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University of Michigan, TCAUP Structures II Slide 1/27

Architecture 324Structures II

Deflection of Structural Members

Slope and Elastic Curve Deflection Limits Diagrams by Parts Symmetrical Loading Asymmetrical Loading Deflection Equations and Superposition

MAISIE GWYNNE, 2013 - USC

University of Michigan, TCAUP Structures II Slide 2/27

Deformation

Based on Hookes Law and the definition of Youngs Modulus, E

And so,

University of Michigan, TCAUP Structures II Slide 3/27

Deflection

Axial fiber deformation in flexure results in normal (vertical) deflection.

The change in lengths, top and bottom, results in the material straining. For a simple span with downward loading, the top is compressed and the bottom stretched.

The material strains result in corresponding stresses. By Hookes Law, these stresses are proportional to the strains which are proportional to the change in length of the radial arcs of the beam fibers.

University of Michigan, TCAUP Structures II Slide 4/27

Deflection

Deflection is the distance that a beam bends from its original horizontal position, when subjected to loads.

The compressive and tensile forces above and below the neutral axis, result in a shortening (above n.a.) and lengthening (below n.a.) of the longitudinal fibers of a simple beam, resulting in a curvature which deflects from the original position.

Axial Stiffness

Flexural Stiffness

University of Michigan, TCAUP Structures II Slide 5/27

Slope

The curved shape of a deflected beam is called the elastic curve

The angle of a tangent to the elastic curve is called the slope, and is measured in radians.

Slope is influenced by the stiffness of the member:

material stiffness E, the modulus of elasticity

sectional stiffness I, the moment of inertia,

as well as the length of the beam, L

University of Michigan, TCAUP Structures II Slide 6/27

Deflection Limits (serviceability)

Various guidelines have been derived, based on usage, to determine maximum allowable deflection limits.

Typically, a floor system with a LL deflection in excess of L/360 will feel bouncy or crack plaster.

Flat roofs require a minimum slope of / ft to avoid ponding. Ponding refers to the retention of water due solely to deflection of relatively flat roofs. Progressive deflection due to progressively more impounded water can lead to collapse.

International Building Code - 2006

roof pondingfrom IRC, Josh 2014

University of Michigan, TCAUP Structures II Slide 7/27

Relationships of Forces and DeformationsThere is a series of relationships involving forces and deformations along a beam, which can be useful in analysis. Using either the deflection or load as a starting point, the following characteristics can be discovered by taking successive derivatives or integrals of the beam equations.

University of Michigan, TCAUP Structures II Slide 8/27

Slope and Deflection in Symmetrically Loaded Beams

Maximum slope occurs at the ends of the beam

A point of zero slope occurs at the center line. This is the point of maximum deflection.

Moment is positive for gravity loads.

Shear and slope have balanced + and - areas.

Deflection is negative for gravity loads.

University of Michigan, TCAUP Structures II Slide 9/27

Cantilever Beams

One end fixed. One end free

Fixed end has maximum moment, but zero slope and deflection.

Free end has maximum slope and deflection, but zero moment.

Slope is either downward (-) or upward (+) depending on which end is fixed.

Shear sign also depends of which end is fixed.

Moment is always negative for gravity loads.

University of Michigan, TCAUP Structures II Slide 10/27

Methods to Calculate Deflection

Integrationcan use to derive equations

Diagramssymmetric load cases

Diagrams (by parts)asymmetric load cases

Diagrams (by shifting baseline)asymmetric load cases

Equationssingle load cases

Superposition of Equationsmultiple load cases

University of Michigan, TCAUP Structures II Slide 11/27

Deflection by Integration

Load

Shear

Moment

University of Michigan, TCAUP Structures II Slide 12/27

Deflection by Integration

Slope

Deflection

University of Michigan, TCAUP Structures II Slide 13/27

Deflection by Diagrams

Load

Shear

Moment

Slope (EI)

Deflection (EI)

University of Michigan, TCAUP Structures II Slide 14/27

Table D-24 I. Engel

University of Michigan, TCAUP Structures II Slide 15/27

Diagrams by Parts marks vertex which must be present for area equations to be valid.

University of Michigan, TCAUP Structures II Slide 16/27

Diagrams by Base Line ShiftAsymmetrically Loaded Beams

The value of the slope at each of the endpoints is different.

The exact location of zero slope (and maximum deflection) is unknown.

Start out by assuming a location of zero slope (Choose a location with a known dimension from the loading diagram)

With the arbitrary location of zero slope, the areas above and below the base line (A and B) are unequal

Adjust the base line up or down by D distance in order to equate areas A and B. Shifting the base line will remove an area a from A and add an area b to B

University of Michigan, TCAUP Structures II Slide 17/27

Asymmetrically Loaded Beams (continued)

Compute distance D with the equation:

so,

With the vertical shift of the base line, a horizontal shift occurs in the position of zero slope.

The new position of zero slope will be the actual location of maximum deflection.

Compute the area under the slope diagram between the endpoint and the new position of zero slope in order to compute the magnitude of the deflection.

University of Michigan, TCAUP Structures II Slide 18/27

Example: Asymmetrical Loading Diagram method

Solve end reactions

Construct shear diagram

Construct moment diagram

Choose point of EI = 0Sum moment areas to each side to find EI end values

Calculate distance D

University of Michigan, TCAUP Structures II Slide 19/27

Example: Asymmetrical Loading Diagram method

Shift the baseline and find point of crossing. Find new areas

Check that areas A and B balance. Any difference is just round-off error. (average the two to reduce error)

University of Michigan, TCAUP Structures II Slide 20/27

Example: Asymmetrical Loading Diagram method

Calculate the deflection in inches for a given member.

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