Structural analysis II

Introduction Statically Indeterminate Structures

Force Method

Instructor: Dr. Sawsan Alkhawaldeh

Department of Civil Engineering

Determinacy of structures

Structures can be classified as: • Statically determinate: All the forces in a structure can be determined

strictly from the equations of equilibrium; r = 3n • Statically indeterminate: Additional equations (Compatibility equations)

needed to solve for the unknown reactions are obtained by relating the applied loads and reactions to the displacement or slope at different points on the structure; r> 3n

Advantages of statically indeterminate structures

• The maximum stress and deflection of an indeterminate structure are generally smaller than those of its statically determinate counterpart for a given loading. • Its tendency to redistribute its load to its redundant supports in cases where faulty design or overloading occurs. Hence, the structure maintains its stability and collapse is prevented.

Analysis of statically indeterminate structures

For analyzing statically indeterminate structure, three requirements should be satisfied in order to ensure their safety:

• Equilibrium, is satisfied when the reactive forces hold the structure at rest.

• Compatibility, is satisfied when the various segments of the structure fit together without intentional breaks or overlaps.

• force-displacement, dependent upon the way the material responds; it is usually assumed to be linear elastic response.

Methods of Analysis

Two different ways are available for analyzing a statically indeterminate structure:

1. The force or flexibility method, consists of writing equations that satisfy the compatibility and force-displacement requirements to determine the redundant forces. Once these forces have been determined, the remaining reactive forces on the structure are determined by satisfying the equilibrium requirements.

Methods of Analysis

2. The displacement or stiffness method, consist of writing force-displacement relations for the members and then satisfying the equilibrium requirements for the structure. The unknowns in the equations are displacements. Once the displacements are obtained, the forces are determined from the compatibility and force displacement equations.

Methods of Analysis

Force method of analysis

General procedure of analysis:

1. Principle of Superposition, converting the statically indeterminate structure to be equal to a series of corresponding statically determinate structures.

2. Compatibility Equations, to find the redundant forces.

3. Equilibrium Equations, to find the remaining unknown reactions.

Illustrative example:

Example (1) Beams

Determine the reaction at the roller support B of the following beam.

Example (2) Beams Draw the shear and moment diagrams for the beam. The support at B settles 1.5 in.

E = 29(103) ksi

I = 750 𝑖𝑛4

Example (3) Beams

Draw the shear and moment diagrams for the beam. EI is constant. Neglect the effects of axial load.

HW (1)

Example (4) Frames The frame, or bent, shown in the photo is used to support the bridge deck. Assuming EI is constant. Determine the support reactions.

Example (4) Trusses Determine the force in member AC of the truss. AE is the same for all the members.