12.2 Shear stresses in beams

Shear stresses in beams are crucial for understanding how beams handle loads. These stresses act parallel to a beam's cross-section and vary across it, with the maximum often occurring at the neutral axis or where the cross-section is thinnest.

Calculating shear stress involves factors like shear force, beam geometry, and material properties. By grasping these concepts, you'll be better equipped to analyze beam behavior and design structures that can withstand applied loads without failing due to shear stresses.

Shear Stress in Beams

Shear Stress Distribution

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• Shear stress acts as an internal force per unit area parallel to the cross-section of a beam, resulting from applied shear forces
• The distribution of shear stress across a beam's cross-section varies non-uniformly depending on the cross-section shape and location along the beam's length
• Shear stress diagrams represent the variation of shear stress across a beam's cross-section at a given location
• The magnitude of shear stress at any point in a beam's cross-section is proportional to the distance from the neutral axis and the acting shear force at that location

Factors Influencing Shear Stress

• Material properties, such as the modulus of rigidity (shear modulus), influence the shear stress distribution by relating shear stress to shear strain
• The maximum shear stress in a beam occurs at the neutral axis of the cross-section, where bending stress is zero
• Symmetrical cross-sections (rectangular or circular beams) experience maximum shear stress at the centroid of the cross-section
• I-beams and cross-sections with thin webs typically have maximum shear stress located at the intersection of the web and flanges, where cross-sectional area is smallest

Maximum Shear Stress Location

Determining Maximum Shear Stress Location

• The location of maximum shear stress in a beam cross-section depends on the cross-section shape and shear stress distribution
• Analyzing the shear stress distribution across the cross-section helps identify the point of highest shear stress
• Symmetrical cross-sections (rectangular or circular beams) have maximum shear stress at the neutral axis, located at the cross-section centroid
• I-beams and cross-sections with thin webs typically have maximum shear stress at the web-flange intersection, where cross-sectional area is smallest

Factors Affecting Maximum Shear Stress Location

• Cross-section geometry plays a crucial role in determining the location of maximum shear stress
• The distribution of shear stress across the cross-section influences the location of maximum shear stress
• Beam loading conditions and support reactions impact the shear force distribution, affecting the location of maximum shear stress
• Changes in cross-section dimensions along the beam's length can shift the location of maximum shear stress

Shear Stress Calculation

Maximum Shear Stress Equation

• The maximum shear stress in a beam is calculated using the equation: $\tau_{max} = \frac{VQ}{It}$
  • $\tau_{max}$: maximum shear stress
  • $V$: shear force at the cross-section
  • $Q$: first moment of area
  • $I$: moment of inertia of the cross-section
  • $t$: width of the cross-section at the location of maximum shear stress

Calculating Maximum Shear Stress

• Determine the shear force ($V$) at the cross-section of interest by analyzing the beam's loading conditions and support reactions
• Calculate the first moment of area ($Q$) by taking the product of the area above or below the point of interest and the distance from that area's centroid to the neutral axis
• Identify the location of maximum shear stress based on cross-section geometry and calculate the width ($t$) at that location
• Determine the moment of inertia ($I$), a geometric property quantifying the cross-section's resistance to bending, based on cross-section shape and dimensions
• Substitute the values of $V$, $Q$, $I$, and $t$ into the maximum shear stress equation to obtain the result

Example Calculation

• For a rectangular beam with a cross-section of width $b$ and height $h$, the maximum shear stress occurs at the neutral axis:
  • $Q = \frac{bh^2}{8}$ (first moment of area)
  • $I = \frac{bh^3}{12}$ (moment of inertia)
  • $t = b$ (width at the location of maximum shear stress)
• Substituting these values into the equation: $\tau_{max} = \frac{VQ}{It} = \frac{V(\frac{bh^2}{8})}{\frac{bh^3}{12}b} = \frac{3V}{2bh}$

Shear Force vs Shear Stress

Relationship between Shear Force and Shear Stress

• Shear force is an external force acting perpendicular to the beam's axis, causing internal shear stress
• The relationship between shear force and shear stress is defined by the equation: $\tau = \frac{VQ}{It}$
  • $\tau$: shear stress at a given point
  • $V$: shear force at the cross-section
  • $Q$: first moment of area
  • $I$: moment of inertia
  • $t$: width of the cross-section at the point of interest
• Shear stress at any point in a beam's cross-section is directly proportional to the shear force acting at that location

Shear Stress Distribution and Shear Force

• The distribution of shear stress across a beam's cross-section varies depending on the cross-section shape and location along the beam's length, even with constant shear force
• Maximum shear stress in a beam occurs where the ratio of $\frac{VQ}{It}$ is highest, typically at the neutral axis for symmetrical cross-sections or at the web-flange intersection for I-beams
• Understanding the relationship between shear force and shear stress is crucial for designing beams that can withstand applied loads without failing due to shear stresses

Example: Shear Force and Shear Stress Distribution

• Consider a simply supported beam with a uniform cross-section subjected to a concentrated load at its midspan
• The shear force diagram will show a constant value from the support to the load and a constant negative value from the load to the other support
• Despite the constant shear force, the shear stress distribution will vary across the cross-section, with maximum values at the neutral axis and zero at the top and bottom surfaces