Mechanics of Materials: Beam Bending

Beam bending sits at the center of mechanics of materials because it connects loads you can see to stresses and deflections you must control. Whether the “beam” is a floor joist, a machine frame member, a crane boom, or a circuit-board support, the workflow is the same: convert applied loads into internal shear forces and bending moments, then translate those internal actions into stresses and deformations.

This article covers the core tools used in introductory and practical beam analysis: shear and moment diagrams, the flexure formula, and common beam deflection calculations.

What a “beam” model assumes

A beam is a slender structural member where length is much larger than cross-section dimensions. Standard beam-bending theory (often called Euler-Bernoulli beam theory) rests on assumptions that work well for many engineering members:

• Material is linear elastic (Hooke’s law) and homogeneous.
• Cross-sections remain plane and perpendicular to the neutral axis after bending.
• Deflections and rotations are small enough that geometry does not change significantly.
• Shear deformation is often neglected for slender beams (acceptable in many cases, less so for deep or short beams).

These assumptions let us link loads, internal forces, stresses, and deflections with relatively simple equations.

From external loading to internal actions

Loads and support reactions

Real beams typically have loads such as:

• Concentrated forces (point loads)
• Distributed loads (uniform or varying)
• Applied couples (moments)

Before drawing internal-force diagrams, you solve for support reactions using static equilibrium:

• $\sum F_x=0$
• $\sum F_y=0$
• $\sum M=0$

For many bending problems, the dominant equilibrium equations are in the vertical direction and moments about a convenient point.

Shear force and bending moment: what they mean

At any cross-section, the beam “resists” the external loads through internal resultants:

• Shear force $V(x)$: the net transverse internal force at a cut section.
• Bending moment $M(x)$: the net internal moment at a cut section.

These are functions of position $x$ along the beam. Their sign convention varies by textbook and industry; the key is to choose a consistent convention and stick with it.

Shear and moment diagrams

Shear and moment diagrams are graphical summaries of how $V(x)$ and $M(x)$ vary along the beam length. They provide immediate insight into where stresses peak and where deflection tends to be largest.

How to build them (a practical method)

1. Find reactions at supports from equilibrium.
2. Move left to right along the beam, tracking changes in shear and moment.
3. Use the following relationships to connect load, shear, and moment:

$$\frac{dV}{dx}=-w(x)$$

$$\frac{dM}{dx}=V(x)$$

Here, $w(x)$ is the distributed load intensity (positive downward in many conventions).

Useful diagram rules

• A point load causes a jump in the shear diagram equal to the load magnitude.
• A distributed load causes the shear diagram to slope. A uniform load produces a straight-line slope in $V(x)$.
• A point moment causes a jump in the moment diagram equal to the applied moment.
• The moment diagram slope equals the shear value. Where $V(x)=0$, $M(x)$ often reaches a local maximum or minimum.

Why these diagrams matter

The maximum bending stress typically occurs where the magnitude of bending moment $|M|$ is greatest. Similarly, large shear stresses occur where $|V|$ is large, especially in web-like sections (I-beams) or near supports.

Bending stress: the flexure formula

Once you know the internal bending moment at a section, the classic flexure formula gives normal stress due to bending:

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

Where:

• $\sigma$ is the bending normal stress at a point in the cross-section
• $M$ is the internal bending moment at the section
• $y$ is the distance from the neutral axis to the point of interest (positive in one direction)
• $I$ is the second moment of area (area moment of inertia) about the neutral axis

Key implications

• Stress varies linearly with $y$. It is zero at the neutral axis and largest at the extreme fibers.
• For symmetric sections under pure bending, the neutral axis passes through the centroid.
• For a given moment $M$, increasing $I$ reduces stress. This is why structural shapes put material far from the neutral axis.

