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 M.
  • Web resists the applied shear V.
• 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 (Dead Load) + Wind Load
  • 0.6 (Dead Load) + 0.7 (Earthquake Load)
• Load combinations to account for uncertainty loads (ASCE 7-05 Standard):
  • 1.4 (Dead Load)
  • 1.2 (Dead Load) + 1.6 (Live Load) + 0.5 (Snow Load)
  • 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 + 3: Determinate (only if geometrically stable)
  • r > n + 3: Indeterminate
    • Where:
      • n = number of internal hinge
      • r = number of reaction
      • m = number of member
      • j = number of joint
• (ii) Frame
  • 3m + r < 3j + n: Statically unstable
  • 3m + r = 3j + n: Determinate (only if geometrically stable)
  • 3m + r > 3j + n: Indeterminate
    • Where:
      • n = number of internal hinge
      • r = number of reaction
      • m = number of member
      • j = number of joint
• (iii) Plane Truss
  • m < 2j – r: Statically unstable
  • m = 2j – r: Determinate (only if geometrically stable)
  • m > 2j – r: Indeterminate
    • Where:
      • n = number of internal hinge
      • r = number of reaction
      • m = number of member
      • j = number of joint
• (iv) Space Truss
  • m < 3j – r: Statically unstable
  • m = 3j – r: Determinate (only if geometrically stable)
  • m > 3j – r: Indeterminate
    • Where:
      • n = number of internal hinge
      • r = number of reaction
      • m = number of member
      • j = 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 (A) are generally calculated in mm^2, as structural member dimensions are in mm, and design stresses are in N/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: \sigma = F / A
  • F: Load
  • A: Cross-sectional area perpendicular to F before loading.
• Strain: \varepsilon = \Delta l / l_o \times 100 \%%
  • \Delta l: Change in length
  • l_o: Original length
• Stress / strain = \sigma/ \varepsilon = E (modulus of elasticity)

Shear Stress & Strain

• Shear stress: \tau = F / A_o
  • F is applied parallel to upper and lower faces, each with area A_0.
• Shear strain: \gamma = tan\theta \times 100 \%%
  • \theta is the strain angle.

Torsion

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

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

Nonlinear Elastic Behavior

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

• \Delta_{MAX} = \frac{FL^3}{48EI}
  • Where:
    • F = Force
    • L = Length
    • E = Modulus of Elasticity
    • I = Moment of Inertia