Strength of SCS sandwich composite plates

2.4 Strength of SCS sandwich composite beam and plate structure

2.4.5 Strength of SCS sandwich composite plates

2.4.5.1 Punching shear strength

The punching shear resistance of the SCS sandwich composite slab equals the nominal shear strength multiplied by the area of the critical section. For different design codes, the specifications on shear strength and the critical section are different.

  ‐ 43 -  2.4.5.1.1 Eurocode 2 and CEB-FIP 90 Method

In EC4, it specified that the punching shear resistance of the steel-concrete composite slab should follow the design specifications in EC2. The nominal shear strength for calculating punching shear resistance is specified in EC 2 as follows:

, ,

1

0.75 1.5 1 sin

Rd cs Rd c sw ywd

r

v v d A f

s u d 

  (2.29a)

, 1

Rd cs

V  v u d (2.29b)

where, vRd c, 0.18/c k 100fck1/3k1cpvmink1cp; k 1 200 2.0

  d  d in mm;

x y 0.02

    ; x and y are the ratio of flexural reinforcement in the slab;

 / 2

cp x y

    ;x and y are the normal concrete stress in the critical section in x and

y direction (MPa, ‘+’ in compression); c y, Ed x,

cy

N

  A and c x, Ed y,

cz

N

  A . NEdx , NEdy are longitudinal forces in x, y direction in the slab;  =the angle between the shear reinforcement and the plane of the slab; sr =the spacing of the radial spacing of perimeters of shear reinforcement, mm; d = the effective depth of the slab, mm; u1 = control perimeter of the critical section, in mm; A fsw ywd= yield strength of the shear reinforcement. The control perimeter is shown in Fig. 2.12.

2.4.5.1.2 ACI 318-05 method

In ACI 318-05, the punching shear resistance of the slab is governed by

c s

v v v (2.30)

vc is the lesser of the minimum value of the following three strengths

‐ 44 -     0.083 2 4

c ck

c

v  f



 

   

  (2.31a)

0.083 2

c s ck

v d f

 u 

 

   

  (2.31b)

c 0.33 ck

v   f (2.31c)

where, c=ratio of longer to shorter dimension of the loaded area; = factor to account for concrete density (1.0 for normal density concrete); s= 40 for interior columns, 30 for edge columns, 20 for corner columns;  =0.75 partial safety factor for shear; u = perimeter of the control section, in mm; fck in MPa.

In presence of the shear reinforcement, the shear strength is defined as

yv v s

v f A

 us (2.32)

where, s= the spacing of the shear reinforcement; u=critical perimeter.

Therefore, the punching shear resistance of the slab will be

V vud (2.33)

u= the critical perimeter specified in Fig. 2.12, mm.

2.4.5.1.3 CSA A23.3-2004

The concrete shear strength is specified as the minimum value of the following three strengths

0.19 1 2

c c ck

c

v  f



 

   

  (2.34a)

  ‐ 45 - 

c s 0.19 c ck

v d f

 u 

 

    (2.34b)

c 0.38 c ck

v   f (2.34c)

Equations used to calculate the punching shear strength of the slab are similar to Eqns.

(2.30), (2.32) and (2.33). The perimeter of the critical is the same as specified in the ACI 318-05 as shown in Fig. 2.12.

2.4.5.2 Flexural strength of SCS sandwich composite plate

The design procedures of designing steel-concrete sandwich system can be found in EC4 (2004) and design textbook by Johnson (2004). The flexural strength of this type of structure has been also studied by Solomn (1976), Ong et al. (1982), Oehelers and Braford (1999), Kumar (2000), and Sohel and Liew (2011). Among them, the Yield Line Method was widely used to calculate the load carrying capacity of the RC slabs, steel- concrete sandwich plates, and SCS sandwich plates.

2.4.5.2.1 Yield-Line Method

This method includes two steps to calculate the load carrying capacity of the SCS sandwich plates. The first step is to calculate the moment capacities of the section including both sagging and hogging moment capacity per unit length. In the second step, the yield line patter developed in the SCS sandwich plates is firstly assumed. Then, based on the assumed pattern, the load carrying capacity of the SCS sandwich plate will be calculated.

2.4.5.2.2 Calculating bending moment capacity of the plate

The moment carrying capacity of the SCS sandwich plate is similar to the strength of SCS sandwich beams as introduced in section 2.4.2.

