Structural analysis of two-dimensional (2D) framed structures is a fundamental skill for civil engineers working on building and infrastructure projects across Kenya. Understanding how forces distribute within these frames ensures safety, serviceability, and economic use of materials. This chapter focuses on the mathematical analysis of 2D framed structures, emphasizing methods to determine internal forces and reactions in pin-jointed and rigid-jointed frames commonly found in buildings, bridges, and other civil engineering works.

2.1 Characteristics of 2D Framed Structures

2D framed structures consist of interconnected elements arranged in a plane, designed to support loads and transfer them safely to foundations. These frames are prevalent in Kenyan construction, from multistory buildings in Nairobi to agricultural storage sheds in rural counties. Understanding their characteristics enables engineers to select appropriate analysis methods and design safe, efficient structures.

2.1.1 Differences between pin-jointed and rigid-jointed frames

Joints are connection points which may be pinned or rigid, influencing force transmission. In a pin-jointed frame, members are connected by pins (or connections that act like pins) that allow the members to rotate freely, so no bending moment passes through the joint and, when loads act only at the joints, each member carries a pure axial force of tension or compression. In a rigid-jointed frame, members are connected so firmly, by welding, bolted end plates or monolithic concrete, that the angle between them stays the same under load, so the joint transfers bending moment and shear as well as axial force. This difference decides how the structure behaves, what it can be made of and how it is analysed. Example: the steel roof trusses of a cereal store for a farmers' cooperative are treated as pin-jointed, while the reinforced concrete frame of its office block is rigid-jointed.

Pin-jointed and rigid-jointed frames compared

Static determinacy and indeterminacy

A frame is statically determinate if all internal forces and reactions can be found using equilibrium equations alone. If additional compatibility conditions are needed, the frame is statically indeterminate. This classification influences the choice of analysis method. The checks use m = number of members, r = number of support reactions and j = number of joints.

• Determinate frames: Simpler to analyze but less common in modern construction.
• Indeterminate frames: Provide greater stiffness and redundancy but require advanced methods.
• Pin-jointed check: If m + r = 2j the frame is determinate; if greater, it is indeterminate; if less, it is a mechanism and unstable.
• Rigid-jointed check: Degree of indeterminacy = (3m + r) − 3j.
• Redundancy: Extra members or supports in an indeterminate frame give alternative load paths if one part fails.

Worked Examples

Example 1: Pin-jointed truss

Given: A truss has 9 members and 6 joints, with a pin support (2 reactions) and a roller support (1 reaction). Check its determinacy.

m + r = 9 + 3 = 12

2j = 2 × 6 = 12

Answer: m + r = 2j, so the truss is statically determinate.

Example 2: Rigid portal frame

Given: A single-bay portal frame has 3 members (two columns and a beam), 4 joints (two column bases and two knees) and two fixed bases, each giving 3 reactions. Find its degree of indeterminacy.

3m + r = 3 × 3 + 6 = 15

3j = 3 × 4 = 12

Degree of indeterminacy = 15 − 12

Answer: The frame is indeterminate to the third degree.

2.1.2 Examples in trusses and frames

Trusses and frames are the two main families of 2D framed structures. Trusses are triangulated assemblies of straight members, assumed pin-jointed, that span long distances with little material and are used for roofs, bridges and towers. Frames are rigid-jointed assemblies of beams and columns that resist load by bending and are used for buildings, warehouses and portal-framed sheds. Example: a new vocational training centre has a Fink timber truss roof over its classrooms and a steel portal frame over its workshop, where a clear floor is needed for machines.

Common types of trusses

Common types of rigid frames

• Single-bay portal frame: Two columns rigidly joined to a beam or pitched rafters, giving a clear span for warehouses, garages and factories.
• Multi-bay portal frame: Several portal frames side by side sharing internal columns, used for large stores and markets.
• Multi-storey frame: Beams and columns connected rigidly at every floor, the usual structure for office and apartment blocks.
• Gable frame: A portal frame with sloping rafters forming a pitched roof.
• Braced frame: A rigid or simple frame with diagonal bracing added to resist lateral loads such as wind.

Choosing between a truss and a frame

• Span: Trusses span long distances economically; portal frames give clear spans with less depth.
• Headroom: Frames leave the roof space clear, while trusses take up roof depth.
• Loads: Trusses carry mainly roof loads at the joints; frames carry floors and moments.
• Material: Trusses suit timber and light steel; frames suit hot-rolled steel and reinforced concrete.
• Cost and speed: Prefabricated trusses are quick and cheap for roofs; frames suit multi-storey work.

2.1.3 Geometry and components of 2D frames

2D frames are composed of straight members connected at nodes, forming a planar system. The geometry defines the shape and size of the frame, affecting stiffness and load distribution. The primary components include:

• Beams: Horizontal members that resist bending and shear loads.

Three types of structural beams in carpentry (Source: Wikimedia Commons)

• Columns: Vertical members designed to carry axial compression and bending.
• Bracing: Diagonal members added to enhance stability against lateral loads.
• Supports: Points where the frame transfers loads to the ground, typically pinned or fixed.

2.1.4 Load types acting on 2D frames

Frames in civil engineering are subjected to various loads that induce internal forces, and each load type affects the frame differently, requiring careful analysis to ensure safety:

• Dead Loads: Permanent loads such as self-weight of members and fixed equipment.
• Live Loads: Transient loads including occupants, furniture, and movable equipment.
• Wind Loads: Lateral forces caused by wind pressure, critical in tall or exposed structures.
• Seismic Loads: Dynamic forces during earthquakes, important in regions of seismic activity such as the Rift Valley.
• Snow and ice loads: Vertical loads from snow accumulation; in Kenya these apply only to structures at very high altitude, while rainwater ponding and hail are the more common extra roof loads.

2.1.5 Stability and possible failure modes

Stability ensures the frame maintains its geometry under load without collapse, and engineers must evaluate the following risks during design and analysis:

• Buckling of Columns: Sudden lateral deflection under compression.
• Member Yielding: Permanent deformation when stresses exceed material limits.
• Joint Failure: Loss of connection integrity, especially in pinned joints.
• Overall Collapse: Progressive failure due to instability or overload.
• Sway instability: Large sideways movement of unbraced frames under lateral load.

Chapter Summary

This chapter explored the characteristics of two-dimensional framed structures, highlighting the fundamental differences between pin-jointed and rigid-jointed frames and illustrating these concepts through examples in trusses and frames. It then examined the key assumptions and methods used in structural analysis, focusing on the method of joints and the method of sections as primary techniques for evaluating internal forces. The chapter proceeded to analyze real-world 2D frames by demonstrating how to calculate forces in truss members and determine support reactions for various frame types. Practical application was emphasized through detailed examples involving roof trusses and portal frames, showing how theoretical principles translate into structural design and assessment. Throughout the chapter, the integration of theory with practice reinforced the importance of accurate force analysis in ensuring the stability and safety of framed structures.