Stresses in Beams - Theory & Concepts

Stresses in Beams

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 and Bending

Flexural Stress (Bending)

Flexural Stress: Flexural stress results from the bending moment in the beam. It varies linearly from zero at the neutral axis (NA) to a maximum at the extreme fibers.

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

where:

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

Maximum Flexural Stress

Occurs at the farthest distance (

c

) from the neutral axis:

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

where

S = I/c

is the 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

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

where:

$M_y, M_z$ are the moment components about the $y$ and $z$ principal axes.
$I_y, I_z$ are the moments of inertia about the principal axes.
$y, z$ are the coordinates of the point where stress is being calculated.

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

Shear Stress in Beams

Shearing Stress in Beams

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

\tau = \frac{VQ}{Ib}

where:

$\tau$ is the shear stress at a specific layer.
$V$ is the vertical shear force at the section.
$Q$ is the First Moment of Area of the portion of the section above (or below) the layer where shear is being calculated ( $Q = A'\bar{y}'$ ).
$I$ is the moment of inertia of the entire section.
$b$ is the 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

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

This maximum value occurs at the Neutral Axis.

Built-Up Beams and Shear Flow (q)

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 design the fasteners or glue joints.

q = \frac{VQ}{I}

Where

q

is the shear flow in units of force per length (e.g., N/mm, kN/m). For fasteners like bolts or nails spaced at a distance

s

, the shear force carried by one fastener is

F = q \times s

Shear Center

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.

The Shear Center (or center of flexure) is the specific point in the cross-section through which the transverse load must pass to cause pure bending without any torsion.

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 & Composite Beams

Principal Stresses in Beams

While the maximum flexural stress (

\sigma_{max}

) occurs at the extreme outer fibers (where

\tau = 0

), and the maximum shear stress (

\tau_{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:

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

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

where

\sigma

is the flexural stress and

\tau

is the transverse shear stress at that specific point.

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.

Modular Ratio ( $n$ ):

n = \frac{E_2}{E_1}

where

E_2

is the modulus of elasticity of the material being transformed, and

E_1

is the modulus of the base material. The width of Material 2 is then multiplied by

n

to find its equivalent width in Material 1:

b_{equivalent} = n \cdot b_2

Once the entire section is transformed to Material 1, you can calculate the new neutral axis location (

\bar{y}

) and the transformed moment of inertia (

I_{tr}

). The stress in the base material is

\sigma_1 = \frac{M y}{I_{tr}}

, while the stress in the transformed material is

\sigma_2 = n \frac{M y}{I_{tr}}

Interactive Visualization

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.

Flexural Stress Distribution

Beam Cross-Section

200mm

100mm

Compression

Tension

Stress Profile (Depth vs Stress)

Loading chart...

Bending Moment (

M

): 15 kN·m

Width (

b

): 100 mm

Depth (

h

): 200 mm

Moment of Inertia (

I

)

66.67 × 10⁶ mm⁴

Section Modulus (

S

)

666.67 × 10³ mm³

Max Flexural Stress (

\sigma_{max}

)

22.50 MPa

Key Takeaways

Flexural Stress ( $\sigma = My/I$ ): Proportional to the Bending Moment ( $M$ ) and distance from Neutral Axis ( $y$ ). Maximum at the extreme fibers.
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.
Shear Flow ( $q = VQ/I$ ): Essential for designing fasteners 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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