending moment.

Worked Example 2: Shear stress in a rectangular beam

A 100 mm × 200 mm rectangular beam carries an internal shear force of 30 kN. Determine the average and maximum shear stresses.

The area is A = (100)(200) = 20,000 mm². Thus,

τ_avg = V/A = 30,000/20,000 = 1.50 MPa.

For a rectangle,

τ_max = 1.5τ_avg = 2.25 MPa.

The maximum occurs on the neutral axis and the stress is zero at the top and bottom surfaces.

5. Combined Bending and Shear

At a general point in a beam, the stress state may include longitudinal normal stress from bending and shear stress from transverse force. The normal stress is found from σₓ = −My/I, while the shear stress is found from τ = VQ/(Ib). These stresses act on different planes but may combine to produce principal stresses that are important near supports, web openings, and connections.

For a state containing σₓ and τₓᵧ, the principal stresses are

and the maximum in-plane shear stress is

These equations apply to plane stress when the other normal stress is negligible. They are useful for ductile-material yielding checks, but a suitable failure criterion must still be selected.

6. Composite Beams

A composite beam is made from two or more materials that act together through a reliable interface. Examples include timber–steel members, reinforced concrete beams, and metal sandwich structures. Because different materials have different elastic moduli, equal strain does not produce equal stress.

6.1 Transformed-section method

Choose a reference material with modulus E_r. A material with modulus E_i is transformed into an equivalent area of the reference material using the modular ratio

Its transformed width or area is

The transformed section is analyzed as a homogeneous section. Its centroid and second moment are calculated using the transformed areas. The reference-material stress is then obtained from the flexure formula. The stress in material i is

provided the materials share the same longitudinal strain at the bonded interface and remain elastic.

Worked Example 3: Two-material composite strip

A composite strip consists of a 20 mm × 100 mm aluminum layer bonded to a 10 mm × 100 mm steel layer. Take E_steel = 200 GPa and E_aluminum = 70 GPa, and use steel as the reference material. The modular ratio for aluminum is n = 70/200 = 0.35. Thus the transformed aluminum area is 0.35 times its actual area. The transformed section centroid is found from

ȳ = ΣA_i'y_i/ΣA_i'.

The transformed second moment follows from

I' = Σ(I_i' + A_i'd_i²).

For a specified moment M, steel stress is σ_s = −My/I', and aluminum stress at the same distance from the neutral axis is σ_a = (E_a/E_s)σ_s = 0.35σ_s. The aluminum stress is lower at equal strain because its modulus is lower. In practice, the adhesive or mechanical interface must transfer longitudinal shear; the transformed-section method does not by itself prove that the bond is adequate.

7. Built-Up Beams and Shear Flow

Built-up beams consist of components connected by welds, bolts, rivets, screws, or adhesive. Even when each component is strong enough individually, the connections must transfer the interface shear required for composite action.

The shear flow is the longitudinal force transferred per unit length along an interface:

where q has units N/mm or N/m. The corresponding average connector force for spacing s is

This relation assumes the connection line is arranged so that q is distributed appropriately and that local effects are not dominant. For multiple connection lines, the flow must be allocated according to stiffness, geometry, and connection layout.

In a thin-walled open section, shear flow is often the primary tool for determining web and flange connection forces. The shear stress is related to flow by

where t is local wall thickness. Shear flow is zero at a free edge and changes along the wall according to the distribution of Q.

Practical design considerations

• Check connector shear and bearing strength.
• Check tear-out, block shear, net-section rupture, and local buckling.
• Provide sufficient spacing to prevent excessive slip.
• Account for eccentricity between the shear center and applied load.
• Consider fatigue where loads are repeated.
• Verify weld quality, bolt pretension, corrosion protection, and inspection requirements.

8. Unsymmetrical Bending

Unsymmetrical bending occurs when the cross-section is not symmetric about the bending direction or when the applied moment is not aligned with a principal centroidal axis. The neutral axis is then generally inclined, and stress varies with both cross-sectional coordinates.

