11.1 Types of beams and loads

Beams and loads are fundamental concepts in structural engineering. Understanding different beam types and support conditions is crucial for analyzing how structures transfer forces and moments. This knowledge forms the basis for creating shear force and bending moment diagrams.

Loads on beams come in various forms, from concentrated forces to distributed pressures. Recognizing these load types and their effects on beam behavior is essential for accurately determining internal forces and designing safe, efficient structures.

Beam types based on supports

Classification and statical determinacy

Top images from around the web for Classification and statical determinacy

• 

  Euler–Bernoulli beam theory - Wikipedia View original

  Is this image relevant?

• 

  File:Statically Indeterminate Beam.svg - Wikimedia Commons View original

  Is this image relevant?

• 

  Beam (structure) - Wikipedia View original

  Is this image relevant?

• 

  Euler–Bernoulli beam theory - Wikipedia View original

  Is this image relevant?

• 

  File:Statically Indeterminate Beam.svg - Wikimedia Commons View original

  Is this image relevant?

1 of 3

Top images from around the web for Classification and statical determinacy

• 

  Euler–Bernoulli beam theory - Wikipedia View original

  Is this image relevant?

• 

  File:Statically Indeterminate Beam.svg - Wikimedia Commons View original

  Is this image relevant?

• 

  Beam (structure) - Wikipedia View original

  Is this image relevant?

• 

  Euler–Bernoulli beam theory - Wikipedia View original

  Is this image relevant?

• 

  File:Statically Indeterminate Beam.svg - Wikimedia Commons View original

  Is this image relevant?

1 of 3

• Beams are classified based on their support conditions, which determine how they are constrained and how they transfer loads to the supports
• The support conditions also determine whether a beam is statically determinate or indeterminate
  • Statically determinate beams have enough equilibrium equations to solve for all unknown reactions (simply supported beams)
  • Statically indeterminate beams have more unknown reactions than available equilibrium equations (fixed beams, continuous beams)

Simply supported and cantilever beams

• A simply supported beam is supported at both ends by pinned or roller supports, allowing for rotation but not translation
  • Pinned supports resist vertical and horizontal translation but allow rotation
  • Roller supports resist translation perpendicular to the support surface but allow rotation and translation along the surface
• A cantilever beam is fixed at one end and free at the other end, with the fixed end providing both translational and rotational resistance
  • The free end can deflect and rotate under load, while the fixed end remains constrained

Fixed, overhanging, and continuous beams

• A fixed beam, or built-in beam, is fixed at both ends, preventing both translation and rotation at the supports
  • Fixed supports provide both translational and rotational resistance, making the beam statically indeterminate
• Overhanging beams extend beyond one or both supports, with the overhanging portion being free
  • The overhanging portion behaves like a cantilever beam, while the supported portion behaves according to its support conditions
• Continuous beams are supported at more than two points, creating multiple spans
  • Each intermediate support introduces additional unknown reactions, making the beam statically indeterminate

Truss beams

• Truss beams are composed of interconnected triangular structures, with each member carrying either tension or compression
  • The triangular arrangement allows truss beams to efficiently transfer loads through axial forces in the members
  • Truss beams are commonly used in bridges, roofs, and other large-scale structures

Loads on beams

Concentrated and moment loads

• Concentrated loads, also known as point loads, are forces applied at a single point on the beam
  • Examples include the weight of a person standing on a beam or a column resting on a beam
• Moment loads are concentrated moments applied at a specific point, causing rotation without translation
  • Moment loads can be created by offset forces or by external moments applied to the beam

Uniformly and non-uniformly distributed loads

• Distributed loads are forces applied over a length of the beam, either uniformly or non-uniformly distributed
  • The intensity of a distributed load is measured in force per unit length $(q = \frac{F}{L})$
• Uniformly distributed loads have a constant intensity along the beam's length
  • Examples include the weight of a beam itself or a constant pressure applied along the beam (snow load on a roof)
• Non-uniformly distributed loads have a varying intensity along the beam's length, often described by a function
  • Examples include hydrostatic pressure on a dam or wind pressure on a tall building

Linearly varying loads

• Linearly varying loads are distributed loads with a linear variation in intensity along the beam's length
  • The load intensity varies from a minimum value at one end to a maximum value at the other end
  • Linearly varying loads can be represented by a trapezoidal or triangular load distribution
  • Examples include the soil pressure on a retaining wall or the water pressure on a submerged gate

Reactions at beam supports

Equilibrium equations and statical determinacy

• Support reactions are the forces and moments provided by the supports to maintain equilibrium in the beam
• For statically determinate beams, the support reactions can be calculated using the equations of equilibrium: sum of forces and sum of moments equal to zero
  • $\sum F_x = 0, \sum F_y = 0, \sum M = 0$
• The number of equilibrium equations available must be equal to the number of unknown reaction forces and moments for the beam to be statically determinate

Reaction forces and moments at supports

• Pinned supports provide a vertical reaction force, while roller supports provide a reaction force perpendicular to the direction of possible translation
  • Pinned supports resist both vertical and horizontal translation, so they provide reaction forces in both directions
  • Roller supports only resist translation perpendicular to the support surface, so they provide a reaction force in that direction
• Fixed supports provide both a vertical reaction force and a reaction moment
  • The fixed support resists both translation and rotation, introducing a reaction moment in addition to the reaction force

Sign convention for reactions

• The sign convention for reaction forces and moments must be consistently applied, typically with upward forces and counterclockwise moments being positive
  • Positive reaction forces act in the opposite direction of gravity (upward)
  • Positive reaction moments act in the counterclockwise direction when viewed from the left side of the beam
• When solving for reactions, assume the direction of unknown forces and moments, and solve the equilibrium equations
  • If the resulting value is positive, the assumed direction is correct; if negative, the actual direction is opposite to the assumed direction

Distributed loads on beams

Load intensity and resultant force

• Distributed loads are forces applied over a length of the beam, as opposed to concentrated loads applied at a single point
• The intensity of a distributed load is measured in force per unit length $(q = \frac{F}{L})$
  • Common units include N/m (SI) or lb/ft (US customary)
• The resultant force of a distributed load acts at the centroid of the load distribution area
  • For a uniformly distributed load, the resultant force is equal to the product of the load intensity and the length $(F_R = q \cdot L)$
  • For a linearly varying load, the resultant force is equal to the average load intensity multiplied by the length $(F_R = \frac{q_1 + q_2}{2} \cdot L)$

Effect on shear force and bending moment diagrams

• Distributed loads affect shear force and bending moment diagrams differently than concentrated loads, creating sloped or curved segments instead of instantaneous jumps
  • Uniformly distributed loads create linearly sloped segments in the shear force diagram and parabolic segments in the bending moment diagram
  • Linearly varying loads create parabolically curved segments in the shear force diagram and cubic segments in the bending moment diagram
• When calculating reactions or drawing diagrams, distributed loads can be replaced by their equivalent resultant force acting at the centroid of the load distribution
  • This simplification allows for easier calculation of reactions and construction of diagrams
  • However, the actual distributed nature of the load must be considered when analyzing the internal forces and moments along the beam