This analytical model is developed by modifying the formulae calculating punching shear strength of the slab in EC4 and EC2. In this section, the control perimeter for calculating the punching shear strength is firstly modified based on the experimental investigations on the curved SCS sandwich composite structure. The stress acted on the control perimeter is calculated using methods in EC2. After that, new formulae developed for the curved SCS sandwich composite structure on calculating punching shear strength are proposed.
7.3.1 Modified controlled perimeter for SCS sandwich composite shell
From the literature review, it can be observed that a control perimeter with a distance of two times the depth of the slab away from the periphery of the loading area was used to calculate the punching shear resistance of the RC slab. The inclination angle between punched shear failure surface and free concrete surface is assumed to be 26.6°. In this section, this inclination angle will be modified based on the test results of nine SCS sandwich composite shells. Based on the proposed inclination angle, the control perimeter of the punching shear failure will be modified.
From the obtained inclination angle of the shear failure face for the punched cone of the core material, it can be observed that the inclination angle in the arch direction varies from 38.5°~41.8° with an average value of 40° (shown in Table 6.4). It is also observed that this angle varied with the curvature. This angle increases from 31.1° to 38.5° and 45.3° when the curvature ratio-L/R increases from 0.53 to 1.41 and 2, respectively. The relationship between the inclination angle of shear failure surface along the arch direction
‐ 303 -
a and the curvature can be expressed as following
0.264 /
26.75 L R
a e
(7.11)
where, L and R are the clear span and radius of inner shell, respectively; a is in degree.
The reasons choosing the exponential function are that 1) when L/R=0 the ashould be a value about 26.6 degree; 2)
This trend is shown in Fig. 7.1. From this figure, it can be found that the proposed formula can describe the relationship between angle of shear failure surface and curvature.
For inclination angle along the longitudinal width direction, an average value of 25- degree is proposed. Considering the inclination angle of 26.6 degree is used in EC2, the
b 26.6
is used for calculating the critical perimeter in the width direction.
Therefore, the critical failure surface can be calculated by the following formulae:
U0 (Laa)2(LbLa) 4 a (7.12) where, La 2 sin ;R Lb 2 cothc b a; 1 tan / 2
cos a ccos
a a
R a h
R
1 tan / 2
0 cos R a a hccos a 90
R
.
The formulae of the lengths for the control section that are used to calculate the shear strength along the arch direction and the longitudinal direction are specified as following (as shown in Fig. 7.2)
4
a a
L a
L a
(7.13a)
‐ 304 -
4
a
b b a
L a
L L L a
(7.13b)
The curvature of the shell not only changes the inclination angle of the shear failure surface but also changes the actual effective depth of the control section as shown in Fig.
7.2. As shown in this figure, the height of the section along the longitudinal direction remains the same thickness hc whilst the height of the shear failure section along the arch direction increases to a larger value due to the curvature of the shell. The height for control section are modified as following for the arch and longitudinal direction
2 2 / 2 2 1 cos
arch s c s c s
long c
h R t h R t h a R t
h h
(7.14)
where, harch and hlong are the height of the control section in arch and longitudinal direction, respectively.
7.3.2 Punching shear strength of the core material
The punching shear resistance of the concrete core is similar to the specifications in EC 2 as following
, ,
0.75 Rd c Rd s
V V V (7.15)
The punching shear strength contributed by the concrete comprises two main components that are shear resistances along the arch direction and shear resistances along the longitudinal direction. These components are calculated as following
, 2
Rd c arch long
V V V (7.16a)
arch c a long
V v L h (7.16b)
‐ 305 -
long c b arch
V v L h (7.16c)
where, vc 0.18 /ck100fck1/3k1cp ; k 1 200 /hc 2.0 hc in mm ;
x y 0.02
, , x y relate to bonded tension steel in x- and y- directions respectively; cp cxcy/ 2, , cx cy are the normal concrete stresses in the critical section in x- and y- direction (in MPa, positive in compression).
Another issue that needs to be carefully considered is that the steel face shell of the SCS sandwich structure changes the effective depth of the section. From the strains measured in the steel face shell during the tests on the seven specimens, it is observed that the strains developed in the inner bottom steel shells are much smaller than those in the outer top surface skin shell when the structure achieves the maximum punching shear strength (as shown in Table 6.3). Ebead and Marzouk (2002) used concept of the effective depth to calculate the punching shear strength of the RC slab stiffened by the steel plates. In this study, only the top steel shell is considered to modify the effective depth of the SCS sandwich composite shell, which is based on the observations of the strains developed in the two steel face shells. The modified depths of the structure are
long= /long s s c
h h t E E (7.17a)
arch= /arch s s c
h h t E E (7.17b)
where, hlong and harch are the effective depth of the SCS sandwich composite shell in width and arch direction, respectively.
Considering the shear connectors are used in the structure, the shear resistance provided by the connectors can be calculated by
‐ 306 -
, c t sin
Rd s c
V n F h
s
(7.18)
where, Ft = the tension capacity of the connector embedded in the core material; hc = thickness of the shell section; s= average spacing of the connectors; =angle between the shear stud and the plane vertical to the radius crossing the center of the loading area;
nc=quantity of the connectors linking shear cracks in the concrete.
Tension capacity of the shear connectors can be calculated by the recommended design approaches in Chapter 4, section 4.3.
7.3.3 Punching shear strength of the steel face shell
The core concrete material fails prior to the skin steel shell that can be judged from the strain compatibility between the steel face shell and the core concrete. After the core concrete is punched through when the structure is subjected to patch loading, the SCS sandwich composite shell exhibits a certain degree of residual strength benefiting from the local tension membrane effect of the top steel shell after the core material is punched.
This local tension membrane behavior occurs in the periphery of the load point. The working principle is shown in Fig. 7.3. The applied patch loading is transferred to the support and redistributed to a larger control perimeter through the tension membrane effect of the top steel shell. If the punching shear strength calculated by the new redistributed perimeter is larger than the punching shear strength of the outer steel shell in the vicinity of the loading area, the outer steel shell will be punched through. Otherwise, a larger perimeter core material will be punched. Moreover, if the width of the arch is limited, a larger control perimeter cannot be developed and the structure probably fails in global buckling.
‐ 307 - For the residual strength provided by the outer top steel shell, the punching resistance of the SCS sandwich structure is governed by
Punch 0.6 u s p
V f t U (7.19)
where, =2Up （a b ) is the perimeter of the patch loading area; a and b are the length of the longer and shorter side of the rectangular patching loading area, respectively.