Elementary Structures Study Notes

Structures and Statically Determinate Beams

1.0 Introduction

• The aim of this course is to understand the behaviour of simple structures. Topics include:

  • Basic principles of rigid statics.
  • Statically determinate structures:
    • Simply supported beams.
    • Cantilevers.
    • Pin-jointed frames/trusses (methods of joints and method of sections).
    • Arches.
    • Cables.
    • Influence lines.
    • Analysis of mass structures (earth dam).

• Definition of a structure: An object that transmits a set of loads or forces from one place in space to the ground without collapsing and without excessive deformation.

• Focus on one-dimensional structures where the length is large compared to cross-sectional dimensions.

• The form of a structure is dependent on several factors:

  • Functional requirements.
  • Aesthetic requirements.
  • Foundation conditions.
  • Availability of materials.
  • Economic limitations.

• Common structural forms include:

  • Pin-jointed frames,
  • Moment frames,
  • Cable structures,
  • Arch structures,
  • Surface structures.

1.1 Basic Principles of Forces and Static Equilibrium

Definition of a Force

• A force is defined as a vector with several components:
  • Magnitude.
  • Direction.
  • Point of application.
  • Line of action.
  • Sense.

Equilibrium of Forces

• Concurrent Forces in a Plane: Forces that intersect at one point (e.g., forces P1, P2, P3).
• Resultant R: For equilibrium,
  • $ext{R} = ext{P1} + ext{P2} + ext{P3}$
• The equilibrium condition states that the sum of forces acting in any direction must equal zero:
  • $ext{ΣF}<em>{x} = 0, ext{ΣF}</em>{y} = 0$

Non-concurrent Force Systems

• Non-concurrent forces do not meet at a single point and result in moments within structures:

  • Moment of a force: $M = F imes d$

• Equilibrium equations for a 2-D planar system:

  • $ext{ΣF}<em>{x} = 0, ext{ΣF}</em>{y} = 0, ext{ΣM} = 0$

• In 3-dimensions additional conditions such as $ext{ΣF}_{z} = 0$ apply.

1.2 Types of Support

Stability and Determinacy of Reactions

• Loads applied to structures transmit reactions through supports.
• Types of support:
  • Hinge or Pin Support: Prevents movement in vertical and horizontal directions but allows rotation, resulting in two reactions (vertical and horizontal).
  • Roller Support: Allows movement in one direction (perpendicular to the support surface) providing one reaction.
  • Encastre/Fixed Support: Prevents vertical and horizontal movement in addition to rotation, resulting in three reactions (two directions and a moment).
  • Link Support: Only allows movement along the link direction, generating one reaction.

Statical Classification of Structures

• A structure can be classified as:
  • Statically Determinate: The number of unknown reactions equals the number of equations for equilibrium.
  • Statically Indeterminate: More unknown reactions than equations available.
  • Unstable: More equilibrium conditions than unknowns.

1.3 Calculation of Reactions

• Reactions can be calculated by considering the structure's loads and support conditions.

• Types of Loads:

  • Point Load (P).
  • Uniformly distributed load (w, kN/m).
  • Triangular Load.
  • Trapezoidal Load.

• Sign Convention: Positive moments typically are counterclockwise, while reactions and loads are positive downward.

  • Example Calculation:
    • $ext{ΣM}_{A} = 0; ext{For a system:} ext{VA} + ext{VB} + ext{ΣL}= 0$

1.4 Internal Forces in Structures

• External forces are transmitted as internal forces including:
  • Axial forces,
  • Shear forces,
  • Bending moments.

Shear Force and Bending Moment Definitions

• Shear Force (V): The algebraic sum of external forces perpendicular to the beam at a given section.
• Bending Moment (M): The moment about a section due to the external forces acting to the left or right.

Shear Force Diagrams (SFD) and Bending Moment Diagrams (BMD)

• Methods for constructing SFD and BMD involve plotting changes in shear and moment along the length of the beam in response to loading.

2.0 Statically Determinate Pin-Jointed Frames

Introduction

• Many steel structures use statically determinate braced frames where forces in members can be calculated using static equilibrium laws alone.
• Assumptions for Pin-Jointed Frames:
  • Members are pin-jointed at the ends.
  • Loads applied only at joints.
  • Self-weight of members neglected.
  • Uniform cross-section assumed.

Types of Pin-Jointed Frames

• Perfect Frame: Just enough members to maintain shape under loads.
• Deficient Frame: Fewer members than necessary, cannot maintain shape.
• Redundant Frame: More members than necessary, leading to statical indeterminacy.

Conditions for Statically Determinate Structures

• Condition: $U = m + r$; where:
  • $U$ = unknowns,
  • $m$ = members,
  • $r$ = restraints.
• Analyze Forces:
  • $E = 2j$ where $j$ = joints.
• Stability conditions lead to determining if redundancies exist.

Analysis by Equilibrium at Joints

• For 2D frames, use static equilibrium equations at each joint:
  • $ext{ΣF}<em>{x} = 0, ext{ΣF}</em>{y} = 0$.

Example Problem (Wind Loads)

• Calculation includes determining reactions at supports and internal member forces.

Summary and Application

• Thorough understanding of these concepts is essential for structural analysis and design in engineering and real-world applications, including buildings, bridges, and trusses. Use these principles to analyze and ensure the safety and functionality of structures.