8.1 Principal axes

Principal axes are centroidal axes for which the product of inertia is zero. If moments are resolved about principal axes x and y, the normal stress is

where Mₓ and Mᵧ are moment components about the principal axes. The neutral axis is obtained by setting σₓ = 0:

This equation shows that the neutral axis need not be perpendicular to the applied moment vector. The formula applies to elastic bending about centroidal principal axes and uses a consistent sign convention.

If the selected axes are not principal axes, the product of inertia must be included. A robust engineering practice is to determine the principal axes first, especially for unequal-angle sections, channels, and asymmetric built-up sections.

8.2 Design significance

Unsymmetrical bending can cause twisting as well as bending when the load does not pass through the shear center. The shear center is the point through which a transverse load must act to produce bending without torsion. For doubly symmetric sections it coincides with the centroid; for channels and angles it generally does not.

9. Stress Concentrations

The elementary beam equations predict nominal stresses based on smooth, idealized geometry. Holes, notches, keyways, abrupt changes in section, weld toes, sharp corners, and load-introduction regions disturb the stress field and create localized peaks called stress concentrations.

The theoretical stress-concentration factor is

where σ_nom is a nominal stress calculated using a specified gross or net area and σ_max is the local elastic peak stress. The appropriate definition of nominal stress must be stated because different references use different conventions.

For shear,

Stress concentration factors depend on geometry, loading mode, and dimensions, and are normally obtained from validated charts, standards, finite-element analysis, or experimental data. They are not material constants.

For ductile materials under static loading, local yielding may redistribute stress, so the elastic peak does not necessarily equal the final failure stress. For brittle materials, fatigue loading, welds, and crack-like defects, the local peak may be much more critical. Fatigue design commonly uses a fatigue notch factor K_f, which may be less than K_t because of material notch sensitivity.

Mitigation measures

• Use generous fillet radii and gradual section transitions.
• Avoid sharp re-entrant corners.
• Place holes away from regions of high moment or shear where possible.
• Use suitable hole-edge distances and reinforcement.
• Improve weld geometry and remove severe weld-toe irregularities.
• Use local finite-element refinement when geometry is complex.
• Distinguish a finite-radius notch from a crack; fracture mechanics may be required for cracks.

10. Detailed Applied Case: Built-Up Crane Girder

Background

Consider an illustrative, fictional case involving a welded crane girder fabricated from two flange plates and a vertical web. The girder carries a moving trolley, so its bending moment and shear force vary with position. The designer must check flange bending stress, web shear, and weld shear flow.

Problem

At a critical section, the girder carries M = 600 kN·m and V = 180 kN. The calculated section properties about the horizontal centroidal axis are I = 1.20 × 10⁹ mm⁴, c = 450 mm, and the first moment of the top flange about the neutral axis is Q = 2.40 × 10⁶ mm³. The web thickness at the interface is 12 mm.

Analysis

The extreme-fiber flexural stress is

σ_max = Mc/I = (600 × 10⁶)(450)/(1.20 × 10⁹) = 225 MPa.

The shear flow at the flange–web interface is

q = VQ/I = (180,000)(2.40 × 10⁶)/(1.20 × 10⁹) = 360 N/mm.

The corresponding average shear stress in the 12 mm web at that location is

τ = q/t = 360/12 = 30 MPa.

If weld pairs share the interface flow equally, each longitudinal weld line must transfer approximately 180 N/mm, subject to the actual weld arrangement and code requirements. Connector spacing, weld throat, fatigue category, local buckling, and load combinations must be checked separately.

Outcome and learning points

The flange stress is governed by bending, whereas the interface and web are governed by shear transfer. The case demonstrates why a beam cannot be assessed from M/S alone. A safe design also requires connection design, stability checks, fatigue evaluation for repeated trolley movement, and compliance with the applicable structural welding and crane standards.

