# Instantaneous and Time Dependent Deflection Example 1

In the structure whose plan view is given in the image below, the instantaneous deflection calculation will be made for the K06 beam and compared with the ideCAD results. For the K06 beam, instantaneous deflection under the effect of live loads (Q) and instantaneous deflection and time-dependent deflection calculations for Dead Loads (G) will be made. The results will be compared with **ideCAD v10** by calculating the total deflection from the deflections consisting of fixed and live loads.

**Important Note:** Deflection in reinforced concrete elements is calculated with Effective Moments of Inertia in ideCAD software . Then, taking into account these effective moments of inertia, a new analysis is made for the deflection control of the structure only. However, in this example, approximate formulas will be used to calculate the beam deflection calculation easier by hand and only deformations due to bending will be considered. In ideCAD software, the deformation effects of all internal forces are taken into account. For this reason, manual calculation and ideCAD results may differ slightly.

**Project File**

Instantaneous and Time Dependent Deflection Example 1.rar

**K06 Beam Section Properties**

Modulus of elasticity of concrete,

*E*= 30250 MPa_{c}Modulus of elasticity of Reinforcement

*E*200000 MPa_{s}=Design tensile strength of concrete,

*f*= 1.167 MPa_{ctd}T-beam width 69.9 cm

Span length of beam

*l*= 5.60 m_{n}

**Calculation of Section Center of Gravity and Gross Concrete Section Moment of Inertia**

Moment of inertia of gross concrete section *I _{c}*, is calculated about controidal axis. Therefore, the axis of the center of gravity of the section, distance from centroidal axis of gross section, neglecting reinforcement, to top surface and distance from centroidal axis of gross section, neglecting reinforcement, to bottom surface of the section should be calculated.

y_{top} : distance from centroidal axis of gross section, neglecting reinforcement, to top surface

y_{bot} : distance from centroidal axis of gross section, neglecting reinforcement, to bottom surface

Moment of inertia of gross concrete section about controidal axis, *I _{c}* is calculated as follows

**Section Cracking Control**

The span, right and left moment values (*M _{max}*) for the K06 beam are compared with the cracking moment

*M*of the member.

_{cr}If

*moment of inertia of gross concrete section*

**M**_{max}≤ M_{cr}*is used for deflection.*

**I**_{c}If

*M*Effective moment of inertia

_{max}> M_{cr}*is used for deflection.*

**I**_{ef}_{ }*M _{cr}* is calculated in accordance with

**TS500 13.2**. y is distance from centroidal axis of gross section, neglecting reinforcement, to tension surface. For this reason,

*M*values should be calculated separately both in the span and in the end regions.

_{cr}For the end region, the bending cracking moment M_{cr} is calculated as follows.

For the span region, the bending cracking moment M _{cr} is calculated as follows.

**Beam Cracking Control Under Constant Load (G) and Live Load (Q)**

The moment diagram of the K06 beam for the G+Q loading combination is shown below.

At the end regions, the bending cracking moment is calculated as *M _{cr}* = 6,208 tfm.

Since *M _{cr}* = 6.208 tfm >

*M*= 3.902 tfm, no cracking occurred at the end regions. For this reason, Moment of inertia of gross concrete section

_{max}*I*value is used in the deflection calculation.

_{c}In the span regions, the bending cracking moment is calculated as *M _{cr}* = 3.898 tfm.

Since *M _{cr}* = 3.898 tfm >

*M*= 3.179 tfm, no cracking occurred at the end regions. For this reason, Moment of inertia of gross concrete section

_{max}*I*value is used in the deflection calculation.

_{c}**Instantaneous Deflection Under Live Load (Q) Effects**

The moment diagram of the K06 beam for the Live Load (Q) effects is shown below.

At the end regions, the bending cracking moment is calculated as *M _{cr}* = 6,208 tfm.

Since *M _{cr}* = 6.208 tfm >

*M*= 0.78 tfm, no cracking occurred at the end regions.

_{max}In the span regions, the bending cracking moment is calculated as *M _{cr}* = 3.898 tfm.

Since *M _{cr}* = 3.898 tfm >

*M*= 0.668 tfm tfm, no cracking occurred at the end regions.

_{max}For this reason, Moment of inertia of gross concrete section *I _{c}* value is used in the deflection calculation. Instantaneous deflection due to live loads, δ

_{iQ}

Deflection control;

δ_{iQ} = 0.143 mm < *l _{n}*/360 =5600/360= 15.55 mm

The deflection results of the K06 beam under the Live Load (Q) effect in the *Beam Reinforcement* window are shown below.

**Instantaneous Deflection Under Dead Load (G) Effects**

The moment diagram of the K06 beam for the DEad Load (G) effects is shown below.

At the end regions, the bending cracking moment is calculated as *M _{cr}* = 6,208 tfm.

Since *M _{cr}* = 6.208 tfm >

*M*= 3.122 tfm, no cracking occurred at the end regions.

_{max}In the span regions, the bending cracking moment is calculated as *M _{cr}* = 3.898 tfm.

Since *M _{cr}* = 3.898 tfm >

*M*= 2.51 tfm tfm, no cracking occurred at the end regions.

_{max}For this reason, Moment of inertia of gross concrete section *I _{c}* value is used in the deflection calculation. Instantaneous deflection due to dead loads, δ

_{iG}

The deflection results of the K06 beam under the Dead Load (G) effect in the *Beam Reinforcement* window are shown below.

**Time Dependent Deflection**

Total deflection calculation and permanent deflection coefficient formulas are shown below.

Permanent deflection multiplier * γ_{t}* is taken as 2.0 based on 5 years or more loading time in accordance with

**Table 13.2.**

*A _{s}' *is area of the compression reinforcement and

*ρ'*Compression reinforcement ratio;

When calculating the total deflection δ_{t}, the instantaneous deflection δ_{iQ} consisting of live loads, the instantaneous deflection δ_{iG} consisting of dead loads and the time-dependent deflection δ_{iG}λ are added. In this situation total deflection δ_{t }is calculated as follows.

Deflection control;

δ_{iQ} = 1.61 mm < *l _{n}*/260 =5600/240= 23.33 mm

The total deflection results of the K06 beam in the *Beam Reinforcement* window under the effect of Dead and Live loads are shown below.

The results calculated above can also be checked in the **Beam Deflections and Crackings **Report .