Deflection
Learning Objectives:
Calculate deflection in statically determinate beams and frames
Various Methods
• • • • Double Integration Method Moment-Area Method Elastic Load Method Conjugate Beam Method
Slope at A negative
Slope at B positive
Deflection at point B
Tangential deviation between points A and B
Change in slope
Change in slope and tangential deviation between points A and B
Moment-Area Method
Beam and moment curve
M/EI curve between points A and B
Moment Area Theorems
•The change in slope between any two points on a smooth continuous elastic curve is equal to the area under the M/EI curve between these points •The tangential deviation at a point B on a smooth continuous elastic curve from the tangent line drawn to the elastic curve at the second point A is equal to the moment about B of the area under the M/EI curve between these two points.
Moment-Area Method
Horizontal, therefore the vertical distance between tangent line and elastic curve are displacements
Cantilever, point of tangency at fixed support
Moment-Area Method
Symmetric members with symmetric loading, point of tangency at intersection of axis of symmetry and elastic curve
Moment-Area Method
Point of tangency at left end of member AB t BA L tan θ A = θ A in radians tan θ A =
θA=
t BA L
Caution 2.The theorem is applicable for continuous elastic curve 3.Presence of hinge breaks continuity of elastic curve 4.If hinge is present on beam or a frame – then work on either side of hinge (left or right side)
P 9.6 (3rd Edition)
Derive the equations for slope and deflection for the beam shown. Determine the slope at each support and value of deflection at mid span. Hint: Take advantage of symmetry; slope is zero at midspan.
P 9.13
Compute the slope at support A and the deflection at point B. Treat the rocker at D as a roller. Express the answers in terms of EI.
P 9.15 Determine the slop and deflection of point C
Calculate deflection in statically determinate beams and frames
Various Methods
• • • • Double Integration Method Moment-Area Method Elastic Load Method Conjugate Beam Method
Slope at A negative
Slope at B positive
Deflection at point B
Tangential deviation between points A and B
Change in slope
Change in slope and tangential deviation between points A and B
Moment-Area Method
Beam and moment curve
M/EI curve between points A and B
Moment Area Theorems
•The change in slope between any two points on a smooth continuous elastic curve is equal to the area under the M/EI curve between these points •The tangential deviation at a point B on a smooth continuous elastic curve from the tangent line drawn to the elastic curve at the second point A is equal to the moment about B of the area under the M/EI curve between these two points.
Moment-Area Method
Horizontal, therefore the vertical distance between tangent line and elastic curve are displacements
Cantilever, point of tangency at fixed support
Moment-Area Method
Symmetric members with symmetric loading, point of tangency at intersection of axis of symmetry and elastic curve
Moment-Area Method
Point of tangency at left end of member AB t BA L tan θ A = θ A in radians tan θ A =
θA=
t BA L
Caution 2.The theorem is applicable for continuous elastic curve 3.Presence of hinge breaks continuity of elastic curve 4.If hinge is present on beam or a frame – then work on either side of hinge (left or right side)
P 9.6 (3rd Edition)
Derive the equations for slope and deflection for the beam shown. Determine the slope at each support and value of deflection at mid span. Hint: Take advantage of symmetry; slope is zero at midspan.
P 9.13
Compute the slope at support A and the deflection at point B. Treat the rocker at D as a roller. Express the answers in terms of EI.
P 9.15 Determine the slop and deflection of point C