01 - Analysis of Internal Forces-output

Mechanics of Deformable Bodies

Overview

• Focus on the analysis of internal forces within deformable bodies.

• Prepared by: Engr. Buluran

Sign Convention

Force Direction

• Upward: Positive

• Downward: Negative

• To the Right: Positive

• To the Left: Negative

Components of a Force in 2D Plane

• Forces can be resolved into components based on the convention.

Moments

Sign Convention for Moments

• Clockwise Moment: Positive

• Counter-Clockwise Moment: Negative

Magnitude of the Moment of a Point

• Mo = F * d

  • Mo: Magnitude of the Moment (N-m/kN-m or lb-ft/lb-in/kips-ft)

  • F: Force applied (N/kN or lbs/kips)

  • d: Moment Arm or Perpendicular distance from the Axis of Point O (m/mm or ft/in)

Principles of Moments

Varignon’s Theorem

• The moment of a force about any point equals the sum of the moments of the components of that force about the same point.

Couples

Definition and Properties

• A couple consists of two parallel, non-collinear forces that are equal in magnitude and opposite in direction.

• Resultant Force of a Couple: Zero

• Couple Moment: The only effect of a couple is to produce rotation.

Load Types

Common Loadings Along a Single Axis

• Point Load: Concentrated force applied at a specific point.

• Uniformly Distributed Load: Load spread evenly across a length.

• Triangular Distributed Load: Load varies linearly along the length.

• Trapezoidally Distributed Load: Load distribution is non-uniform but forms a trapezoidal shape.

Equilibrium

Definition

• A body is in equilibrium if the resultant force and resultant couple are both zero.

Equations of Equilibrium

• Used when forces lie in the xy-plane.

Analysis of Force Systems

Categories

1. Single Bodies

   • Involves free-body diagrams (FBDs) to identify all forces acting on a body.

2. Composite Bodies

   • Analyze forces at internal connections called internal reactions.

3. Plane Trusses

   • Included as a separate analysis category.

Free-Body Diagrams (FBDs)

General Procedure

1. Sketch the body as if all supports have been removed.

2. Label applied forces, considering weight as acting on the center of gravity.

3. Draw and label support reactions, assume sense if unknown.

4. Include relevant angles and dimensions in the sketch.

Connections and Reactions

Types of Connections and Corresponding Reactions

• Cable: One unknown (F)

• External Pin: Two unknowns (Fx, Fy)

• Internal Pin: One unknown (F)

• Roller and Smooth Support: One unknown (vertical or horizontal force)

• Fixed Support: Three unknowns (Fx, Fy, M)

Internal Forces in Members

Summary

• Strong internal forces develop due to external loads acting on members at specific points.

Concepts of Centroid and Moment of Inertia

Common Shapes

• Centroid Location and Area Elements: Includes rectangular, triangular, trapezoidal, circular shapes.

• Moment of Inertia: Important for structural analysis and determining resistance to bending or twisting.

Moment of Inertia of Common Shapes

• Formulas provided for calculating moment of inertia for various shapes:

  • Rectangle, Isosceles Triangle, Circle, etc.

Sample Problems

• Problems illustrate calculations for determining internal forces, reactions, and centroid locations using FBDs and equilibrium principles.

• Examples include cantilevered beams, reinforced bodies, and axial load scenarios.

References

• Mechanics of Materials by Kiusalaas

• Mechanics of Materials by Hibbeler