Deflection of Structures - Theory & Concepts

Deflection of Structures

Calculating the elastic deformation of beams and frames.

In addition to strength (capacity to resist internal forces without failing), a structure must also satisfy serviceability requirements. A primary serviceability criterion is limiting deflection—the vertical or horizontal displacement of a structure under load. Excessive deflection can damage non-structural elements (like plaster ceilings or glass partitions), cause poor drainage on flat roofs, or simply make occupants feel unsafe.

Why Calculate Deflection?

Serviceability Limits: Building codes specify maximum allowable deflections (e.g., span/360 for live loads) to prevent damage to finishes.
Analysis of Indeterminate Structures: Deflection calculations are a prerequisite for analyzing statically indeterminate structures, as compatibility equations (which relate forces to deformations) must be used.
Aesthetics and Functionality: Preventing sagging floors or misaligned moving parts (like crane rails).

Key Takeaways

Calculating deflection is necessary to satisfy serviceability requirements and analyze indeterminate structures.

The Elastic Curve

The deflected shape of a beam.

When a beam bends under a transverse load, its longitudinal centroidal axis forms a curve known as the elastic curve. The equation of this curve defines the deflection

v

at any position

x

along the beam.

$$
EI \\frac{d^2v}{dx^2} = M(x)
$$

This fundamental differential equation is the basis for several geometric methods of calculating deflection.

Key Takeaways

The governing differential equation for the elastic curve is $EI v'' = M(x)$ .

Standard Boundary Conditions

Solving differential equations for deflection requires applying known geometric conditions at the supports (boundary conditions).

Common Boundary Conditions

Fixed Support: Prevents both translation and rotation. Therefore, deflection ( $v = 0$ ) and slope ( $\theta = 0$ ).
Pinned or Roller Support (End): Prevents vertical translation but allows rotation. Deflection ( $v = 0$ ), but slope ( $\theta \neq 0$ ).
Free End (Cantilever): Allows both translation and rotation. Neither deflection nor slope is zero ( $v \neq 0$ , $\theta \neq 0$ ). Internal moment ( $M = 0$ ) and shear ( $V = 0$ ) unless loaded.
Internal Hinge: Allows rotation but maintains continuity of displacement. Deflection is non-zero but continuous ( $v_{left} = v_{right}$ ). Slope is discontinuous ( $\theta_{left} \neq \theta_{right}$ ). Internal moment ( $M = 0$ ).

Geometric Methods for Calculating Deflection

Different approaches to solving the elastic curve equation for beams.

1. The Double Integration Method

Direct mathematical integration of the moment equation.

This method involves integrating the fundamental equation

EI v'' = M(x)

twice to find the deflection equation

v(x)

Procedure

STEP-BY-STEP

Determine the Moment Equation: Find the algebraic equation for the bending moment $M(x)$ as a function of position $x$ . If the loading changes along the span, multiple $M(x)$ equations (and thus multiple integrations) will be needed for each segment.
First Integration (Slope): Integrate $M(x)$ once to obtain the equation for the slope, $\theta$ (or $v'$ ). This introduces a constant of integration, $C_1$ .
Second Integration (Deflection): Integrate the slope equation to obtain the equation for deflection, $v$ . This introduces a second constant, $C_2$ .
Apply Boundary Conditions: Use known conditions at the supports (e.g., deflection is zero at a pin or roller; slope and deflection are zero at a fixed end) to solve for the constants $C_1$ and $C_2$ .

Slope Equation

The first integral of the moment equation.

$$
EI \theta = EI \frac{dv}{dx} = \int M(x) dx + C_1
$$

Deflection Equation

The second integral of the moment equation.

$$
EI v = \iint M(x) dx + C_1 x + C_2
$$

2. The Moment-Area Method

A semi-graphical method using the area under the M/EI diagram.

This method uses two theorems to relate the geometry of the elastic curve to the area under the

M/EI

diagram. It is particularly useful for beams with varying cross-sections (varying

I

) or when finding the deflection at a specific point relative to a tangent.

First Moment-Area Theorem (Slope)

The change in slope (angle) between the tangents at any two points

A

and

B

on the elastic curve is equal to the area of the

M/EI

diagram between those two points.

\theta_{B/A} = \theta_B - \theta_A = \int_{A}^{B} \frac{M}{EI} dx = \text{Area of } \frac{M}{EI} \text{ diagram}

Second Moment-Area Theorem (Tangential Deviation)

The tangential deviation of point

B

from the tangent drawn to the elastic curve at point

A

(

t_{B/A}

) is equal to the "moment" (first moment of area) of the

M/EI

diagram area between

A

and

B

, taken about point

B

t_{B/A} = \int_{A}^{B} x \frac{M}{EI} dx = \left( \text{Area of } \frac{M}{EI} \text{ diagram} \right) \times \bar{x}

Where

\bar{x}

is the distance from point

B

to the centroid of the

M/EI

area.

