Structural Analysis Vocabulary

Introduction to Structural Analysis and Structural Behaviour

  • Structure Definition: A structure is a system of connected parts used to support a load (Hibbeler, 2009).
  • Civil engineering structures include buildings, bridges, towers, and tunnels.
  • Other engineering branches: ships, aircraft frames, tanks, pressure vessels, mechanical systems, and electrical supporting structures.
  • Alternative Definition: A structure is a part of construction with one or more elements subjected to various loads that it must resist without collapsing or deforming excessively (Bambang Prihartanto et al., 2008).
  • Design Considerations:
    • Safety
    • Aesthetics
    • Serviceability
    • Economic factors
    • Environmental impact
    • Materials availability

Classification of Structures

  • Structural Elements:
    • Tie Rods
    • Beams
    • Columns

Tie Rods

  • Tie rods or bracing struts are structural members designed to resist tensile forces.
  • They are slender and often made from rods, bars, angles, or channels.

Beams

  • Beams are straight, horizontal members primarily used to carry vertical loads.
  • Common materials: steel, aluminum, concrete, and timber.
  • Beams are classified by their support methods:
    • Simply Supported Beam
    • Cantilevered Beam
    • Fixed-supported Beam (fixed at both ends)
    • Continuous Beam
    • Beam fixed at one end and simply supported at the other end
    • Overhanging Beam
  • Beams are designed to resist bending moments and internal shear forces.
  • Metal beams (steel, aluminum) often have I-shaped cross-sections.
    • Flanges resist the applied moment MM.
    • Web resists the applied shear VV.
  • Beam cross-sections are typically rolled in lengths up to 23m.
  • For long spans or large loads, plate girders are used.
    • Fabricated from a large plate for the web with welded or bolted flange plates.
    • Often transported in segments and spliced together where internal moment is small.
  • Concrete beams generally have rectangular cross-sections.
    • Reinforcing steel is combined/cast with concrete to resist tension.
  • Precast concrete beams and girders are fabricated off-site and transported to the construction site.
  • Timber beams are made from sawn or laminated timber.
    • Laminated beams are constructed from solid wood sections fastened together.

Columns

  • Columns are generally vertical members that resist axial compressive loads.
  • Metal columns commonly use tubes and wide-flange cross-sections.
  • Concrete columns use circular and square cross-sections with reinforcing rods.
  • Beam-columns are subjected to both axial loads and bending moments.

Types of Structures

  • Trusses
  • Cables and Arches
  • Frames
  • Surface Structures
  • Structural Forms
    • Tension & Compression Structure
    • Flexural Beam & Frame Structure
    • Surface Structures
  • A structural system is a combination of structural elements and materials.
  • Basic types of structures: trusses, cables and arches, frames, and surface structures.

Trusses

  • A truss is a structure with a large span where depth is not a critical design factor, consisting of slender elements arranged in a triangular pattern.
  • Trusses support loads through compression and tension in their members.
  • Trusses use less material to support a given load compared to beams.
  • Truss spans economically range from 9m to 122m.

Cables

  • Cables are flexible structures that carry loads in tension and can span long distances (greater than 46 m).
  • Cables are used to support bridges and building roofs.
  • Cables are stable because they are always in tension.
  • Reducing sag, weight, and anchorage methods increases costs and depth of truss construction as span increases.

Arches

  • An arch is a rigid, reverse curvature structure that primarily resists loads through compression.
  • Shear and moment loadings must be considered in the design process to maintain shape.
  • Arches are used in bridge structures, dome roofs, and masonry wall openings.

Frames

  • Frames are composed of beams and columns connected by pins or fixed joints, commonly used in buildings.
  • Frame strength is derived from moment interactions between beams and columns at rigid joints.
  • Economic frame construction involves using smaller beam sizes and increasing column sizes due to beam-column action caused by bending at the joints.

Surface Structures

  • A surface structure is made from rigid materials like reinforced concrete, folded plates, cylinders, or hyperbolic paraboloids; referred to as thin plates or shells.
  • It acts like cables or arches, supporting tension or compression loads with minimal bending.

Structural Forms

  • Tension & Compression Structures
    • Column, strut (compression)
    • Cable-supported structure (tension)
    • Arch (compression)
    • Truss (compression & tension)
  • Flexural Beam & Frame Structures
    • Beam
    • Frame
    • Combination (bridges, buildings)

Structural Design

  • Structural design must consider both material and load uncertainties.
  • Material Uncertainties:
    • Variability in material properties
    • Residual stress in materials
    • Measurement differences from fabricated sizes
    • Accidental loadings due to vibration or impact
    • Material corrosion or decay
  • Typical Load Combinations (ASCE 7-05 Standard):
    • Dead Load
    • 0.6(DeadLoad)+WindLoad0.6 (Dead Load) + Wind Load
    • 0.6(DeadLoad)+0.7(EarthquakeLoad)0.6 (Dead Load) + 0.7 (Earthquake Load)
  • Load combinations to account for uncertainty loads (ASCE 7-05 Standard):
    • 1.4(DeadLoad)1.4 (Dead Load)
    • 1.2(DeadLoad)+1.6(LiveLoad)+0.5(SnowLoad)1.2 (Dead Load) + 1.6 (Live Load) + 0.5 (Snow Load)
    • 1.2(DeadLoad)+1.5(EarthquakeLoad)+0.5(LiveLoad)1.2 (Dead Load) + 1.5 (Earthquake Load) + 0.5 (Live Load)

What is Structural Analysis?

