Slope Deflection Derivation - Page 1 of 7

Slope Deflection Derivation Objectives:

1. Calculate the FEM for a beam 2. Write the slope deflection equation 3. Define the four terms of the slope deflection equation 4. Explain the sign convention for the slope deflection terms

Slope deflection is the first stiffness method Slope deflection is based on the superposition of moments on the end of a member. The moment at the end of any member can be broken up into four parts:

Total moment at the end of a beam (Ma for the a end) is the sum of

1) The moment caused by the external load assuming the ends of the element are fixed, called the fixed end moments (FEM),

2) The moment due to a rotation of the end in consideration (Θa), 3) The moment due to a rotation of the opposite end of the elements (Θb) and 4) The moment due to a relative displacement of the two ends (δ).

Slope Deflection Derivation - Page 2 of 7

From the figure, you can see that the moment at end a is:

M + M + M + M = M aaaaa baFEM δθθ

Axial deformations are ignored. First, we look at the moments due to applied loads within the span of a fixed beam. These are called the Fixed End Moments (FEM). Find by consistent deformations;

Remove the right hand support entirely leaving a cantilever base structure. Apply the unit forces and do the integration, we generate two equations for the two unknowns. Solving we get:

Slope Deflection Derivation - Page 3 of 7

L* W* coeff. = FEM

Where: Coeff. from diagram.

W is total load applied. L is length of beam.

Slope Deflection Derivation - Page 4 of 7

Next, find the moment, MΘ, due to rotation.

θ

θ

Using consistent deformations, we can find the support reaction. The virtual load case is:

Integrating for the r10 and f11 terms and substituting into the compatibility equation, we get the resulting determinate structure.

Slope Deflection Derivation - Page 5 of 7

The structure is determinate with two loads. Solving for the angle Θa , the virtual load case for finding the rotation is:

Virtual Load for Moment-Rotation Equation

1 (Unit Moment)

Using virtual work and integrating we find that:

θθ aa LEI 4 = M

a

We can now find M b aθ

from statics and the determinate system. Summing moments about point b and substituting for M a aθ

we get: The moment at the opposite end of a rotation is equal to half the moment at the rotated end. Finally, the moment due to a relative displacement between the two ends. Only bending strain energy is considered. Hence, a beam displacement is compised of three parts;

Slope Deflection Derivation - Page 6 of 7

The first two parts are a rigid body displacement and a rigid body rotation with no strain. The third part is an equal rotation at both ends. For small angles, the angle at each end is:

L = = baδ

θθ

Substituting into the M-θ equation: Summing the four contributions gives: The sign convention followed is that all rotations are positive counter clockwise (this follows the right hand rule). The convention for positive and negative displacement is shown below.

Slope Deflection Derivation - Page 7 of 7

Another way: Draw a chord between the two ends of a deformed element. Counter clockwise from the chord; the displacement is defined as positive.