Lecture 35 - Beam DeflectionNovember 22, 2002CVEN 444

Lecture GoalsExample on moment of inertiaServiceabilityDeflection calculationDeflection example

Reinforced Concrete Sections - ExampleGiven a doubly reinforced beam with h = 24 in, b = 12 in., d = 2.5 in. and d = 21.5 in. with 2# 7 bars in compression steel and 4 # 7 bars in tension steel. The material properties are fc = 4 ksi and fy= 60 ksi.Determine Igt, Icr , Mcr(+), Mcr(-), and compare to the NA of the beam.

Reinforced Concrete Sections - ExampleThe components of the beam

Reinforced Concrete Sections - Example The compute the n value and the centroid, I uncracked

Sheet1

Exposure ConditionTolerable

Crack Width

Dry air or protective membrane-0.016 in.

Humidity, moist air, soil-0.012 in.

Deicing chemicals-0.007 in.

Seawater and seawater spray-0.006 in.

wetting and drying

Water-retaining structures-0.004 in.

(excluding nonpressure pipes)

Sheet2

nd (in)

7.041.28.4482.521.12--9.756804.10

7.042.416.89621.5363.26-9.2441443.75

1288288123456.0013824-0.25618.89

313.3443840.38138242266.74

Sheet3

Reinforced Concrete Sections - ExampleThe compute the centroid and I uncracked

Reinforced Concrete Sections - Example The compute the centroid and I for a cracked doubly reinforced beam.

Reinforced Concrete Sections - ExampleThe compute the centroid for a cracked doubly reinforced beam.

Reinforced Concrete Sections - Example The compute the moment of inertia for a cracked doubly reinforced beam.

Reinforced Concrete Sections - Example The critical ratio of moment of inertia

Reinforced Concrete Sections - ExampleFind the components of the beam

Reinforced Concrete Sections - ExampleFind the components of the beamThe neutral axis

Reinforced Concrete Sections - ExampleThe strain of the steelNote: At service loads, beams are assumed to act elastically.

Reinforced Concrete Sections - ExampleUsing a linearly varying e and s = Ee along the NA is the centroid of the area for an elastic centerThe maximum tension stress in tension is

Reinforced Concrete Sections - ExampleThe uncracked moments for the beam

Calculate the Deflections(1) Instantaneous (immediate) deflections(2) Sustained load deflectionInstantaneous Deflectionsdue to dead loads( unfactored) , live, etc.

Calculate the DeflectionsInstantaneous DeflectionsEquations for calculating Dinst for common cases

Calculate the DeflectionsInstantaneous DeflectionsEquations for calculating Dinst for common cases

Calculate the DeflectionsInstantaneous DeflectionsEquations for calculating Dinst for common cases

Calculate the DeflectionsInstantaneous DeflectionsEquations for calculating Dinst for common cases

Sustained Load DeflectionsCreep causes an increase in concrete strainCurvature increasesCompression steel presentIncrease in compressive strains cause increase in stress in compression reinforcement (reduces creep strain in concrete)Helps limit this effect.

Sustained Load DeflectionsSustain load deflection = l Di Instantaneous deflectionACI 9.5.2.5at midspan for simple and continuous beams at support for cantilever beams

Sustained Load Deflectionsx = time dependent factor for sustained loadAlso see Figure 9.5.2.5 from ACI code

Sustained Load DeflectionsFor dead and live loadsDL and LL may have different x factors for LT ( long term ) D calculations

Sustained Load DeflectionsThe appropriate value of Ic must be used to calculate D at each load stage.

Serviceability Load Deflections - ExampleShow in the attached figure is a typical interior span of a floor beam spanning between the girders at locations A and C. Partition walls, which may be damaged by large deflections, are to be erected at this level. The interior beam shown in the attached figure will support one of these partition walls. The weight of the wall is included in the uniform dead load provided in the figure. Assume that 15 % of the distributed dead load is due to a superimposed dead load, which is applied to the beam after the partition wall is in place. Also assume that 40 % of the live load will be sustained for at least 6 months.

