Structural Engineering Concepts: Supports, Stress, Vibration
Types of Structural Supports
1. Pinned Support
A pinned support, also known as a hinge support, allows rotation but restricts translational movement in any direction.
Characteristics:
- Resists horizontal and vertical forces
- Allows rotation
- Commonly represented in diagrams as a triangle or a hinge symbol
Applications:
- Trusses: Pinned supports are often used in truss structures, such as bridges, to allow for some degree of rotation while ensuring stability.
- Beams: In structural frames, pinned supports provide stability while accommodating slight rotations due to loads.
2. Fixed Support
Fixed supports restrict all forms of movement: horizontal, vertical, and rotational. This type of support provides the highest degree of restraint and is capable of resisting moments as well as forces.
Characteristics:
- Resists horizontal, vertical, and rotational forces
- No movement is allowed
- Represented by a filled triangle or a solid block in diagrams
3. Roller Support
Roller supports allow movement in one direction while restricting movement in the perpendicular direction. They cannot resist horizontal forces or moments but can resist vertical forces.
Characteristics:
- Resists vertical forces only
- Allows horizontal movement
- Commonly represented as a roller or wheel symbol in diagrams
Applications:
- Bridges: Roller supports are often used in bridges to allow for thermal expansion and contraction.
- Long Beams: Structures experiencing temperature changes or shrinkage often incorporate roller supports.
4. Simple Support
Simple supports are a combination of pinned and roller supports.
Characteristics:
- Resists vertical forces
- Allows rotation and horizontal movement
- Represented as a triangle
Applications:
- Beam Structures: Simple supports are often used in idealized beam analyses for simplicity.
Shear Force and Bending Moment
Conceptual Overview
Shear Force (V):
- Definition: The internal force acting parallel to the cross-section of a beam, tending to cause one part of the beam to slide past the other.
- Sign Convention:
- Positive shear force: Tends to cause clockwise rotation of the beam segment.
- Negative shear force: Tends to cause counter-clockwise rotation of the beam segment.
Bending Moment (M):
- Definition: The internal moment acting on the cross-section of a beam, tending to bend the beam.
- Sign Convention:
- Positive bending moment: Causes the beam to bend in a “U” shape (concave downwards).
- Negative bending moment: Causes the beam to bend in an “inverted U” shape (concave upwards).
Relationship Between Loading, Shear Force, and Bending Moment:
- Loading and Shear Force:
- A point load causes a sudden jump in the shear force diagram.
- A uniformly distributed load (UDL) causes a linear variation in the shear force diagram.
- Shear Force and Bending Moment:
- The rate of change of shear force along the length of the beam is equal to the intensity of the distributed load.
- The area under the shear force diagram at any point gives the magnitude of the bending moment at that point.
SFD and BMD with Different Loads
- BMD: Parabolically varying, with maximum values at the points of maximum deflection (usually at the center).
Key Points
- The shape of the SFD and BMD diagrams depends on the type of loading and support conditions.
- Understanding these diagrams is crucial for determining the maximum shear force and bending moment, which are used in structural design.
SFD and BMD for Different Conditions
1. Cantilever Beam:
- Point Load:
- SFD: Constant and equal to the load magnitude between the load and the fixed support, zero elsewhere.
- BMD: Linearly varying from zero at the free end to a maximum value at the fixed support.
- UDL:
- SFD: Linearly varying from zero at the free end to a maximum value at the fixed support.
- BMD: Parabolically varying, with the maximum value at the fixed support.
2. Simply Supported Beam:
- Point Load:
- SFD: Constant between supports, with jumps at the load points.
- BMD: Linearly varying between supports, with maximum values at the load points.
- UDL:
- SFD: Linearly varying between supports.
- BMD: Parabolically varying, with maximum values at the center.
Moment of Inertia (I)
- Definition: A measure of a cross-section’s resistance to bending. It depends on the shape and dimensions of the cross-section.
