Slope-Deflection Method

Slope-Deflection Method

An analytical displacement-based method for determining end moments in continuous beams and rigid frames.

Introduction

The Slope-Deflection Method is a classical displacement method for analyzing statically indeterminate beams and frames. It expresses the unknown moments at the ends of members in terms of the unknown joint rotations (slopes) and translations (deflections). By applying equilibrium equations at the joints, these unknown displacements can be solved.

Key Advantages

It serves as the conceptual foundation for more advanced iterative (Moment Distribution) and matrix (Direct Stiffness) methods.
It directly yields the joint rotations and moments.
It easily accounts for support settlements.

Sign Convention

Before establishing the equations, a strict sign convention must be adopted:

Moments: Clockwise moments acting on the ends of the member are considered positive.
Rotations ( $\theta$ ): Clockwise rotations of the joints are considered positive.
Chord Rotations ( $\psi$ ): Clockwise rotation of the chord (the line connecting the two ends of the member) is positive. If one end deflects relative to the other by a distance $\Delta$ , the chord rotation is $\psi = \frac{\Delta}{L}$ .

The Slope-Deflection Equation

The general slope-deflection equation relates the moment at the near end (

M_{AB}

) of a member AB to the rotations at its ends (

\theta_A

\theta_B

), its relative displacement (

\Delta

), and its loading.

M_{AB} = 2E \left( \frac{I}{L} \right) \left[ 2\theta_A + \theta_B - 3\left(\frac{\Delta}{L}\right) \right] + FEM_{AB}

Alternatively, using stiffness

K = \frac{I}{L}

and chord rotation

\psi = \frac{\Delta}{L}

M_{AB} = 2EK(2\theta_A + \theta_B - 3\psi) + FEM_{AB}

Where:

$M_{AB}$ = Final internal moment at end A of member AB.
$E$ = Modulus of elasticity.
$I$ = Moment of inertia of the member.
$L$ = Length of the member.
$\theta_A$ , $\theta_B$ = Rotations of joints A and B.
$\Delta$ = Relative linear displacement between joints A and B perpendicular to the member's axis.
$FEM_{AB}$ = Fixed-End Moment at end A due to applied loads (assuming joints A and B are perfectly fixed).

Derivation of the Equation

The fundamental slope-deflection equation is derived using the principle of superposition and the moment-area theorems.

Superposition of Effects

The final moment at the end of a member (

M_{AB}

) is the algebraic sum of four distinct effects acting on a perfectly fixed beam of the same length and stiffness:

Effect 1: The moment required to produce a rotation $heta_A$ at end $A$ while end $B$ remains fixed ( $M = rac{4EI}{L} heta_A$ ).
Effect 2: The carry-over moment at end $A$ caused by a rotation $heta_B$ at end $B$ ( $M = rac{2EI}{L} heta_B$ ).
Effect 3: The moment caused by a relative transverse displacement $\Delta$ between the two ends without any joint rotation ( $M = - rac{6EI}{L^2}\Delta$ ).
Effect 4: The Fixed-End Moment ( $FEM_{AB}$ ) caused by the actual external loads applied along the span, assuming both ends are completely fixed against rotation and translation.

Summing these four effects yields the general slope-deflection equation.

Application to Frames

The method is highly effective for analyzing rigid frames, but the procedure differs depending on whether the frame can sway laterally.

Frames without Sidesway

Braced Frames

A frame will not undergo sidesway if it is adequately braced, or if both the frame geometry and the applied loading are perfectly symmetrical.

In these cases, the relative lateral displacement ( $\Delta$ ) between the ends of the columns is zero.
The chord rotation term ( $\psi = rac{\Delta}{L}$ ) is eliminated from all slope-deflection equations.
The only unknowns are the joint rotations ( $heta$ ). The number of simultaneous equations to solve is equal to the number of joints that can rotate.

Frames with Sidesway

Unbraced Frames

Frames will sway laterally if they are unbraced and subjected to lateral loads (like wind), or if they are unsymmetrical in geometry or vertical loading.

