Stresses in Beams - Theory & Concepts

Stresses in Beams

$Learning Objectives$

Understand how flexural stress is caused by bending and how to calculate it.
Determine maximum flexural stresses and understand the role of the section modulus.
Evaluate stresses under unsymmetrical or biaxial bending conditions.
Comprehend how vertical shear forces create shearing stresses in beams.
Calculate maximum shear stress for a given cross-section.
Define and apply the concept of shear flow in built-up beams.
Explain the concept and significance of the shear center.
Calculate principal stresses using Mohr's Circle or principal stress equations.
Analyze composite beams using the Transformed Section Method and Modular Ratio.

When a beam is subjected to transverse loads and bends, it must resist both the external bending moments and the vertical shear forces. It does this by developing internal stress fields across its cross-section. The dominant stresses to analyze are Flexural Stresses (normal stresses caused by bending) and Shearing Stresses (tangential stresses caused by the shear force).

$Flexural Stress$

Normal stress caused by bending moment in the beam. It varies linearly from zero at the neutral axis (NA) to a maximum at the extreme fibers.

$Flexure Formula Assumptions$

The standard flexure formula is based on several critical kinematic and material assumptions:

Plane sections remain plane: The cross-section remains flat and perpendicular to the longitudinal axis during bending.
Linear elastic material: The material obeys Hooke's Law ( $\sigma = E\epsilon$ ).
Homogeneous material: The modulus of elasticity ( $E$ ) is constant throughout the cross-section.
Symmetric cross-section: The beam has a longitudinal plane of symmetry, and the bending moment acts in this plane (avoids torsional twisting).
No lateral buckling: The beam is properly braced against lateral-torsional buckling.

$Flexural Stress (Bending)$

Flexural stress must be resisted by the beam to balance external bending moments. The flexure formula calculates this stress at any given point across the cross-section.

Flexural Stress Formula

Calculates normal stress caused by bending moment in the beam.

\sigma = -\frac{My}{I}

Variables

Symbol	Description	Unit
$\sigma$	flexural stress at distance y from the neutral axis	-
$M$	bending moment at the section	-
$y$	perpendicular distance from the neutral axis to the point of interest	-
$I$	Moment of Inertia of the cross-sectional area about the neutral axis	-

$Section Modulus$

A geometric property of a cross-section indicating flexural strength, defined as the moment of inertia divided by the farthest distance from the neutral axis ( $S = I/c$ ).

Maximum Flexural Stress

Occurs at the farthest distance ( $c$ ) from the neutral axis:

Maximum Flexural Stress Formula

Calculates maximum normal stress caused by bending moment in the beam.

\sigma_{\text{max}} = \frac{Mc}{I} = \frac{M}{S}

Variables

Symbol	Description	Unit
$\sigma_{\text{max}}$	maximum flexural stress	-
$M$	bending moment at the section	-
$c$	farthest distance from the neutral axis	-
$I$	Moment of Inertia	-
$S$	Section Modulus	-

$Unsymmetrical (Biaxial) Bending$

The standard flexure formula ( $\sigma = -My/I$ ) assumes the bending moment is applied about a principal axis of the cross-section.

If the bending moment is applied at an angle, or if the load does not pass through the shear center of an unsymmetrical section, biaxial bending occurs. The stress must be evaluated by breaking the moment into components along the principal axes ( $y$ and $z$ ):

Unsymmetrical Bending Stress Formula

Calculates stress due to biaxial bending.

\sigma = \frac{M_y z}{I_y} - \frac{M_z y}{I_z}

Variables

Symbol	Description	Unit
$\sigma$	flexural stress	-
$M_y, M_z$	moment components about the y and z principal axes	-
$I_y, I_z$	moments of inertia about the principal axes	-
$y, z$	coordinates of the point where stress is being calculated	-

$Sign Convention$

Sign convention is critical here. Tension is positive, compression is negative.

$Shearing Stress in Beams$

In addition to bending, beams resist shear forces. This creates shearing stresses acting horizontally and vertically.

$First Moment of Area (Q)$

A geometric property representing the moment of an area about the neutral axis. Calculated as $Q = A'\bar{y}'$ , where $A'$ is the area of the cross-section either completely above or completely below the layer of interest, and $\bar{y}'$ is the distance from the neutral axis to the centroid of that area $A'$ .

Shearing Stress Formula

Calculates transverse shear stress.

\tau = \frac{VQ}{Ib}

Variables

Symbol	Description	Unit
$\tau$	shear stress at a specific layer	-
$V$	vertical shear force at the section	-
$Q$	First Moment of Area of the portion of the section above (or below) the layer where shear is being calculated	-
$I$	moment of inertia of the entire section	-
$b$	width of the beam at the layer where shear is calculated	-

$Maximum Shear Stress (Rectangular Section)$

For a rectangular beam of width $b$ and height $h$ :

Maximum Shear Stress Formula (Rectangular Section)

Calculates maximum shear stress for a rectangular section.