- Formulas:
- Rectangle: I = (b * h³) / 12, where b is the width and h is the height
- Circle: I = (π * d⁴) / 64, where d is the diameter
- Example: Rectangular Section
- Determine the maximum bending moment (M). This depends on the loading and support conditions of the beam.
- Calculate the moment of inertia (I) of the rectangular section.
- Determine the distance from the neutral axis to the outermost fiber (y_max). For a rectangle, y_max = h/2.
- Apply the flexure formula: σ_max = M * (h/2) / [(b * h³) / 12]
- Example: Circular Section
- Determine the maximum bending moment (M).
- Calculate the moment of inertia (I) of the circular section.
- Determine the distance from the neutral axis (y_max). For a circle, y_max = d/2.
Load, Stress, and Strain
- Load: An external force applied to a body.
- Stress: The internal resistance force per unit area developed within a body to resist the applied load.
- Formula: Stress (σ) = Force (F) / Area (A)
- Strain: The deformation or change in dimensions of a body caused by the applied load. It is a dimensionless quantity.
- Formula: Strain (ε) = Change in dimension (ΔL) / Original dimension (L)
Types of Stress and Strain
- Tensile Stress and Strain: Occur when a force tends to pull a body apart.
- Compressive Stress and Strain: Occur when a force tends to push a body together.
- Shear Stress and Strain: Occur when forces act parallel to the surface of a body, causing it to deform by sliding or shearing.
Material Properties
- Ductile Materials: Can undergo significant plastic deformation before failure. Examples: Steel, copper.
- Brittle Materials: Fail suddenly with little or no plastic deformation. Examples: Glass, concrete.
Elastic and Plastic Properties
- Elasticity: The ability of a material to return to its original shape and size after the removal of an applied load.
- Plasticity: The ability of a material to undergo permanent deformation without failure.
Hooke’s Law
- States that within the elastic limit, stress is directly proportional to strain.
- Formula: Stress (σ) = E * Strain (ε)
- E is the Young’s Modulus of Elasticity, a constant of proportionality.
Young’s Modulus of Elasticity
- A measure of a material’s stiffness or resistance to elastic deformation.
- It is the slope of the stress-strain curve in the elastic region.
Nominal Stress
- Stress calculated based on the original cross-sectional area of the material, even after deformation.
Yield Point
- Stress at which a material begins to exhibit plastic deformation.
Types of Mechanical Vibration
- Free Vibration, Forced Vibration, Resonance
Key Concepts in Mechanical Vibration:
- Natural Frequency: The frequency at which a system vibrates freely after being disturbed.
- Damping: The dissipation of energy in a vibrating system, which reduces the amplitude of vibration over time.
- Resonance: A phenomenon that can lead to excessive vibrations and potential damage to the system.
- Vibration Isolation: Techniques used to reduce the transmission of vibrations, such as using isolators or vibration absorbers.
Applications of Vibration Analysis:
- Structural Engineering: To assess the dynamic response of structures to earthquakes, wind loads, and other excitations.
- Mechanical Design: To design machines and components that minimize vibration and noise.
- Condition Monitoring: To detect faults and predict failures in machinery based on vibration measurements.
- Biomedical Engineering: To study the dynamics of biological systems, such as the human body.
Stress, Deformation, Stiffness, and Angle of Twist (Springs)
- Stress: Primarily shear stress within the spring wire due to the twisting action caused by the applied load. This stress is a function of the applied force, wire diameter, and spring geometry.
- Deformation: The change in length of the spring under load. It is directly proportional to the applied force and inversely proportional to the spring stiffness.
- Stiffness (k): A measure of the spring’s resistance to deformation. It is defined as the force required to produce a unit deflection (k = Force/Deflection).
- Angle of Twist: The angular deformation of the spring wire as it coils or uncoils under load. This angle is related to the shear stress and the material properties of the spring wire.
Strain Energy
Strain energy is the potential energy stored within the spring due to its deformation. It is equal to the work done in deforming the spring and can be calculated as:
Strain Energy = (1/2) * Force * Deflection