Design of shear reinforcement for thick plates using a strut-and-tie model
aDepartment of Civil Engineering, Faculty of Engineering, Architecture and Science, Ryerson University, Toronto, ON M5B 2K3, Canada.
bFaculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL A1B 3X5, Canada.
cTiller Engineering Inc, Box 403, 50 Hamlyn Rd., St. John’s, NL A1E 5X7, Canada.
Published on the web 10 February 2010.
Received November 24, 2008.
Canadian Journal of Civil Engineering, 2010, 37(2): 181-194, https://doi.org/10.1139/L09-120
Abstract
The strut-and-tie method is a rational approach to structural concrete design that results in a uniform and consistent design philosophy. A strut-and-tie model has been developed to model the punching-shear behaviour of thick concrete plates. This model provides a quick and simple approach to check the punching-shear behaviour. Thick concrete slabs (250–500 mm) without shear reinforcement can exhibit brittle shear failure under a central force and an unbalanced moment. Shear reinforcement has proven to be very effective in preventing such failures. The developed strut-and-tie model has also been used to evaluate the minimum shear reinforcement required to prevent brittle shear failure of two-way slabs in the vicinity of concentrated loads. The strut-and-tie model for symmetric punching consists of a “bottle-shaped” compressive zone in the upper section of the slab depth, leading to a “rectangular-stress” compressive zone in the lower section of the slab depth. Inclined shear cracking develops in the bottle-shaped zone prior to failure in the lower zone. Cracking in the bottle-shaped zone is related to the splitting tensile strength of concrete.
Keywords: thick plates, strut-and-tie model, shear reinforcement, size effect, splitting bond stress, punching, bottle-shaped strut
References
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| A | width of the diagonal compression strut and taken as being equal to the truss width |
| Av | area of the stirrups legs (or shear studs) |
| Az,min | area of minimum shear reinforcement |
| bef | effective width of a bottle-shaped strut |
| bmin | width of the bearing area |
| bo | perimeter of critical section for shear and slap footings |
| c | side length of square column |
| C | maximum load on unreinforced strut |
| CT | total compression force acting on the thickness of the conical shell–strut around the periphery of a circular column |
| d | effective depth of slab |
| dv | effective shear depth, taken as the greater of 0.9d or 0.72h |
| D | diameter of column |
| Ec | modulus of elasticity of concrete |
| Es | modulus of elasticity of steel |
 | uniaxial compressive strength of concrete (cylinder strength) |
| fck | characteristic compressive strength of concrete |
| fct | tensile stress in concrete |
| fctm | mean concrete tensile strength at the time that the crack forms |
| fcu | limiting compressive stress in concrete strut |
| fc2,max | maximum allowable concrete strength in biaxial compression-compression |
| fsp,t | splitting bond stress |
| fy | yield stress of steel |
| Gf | fracture energy |
| h | slab height |
| l | length of the strut from face to face of the nodes |
| lch | characteristic length |
| n | modular ratio, Es/Ec |
| P | support reaction |
| Peq | equivalent point load |
| Ptest | experimental failure load |
| Pult | corresponding ultimate punching-shear capacity failure mechanism |
| s | spacing between peripheral lines of vertical members |
| t | thickness of the strut |
| T | transverse tension force |
| ucrit | length along control perimeter |
| vc | shear stress resistance provided by concrete |
| vf | maximum factored shear stress |
| vr | factored shear stress resistance |
| vu | shear strength |
| VRd,ct | punching shear capacity of point-supported, reinforced concrete |
| Vtest | punching failure load |
| Vu | ultimate punching shear |
| y | depth of flexural compression zone in slab (depth of neutral plane) |
| y1 | distance from the neutral axis to the centre of the lower tensile force |
| α | sensitivity exponent |
| ε1 | principal tensile strain in cracked concrete due to factored loads |
| εs | tensile strain |
| ϕc | resistance factor of concrete |
| λ | modification factor of lightweight concrete |
| ρ | flexural reinforcement ratio |
| ρ′ | compression steel reinforcement ratio |
| ρe | ratio of reinforcement for a basic yield strength of 500 MPa |
| ρl | ratio of longitudinal reinforcement |
| ρz,min | minimum shear reinforcement ratio |
| θ | angle of inclination of normal to crack to x reinforcement |
| θs | smallest angle between strut and adjoining ties |
| φ | angle of compression field |
Appendix A. Design examples
Design example 1 (thick slab)
Input the following slab information (Rizk and Marzouk 2010*) (slab HS3): slab height h = 350 mm, square column dimension c = 400 mm, equivalent circular column diameter D = 509 mm, structural depth d = 262.5 mm, 
Symmetric punching
Neutral axis depth 
Crack zone of strut-and-tie model
Crack zone length 
Ultimate failure zone of strut-and-tie model
The ultimate punching shear based on strut-and-tie model, Pult:
The required minimum shear reinforcement ratio is given as follows: 
Use three number 15M studs per each peripheral line (Fig. A1).
Fig. A1.
Arrangement of T-headed minimum shear reinforcement for slab HS3 (slab height 350 mm).
Design example 2 (thin slab)
Input the following slab information (Marzouk and Hussein 1991) (slab HS10): square column dimension c = 150 mm, equivalent circular column diameter D = 191 mm, structural depth d = 120 mm, 
Symmetric punching
Neutral axis depth 
Crack zone of strut-and-tie model
Crack zone length 
Ultimate failure zone of strut-and-tie model
The ultimate punching shear based on the strut-and-tie model, Pult: 
Instructions to authors
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