Serviceability Load Deflections - Examplefc = 5 ksify = 60 ksi

Serviceability Load Deflections - ExamplePart IDetermine whether the floor beam meets the ACI Code maximum permissible deflection criteria. (Note: it will be assumed that it is acceptable to consider the effective moments of inertia at location A and B when computing the average effective moment of inertia for the span in this example.)

Part IICheck the ACI Code crack width provisions at midspan of the beam.

Serviceability Load Deflections - ExampleDeflection before glass partition is installed (85 % of DL)

Serviceability Load Deflections - ExampleCompute the centroid and gross moment of inertia, Ig.

_1068104861.xls

Sheet1

bhAreayiAi * yi

Flange844.537817.756709.5

Web15.5121867.751441.5

5648151

Serviceability Load Deflections - ExampleThe moment of inertia

Serviceability Load Deflections - ExampleThe moment capacity

Serviceability Load Deflections - ExampleDetermine bending moments due to initial load (0.85 DL) The ACI moment coefficients will be used to calculate the bending moments Since the loading is not patterned in this case, This is slightly conservative

Serviceability Load Deflections - ExampleThe moments at the two locations

Serviceability Load Deflections - ExampleMoment at C will be set equal to Ma for simplicity, as given in the problem statement.

Serviceability Load Deflections - ExampleAssume Rectangular Section Behavior and calculate the areas of steel and ratio of Modulus of Elasticity

Serviceability Load Deflections - ExampleCalculate the center of the T-beam

Serviceability Load Deflections - ExampleThe centroid is located at the As < 4.5 in. = tf Use rectangular section behavior

Serviceability Load Deflections - ExampleThe moment of inertia at midspan

Serviceability Load Deflections - ExampleCalculate average effective moment of inertia, Ie(avg) for interior span (for 0.85 DL) For beam with two ends continuous and use Ig for the two ends.

Serviceability Load Deflections - ExampleCalculate instantaneous deflection due to 0.85 DL:Use the deflection equation for a fixed-fixed beam but use the span length from the centerline support to centerline support to reasonably approximate the actual deflection.

Serviceability Load Deflections - ExampleCalculate additional short-term Deflections (full DL & LL)

Serviceability Load Deflections - ExampleCalculate additional short-term Deflections (full DL & LL)

Let Mc = Ma = - 2000 k-in for simplicity see problem statement

Serviceability Load Deflections - ExampleAssume beam is fully cracked under full DL + LL, therefore I= Icr (do not calculate Ie for now).Icr for supports

Serviceability Load Deflections - ExampleClass formula using doubly reinforced rectangular section behavior.

Serviceability Load Deflections - ExampleClass formula using doubly reinforced rectangular section behavior.

Serviceability Load Deflections - ExampleCalculate moment of inertia.

Serviceability Load Deflections - ExampleWeighted Icr

Serviceability Load Deflections - ExampleInstantaneous Dead and Live Load Deflection.

Serviceability Load Deflections - ExampleLong term Deflection at the midspanDead Load (Duration > 5 years)

Serviceability Load Deflections - ExampleLong term Deflection use the midspan informationLive Load (40 % sustained 6 months)

Serviceability Load Deflections - ExampleTotal Deflection after Installation of Glass Partition Wall.

Serviceability Load Deflections - ExampleCheck whether modifying Icr to Ie will give an acceptable deflection:

Serviceability Load Deflections - ExampleCheck whether modifying Icr to Ie will give an acceptable deflection:

Serviceability Load Deflections - ExampleFloor Beam meets the ACI Code Maximum permissible Deflection Criteria. Adjust deflections:

Serviceability Load Deflections - ExampleAdjust deflections:

Serviceability Load Deflections - ExamplePart II: Check crack width @ midspan

Serviceability Load Deflections - ExampleAssumeFor interior exposure, the crack width @ midspan is acceptable.

Homework-12/2/02Problem 8.7