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**Chp-6:Lecture Goals Serviceability Deflection calculation**

Deflection example

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**Structural Design Profession is concerned with:**

Limit States Philosophy: Strength Limit State (safety-fracture, fatigue, overturning buckling etc.-) Serviceability Performance of structures under normal service load and are concerned the uses and/or occupancy of structures

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;

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λ :Amplification factor ζ :Time dependent Factor

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Solution:

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See Example 2.2 Ma: Max. Service Load Moment

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**Reinforced Concrete Sections - Example**

Given 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.

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**Reinforced Concrete Sections - Example**

The components of the beam

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**Reinforced Concrete Sections - Example**

The compute the n value and the centroid, I uncracked

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**Reinforced Concrete Sections - Example**

The compute the centroid and I uncracked

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**Reinforced Concrete Sections - Example**

The compute the centroid and I for a cracked doubly reinforced beam.

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**Reinforced Concrete Sections - Example**

The compute the centroid for a cracked doubly reinforced beam.

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**Reinforced Concrete Sections - Example**

The compute the moment of inertia for a cracked doubly reinforced beam.

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**Reinforced Concrete Sections - Example**

The critical ratio of moment of inertia

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**Reinforced Concrete Sections - Example**

Find the components of the beam

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**Reinforced Concrete Sections - Example**

Find the components of the beam The neutral axis

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**Reinforced Concrete Sections - Example**

The strain of the steel Note: At service loads, beams are assumed to act elastically.

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**Reinforced Concrete Sections - Example**

Using a linearly varying e and s = Ee along the NA is the centroid of the area for an elastic center The maximum tension stress in tension is

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**Reinforced Concrete Sections - Example**

The uncracked moments for the beam

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**Calculate the Deflections**

(1) Instantaneous (immediate) deflections (2) Sustained load deflection Instantaneous Deflections due to dead loads( unfactored) , live, etc.

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**Calculate the Deflections**

Instantaneous Deflections Equations for calculating Dinst for common cases

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**Calculate the Deflections**

Instantaneous Deflections Equations for calculating Dinst for common cases

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**Calculate the Deflections**

Instantaneous Deflections Equations for calculating Dinst for common cases

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**Calculate the Deflections**

Instantaneous Deflections Equations for calculating Dinst for common cases

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**Sustained Load Deflections**

Creep causes an increase in concrete strain Curvature increases Compression steel present Increase in compressive strains cause increase in stress in compression reinforcement (reduces creep strain in concrete) Helps limit this effect.

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**Sustained Load Deflections**

Sustain load deflection = l Di Instantaneous deflection ACI at midspan for simple and continuous beams at support for cantilever beams

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**Sustained Load Deflections**

x = time dependent factor for sustained load 5 years or more 12 months months months Also see Figure from ACI code

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**Sustained Load Deflections**

For dead and live loads DL and LL may have different x factors for LT ( long term ) D calculations

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**Sustained Load Deflections**

The appropriate value of Ic must be used to calculate D at each load stage. Some percentage of DL (if given) Full DL and LL

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**Serviceability Load Deflections - Example**

Show 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.

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**Serviceability Load Deflections - Example**

fc = 5 ksi fy = 60 ksi

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**Serviceability Load Deflections - Example**

Part I Determine 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 II Check the ACI Code crack width provisions at midspan of the beam.

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**Serviceability Load Deflections - Example**

Deflection before glass partition is installed (85 % of DL)

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**Serviceability Load Deflections - Example**

Compute the centroid and gross moment of inertia, Ig.

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**Serviceability Load Deflections - Example**

The moment of inertia

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**Serviceability Load Deflections - Example**

The moment capacity

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**Serviceability Load Deflections - Example**

Determine 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

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**Serviceability Load Deflections - Example**

The moments at the two locations

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**Serviceability Load Deflections - Example**

Moment at C will be set equal to Ma for simplicity, as given in the problem statement.

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**Serviceability Load Deflections - Example**

Assume Rectangular Section Behavior and calculate the areas of steel and ratio of Modulus of Elasticity

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**Serviceability Load Deflections - Example**

Calculate the center of the T-beam

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**Serviceability Load Deflections - Example**

The centroid is located at the A’s < 4.5 in. = tf Use rectangular section behavior

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**Serviceability Load Deflections - Example**

The moment of inertia at midspan

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**Serviceability Load Deflections - Example**

Calculate 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.

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**Serviceability Load Deflections - Example**

Calculate 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.

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**Serviceability Load Deflections - Example**

Calculate additional short-term Deflections (full DL & LL)

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**Serviceability Load Deflections - Example**

Calculate additional short-term Deflections (full DL & LL) Let Mc = Ma = k-in for simplicity see problem statement

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**Serviceability Load Deflections - Example**

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

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**Serviceability Load Deflections - Example**

Class formula using doubly reinforced rectangular section behavior.

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**Serviceability Load Deflections - Example**

Class formula using doubly reinforced rectangular section behavior.

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**Serviceability Load Deflections - Example**

Calculate moment of inertia.

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**Serviceability Load Deflections - Example**

Weighted Icr

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**Serviceability Load Deflections - Example**

Instantaneous Dead and Live Load Deflection.

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**Serviceability Load Deflections - Example**

Long term Deflection at the midspan Dead Load (Duration > 5 years)

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**Serviceability Load Deflections - Example**

Long term Deflection use the midspan information Live Load (40 % sustained 6 months)

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**Serviceability Load Deflections - Example**

Total Deflection after Installation of Glass Partition Wall.

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**Serviceability Load Deflections - Example**

Check whether modifying Icr to Ie will give an acceptable deflection:

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**Serviceability Load Deflections - Example**

Check whether modifying Icr to Ie will give an acceptable deflection:

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**Serviceability Load Deflections - Example**

Floor Beam meets the ACI Code Maximum permissible Deflection Criteria. Adjust deflections:

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**Serviceability Load Deflections - Example**

Adjust deflections:

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**Serviceability Load Deflections - Example**

Part II: Check crack midspan

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**Serviceability Load Deflections - Example**

Assume For interior exposure, the crack midspan is acceptable.

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