STRUCTURAL THEORY
(CETHOS30)
Deflection (Ch. 8)
Engr. Jenalyn M. Columna, M. Eng
Civil Engineering Department | NU Baliwag
At the end of the lesson, the students are expected to:
1. Determine the elastic deflections of a beam;
2. Define Moment Curvature relationships; &
3. Set- up slope and deflection using the method of double integration.
Deflection Diagrams and the Elastic Curve
What causes deflection on
beams?
Deflection Diagrams and the Elastic Curve
Why is it important to check the deflection?
1. To provide integrity and stability of roofs
2. Prevent cracking of attached brittle materials
3. To ensure the occupants of a safe structure
4. Requirement in analyzing statically
Indeterminate Structures
Deflection Diagrams and the Elastic Curve
Deflection Diagrams and the Elastic Curve
Assumption:
- Deflections considered here only applies to structures having linear
elastic material response
Deflection Diagrams and the Elastic Curve
Deflection Diagram
- represents the elastic curve or locus of points which defines
the displaced position of the centroid of the cross section along the
members
Deflection Diagrams and the Elastic Curve
Deflection Diagrams and the Elastic Curve
If the shape of the moment diagram is known, it will be easy to
construct the elastic curve and vice versa.
Deflection Diagrams and the Elastic Curve
Moment Curvature Relationships
Consider a portion of the beam of length L,
For constant bending moment, the elastic curve becomes
an arc of circle with radius ρ,
Moment Curvature Relationships
Recall the arc length formula:
Solving the above equations in terms of θ and equating,
From Hooke's Law, From fiber stress formula,
Moment Curvature Relationships
𝟏 𝑴
=
𝝆 𝑬𝑰
where ρ - radius of curvature at a point (section)
1/ρ - curvature at a point (section)
M - internal bending moment at a section
E - modulus of elasticity of the beam
I - moment of inertia of the cross-sectional area
Moment Curvature Relationships
For a given curve, the curvature (from any calculus book) at any point is
defined by,
Small Displacement Theory applies.
Deflection y and slope dy/dx are very
small quantities.
So, (dy/dx)2 ≈ 0
Differential Equation for
the Elastic Curve of the Beam
Moment Curvature Relationships
Sign Convention
Moment Curvature Relationships
Note:
Double Integration Method
Procedure:
1. Obtain the moment function, M(x).
2. Solve the equation by integrating the it twice.
3. Obtain boundary/ continuity conditions at specific points of the beam
to get constants of integration, C1 and C2.
4. Solve for y(x).
Double Integration Method
Example
Each simply supported floor joist shown in the photo is subjected to a uniform
design loading of 4 kN/m. Determine the maximum deflection of the joist. EI is
constant.
Example
Example
Using Equation Function of Calculator to Solve for X
Using Equation Function of Calculator to Solve for X
Inputting the values for the (A, B, C , and D) constants of the equation to get the
location of max displacement.
𝐸𝐼𝑑𝑦 20𝑥 2 4𝑥 3
= − − 166.67 𝑠𝑎𝑚𝑒 𝑓𝑜𝑟𝑚 𝑤𝑖𝑡ℎ 𝑎𝑋 3 + 𝑏𝑋 2 + 𝑐𝑋 + 𝑑 = 0.
𝑑𝑥 2 6
Using Equation Function of Calculator to Solve for X
X values:
Example
Example
The cantilevered beam shown is subjected to a couple moment Mo at its end.
Determine the equation of the elastic curve. EI is constant.
Example
Example
STRUCTURAL THEORY
(CETHOS30)
Deflection – DIM Part 2
(Ch. 8)
Engr. Jenalyn M. Columna, M. Eng
Civil Engineering Department | NU Baliwag
At the end of the lesson, the students are expected to:
1. Define singularity functions;
2. Know singularity functions of different loadings
3. Set-up moment functions using singularity functions;
4. Set- up slope and deflection using the method of double integration.
Singularity Functions
Singularity functions are used for getting a single expression for moment
function, M(x), that is valid throughout the beam.
Singularity Functions
Properties of Singularity Functions
Singularity Functions
Illustration:
Singularity Functions
Singularity Functions for Different Loadings
Singularity Functions
Singularity Functions for Different Loadings
Note: For 3 & 4, the beam must be loaded all the way up to the end of the beam for M(x) to be
valid.
Example
The beam is subjected to a load P at its end. Determine the displacement
at C. EI is constant.
Example
Example
Example
Determine the equations of the elastic curve using x and specify the slope
at B and deflection at C. EI is constant.
Example
Example
Example
Example
Determine the equations of the elastic curve for the beam using x. Specify
the slope at support R1 and the maximum deflection. EI is constant.
Example
Example
Example
Example
Example
Determine the equations of the elastic curve using x, and specify the slope
and deflection at point B. EI is constant.
Example
Example
Example
