Problem 4.4. Cracking moment

The reinforced concrete cross section given in the Figure is subjected to pure bending. Does the cross section crack for a moment, M = 32 kNm? Determine the stresses in the extreme concrete fibres and in the steel. Give the curvature which belongs to the moment, M. Material and section properties are given in the Figure below. (In case of cracked section neglect the tensile stress in concrete.)

Solve Problem

Solve

Cracking moment Mcr [kNm]=

Stress in extreme concrete fiber, σc [N/mm2]=

Stress in steel, σs [N/mm2]=

Curvature, κ ×1061mm =

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Steps

Step by step

Step 1. Calculate the section properties of the inhomogeneous cross section in elastic stage.

See Example 4.1.

Check uncracked section properties

Concrete is chosen to be the reference material. Ratio of elastic moduli of concrete and steel is denoted by

Eq.(4-4)
α=EsEc=20018.3=10.93

The cross sectional properties of the replacement homogeneous cross section are

See Figure 4.5 and Eqs.(4-6)-(4-8)
Ae=Ac+αAs=bh+α1As=300×500+9.93×1257=1.625×105 mm2 xc=se=SeAe=bh22+α1AsdAe=300×50022+9.93×1257×455 1.625×105 =  265.7 mmIe,I=bxc33+bhxc33+α1Asdxc2=3.609×109 mm4
The moment of inertia of steel bars about their centroidal axis is neglected
Holes in concrete are often neglected and (α-1) is replaced by α.


Step 2. Calculate the cracking moment.

Check cracking moment

Concrete cracks when stress in the bottom extreme fiber reaches the tensile strength of concrete.

Eq.(4-9)

σcb=fct=McrIe,Ihxc      Mcr=fctIe,Ihxc=1.0×3.61×109500265.7=15.41 Nmm=15.41 kNm<M= 32 kNm

Thus the concrete cracks.

Step 3. Calculate the section properties of the cracked cross section.

See Example 4.3.

Check cracked section properties

After cracking of the cross section tensile stress in the concrete is neglected, both the steel and the compressed concrete zone still behave in a linearly elastic manner. Compressed concrete zone and tensile steel bars are replaced again by an equivalent homogeneous cross section. From the concrete cross section only the compressed concrete zone is taken into consideration (xcb), where xc is unknown (see Figure below).

The cross sectional properties of the equivalent homogeneous cross section are

Ae=bxc+αAs=300×xc+10.93×1257xc=se=SeAe=bxc22+αAsdbxcαAs =150xc2+10.93×1257×455300×xc10.93×1257       xc=163.4 mmIe,II=bxc33+αAsdxc2=300×163.433+10.93×1257455163.42=1.604×109 mm4

Step 4. Determine the relevant stress values in the concrete and in the steel bars.

Check stresses

We assume that both materials are in elastic stage.

Relevant stress values are

Eq.(4-9)

in the top extreme concrete fibre:

σct=MIe,IIxc=32×1061.60×109163.4=3.26Nmm2 (compression) < fc=13.33 Nmm2

in the steel bars:σsb=αMIe,IIdxc=10.9332×1061.60×109(455163.4)=63.56Nmm2 (tension) < fy=435Nmm2

Stresses arising in the cross section are lower than the tensile and compressive strength of the materials, thus the materials do not yield.

Step 5. Give the curvature from the given moment.

Check curvature

κII=MEcIe,II=32×10618.3×103×1.60×109=1.09×1061mm

Results

Worked out solution

First elastic materials and uncracked cross section is assumed. The inhomogeneous cross section is replaced by an equivalent homogeneous one.

Follow steps in Example 4.1.

Concrete is chosen to be the reference material. Ratio of elastic moduli of concrete and steel is denoted by

Eq.(4-4)
α=EsEc=20018.3=10.93

The cross sectional properties of the replacement homogeneous cross section are

See Figure 4.5 and Eq.(4-6)-(4-8)
Ae=Ac+αAs=bh+α1As=300×500+9.93×1257=1.625×105 mm2 xc=se=SeAe=bh22+α1AsdAe=300×50022+9.93×1257×455 1.625×105 =  265.7 mmIe,I=bxc33+bhxc33+α1Asdxc2=3.609×109 mm4
The moment of inertia of steel bars about their centroidal axis is neglected
Holes in concrete are often neglected and (α-1) is replaced by α.

Concrete cracks when stress in the bottom extreme fiber reaches the tensile strength of concrete. The cracking moment is:

Eq.(4-9)

σcb=fct=McrIe,Ihxc      Mcr=fctIe,Ihxc=1.0×3.61×109500265.7=15.41 Nmm=15.41 kNm<M= 32 kNm

Thus the concrete cracks. After cracking of the cross section tensile stress in the concrete is neglected, both the steel and the compressed concrete zone still behave in a linearly elastic manner. Compressed concrete zone and tensile steel bars are replaced again by an equivalent homogeneous cross section. From the concrete cross section only the compressed concrete zone is taken into consideration (xcb), where xc is unknown (see Figure below).

The cross sectional properties of the equivalent homogeneous cross section are

See Example 4.3.

Ae=bxc+αAs=300×xc+10.93×1257xc=se=SeAe=bxc22+αAsdbxcαAs =150xc2+10.93×1257×455300×xc+10.93×1257       xc=163.4 mmIe,II=bxc33+αAsdxc2=300×163.433+10.93×1257455163.42=1.604×109 mm4

We assume that both materials are still in elastic stage.

Relevant stress values are

Eq.(4-9)

in the top extreme concrete fibre:

σct=MIe,IIxc=32×1061.60×109163.4=3.26Nmm2 (compression) < fc=13.33 Nmm2

in the steel bars:σsb=αMIe,IIdxc=10.9332×1061.60×109(455163.4)=63.56Nmm2 (tension) < fy=435Nmm2

Stresses arising in the cross section are lower than the tensile and compressive strength of the materials, thus the materials do not yield.

The curvature from the given moment is

κII=MEcIe,II=32×10618.3×103×1.60×109=1.09×1061mm