🔗Statics and Strength of Materials Unit 12 – Stresses in Beams

Key Concepts and Definitions

• Beams are structural elements that resist loads primarily by bending and are characterized by their length being much greater than their width and depth
• Loads applied to beams can be classified as concentrated (applied at a single point), distributed (spread over a length), or moment (a force applied at a distance causing rotation)
• Supports for beams include simple supports (allows rotation but not translation), fixed supports (prevents both rotation and translation), and cantilever (fixed at one end and free at the other)
• Shear force ($V$) represents the internal force that resists the tendency of one part of the beam to slide past another part due to applied loads
• Bending moment ($M$) is the internal moment that resists the tendency of the beam to bend or curve under applied loads
  • Bending moment is calculated by taking the sum of the moments about a point along the beam
• Normal stress ($\sigma$) in beams is the stress that acts perpendicular to the cross-section of the beam and is caused by the bending moment
• Shear stress ($\tau$) in beams is the stress that acts parallel to the cross-section of the beam and is caused by the shear force
• Beam deflection refers to the vertical displacement of a beam under applied loads and is an important consideration in design to ensure serviceability and prevent excessive deformation

Types of Beams and Loads

• Simply supported beams are supported at both ends and are free to rotate and deflect under load
  • Examples include a beam supported on two walls or a bridge spanning between two piers
• Cantilever beams are fixed at one end and free at the other, with the free end able to deflect and rotate under load (diving board, balcony)
• Continuous beams are supported at more than two points and have multiple spans, resulting in a more complex analysis due to the internal forces and moments at the supports
• Truss beams are composed of interconnected triangular elements and are commonly used in bridges and roof structures to efficiently carry loads
• Concentrated loads are applied at a single point along the beam and result in a sudden change in shear force and a local peak in bending moment at the point of application
• Distributed loads are spread over a length of the beam and can be uniform (constant intensity along the length) or non-uniform (varying intensity)
  • Examples of distributed loads include the weight of the beam itself, snow on a roof, or water pressure on a dam
• Moment loads are applied as a force at a distance from the beam, causing a pure bending effect without shear force (a force applied to a lever arm)

Shear Force and Bending Moment

• Shear force and bending moment diagrams are graphical representations of the internal shear forces and bending moments along the length of a beam
• Shear force at a point along the beam is equal to the sum of the vertical forces acting on either side of that point
  • A positive shear force indicates that the portion of the beam to the left of the point tends to slide upward relative to the portion on the right
• Bending moment at a point is equal to the sum of the moments acting on either side of that point, with counterclockwise moments considered positive and clockwise moments negative
• The relationship between shear force and bending moment is given by $\frac{dM}{dx} = V$, meaning that the slope of the bending moment diagram at any point is equal to the shear force at that point
• Concentrated loads cause a sudden change (step) in the shear force diagram and a local peak (kink) in the bending moment diagram at the point of application
• Distributed loads result in a gradual change (slope) in the shear force diagram and a parabolic curve in the bending moment diagram over the loaded length
• The maximum bending moment in a beam occurs where the shear force crosses zero, and the maximum shear force occurs at the supports or at concentrated load locations

Normal Stress in Beams

• Normal stress in beams is caused by the bending moment and varies linearly across the height of the beam, with maximum compression on one side and maximum tension on the other
• The magnitude of the normal stress at a point in the beam is given by the flexure formula: $\sigma = \frac{My}{I}$, where $M$ is the bending moment, $y$ is the distance from the neutral axis, and $I$ is the moment of inertia of the cross-section
• The neutral axis is the line in the cross-section where the normal stress is zero and passes through the centroid of the cross-section
• The moment of inertia ($I$) is a geometric property of the cross-section that measures its resistance to bending and depends on the shape and size of the cross-section
  • For a rectangular cross-section, $I = \frac{bh^3}{12}$, where $b$ is the width and $h$ is the height
• The maximum normal stress occurs at the extreme fibers (top and bottom) of the beam, where $y$ is maximum
• To design a beam for strength, the maximum normal stress must be kept below the allowable stress of the material, which is typically based on the yield strength divided by a factor of safety