\tau_{\text{max}} = \frac{3V}{2A} = 1.5 \frac{V}{bh}

Variables

Symbol	Description	Unit
$\tau_{\text{max}}$	maximum shear stress	-
$V$	vertical shear force at the section	-
$A$	cross-sectional area	-
$b$	width of the beam	-
$h$	height of the beam	-

$Maximum Shear Stress (Circular Section)$

For a solid circular beam of radius $r$ :

Maximum Shear Stress Formula (Circular Section)

Calculates maximum shear stress for a solid circular section.

\tau_{\text{max}} = \frac{4V}{3A}

Variables

Symbol	Description	Unit
$\tau_{\text{max}}$	maximum shear stress	-
$V$	vertical shear force at the section	-
$A$	cross-sectional area	-

$Location of Maximum Shear$

For both rectangular and standard circular sections, the maximum shear stress value occurs at the Neutral Axis.

$Shear Flow$

The internal shearing force per unit of length along the longitudinal axis of a built-up beam. It is essential for designing the fasteners or glue joints that hold the components together.

$Built-Up Beams$

When a beam is built-up from multiple components (like a wooden box beam glued together, or steel plates welded/bolted to form an I-beam), we must determine the shear flow ( $q$ ) to ensure the built-up section acts as a single solid unit.

Shear Flow Formula

Calculates the internal shearing force per unit of length.

q = \frac{VQ}{I}

Variables

Symbol	Description	Unit
$q$	shear flow	-
$V$	vertical shear force at the section	-
$Q$	First Moment of Area	-
$I$	moment of inertia of the entire section	-

$Practical Application of Shear Flow$

Shear flow $q$ has units of force per unit length (e.g., N/mm, kN/m, lb/in). It is primarily used to design the connections between the separate pieces of a built-up beam.

For discrete fasteners (nails, bolts, rivets): If the fasteners are spaced at a longitudinal distance $s$ , the shear force $F$ that must be resisted by a single fastener is calculated as $F = q \times s$ .
For continuous fasteners (glue, continuous welds): The shear flow $q$ directly represents the required shear strength of the glue line or weld per unit length.

$Shear Center$

The specific point in the cross-section through which a transverse load must pass to cause pure bending without any torsion (also known as the center of flexure).

$Concept of the Shear Center$

When a transverse load is applied to a beam, it generates both bending and shear stresses. If the beam's cross-section is symmetrical (like an I-beam or rectangular tube) and the load is applied through the centroid, the beam bends without twisting.

However, for thin-walled open sections that are unsymmetrical (like a C-channel, an angle iron, or a Z-section), applying a vertical load through the centroid will cause the beam to severely twist as it bends. This happens because the internal shear flows along the thin flanges and web create a net torsional moment.

Key Principles of the Shear Center:

It is a geometric property of the cross-section, entirely independent of the applied load.
If the cross-section has an axis of symmetry, the shear center always lies on that axis.
If a section consists of intersecting thin straight elements (like a T-section, angle section, or cross), the shear center is located at the exact intersection point of the elements.
For a C-channel loaded vertically, the shear center is located outside the web, opposite the flanges. The load must be applied on an outrigger bracket to prevent twisting.

$Principal Stresses in Beams$

While the maximum flexural stress ( $\sigma_{\text{max}}$ ) occurs at the extreme outer fibers (where $\tau = 0$ ), and the maximum shear stress ( $\tau_{\text{max}}$ ) occurs at the neutral axis (where $\sigma = 0$ ), points inside the beam web experience a combination of both normal and shear stresses.

To evaluate the absolute critical stress at these intermediate points (often near the web-flange junction of an I-beam), we must use Mohr's Circle or the Principal Stress equations:

Principal Normal Stresses Formula

Calculates the maximum and minimum normal stresses at a point.

\sigma_{1,2} = \frac{\sigma}{2} \pm \sqrt{\left(\frac{\sigma}{2}\right)^2 + \tau^2}

Variables

Symbol	Description	Unit
$\sigma_{1,2}$	principal normal stresses	-
$\sigma$	flexural stress at that specific point	-
$\tau$	transverse shear stress at that specific point	-

Principal Shear Stress Formula

Calculates the maximum shear stress at a point.

