CE 221: MECHANICS OF SOLIDS I CHAPTER 9: DEFLECTIONS OF BEAMS By Dr. Krisada Chaiyasarn Department of Civil Engineering, Faculty of Engineering Thammasat university

Outline •  The elastic curve • Slope and displacement by integration

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The Elastic Curve •  The deflection of a beam or shaft must be limited to provide integrity and stability

and prevent cracking in brittle materials. •  Members must not vibrate or deflect severely to safely support intending loading •  Hence deflections at specific point in a beam must be found, especially in

statically indeterminate structures •  The deflection curve of the longitudinal axis passing through the centroid is called

the •  In general, supports that resist a force restrict displacement and fixed support

restrict rotation.

The Elastic Curve •  The elastic curve can be related by the moment diagram based

on the sign convention, a positive moment bends the beam upwards and a negative moment bends the beam downwards

•  Once the moment diagram is known, the elastic curve can be found

•  The inflection point is when the curve changes the direction, when moment is zero

Moment-curvature relationship •  The relationship between the internal moment and the radius of

curvature ρ •  The analysis is limited to a initially straight beam and elastically

deformed by loads applied perpendicular to the beam. •  The deformation is caused by both the internal shear force and

bending moment

Moment-curvature relationship •  The internal moment M deforms the element and cause the

angle dθ •  EI is called the flexural rigidity

Moment-curvature relationship •  The internal moment M deforms the element and cause the

angle dθ •  EI is called the flexural rigidity

Slope and Displacement by Integration •  The elastic curve can be expressed as v = f(x) •  For most engineering applications and design codes, the

square of dv/dx is very small, hence the denominator can be ignored.

Slope and Displacement by Integration •  For each integration, it requires a constant of integration •  If the distributed w is used, require 4 constants •  Generally, we start with the internal moment M and only two constants are

required •  If there are discontinuity in the moment, several equations are required for each

region of discontinuity

Sign conventions and coordinates

Boundary and Continuity Conditions •  We determine the functions for shear, moment,

slope and displacement at the the place where the value of the function is known, this is called the boundary conditions.

•  If the elastic curve cannot be expressed using a single coordinate, the continuity conditions must be used

Example