Beam Bending and Loading Systems

Introduction to Beam Bending

• This module covers the topic of beam bending, building on the previous modules that examined the effects of axial forces and the resulting stresses and strains in bars
• The previous modules discussed the effects of torsional moments on bars, and this module will examine other aspects of loading systems acting on beams
• After this module, the concept of beams will be well-understood, including the definition of a beam and the various forces acting on it
• Different types of beam supports and beam configurations will also be covered
• The concepts of shear force and bending moment will be explained, and examples will be provided on evaluating the reactive forces for different beam loading conditions

Understanding Beams

• A beam is a structural member that is loaded transversely (perpendicular) to its longitudinal axis
• Beams are different from bars which were loaded axially (along the length) in previous modules
• While bars experienced axial forces, beams experience transverse loads and the resulting bending moments
• The vectorial direction of the loads and moments acting on beams is perpendicular to the beam's longitudinal axis

Beam Supports

• Beams can be supported in different ways, such as:

  • Pinned or hinged support - allows rotation but no translation
  • Roller support - allows translation in one direction but no rotation
  • Fixed support - completely restricts translation and rotation

• The type of support affects the reactive forces developed at the supports

• Pinned and roller supports have 2-3 unknown reactive forces, which can be determined using the equations of static equilibrium

• Fixed supports have 3 unknown reactive forces that need to be evaluated

Cantilever and Overhanging Beams

• A cantilever beam is supported at one end and free at the other
• It develops 3 unknown reactive forces - a vertical force, a horizontal force, and a moment
• An overhanging beam is similar, with one end supported and the other end extending past the support
• The overhang can induce bending moments and rotation, which the roller support cannot resist

Types of Beam Loading

• Concentrated (or point) load - a load applied over a small area

• Distributed (or linear) load - a load that varies linearly along the beam length

• The beam may experience a combination of different loading types along its length### Overview of Beams and Supports

• Beams can experience various types of loading and support conditions, which result in different internal forces and moments

• Axial force, shear force, and bending moment are the key internal forces that develop in beams due to external loads

• Beams supported at both ends (simply supported) can experience different loading patterns like concentrated, linearly varying, or uniformly distributed loads

• Cantilever beams are fixed at one end and free at the other, and can experience concentrated or moment loads

Analyzing Internal Forces in Beams

• To analyze the internal forces, we first draw the free-body diagram of the beam section of interest
• The support reactions (e.g. RA, HA, RB) are determined using the equilibrium equations: ΣFx = 0, ΣFy = 0, and ΣM = 0
• The shear force (V) and bending moment (M) diagrams are then plotted along the length of the beam
• The shear force represents the transverse internal force, while the bending moment counters the external bending effects

Example Problems

• For a simply supported beam with an overhang, the bending moment at the midspan is zero when the overhang length is b/L = 0.5
• To find the shear force and bending moment at a section 2 m from the left support, we need to determine the support reactions first
• The support reactions can be calculated using the equilibrium equations, which then allows us to find the desired internal forces at the section of interest

Key Takeaways

• Beams can be supported in different ways (pinned, roller, fixed) which affects the internal force distributions
• The sign conventions for shear force, axial force, and bending moment need to be clearly understood
• Analyzing the internal forces is a crucial step before evaluating the stresses in beam members

Introduction to Beam Bending

• Beam bending builds on previous modules that examined axial forces and resulting stresses and strains in bars
• This module examines other aspects of loading systems acting on beams

Understanding Beams

• A beam is a structural member loaded transversely (perpendicular) to its longitudinal axis
• Beams experience transverse loads and resulting bending moments, unlike bars which experienced axial forces
• The vectorial direction of loads and moments acting on beams is perpendicular to the beam's longitudinal axis

Beam Supports

• Beams can be supported in different ways:
  • Pinned or hinged support: allows rotation but no translation
  • Roller support: allows translation in one direction but no rotation
  • Fixed support: completely restricts translation and rotation
• The type of support affects the reactive forces developed at the supports

Cantilever and Overhanging Beams

• A cantilever beam is supported at one end and free at the other, developing 3 unknown reactive forces
• An overhanging beam is similar, with one end supported and the other end extending past the support, inducing bending moments and rotation

Types of Beam Loading

• Concentrated (or point) load: a load applied over a small area
• Distributed (or linear) load: a load that varies linearly along the beam length

Overview of Beams and Supports

• Beams experience various types of loading and support conditions, resulting in different internal forces and moments
• Axial force, shear force, and bending moment are the key internal forces that develop in beams due to external loads

Analyzing Internal Forces in Beams

• To analyze internal forces, draw the free-body diagram of the beam section of interest
• Determine support reactions (e.g. RA, HA, RB) using the equilibrium equations: ΣFx = 0, ΣFy = 0, and ΣM = 0
• Plot shear force (V) and bending moment (M) diagrams along the beam length
• Shear force represents the transverse internal force, while bending moment counters the external bending effects