1. Structural Analysis II
Course Code: CIVL322
CH. 5
Dr. Aeid A. Abdulrazeg

2. Contents: Preliminary remarks Beam-member stiffness matrix
Beam-structure stiffness matrix
Frame-member stiffness matrix
Frame-structure stiffness matrix
Application of the stiffness method for beam and frame analysis

3. Preliminary remarks Member & node identification
In general each element must be free from load & have a prismatic cross section
The nodes of each element are located at a support or at points where members are connected together or where the vertical or rotational displacement at a point is to be determined

4. Preliminary remarks Global & member coordinates
The global coordinate system will be identified using x, y, z axes that generally have their origin at a node & are positioned so that the nodes at other points on the beam are +ve coordinates,

5. Preliminary remarks Kinematic indeterminacy
The consider the effects of both bending & shear
Each node on the beam can have 2 degrees of freedom, vertical displacement & rotation
These displacement will be identified by code numbers
The lowest code numbers will be used to identify the unknown displacement & the highest numbers are used to identify the known displacement

6. Preliminary remarks Kinematic indeterminacy
First we must establish the stiffness matrix for each element
These matrices are combined to form the beam or structure stiffness matrix
We can then proceed to determine the unknown displacement at the nodes
This will determine the reactions at the beam & the internal shear & moments at the nodes

7. Beam-member stiffness matrix
We will develop the stiffness matrix for a beam element or member having a constant cross-sectional area & referenced from the local x’, y’, z’ coordinate system,

8. Beam-member stiffness matrix
y’ displacement
A positive displacement dNy’ is imposed while other possible displacement are prevented
The resulting shear forces & bending moments that are created are shown in Figure

9. Beam-member stiffness matrix
z’ rotation
A positive rotation dNz’ is imposed while other possible displacement are prevented
The required shear forces & bending moments necessary for the deformation are shown in Figure
Likewise, when dNz’ is imposed, the resultant loadings are shown in Figure

10. Beam-member stiffness matrix

11. Beam-member stiffness matrix

12. Frame-member stiffness matrix
In this section, we will develop the stiffness matrix for a prismatic frame member reference from the local x’, y’, z’ coordinate system
The member is subjected to axial loads qNx’. qFy’, shear loads qNy’, qFy’ and bending moment qNz’, qFz’ at its far end respectively
These loadings all act in the +ve coordinate directions along with their associated displacement

13. Frame-member stiffness matrix

14. Displacement & force transformation matrices

15. Displacement & force transformation matrices

16. Frame-member global stiffness matrix

17. Frame-member global stiffness matrix

18. Application of the stiffness method for beam and frame analysis
Once the stiffness matrix is determined, the loads at the nodes of the beam can be related to the displacement using the structure stiffness equation.
Partitioning the stiffness matrix into the known & unknown elements of load & displacement, we have

19. Non- Nodal Loads

20. Example 1 (Matrix Analysis for Beam)
Determine the reactions at the supports of the beam shown in Figure. EI is constant