Indeterminate Trusses & Industrial Frames - 1 - Structural Analysis - Civil

Instructional Objectives:

After reading this chapter the student will be able to

1. Make suitable approximations so that an indeterminate structure is reduced to a determinate structure.
2. Analyse indeterminate trusses by approximate methods.
3. Analyse industrial frames and portals by approximate methods.

Introduction

Indeterminate trusses and industrial frames are vital elements in structural engineering, offering versatile solutions for supporting various loads and providing stability in construction. Indeterminate trusses, with their complex load distribution and redundancy, demand sophisticated analysis methods. Industrial frames, comprising columns, beams, and braces, cater to the heavy loads and dynamic forces encountered in industrial settings. Understanding the principles and design considerations of these structural systems is essential for ensuring the integrity and efficiency of diverse construction projects.

Indeterminate Trusses

• Complex Structural Systems: Comprised of interconnected members with redundant supports or members.
• Redundancy: Contains more unknowns (reactions or internal forces) than the number of equilibrium equations available.
• Analysis Methods: Requires advanced techniques such as flexibility or stiffness methods for accurate analysis.
• Applications: Commonly used in bridges, large-span roofs, and structures requiring precise load distribution.
• Behavior: Exhibits non-linear load-displacement characteristics due to redundancy, allowing for redistribution of loads.

Industrial Frames

• Structural Frameworks: Consist of columns, beams, and braces arranged in a rigid framework.
• Purpose: Designed to support heavy loads and provide stability in industrial buildings, factories, and warehouses.
• Load-Bearing Capacity: Engineered to withstand dynamic forces, environmental loads, and heavy equipment.
• Types: Include portal frames, rigid frames, and braced frames, each with specific load-resisting characteristics.
• Analysis: Analyzed using traditional methods such as moment distribution and finite element analysis (FEA).
• Durability: Constructed to endure harsh industrial environments and ensure long-term structural integrity.

Important Terms

1. Truss:

   • A truss is a structural framework composed of members joined together to form triangular units. It is designed to carry loads primarily through axial forces in its members.

2. Frame:

   • A frame is a structural system composed of interconnected members forming a rigid framework. Frames are designed to support loads and provide structural stability, commonly used in buildings and other structures.

3. Indeterminate Structure:

   • An indeterminate structure is a structural system where the number of unknown reactions or internal forces exceeds the number of available equilibrium equations. This results in redundant members or supports, requiring more advanced analysis methods.

4. Redundant Member:

   • A redundant member is a structural member in an indeterminate structure that does not directly contribute to the equilibrium of the structure but affects its behavior under loading conditions.

5. Support Reactions:

   • Support reactions are the forces exerted by a structure's supports to maintain its equilibrium. They include reactions such as vertical reactions, horizontal reactions, and moments.

6. Method of Joints:

   • The method of joints is a structural analysis technique used to determine the internal forces in the members of a truss by analyzing the equilibrium of joints.

7. Method of Sections:

   • The method of sections is a structural analysis technique used to determine the internal forces in the members of a truss by cutting through the structure and analyzing the equilibrium of sections.

8. Flexibility Method:

   • The flexibility method is a matrix-based structural analysis technique used to analyze indeterminate structures by considering the flexibility of the structure's members.

9. Stiffness Method:

   • The stiffness method is a matrix-based structural analysis technique used to analyze indeterminate structures by considering the stiffness of the structure's members.

10. Portal Frame:

    • A portal frame is a type of industrial frame commonly used in buildings and warehouses. It consists of vertical columns and horizontal beams rigidly connected to form a portal structure.

Indeterminate Trusses: Parallel-chord trusses with two diagonals in each panel

Consider an indeterminate truss, which has two diagonals in each panel as shown in Fig. 35.1. This truss is commonly used for lateral bracing of building frames and as top and bottom chords of bridge truss.

This truss is externally determinate and internally statically indeterminate to 3^rd degree. As discussed in lesson 10, module 2, the degree of static indeterminacy of the indeterminate planar truss is evaluated by

i = (m + r) - 2j        (reproduced here for convenience)

Where m, j and r respectively are number of members, joints and unknown reaction components. Since the given truss is indeterminate to 3^rd degree, it is required to make three assumptions to reduce this frame into a statically determinate truss. For the above type of trusses, two types of analysis are possible.

1. If the diagonals are going to be designed in such a way that they are equally capable of carrying either tensile or compressive forces. In such a situation, it is reasonable to assume, the shear in each panel is equally divided by two diagonals. In the context of above truss, this amounts to 3 independent assumptions (one in each panel) and hence now the structure can be solved by equations of static equilibrium alone.

2. In some cases, both the diagonals are going to be designed as long and slender. In such a case, it is reasonable to assume that panel shear is resisted by only one of its diagonals, as the compressive force carried/resisted by the other diagonal member is very small or negligible. This may be justified as the compressive diagonal buckles at very small load. Again, this leads to three independent assumptions and the truss may be solved by equations of static alone.

Generalizing the above method, it is observed that one need to make n independent assumptions to solve n^th order statically indeterminate structures by equations of statics alone. The above procedure is illustrated by the following examples.

Example 
Evaluate approximately forces in the truss members shown in Fig. 35.2a, assuming that the diagonals are to be designed such that they are equally capable of carrying compressive and tensile forces.

Solution:
The given frame is externally determinate and internally indeterminate to order 3. Hence reactions can be evaluated by equations of statics only. Thus,

                (1)

Now it is required to make three independent assumptions to evaluate all bar forces. Based on the given information, it is assumed that, panel shear is equally resisted by both the diagonals. Hence, compressive and tensile forces in diagonals of each panel are numerically equal. Now consider the equilibrium of free body diagram of the truss shown left of A-A. This is shown in Fig. 35.2b.

For the first panel, the panel shear is 23.33 kN . Now in this panel, we have

                    (2)

Considering the vertical equilibrium of forces, yields

                (3)

2 sinθ = 23.33

                         (4)

Thus,

Considering the joint , L0,

                       (5)

                         (6)

Similarly,

Now consider equilibrium of truss left of section C-C (ref. Fig. 35.2d)

              (7)

It is given that

2F sinθ = 3.33

Thus,

Taking moment about U₁ of all the forces,

                                  (8)

Taking moment about L₁ of all the forces,

                         (9)

Considering the joint equilibrium of L₁ (ref. Fig. 35.2e),

                             (10)

Consider the equilibrium of right side of the section B - B (ref. Fig. 35.2f) the forces in the 3^rd panel are evaluated.

We know that,  

                             (11)

                                  (12)

Considering the joint equilibrium of L₃ (ref. Fig. 35.2g), yields

The bar forces in all the members of the truss are shown in Fig. 35.2h. Also in the diagram, bar forces obtained by exact method are shown in brackets.