\tau_{\text{max\_principal}} = \sqrt{\left(\frac{\sigma}{2}\right)^2 + \tau^2}

Variables

Symbol	Description	Unit
$\tau_{\text{max\_principal}}$	maximum principal shear stress	-
$\sigma$	flexural stress at that specific point	-
$\tau$	transverse shear stress at that specific point	-

$Modular Ratio$

The ratio of the modulus of elasticity of the material being transformed ( $E_2$ ) to the modulus of elasticity of the base material ( $E_1$ ), defined as $n = E_2 / E_1$ .

$Composite Beams (Transformed Section Method)$

When beams are made of two or more different materials bonded together (e.g., steel and timber, or reinforced concrete), the neutral axis does not coincide with the geometric centroid. To use the flexure formula, the beam cross-section must be transformed into an equivalent section of a single material using the modular ratio.

Modular Ratio Formula

Calculates the ratio of the modulus of elasticity of two materials.

n = \frac{E_2}{E_1}

Variables

Symbol	Description	Unit
$n$	modular ratio	-
$E_2$	modulus of elasticity of the material being transformed	-
$E_1$	modulus of elasticity of the base material	-

$Equivalent Width Calculation$

The width of Material 2 is multiplied by $n$ to find its equivalent width in Material 1:

Equivalent Width Formula

Calculates the transformed width of a material.

b_{\text{equivalent}} = n \cdot b_2

Variables

Symbol	Description	Unit
$b_{\text{equivalent}}$	equivalent width in Material 1	-
$n$	modular ratio	-
$b_2$	original width of Material 2	-

$Calculating Stresses in Composite Beams$

STEP-BY-STEP

Transform the entire section to Material 1 to calculate the new neutral axis location ( $\bar{y}$ ) and the transformed moment of inertia ( $I_{\text{tr}}$ ).
Calculate the stress in the base material: $\sigma_1 = \frac{M y}{I_{\text{tr}}}$
Calculate the stress in the transformed material: $\sigma_2 = n \frac{M y}{I_{\text{tr}}}$

$Interactive Simulation$

Use the simulation below to review how moment and shear vary along a beam, which directly dictates where the maximum flexural and shear stresses will occur.

Controls

15 kN·m

20 kN

100 mm

200 mm

0.00

Inspection height y

0.0 mm

Second moment I

66.667 ×10⁶ mm⁴

Normal stress at y

0.000 MPa

Positive tension; negative compression.

Shear stress at y

1.500 MPa

$Inspect both stress distributions$

Move the inspection height from the bottom to the top surface. Bending stress is zero at the neutral axis, where rectangular-section transverse shear reaches its maximum. Reverse the moment or shear force to check their independent sign conventions.

Mechanics Visual Synthesis

Use these deterministic figures to connect the lesson equations to the corresponding free-body, section, transformation, or response geometry. Each visual addresses a separate mechanics question and is intentionally static so it can be compared directly with the interactive model above.

Linear flexural stress through beam depth

Elastic bending stress changes linearly with distance from the neutral axis, changing sign across the neutral axis and reaching extrema at the outer fibers.

Transverse shear profile in a rectangular section

For a rectangular section the elastic VQ/Ib distribution is parabolic, zero at free surfaces, and maximum at the neutral axis.

Moment of inertia, extreme fiber, and section modulus

Section modulus links geometry to extreme-fiber bending stress through S = I/c; increasing I without unnecessarily increasing c improves bending efficiency.

Beam Stress Evaluation Workflow

Key Takeaways

Flexural Stress ( $\sigma = My/I$ ): Proportional to the Bending Moment ( $M$ ) and distance from Neutral Axis ( $y$ ). Maximum at the extreme fibers. It assumes plane sections remain plane and a linear elastic material.
Section Modulus ( $S = I/c$ ): A geometric property indicating flexural strength. Higher $S$ means higher bending capacity.
Unsymmetrical Bending: Use superposition of the components about the principal axes.
Shear Stress ( $\tau = VQ/Ib$ ): Usually maximum at the Neutral Axis. For rectangular sections, $\tau_{\text{max}} = 1.5 V/A$ . For solid circular sections, $\tau_{\text{max}} = 4 V/3A$ .
First Moment of Area ( $Q$ ): A geometric property ( $A'\bar{y}'$ ) necessary for calculating shear stress and shear flow.
Shear Flow ( $q = VQ/I$ ): Force per unit length, essential for designing the spacing of fasteners (nails/bolts, where $F = qs$ ) or continuous welds in built-up shapes.
Shear Center: The point where a transverse load must be applied to produce bending without torsion. For thin-walled unsymmetrical open sections (like C-channels), it often lies outside the material itself.
Principal Stresses: Internal points experience combined normal and shear stresses, so principal stresses must be evaluated to find the critical state.
Composite Sections: Transform multiple materials into a single equivalent material using the Modular Ratio ( $n = E_2/E_1$ ).

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