Normal stresses in beams are crucial for understanding how structures handle loads. These stresses vary linearly from the neutral axis, with maximum values at the extreme fibers. Knowing their distribution helps engineers design safer, more efficient structures.

The flexure formula is key for calculating normal stresses in beams. It relates stress to bending moment, distance from the neutral axis, and moment of inertia. Understanding this relationship is essential for analyzing beam behavior under different loading conditions.

Normal Stress in Beams

Stress Distribution

• Normal stress in a beam varies linearly from the neutral axis to the extreme fibers of the cross-section
• The distribution of normal stress in a beam depends on the magnitude and direction of the applied bending moment
• Compressive stresses develop on the side of the beam where the bending moment causes the fibers to shorten (top side in a simply supported beam with a downward load)
• Tensile stresses occur on the side where the fibers elongate (bottom side in a simply supported beam with a downward load)
• The magnitude of normal stress at any point in the beam cross-section is directly proportional to its distance from the neutral axis

Maximum Stress Location

• The maximum normal stresses occur at the extreme fibers of the beam cross-section, which are furthest from the neutral axis
• In a rectangular beam, the maximum normal stresses occur at the top and bottom surfaces
• For an I-beam, the maximum normal stresses are located at the top and bottom flanges
• The maximum tensile and compressive stresses have equal magnitudes but opposite signs
• Identifying the location of maximum normal stress is crucial for determining the critical points in a beam design

Neutral Axis Location

Symmetric Cross-Sections

• The neutral axis is the line in the beam cross-section where the normal stress is zero
• For symmetric cross-sections, the neutral axis passes through the centroid of the cross-section
• Examples of symmetric cross-sections include rectangular, circular, and I-shaped beams
• In a rectangular beam, the neutral axis is located at the geometric center of the cross-section
• For an I-beam, the neutral axis coincides with the horizontal centerline of the web

Unsymmetric Cross-Sections

• In unsymmetric cross-sections, the neutral axis location can be determined using the first moment of area concept
• Examples of unsymmetric cross-sections include T-shaped and L-shaped beams
• The first moment of area is calculated by multiplying each cross-sectional area by its distance from an arbitrary axis and summing the results
• The neutral axis location is found by setting the first moment of area equal to zero and solving for the distance from the arbitrary axis
• For a T-beam, the neutral axis is typically located below the geometric center due to the larger area of the flange compared to the web
• In an L-shaped beam, the neutral axis is located closer to the corner with the larger moment of area

Maximum Normal Stress Calculation

Flexure Formula

• The maximum normal stress in a beam can be calculated using the flexure formula: $\sigma = My / I$
• $\sigma$ represents the normal stress at a given point
• $M$ represents the bending moment at the cross-section
• $y$ represents the distance from the neutral axis to the point of interest
• $I$ represents the moment of inertia of the cross-section about the neutral axis
• To find the maximum normal stress, substitute the maximum distance from the neutral axis ($y_{max}$) and the corresponding bending moment ($M$) into the flexure formula

Moment of Inertia

• The moment of inertia ($I$) is a geometric property that depends on the shape and dimensions of the beam cross-section
• For common cross-sectional shapes, the moment of inertia can be found using standard formulas
• Example formulas for moment of inertia:
  • Rectangle: $I = \frac{bh^3}{12}$
  • Circle: $I = \frac{\pi r^4}{4}$
  • I-shape: $I = \frac{bh^3}{12} - \frac{b_1h_1^3}{12}$
• For irregular cross-sections, the moment of inertia can be calculated using the parallel axis theorem or by integration
• The parallel axis theorem states that the moment of inertia about any axis parallel to the centroidal axis is equal to the moment of inertia about the centroidal axis plus the product of the area and the square of the distance between the axes

Bending Moment and Normal Stress

Proportional Relationship

• The bending moment in a beam is directly proportional to the normal stress at any given point in the cross-section
• As the bending moment increases, the normal stress at any point in the beam cross-section also increases proportionally
• Doubling the bending moment will result in a doubling of the normal stress at any given point
• The proportional relationship between bending moment and normal stress is described by the flexure formula ($\sigma = My / I$)

Stress Distribution along the Beam

• The direction of the bending moment (positive or negative) determines the nature of the normal stress (tensile or compressive) at a given point in the beam cross-section
• A positive bending moment induces tensile stress at the bottom fibers and compressive stress at the top fibers of the beam
• A negative bending moment induces compressive stress at the bottom fibers and tensile stress at the top fibers of the beam
• The distribution of bending moment along the length of the beam influences the distribution of normal stress in the beam
• Points of maximum bending moment in a beam correspond to the locations of maximum normal stress, which are critical for design considerations and failure analysis
• In a simply supported beam with a uniformly distributed load, the maximum bending moment and normal stress occur at the midspan of the beam
• For a cantilever beam with a concentrated load at the free end, the maximum bending moment and normal stress are located at the fixed support