Important Distinction

The second theorem gives the tangential deviation (vertical distance from a point on the curve to a tangent line), not the actual deflection from the undeformed axis. Finding the actual deflection usually requires geometry and similar triangles.

3. The Conjugate Beam Method

Translating geometric problems into statics problems.

This is an elegant method that transforms the problem of finding slope and deflection into a familiar statics problem of finding shear and moment on an imaginary "conjugate" beam.

Procedure

STEP-BY-STEP

Create the Conjugate Beam: Draw a beam with the same length as the real beam. The supports of the conjugate beam are determined by the boundary conditions of the real beam such that:

Real pin/roller (end) → Conjugate pin/roller (end)
Real fixed end → Conjugate free end
Real free end → Conjugate fixed end
Real internal hinge → Conjugate internal roller
Real internal roller → Conjugate internal hinge

Apply the "Elastic Load": Load the conjugate beam with the $M/EI$ diagram of the real beam. The $M/EI$ diagram is treated as a distributed load (positive $M/EI$ acts upward).
Solve by Statics: Use equations of equilibrium ( $\sum F = 0, \sum M = 0$ ) on the conjugate beam.

The shear force ( $V'$ ) at any point on the conjugate beam equals the slope ( $\theta$ ) at that same point on the real beam.
The bending moment ( $M'$ ) at any point on the conjugate beam equals the deflection ( $v$ ) at that same point on the real beam.

Conjugate Beam Transformation

Load Magnitude (P)10 kN

Load Position (x)5 m

Notice how the real beam's fixed support becomes free on the conjugate beam, and the free end becomes fixed. The M/EI diagram is applied as a distributed load to the conjugate beam.

Key Takeaways

The Double Integration method integrates the moment equation twice, solving for constants using boundary conditions.
The Moment-Area method uses the area and the first moment of area of the $M/EI$ diagram to find changes in slope and tangential deviations.
The Conjugate Beam method converts deflection problems into statics problems by loading a fictitious beam with the $M/EI$ diagram.

Maxwell-Betti Reciprocal Theorem

A fundamental theorem for linear elastic structures which states: The deflection at point

A

caused by a load at point

B

is exactly equal to the deflection at point

B

caused by the same load placed at point

A

. This symmetry principle is crucial for simplifying complex deflection calculations and forming stiffness/flexibility matrices.

Virtual Work Formulas

The Principle of Virtual Work utilizes a "dummy" or virtual unit load to calculate real deflections:

For Trusses (Axial Deformation): $\Delta = \sum \frac{n N L}{A E}$ where $n$ is the internal virtual force, and $N$ is the internal real force.
For Beams (Bending Deformation): $\Delta = \int_0^L \frac{m M}{E I} dx$ where $m$ is the internal virtual moment, and $M$ is the internal real moment.

Energy Methods

Calculating deflections using work and strain energy concepts.

While geometric methods are excellent for beams, they become very complex when applied to trusses or rigid frames. Energy methods, based on the principle of conservation of energy, offer a more universal approach to finding deflections in any type of structure.

1. The Principle of Virtual Work (Unit Load Method)

The most versatile method for finding specific deflections.

This method applies a virtual (imaginary) unit load to the structure at the point and in the direction of the desired deflection. It equates the external virtual work done by this unit load to the internal virtual strain energy stored in the members.

Virtual Work for Trusses

To find the deflection

\Delta

at a joint in a truss:

Apply a virtual unit load (e.g., 1 kN) at the joint in the direction of $\Delta$ . Calculate the virtual internal force $u$ in each member.
Apply the real loads to the truss. Calculate the real internal axial force $N$ in each member.
Calculate the deflection by summing over all members: $\Delta = \sum \frac{u N L}{A E}$

Virtual Work for Beams and Frames

To find the deflection

\Delta

or rotation

\theta

at a point:

Apply a virtual unit load (for deflection) or virtual unit moment (for rotation) at the point of interest. Find the virtual internal moment equation $m$ .
Apply the real loads to find the real internal moment equation $M$ .
Integrate over the length of the structure: $\Delta = \int \frac{m M}{EI} dx$

2. Castigliano's Second Theorem

Using partial derivatives of strain energy.

Castigliano's theorem states that the deflection at a point caused by a specific load

P

is equal to the partial derivative of the total internal strain energy (

U

) of the structure with respect to that load

P

Procedure

STEP-BY-STEP

Express the internal forces (axial $N$ , moment $M$ ) in terms of an actual applied load $P$ at the point of interest. (If no load exists there, apply a fictitious load $P$ and set it to zero at the end).
For trusses, calculate $\Delta = \sum N \left(\frac{\partial N}{\partial P}\right) \frac{L}{AE}$ .
For beams/frames, calculate $\Delta = \int M \left(\frac{\partial M}{\partial P}\right) \frac{1}{EI} dx$ .

Key Takeaways

Energy methods are universally applicable to beams, trusses, and frames.
The Virtual Work (Unit Load) method involves applying a fictitious unit load to find specific deflections.
Castigliano's Second Theorem relates deflection to the partial derivative of strain energy with respect to an applied load.

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