The lateral displacement ( $\Delta$ ) becomes an additional unknown degree of freedom.
Consequently, an additional equilibrium equation is required to solve the system. This is called the shear equation.
The shear equation is formulated by passing a horizontal section through the columns of a story and applying global lateral equilibrium ( $\sum F_x = 0$ ). The shear in each column is expressed in terms of its end moments, linking the joint equilibrium equations to the global sway of the frame.

Analysis Procedure

Procedure

STEP-BY-STEP

Identify the unknown degrees of freedom (joint rotations $\theta$ and independent joint translations $\Delta$ ).
Calculate the Fixed-End Moments (FEMs) for all loaded members, adhering to the clockwise-positive sign convention.
Write the slope-deflection equation for the moment at each end of every member in terms of the unknown displacements.
Apply joint equilibrium equations: $\sum M_{joint} = 0$ . For example, at joint B, $M_{BA} + M_{BC} = 0$ . Substitute the slope-deflection equations from Step 3 into these equilibrium equations.
If the frame can sway (sidesway), formulate an additional shear equation based on global lateral equilibrium ( $\sum F_x = 0$ ) and the column end moments.
Solve the resulting system of simultaneous linear equations for the unknown displacements ( $\theta$ , $\Delta$ ).
Substitute the calculated displacements back into the original slope-deflection equations (Step 3) to determine the final end moments.
Use statics to find the remaining member shears, axial forces, and support reactions.

Interactive Simulation: Slope-Deflection Simulator

Visualize how joint rotations, support settlements, and applied loads combine to generate end moments in a continuous beam using the Slope-Deflection equations.

Slope-Deflection Method: Propped Cantilever

Span Length (L): 10 m

Uniform Load (w): 10 kN/m

Support A Settlement (Δ): 0 mm

Results

Rotation at A (θ_A): 0.000 x 10^-3 rad
Rotation at B (θ_B): 0.000 rad (Fixed)
Moment at A (M_AB): 0.00 kNm (Pinned)
Moment at B (M_BA): 0.00 kNm

Loading chart...

Modified Slope-Deflection Equation

Pined-End Simplification

If the far end of a member is pinned or roller-supported (meaning the moment there is zero), the standard slope-deflection equation can be modified to eliminate the far-end rotation from the unknowns. This reduces the number of simultaneous equations required.

For a member AB where end B is pinned:

M_{AB} = \frac{3EI}{L} \left( \theta_A - \frac{\Delta}{L} \right) + \text{FEM}_{AB} - \frac{1}{2}\text{FEM}_{BA}

Sway vs. Non-Sway Frames

The application of the slope-deflection method depends heavily on whether the frame can translate laterally (sidesway).

Frames Without Sidesway

A frame is considered non-sway if horizontal translation of its joints is prevented by adequate supports, symmetrical geometry, and symmetrical loading.

The chord rotation $\psi = \Delta / L$ is zero for all members.
The unknown degrees of freedom are purely joint rotations ( $\theta$ ).

Frames With Sidesway

If a frame is unsymmetrical, subjected to lateral loads, or has unsymmetrical supports, it will sway laterally.

The joints will translate, creating an unknown lateral displacement $\Delta$ , which produces an unknown chord rotation $\psi = \Delta / L$ .
An additional equilibrium equation, the "shear equation" (based on $\sum F_x = 0$ for the entire frame or a story), must be formulated to solve for this additional unknown $\Delta$ .

Key Takeaways

The Slope-Deflection Method is a displacement-based approach that formulates end moments in terms of joint rotations and translations.
A strict sign convention (usually clockwise positive) is crucial for correct application.
The general equation relates the final moment to the member's stiffness, end rotations, chord rotation, and Fixed-End Moments.
Solving involves setting up and solving simultaneous equilibrium equations at the joints.
This method forms the theoretical basis for modern matrix structural analysis.
The Slope-Deflection Method is a classic displacement method where joint rotations and translations are the primary unknowns.
The fundamental equation relates member end moments to joint rotations ( $\theta$ ), chord rotations ( $\Delta/L$ ), and Fixed-End Moments (FEM).
Frames are classified as either without sidesway (braced) or with sidesway (unbraced), fundamentally altering the kinematic unknowns.
Equilibrium equations are formulated at each joint ( $\sum M = 0$ ) and for the entire frame (shear/story equations) to solve for the unknown displacements.

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