Structural Analysis of Truss Structures using Stiffness Matrix. Dr. Nasrellah Hassan Ahmed

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1 Structural Analysis of Truss Structures using Stiffness Matrix Dr. Nasrellah Hassan Ahmed

2 FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS In general, there are three types of relationships: Equilibrium equations, compatibility conditions, and constitutive relations. Equilibrium Equations : A structure is considered to be in equilibrium if, initially at rest, it remains at rest when subjected to a system of forces and couples.

3 FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS These equations are referred to as the equations of equilibrium for plane structures

4 FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS These equations are referred to as the equations of equilibrium of a space (three-dimensional)

5 FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS Compatibility Conditions: The compatibility conditions relate the deformations of a structure so that its various parts (members, joints, and supports) fit together without any gaps or overlaps. These conditions (also referred to as the continuity conditions) ensure that the deformed shape of the structure is continuous (except at the locations of any internal hinges)

6 FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS

7 FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS Constitutive Relations: The constitutive relations (also referred to as the stress-strain relations) describe the relationships between the stresses and strains of a structure. The constitutive relations provide the link between the equilibrium equations and compatibility conditions that is necessary to establish the loaddeformation relationships for a structure or a member.

8 Global and Local Coordinate Systems:

9 FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS Degrees Of Freedom : The degrees of freedom of a structure, in general, are defined as the independent joint displacements (translations and rotations) that are necessary to specify the deformed shape of the structure when subjected to an arbitrary loading.

10 FUNDAMENTAL RELATIONSHIPS FOR STRUCTURAL ANALYSIS in which d is called the joint displacement vector Joint Load Vector Reaction Vector

11 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM In the stiffness method of analysis, the joint displacements, d, of a structure due to an external loading, P, are determined by solving a system of simultaneous equations, expressed in the form. in which S is called the structure stiffness matrix. It will be shown subsequently that the stiffness matrix for the entire structure, S, is formed by assembling the stiffness matrices for its individual members.

12 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM member stiffness relations for plan truss:

13 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM

14 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM The above equations can be expressed in matrix form as:

15 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM in which Q and u are the member end force and member end displacement vectors, respectively, in the local coordinate system; and k is called the member stiffness matrix in the local coordinate system.

16 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM

17 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM

18 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM By using a similar approach, it can be shown that the stiffness coefficients equal

19 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM

20 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM

21 MEMBER STIFFNESS RELATIONS IN THE LOCAL COORDINATE SYSTEM

22 COORDINATE TRANSFORMATIONS When members of a structure are oriented in different directions, it becomes necessary to transform the stiffness relations for each member from its local coordinate system to a single global coordinate system selected for the entire structure.

28 Example:

33 Example:

37 Example:

Stress analysis of a stepped bar

Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.