  • Specific structures should be analyzed to ensure strength and rigidity after preliminary design.
  • Structural analysis is the process of determining the reaction of the structure under specified loads or actions (Bambang Prihartanto et al., 2008).
  • Reactions are measured by establishing forces and deformations throughout the structure.
  • Results from structural analysis are used to:
    • Redesign the structure
    • Determine accurate weight of the members
    • Account for the size of the members
  • Steps for Proper Structural Analysis:
    1. Identify the structures
    2. Idealization of the structures
    3. Loading determination (codes and local specifications)
    4. Structure members forces and displacements (using theory of structural analysis)

Idealization of Structure

  • Idealization of Structures involves creating an idealized model from the actual structure in a line diagram for structural analysis.
  • Structures are often idealized from 3D to 2D.

Stability & Determinacy

Stability

  • Types of Structural Instability:
    • (i) Kinematics Unstable (Partial Constraints):
      • Number of reactions at the support or number of members is less than the minimum requirement.
      • Structure members may have fewer reactive forces than equations of equilibrium, leading to partial constraint.
    • (ii) Geometry Unstable (Improper Constraints):
      • The location or the arrangement of the support or member is improper.

Determinacy

  • (i) Beam
    • r < n + 3: Statically unstable
    • r=n+3r = n + 3: Determinate (only if geometrically stable)
    • r > n + 3: Indeterminate
      • Where:
        • nn = number of internal hinge
        • rr = number of reaction
        • mm = number of member
        • jj = number of joint
  • (ii) Frame
    • 3m + r < 3j + n: Statically unstable
    • 3m+r=3j+n3m + r = 3j + n: Determinate (only if geometrically stable)
    • 3m + r > 3j + n: Indeterminate
      • Where:
        • nn = number of internal hinge
        • rr = number of reaction
        • mm = number of member
        • jj = number of joint
  • (iii) Plane Truss
    • m < 2j – r: Statically unstable
    • m=2j–rm = 2j – r: Determinate (only if geometrically stable)
    • m > 2j – r: Indeterminate
      • Where:
        • nn = number of internal hinge
        • rr = number of reaction
        • mm = number of member
        • jj = number of joint
  • (iv) Space Truss
    • m < 3j – r: Statically unstable
    • m=3j–rm = 3j – r: Determinate (only if geometrically stable)
    • m > 3j – r: Indeterminate
      • Where:
        • nn = number of internal hinge
        • rr = number of reaction
        • mm = number of member
        • jj = number of joint

Structural Behaviour

  • Introduction
  • Stress and Strain
  • Elastic Deformation
  • Moment of Inertia
  • Modulus of Elasticity

Introduction

  • When designing beams in bending or columns in buckling, it is necessary to consider basic geometrical properties of the cross sections of structural members.
  • Area
    • Cross-section areas (AA) are generally calculated in mm2mm^2, as structural member dimensions are in mm, and design stresses are in N/mm2N/mm^2.
  • Centre of Gravity or Centroid
    • This is the point about which the area of the section is evenly distributed. The centroid may be outside the actual cross-section.
  • Reference Axes
    • Reference axes of structural sections pass through the centroid. The x-x axis is perpendicular to the greatest lateral dimension, and the y-y axis is perpendicular to the x-x axis, intersecting at the centroid.

Stress & Strain

  • Stress: σ=F/A\sigma = F / A
    • FF: Load
    • AA: Cross-sectional area perpendicular to FF before loading.
  • Strain: ε=Δl/lo×100%\varepsilon = \Delta l / l_o \times 100 \%%
    • Δl\Delta l: Change in length
    • lol_o: Original length
  • Stress / strain = σ/ε=E\sigma/ \varepsilon = E (modulus of elasticity)

Shear Stress & Strain

  • Shear stress: τ=F/Ao\tau = F / A_o
    • FF is applied parallel to upper and lower faces, each with area A0A_0.
  • Shear strain: γ=tanθ×100%\gamma = tan\theta \times 100 \%%
    • θ\theta is the strain angle.

Torsion

  • Torsion is like shear.
  • Load: Applied torque, TT (a twisting force that tends to cause rotation).
  • Strain: Angle of twist, ϕ\phi.

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Stress-Strain Behavior (Tension)

  • Elastic deformation
    • Reversible: For small strains, stress removed returns material to original size.
  • Plastic deformation
    • Irreversible: Stress removed does not return material to original dimensions.

Elastic Deformation

  • EE = Young's modulus or modulus of elasticity (same units as σ\sigma, N/m2N/m^2 or Pa)
  • Hooke's law: σ=Eε\sigma = E \varepsilon
  • Higher EE implies higher “stiffness”

Nonlinear Elastic Behavior

  • In some materials (polymers, concrete), elastic deformation is not linear but still reversible.
  • Definitions of E
    • Δσ/Δε\Delta\sigma/\Delta\varepsilon = tangent modulus at σ2\sigma_2
    • Δσ/Δε\Delta\sigma/\Delta\varepsilon = secant modulus between origin and σ1\sigma_1

Moment of Inertia

  • Area moment of inertia, or second moment of area, measures area distribution around a particular axis of a cross section and is important for bending resistance.
  • Material strength is also important for bending resistance.

Modulus of Elasticity

  • Modulus of Elasticity, also known as Elastic Modulus or Young’s Modulus, measures how a material or structure deforms and strains under stress.
  • Materials deform differently when loads and stresses are applied.
  • The ratio of stress to strain is known as the coefficient of elasticity, elasticity modulus, or elastic modulus.
  • Stiffness of an object is related to material chemical properties.
  • Higher modulus of elasticity produces greater resistance to deformation.

Calculating Beam Deflection

  • ΔMAX=FL348EI\Delta_{MAX} = \frac{FL^3}{48EI}
    • Where:
      • FF = Force
      • LL = Length
      • EE = Modulus of Elasticity
      • II = Moment of Inertia