Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by...

14
Analysis of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method Theory of Structure - I

Transcript of Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by...

Page 1: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

Department of Civil Engineering

University of Engineering and Technology, Taxila, Pakistan

Analysis of Statically

Indeterminate Structures

by the Displacement Method:

Slope-Deflection Method

Theory of Structure - I

Page 2: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

2

Introduction

As pointed out earlier, there are two distinct methods of analysis for statically indeterminate structures depending on how equations of equilibrium, force-displacement and compatibility conditions are satisfied: 1) force method of analysis, and 2) displacement method of analysis. In the force method of analysis, primary unknowns are forces and compatibility of displacements is written in terms of pre-selected redundant reactions and flexibility coefficients using force-displacement relations. Solving these equations, the unknown redundant reactions are evaluated. The remaining reactions are obtained from equations of equilibrium.

Page 3: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

3

Introduction

As the name itself suggests, in the displacement method of analysis, the primary unknowns are displacements. Once the structural model is defined for the problem, the unknowns are automatically chosen unlike the force method. Hence this method is more suitable for computer implementation. In the displacement method of analysis, first equilibrium equations are satisfied. The equilibrium of force is written by expressing the unknown joint displacements in terms of force using force-displacement relations. These equilibrium equations are solved for unknown joint displacements. In the next step, the unknown reactions are computed from compatibility equations using force-displacement relations.

Page 4: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

4

Degrees of Freedom

When a structure is loaded, specified points on it, called nodes, will undergo unknown displacements. These displacements are referred to as the degrees of freedom for the structure, and in the displacement method of analysis it is important to specify these degrees of freedom since they become the unknowns when the method is applied. The number of these unknowns is referred to as the degree in which the structure is kinematically indeterminate.

To determine the kinematic indeterminacy we can imagine the structure to consist of a series of members connected to nodes, which are usually located at joints, supports, at the ends of a member or where the members have a sudden change in cross section.

Page 5: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

5

Degrees of Freedom

In three dimensions, each node on a frame or beam can have at most three linear displacements and three rotational displacements; and in two dimensions, each node can have at most two linear displacements and one rotational displacement. Furthermore, nodal displacements may be restricted by the supports, or due to assumptions based o the behavior of the structure. For example, if the structure is a beam ad only deformation due to bending is considered, then there can be no linear displacement along the axis of the beam since this displacement is caused by axial-force deformation.

Page 6: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

6

Degrees of Freedom

Kinematically indeterminate to the first degree.

Kinematically indeterminate to the fourth degree.

Kinematically indeterminate to the third degree.

Page 7: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

7

Degrees of Freedom

In summary, specifying the kinematic indeterminacy or the number of unconstrained degrees of freedom for the structure is a necessary first step when applying a displacement method of analysis. It identifies the number of unknowns in the problem, based on the assumptions made regarding the deformation behavior of the structure. Furthermore, once these nodal displacements are known, the deformation of the structural members can be completely specified, and the loadings within the members obtained.

Page 8: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

8

Slope-Deflection Equations

The slope-deflection method was introduced by George A. Maney of the University of Minnesota in the year 1915. In this method, the moment at the end of each method is expressed in terms of the 1) fixed-end moment due to external loads, 2) the rotation of the tangent at the end of each elastic curve, and 3) the rotation of the chord joining the ends of the elastic curve.

Page 9: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

9

Slope-Deflection Equations

In this method, the moment at the end of each method is expressed in terms of the 1) fixed-end moment due to external loads, 2) the rotation of the tangent at the end of each elastic curve, and 3) the rotation of the chord joining the ends of the elastic curve.

Page 10: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

10

Slope-Deflection Equations

Page 11: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

11

Slope-Deflection Equations

SIGN CONVENTION

The member fixed-end moments, end (tangent) rotations, and chord rotation are

positive (+) when clockwise.

Page 12: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

12

Page 13: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

13

Page 14: Analysis of Statically Indeterminate Structures by the ... of Statically Indeterminate Structures by the Displacement Method: Slope-Deflection Method ... Slope-Deflection Equations

14