11. Limitations, Standards, and Professional Practice

Beam formulas are models, not universal descriptions of stress. For deep beams, shear deformation may be important and Timoshenko beam theory may be preferable. Near supports and concentrated loads, Saint-Venant beam assumptions may not describe local three-dimensional stresses. Local buckling can control thin webs and flanges before material strength is reached. Composite action may be reduced by slip, cracking, creep, temperature change, or imperfect bonding.

Engineering design should therefore combine hand calculations with applicable national standards, verified material properties, connection rules, stability checks, and, where justified, finite-element analysis. Software results require mesh convergence, realistic boundary conditions, appropriate material models, and independent engineering review. AI-assisted design tools may help identify alternatives or automate calculations, but they do not replace verification, traceability, professional judgment, or responsibility for safety.

Ethically, the engineer must not report nominal beam stresses as complete proof of safety when stress concentrations, fatigue, instability, fire, corrosion, or connection failure have not been addressed. Assumptions, units, load combinations, and uncertainty should be documented so that another engineer can reproduce and audit the analysis.

Best-Practice Checklist

1. Draw the free-body diagram and establish the sign convention.
2. Determine M and V at the critical sections.
3. Locate the centroid and calculate I about the correct axis.
4. Check flexural stress using M/S or My/I.
5. Check transverse shear using VQ/(Ib), not merely V/A unless the approximation is justified.
6. Examine combined stress and principal stress where appropriate.
7. For composite members, verify compatibility and interface transfer.
8. For built-up members, calculate shear flow and design connectors.
9. Identify stress raisers, fatigue-sensitive details, and local instability.
10. Confirm results against standards, independent calculations, and realistic service conditions.

Chapter Summary

Beam loading produces internal bending moments and shear forces. Bending moment produces a linear longitudinal stress distribution in an elastic, homogeneous beam, governed by σ = −My/I. The maximum elastic stress is M/S. Transverse shear produces a nonuniform distribution governed by τ = VQ/(Ib). Rectangular sections have parabolic shear distributions with maximum stress 1.5 times the average.

Centroid location, second moment of area, product of inertia, and section modulus are essential section properties. Composite beams are analyzed through compatibility and transformed sections, while built-up beams require shear-flow calculations to design their connectors. Unsymmetrical bending is most clearly handled using principal centroidal axes. Stress concentrations arise from abrupt geometric or connection details and must be addressed using concentration factors, notch sensitivity, fatigue assessment, or refined analysis.

The equations in this chapter are powerful but conditional. Reliable design requires attention to material behavior, local buckling, fatigue, shear center, connections, standards, construction quality, and the limitations of ideal beam theory.

Key Terms

Centroid: Geometric center of a plane area.

Neutral axis: Line in a cross-section where elastic bending stress is zero.

Second moment of area: Geometric measure of resistance to bending about an axis.

Product of inertia: Cross-sectional property used in unsymmetrical bending analysis.

Section modulus: Ratio I/c used to calculate extreme-fiber bending stress.

Flexural stress: Longitudinal normal stress caused by bending moment.

Transverse shear stress: Stress caused by internal shear force acting across a beam section.

First moment of area: Quantity Q used in calculating beam shear stress and shear flow.

Composite beam: Beam made of different materials acting together.

Modular ratio: Ratio of the elastic modulus of a material to that of a reference material.

Transformed section: Equivalent homogeneous section used to analyze composite action.

Built-up beam: Beam assembled from components joined by mechanical or welded connections.

Shear flow: Longitudinal force transferred per unit length along an interface.

Principal axis: Centroidal axis about which the product of inertia is zero.

Unsymmetrical bending: Bending in which the moment or section geometry is not aligned with a principal axis.

Shear center: Point through which a transverse load produces bending without twisting.

Stress concentration: Local increase in stress caused by a geometric or material discontinuity.

Stress-concentration factor: Ratio of local peak elastic stress to nominal stress.