Section modulus (a design-friendly form)

Engineers often use the section modulus $S=I/c$, where $c$ is the distance from the neutral axis to the extreme fiber. Then:

$${\sigma }_{max}=\frac{M}{S}$$

This makes it easy to compare cross-sections for bending strength.

Shear stress in beams (context you should not ignore)

Although the prompt focus is bending, shear often controls details like web thickness, adhesive joints, or fasteners. For many prismatic beams, the transverse shear stress can be estimated with:

$$\tau =\frac{VQ}{Ib}$$

Where:

• $\tau$ is shear stress at the location of interest
• $V$ is internal shear force
• $Q$ is the first moment of area of the portion of the cross-section above (or below) the point
• $b$ is the local width of the cross-section at the point

A common practical takeaway: shear stress is typically highest near the neutral axis for rectangular sections and concentrated in the web for I-shaped sections.

Beam deflection: stiffness and serviceability

Strength checks (stress) prevent failure; deflection checks protect performance. Excessive deflection can cause cracked finishes, misalignment of machinery, ponding on roofs, or unacceptable vibration.

The moment-curvature relationship

For Euler-Bernoulli bending in linear elasticity:

$$\kappa =\frac{1}{\rho }=\frac{M}{EI}$$

Curvature $\kappa$ relates to deflection $v(x)$ by:

$$\frac{d^{2}v}{d{x}^2}=\frac{M(x)}{EI}$$

Where:

• $E$ is Young’s modulus
• $I$ is the second moment of area about the neutral axis
• $v(x)$ is transverse deflection

This is the foundation of most beam deflection calculations.

Common solution approaches

Double integration method

1. Determine $M(x)$ piecewise from the moment diagram.
2. Integrate twice: $v^{\prime \prime }(x)=M(x)/(EI)$ to get slope $v^{\prime }(x)$ and deflection $v(x)$.
3. Apply boundary conditions, such as:

• Simply supported end: deflection $v=0$ at the support.
• Fixed end: deflection $v=0$ and slope $v^{\prime }=0$.

Singularity functions (Macaulay method)

A compact way to write piecewise loading and integrate without splitting into many intervals. Common in hand analysis for multiple point loads.

Superposition

For linear elastic beams, deflections from different loads add. This is especially helpful when using standard deflection formulas (for example, simply supported beam with uniform load plus a midspan point load).

What controls deflection most

• Span length has a strong effect; deflection often scales with $L^3$ or $L^4$ depending on loading.
• Stiffness is governed by $EI$. Materials with higher $E$ (steel vs wood) and sections with higher $I$ deflect less.
• Loading distribution matters; a uniform load tends to produce a smoother moment profile, while concentrated loads can create sharper peaks.

Putting it together: a typical beam-bending workflow

1. Define geometry and supports (simply supported, cantilever, fixed-fixed).
2. Model loads (point, distributed, applied moment) and compute reactions.
3. Draw shear and moment diagrams to locate peak $V$ and $M$ and understand where changes occur.
4. Compute bending stresses using $\sigma =My/I$ (or ${\sigma }_{max}=M/S$) at critical sections.
5. Check shear stress where relevant, especially near supports and in webs.
6. Compute deflection using $v^{\prime \prime }=M/(EI)$ with proper boundary conditions, then compare to allowable deflection criteria for the application.
7. Iterate design by changing material (affecting $E$ and allowable stress), cross-section (affecting $I$ and $S$), or support conditions.

Practical insight: what engineers watch for

• Maximum moment locations often control bending stress; on a simply supported beam with symmetric loading, that is frequently near midspan.
• Discontinuities in loading (point loads, point moments) create abrupt changes in $V$ or $M$ and deserve careful section checks.
• Serviceability limits can govern even when stresses are low, particularly in long, slender beams.
• Beam theory limits show up in deep beams, short spans, or when shear deformation is significant. In those cases, refined methods (such as Timoshenko beam theory or finite element analysis) may be appropriate.

Beam bending is powerful because it is systematic: equilibrium