Shear Stress in Beams

• Shear stress in beams is caused by the shear force and acts parallel to the cross-section of the beam
• The distribution of shear stress across the cross-section is not uniform and depends on the shape of the cross-section
  • For a rectangular cross-section, the shear stress varies parabolically, with maximum values at the neutral axis and zero at the top and bottom surfaces
• The maximum shear stress in a rectangular beam is given by $\tau_{max} = \frac{3V}{2A}$, where $V$ is the shear force and $A$ is the cross-sectional area
• For I-beams and other non-rectangular cross-sections, the shear stress distribution is more complex and requires the use of the shear flow formula: $q = \frac{VQ}{Ib}$, where $q$ is the shear flow, $Q$ is the first moment of area, $I$ is the moment of inertia, and $b$ is the width at the point of interest
• Shear stress is generally less critical than normal stress in beam design, but it can be important in short, deep beams or near supports and concentrated loads
• To prevent shear failure, the maximum shear stress must be kept below the allowable shear stress of the material, which is typically based on the shear strength divided by a factor of safety

Beam Deflection

• Beam deflection is the vertical displacement of a beam under applied loads and is an important consideration in design for serviceability and functionality
• The deflection of a beam depends on the load, the beam's geometry, and the material properties (modulus of elasticity)
• The differential equation governing beam deflection is given by $\frac{d^2}{dx^2}(EI\frac{d^2v}{dx^2}) = w(x)$, where $E$ is the modulus of elasticity, $I$ is the moment of inertia, $v$ is the deflection, and $w(x)$ is the distributed load
• For simple cases, such as a simply supported beam with a concentrated load at midspan, the maximum deflection can be calculated using the formula $\delta_{max} = \frac{PL^3}{48EI}$, where $P$ is the load and $L$ is the beam length
• More complex loading and support conditions require the use of the moment-area method or integration of the differential equation to determine the deflection
  • The moment-area method involves calculating the area and centroid of the bending moment diagram to find the slope and deflection at various points along the beam
• Excessive deflection can cause problems such as cracking of finishes, improper drainage, or interference with connected elements, so building codes and design standards often specify maximum allowable deflection limits (span/360)

Design Considerations and Applications

• The choice of beam material depends on factors such as strength, stiffness, durability, cost, and aesthetics
  • Common materials include steel, reinforced concrete, timber, and aluminum
• Cross-sectional shape is selected based on the required moment of inertia, section modulus, and shear area, as well as constructability and economy
  • I-beams, wide-flange beams, and box beams are efficient shapes for resisting bending and shear
• Lateral-torsional buckling is a failure mode in which a beam twists and deflects laterally under load, and it is a concern for long, slender beams with insufficient lateral support
  • Providing lateral bracing or reducing the unbraced length can increase the buckling resistance
• Composite beams, such as steel-concrete or timber-concrete, can be used to optimize the strengths of each material and improve overall performance
• Prestressed concrete beams are designed with a compressive force applied before loading to counteract the tensile stresses caused by bending, allowing for longer spans and thinner sections
• Continuous beams offer advantages over simply supported beams, such as reduced maximum moments and deflections, but require careful design of the supports and consideration of the negative moments over the supports
• Beams are critical elements in a wide range of structures, including buildings, bridges, towers, and machines, and their design directly impacts the safety, functionality, and economy of these systems

Problem-Solving Techniques

• Identify the beam type, support conditions, and loading scenario, and create a clear diagram with all relevant dimensions and labels
• Determine the reactions at the supports by applying the equations of equilibrium ($\Sigma F_x = 0$, $\Sigma F_y = 0$, $\Sigma M = 0$)
  • For statically indeterminate beams, additional compatibility equations or approximations may be needed
• Divide the beam into segments based on changes in loading or cross-section, and establish the coordinate system and sign conventions for each segment
• Calculate the shear force and bending moment at critical points (supports, load locations, and cross-section changes) and plot the shear force and bending moment diagrams
  • Use the relationships $\frac{dV}{dx} = -w(x)$ and $\frac{dM}{dx} = V$ to check the diagrams for consistency and accuracy
• Apply the flexure formula ($\sigma = \frac{My}{I}$) and shear stress formula ($\tau = \frac{VQ}{Ib}$) to determine the maximum normal and shear stresses in the beam
  • Compare these values to the allowable stresses for the material and adjust the design if necessary
• Use the differential equation ($\frac{d^2}{dx^2}(EI\frac{d^2v}{dx^2}) = w(x)$) or moment-area method to calculate the beam deflection at critical points and check against serviceability limits
• Consider other factors such as lateral-torsional buckling, shear deformation, and local stability, and apply appropriate analysis methods or design provisions as needed
• Verify the results using alternative methods, such as finite element analysis or experimental testing, and iterate the design process as necessary to achieve an optimal solution