06CV761 NUMERICAL METHODS IN CIVIL ENGINEERING

By Radhika Bhandary P Civil Engineering Dept.

Indeterminate Structures Slope-Deflection Method

IntroductionThe slope deflection method is a structural analysis method for beams and frames introduced in 1915 by George A. Maney. The slope deflection method was widely used for more than a decade until the moment distribution method was developed.

In the slope-deflection method, the relationship is established between moments at the ends of the members and the corresponding rotations and displacements. The basic assumption used in the slope-deflection method is that a typical member can flex but the shear and axial deformation are negligible. It is no different from that used with the force method.

Kinematically indeterminate structures versus statically indeterminate structuresRedundants: 1 (By) Kinematics: 1 ( B)

Redundants: 1 (By) Kinematics: 3 ( A, B,

C)

Redundants: 2(By,Cy) Kinematics: 2 ( B, C)

Redundants: 1(Cy) Kinematics: 3 ( B, C,x)

Analysis of Beams Slope-Deflection Method (General Procedure)Step 1: Scan the beam and identify the number of (a) segments (b) kinematic unknowns. A segment is the portion of the beam between two nodes. Kinematic unknowns are those rotations and displacements that are not zero and must be computed. The support or end conditions of the beam will help answer the question. Step 2: For each segment, generate the two governing equations.Check the end conditions to see whether one of the end rotations is zero or not.

Step 3: For each kinematic unknown, generate an equilibrium condition using the free-body diagram. Step 4: Solve for all unknowns by combining all the equations from steps 2 and 3. Now the equations are entirely in terms of the kinematic unknowns. Step 5: Compute the support reactions with appropriate FBDs.

Basic Idea of Slope Deflection MethodThe basic idea of the slope deflection method is to write the equilibrium equations for each node in terms of the deflections and rotations. Solve for the generalized displacements, Using moment displacement relations, moments are then known. The structure is thus reduced to a determinate structure.

The slope deflection equations of member ab of flexural rigidity EIab and length Lab are:

where

a,

b

are the slope angles of ends a and b respectively,

is the relative lateral displacement of ends a and b. The absence of cross-sectional area of the member in these equations implies that the slope deflection method neglects the effect of shear and axial deformations.

Equilibrium conditionsJoint equilibrium Joint equilibrium conditions imply that each joint with a degree of freedom should have no unbalanced moments i.e. be in equilibrium. Therefore,

Here, Mmember are the member end moments, Mf are the fixed end moments, and Mjoint are the external moments directly applied at the joint. Shear equilibrium When there are chord rotations in a frame, additional equilibrium conditions, namely the shear equilibrium conditions need to be taken into account.

The statically indeterminate beam shown in the figure is to be analysed.

Members AB, BC, CD have the same length L= 10m . Flexural rigidities are EI, 2EI, EI respectively. Concentrated load of magnitude P= 10kN acts at a distance a = 3m from the support A. Uniform load of intensity q = 1 kN/m acts on BC. Member CD is loaded at its midspan with a concentrated load of magnitude . In the following calcuations, clockwise moments and rotations are positive. Degrees of freedom Rotation angles A, B, C, D of joints A, B, C, D respectively are taken as the unknowns. There are no chord rotations due to other causes including support settlement.

Fixed end moments Fixed end moments are :

The slope deflection equations are constructed as follows:

Joint equilibrium equations

Joints A, B, C should suffice the equilibrium condition. Therefore

Here having three equations with three unknowns numerical methods are used to find the unknowns

Rotation angles The rotation angles are calculated from simultaneous equations above.

Member end moments Substitution of these values back into the slope deflection equations yields the member end moments (in kNm):