Abstract
A fictitious crack model is introduced into cracked reinforced concrete beams to assess the changing beam stiffness under loads. Firstly, nonlinear concrete stress distributions near cracks are built based on the model. Then the stress of the steel bar at the cracked section is considered as cohesive stress. The concrete and steel stresses are substituted into the equilibrium equations of forces to solve the concrete stress. Based on the solution, the section inertias are estimated by iterating the calculation of the cracking open displacement, and finally the beam stiffness is assessed. Experimental data from seven concrete beams after cracking are adopted to validate the effectiveness of the proposed method, and the results show that the fictitious cracks ahead of actual cracks increase their depth with the load, which will raise the neutral axis and change the inertias of cracked sections and their neighboring sections. These changes are taken into account in the stiffness assessment, so the results predicted by the proposed method are shown to coincide well with the nonlinear deflections measured in the experiments.
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Abbreviations
- A c :
-
Area of concrete
- A ca :
-
Concrete area above the fictitious crack
- A cb :
-
Areas of the parts below the inflection point
- A cf :
-
Area between the tips of the fictitious and actual cracks
- A cu :
-
Areas of the parts above the inflection point
- a f :
-
Depth of the fictitious crack
- A t :
-
Area of steel bars
- a 0 :
-
Initial depth of the crack
- b :
-
Width of a beam
- c:
-
Thickness of concrete cover
- d t :
-
Distance from the steel centroid to the beam bottom Ds
- E c :
-
Concrete elastic modulus
- F :
-
External load vector
- f ct :
-
Flexural tensile strength of concrete
- f ctm :
-
Mean tensile strength of concrete
- f y :
-
Yield stress of the steel bars
- h :
-
Height of a beam
- I 0 :
-
Section inertia unaffected by cracks, including the steel contribution
- I eq :
-
Inertias of studied sections
- I f :
-
Fitted curve of inertias
- K b :
-
Global stiffness matrix
- K le :
-
Local stiffness matrix
- L :
-
Span of a beam lme
- l r :
-
Length of the effect region
- l s :
-
Spacing between two adjacent cracks
- l sl :
-
Left crack spacings
- l sr :
-
Right crack spacings
- l t :
-
Transfer length
- w b :
-
Crack opening displacement at the crack bottom
- W f :
-
Opening displacement of fictitious crack
- w t :
-
Crack opening displacement at the level of the steel
- w 0 :
-
Maximum crack opening width
- X :
-
Displacement vector of the beam
- y n :
-
Y-axis coordinate of neutral axis
- ε ct :
-
Concrete strain at the steel level
- ε ct0 :
-
Concrete strain at the steel level at the sections unaffected by cracks
- ε t :
-
Strain of steel
- ε tc :
-
Steel strain at the cracked section
- ε t0 :
-
Steel strain at the sections unaffected by cracks Δε(x)= εt − εct
- η :
-
Parameter denoting the nonlinear distribution
- ρ ef :
-
Effective reinforcement ratio
- σ c :
-
Concrete stress
- σ cb0 :
-
Bottom strain of sections unaffected by cracks
- σ ct :
-
Concrete stress at the beam top
- σ ctp :
-
Ctress at the crack tip
- σ t :
-
Stress of the steel bar
- σ w :
-
Cohesive stresses of the concrete
- τ bms :
-
Bond strength between steel and concrete
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Acknowledgments
This research was sponsored by the National Natural Science Foundation of China (51508155) and the Fundamental Research Funds for the Center Universities (2017B12314).
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Fu, C., Zhu, Y. & Tong, D. Stiffness Assessment of Cracked Reinforced Concrete Beams Based on a Fictitious Crack Model. KSCE J Civ Eng 25, 516–528 (2021). https://doi.org/10.1007/s12205-020-2056-0
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DOI: https://doi.org/10.1007/s12205-020-2056-0