Local buckling: Instability of a plate element such as a thin web or flange.

Fatigue notch factor: Effective stress multiplier used for fatigue-sensitive notches.

Review Questions

Short Answer Questions

1. State the assumptions underlying the elementary flexure formula.
2. Define the second moment of area and explain its units.
3. Why is the maximum shear stress in a rectangle 1.5 times its average shear stress?
4. What is the purpose of a transformed section?
5. Define shear flow and state its principal design use.

Long Answer Questions

1. Derive the elastic flexure formula from the strain distribution in a bent beam.
2. Explain how centroid and second moment of area are calculated for a composite cross-section.
3. Derive the rectangular-section shear-stress distribution and identify its maximum value.
4. Explain unsymmetrical bending using principal axes and discuss the role of the shear center.
5. Discuss the causes, evaluation, and mitigation of stress concentrations in beam details.

Analytical / Application Questions

1. A rectangular beam 75 mm wide and 150 mm deep carries M = 3.0 kN·m and V = 18 kN at one section. Calculate the extreme-fiber bending stress and the maximum transverse shear stress. State where each occurs.
2. A built-up beam has V = 90 kN, I = 450 × 10⁶ mm⁴, and Q at a connection line equal to 1.10 × 10⁶ mm³. Determine the shear flow and explain how it would be used to select connector spacing.
3. An unequal-angle beam is subjected to moments about two centroidal axes. Describe the procedure for finding principal axes, calculating the stress at a selected point, and checking whether torsion may also occur.

Multiple Choice Questions with Answers

Question 1

For a homogeneous elastic beam in simple bending, where is longitudinal flexural stress zero?

A. At the extreme tensile fiber

B. At the neutral axis

C. At the top free surface only

D. At the shear center

Correct Answer: B. At the neutral axis

Question 2

What is the SI dimensional unit of the second moment of area?

A. m

B. m²

C. m³

D. m⁴

Correct Answer: D. m⁴

Question 3

A rectangular beam has an average shear stress of 2 MPa. Its maximum elastic shear stress is:

A. 1 MPa

B. 2 MPa

C. 3 MPa

D. 4 MPa

Correct Answer: C. 3 MPa

Question 4

Which equation gives elastic flexural stress at a distance y from the neutral axis?

A. σ = V/A

B. σ = My/I

C. τ = VQ/(Ib)

D. σ = Eρ/y

Correct Answer: B. σ = My/I

Question 5

In a transformed-section method, the modular ratio of material i relative to the reference material is:

A. E_r/E_i

B. E_i/E_r

C. I_i/I_r

D. A_i/A_r

Correct Answer: B. E_i/E_r

Question 6

The shear flow at an interface in a built-up beam is calculated from:

A. q = VQ/I

B. q = Mc/I

C. q = Eε

D. q = V/A

Correct Answer: A. q = VQ/I

Question 7

For bending about centroidal principal axes, the normal stress equation includes:

A. Only the shear force

B. Only the product of inertia

C. Moment components divided by their corresponding second moments of area

D. Only the beam length

Correct Answer: C. Moment components divided by their corresponding second moments of area

Question 8

A stress-concentration factor is defined as the ratio of:

A. Average stress to allowable stress

B. Local peak elastic stress to nominal stress

C. Yield stress to ultimate stress

D. Shear stress to normal stress

Correct Answer: B. Local peak elastic stress to nominal stress

Question 9

For a solid rectangular section under transverse shear, the shear stress is generally:

A. Uniform over the depth

B. Triangular, with maximum at the surface

C. Parabolic, with maximum at the neutral axis

D. Zero throughout the section

Correct Answer: C. Parabolic, with maximum at the neutral axis

Question 10

For a doubly symmetric cross-section, the shear center generally:

A. Coincides with the centroid

B. Lies at an extreme corner

C. Is always outside the section

D. Coincides with the neutral axis at every load level

Correct Answer: A. Coincides with the centroid