‐ 46 -     Firstly, the longitudinal shear strength of the mechanical shear connectors need to be calculated by Eqns. (2.1)~(2.9). Secondly, based on the calculated shear strength of the connectors and quantity of the provided shear connectors, the moment capacity per unit length can be calculated by the following procedures:

1) The spacing between two neighbored rows of connectors welded to the top and bottom steel plates are denoted as STx and SB x in x direction, STy and SB yin y direction, respectively. For each strip of steel plate with a width equal to the spacing of the connectors as shown in Fig. 2.13, the tension or compression forces in the corresponding steel plates will be governed by the following two items:

Compression plate in a width of STx

min ,

2

ty

uTx Tx u c Rd

N S f t n P 

  

  (2.35a)

Tension plate with a width of SB x

min ,

2

By

uBx Bx u t Rd

N  S f t n P 

  (2.35b)

where, nty, nBy= quantity of the connectors attached to the top and bottom steel plate in the whole span along the y direction; PRd= shear strength by Eqns. (2.1)~(2.9);

Therefore, the unit compression and tension force in the top and bottom steel plates are determined as follows: 1) average unit compression force in the top steel plate is

min , /

2

ty

uTx Tx y c Rd Tx

N  S f t n P  S

  (2.36a)

2) Average unit tension force in the bottom steel plate is

  ‐ 47 - 

min , /

2

By

uBx Bx y t Rd Bx

N  S f t n P  S

  (2.36b)

The average forces acted on per unit width of the plate are as shown in Fig. 2.14. Average unit compression force in the concrete stress block is different in different design codes.

In EC 2, this force can be calculated by

  

uCx ck

N  x f (2.36c)

where,

   

0.8, 1.0 50

0.8 50 / 400, 1.0 50 / 200 50 90

ck

ck ck ck

for f MPa

f f for f MPa

 

 

  

        



By equating the compression force to the tension force, it gives

uBx uCx uTx

N N N (2.37)

The neutral axis depth can be obtained

uTx uBx

ck

N N

x f

  (2.38)

Therefore, the unit moment capacity along x direction is

2 2

t c c

x uBx c uCx

t t t

M N h   N x  (2.39) By the same method, the unit moment capacity My can be developed.

2.4.5.2.3 Yield line method analyses on slabs under concentrated load

They are two patterns of yield line for the slabs subjected to a concentrated load. The first type is Fan mechanism or radial yield line method whilst the second type is diagonal yield line method (as shown in Fig. 2.15). The yield line method was used to analyze the plates and slabs (Brổstrup, 1970). This theory was widely used in the design of the RC concrete structures (Megson, 2005). Later, Kumar (2000) applied this model in the analysis of SCS sandwich plates with headed stud connectors.

‐ 48 -     2.4.5.2.4 Flexural strength based on radial yield line method

The assumed yield lines developed in the SCS sandwich plates are shown in Fig. 2.15. It was assumed that the plate would yield in a circle section with a radius ‘R’ away from the point that the load was applied. The radial yield lines were developed in the yielding circle. A segment was taken out and the internal reaction moment forces are shown in Fig.

2.15. The reaction radial moment per unit length mr acted along the radius of the yield circle while the mcacted along the circumferential direction.

By taking moment on the c-c axis and based on the yield-line theory, the external energy would equal internal energy that acted on this sector segment. We can obtain

External applied energy= Internal Energy

r c 2

m r  m r  P  r

       Therefore, the applied load can be obtained as following

 

2 c r

P  m m (2.40)

where, mc and mr are the moment resistance per unit length along the circumferential and radial direction, respectively. mc and mr can be calculated by Eqn. (2.39).

2.4.5.2.5 Flexural strength based on diagonal yield line method

The second pattern of yield line in the SCS sandwich plates with simple support is shown in Fig. 2.15. The diagonal yield lines were assumed that developed in the sandwich plates linking the corner and the load point. The triangular segment was also taken out and the internal acting moments are shown in Fig. 2.15.

Following the same principle used for the radial yield line method, taking moment about

  ‐ 49 -  the simply-supported edge, it gives

u 4 2 m L  P L

Therefore, the applied concentrated load can be solved. The solution is 8 u

P  m (2.41)

where, mucan be calculated by Eqn. (2.39).

Moreover, Rankin and Long (1987) proposed a formula to calculate the RC concrete plate subjected to the concentrated loading. Sohel and Liew (2011) modified this model and apply it in analysis of the SCS sandwich composite plates with J-hook connectors. This modified formula is shown as follows

8 u Ls 0.172

P m

L c

 

     (2.42)

where, cis the side length of the loading area; Lsis the dimension of the slab specimen;

L is the span between the supports.