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Investigating the use of autoclaved aerated concrete as an infill in reinforced concrete sandwich panels

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Abstract

Reinforced concrete composite sandwich panels are typically made of two concrete wythes separated by an insulating core of rigid polystyrene foam. The strength of the panels relies mostly on the efficiency of the shear connectors between the two concrete wythes. The rigid foam is too weak in shear and provides no contribution to the shear resistance of the connectors. In this paper, autoclaved aerated concrete (AAC) is used as an insulating core in place of foam and its efficiency to transfer shear between the sandwich panel layers is investigated. A total of nine full scale specimens made of rigid foam and AAC are tested under flexural load with varying types of shear connectors. Their ultimate load capacities are compared and their respective modes of failure are discussed. A detailed analysis of the stiffness of the shear connectors is done and used to predict the deflection of the panels under the applied load.

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Abbreviations

A :

Shear connector area

P :

Total applied lateral load

b :

Panel width

q u :

Maximum shear flow at support

c :

Thickness of the inner core

Q :

First moment of area of lower concrete wythe

d :

Distance from center of upper wythe to center of lower wythe

Q 1 :

Shear force in panel associated with bending moment M 1

E A :

Modulus of elasticity of AAC

E c :

Modulus of elasticity of concrete

E s :

Shear connector modulus of elasticity

Q 2 :

Shear force in panel associated with the additional vertical deflection w 2

t :

Thickness of the outer layer of the sandwich panel

F :

Axial connector force

T max :

Maximum tension

G :

Shear modulus of connector

I :

Moment of inertia of the faces about the centroid of the sandwich

I f :

Sum of the moment of inertia of the faces about their own centroids

I T :

Moment of inertia of the entire section

L :

Panel length

M max :

Maximum moment

n :

Number of connector lines across panel width

p :

Shear connector slope

V u :

Shear force at the support

w 1 :

Panel vertical deflection due to bending deformation of the panel

w 2 :

Panel vertical deflection due to shear stress in the core

γ :

Inner core shear deformation

τ :

Shear stress in the core

τ ave :

Average shear stress

τ max :

Maximum shear stress

References

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Author information

Authors and Affiliations

  1. Construction Engineering Department, The American University in Cairo, AUC Avenue, 5th Settlement, New Cairo, Egypt

    M. ElKashef

  2. Structural Engineering Department, Faculty of Engineering, Cairo University, Giza, Egypt

    M. AbdelMooty

Authors
  1. M. ElKashef
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  2. M. AbdelMooty
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Correspondence to M. ElKashef.

Appendix

Appendix

According to [7], the shear force Q 1 associated with bending moment M 1 can be divided into two components.

$$- Q_{1} = Dw_{1}^{\prime \prime \prime } = E_{\text{c}} (I - I_{\text{f}} )w_{1}^{\prime \prime \prime } + E_{\text{c}} (I_{\text{f}} )w_{1}^{\prime \prime \prime }$$
(8)

Where the first term on the right hand side of the equation represents the shear, neglecting the effect of the flexural rigidity of the faces, and the second term takes into account the effect of this flexural rigidity.

E c is the modulus of elasticity of the faces which is made of concrete, I is the moment of inertia of the faces about the centroid of the sandwich and I f is the sum of the moment of inertia of the faces about their own centroids,

$$I = {{bt^{3} } \mathord{\left/ {\vphantom {{bt^{3} } 6}} \right. \kern-0pt} 6} + {{btd^{2} } \mathord{\left/ {\vphantom {{btd^{2} } 2}} \right. \kern-0pt} 2}$$
$$I_{f} = {{bt^{3} } \mathord{\left/ {\vphantom {{bt^{3} } 6}} \right. \kern-0pt} 6}$$

The first term in Eq. 8 which represents a state of constant shear stress τ across the thickness of the core, assumed to be relatively weak, can be replaced by\(- bd\tau\), thus Eq. 8 can be written in the form,

$$- Q_{1}^{{}} = - bd\tau + E_{\text{c}} (I_{\text{f}} )w_{1}^{\prime \prime \prime }$$
(9)

The shear stress τ will result in an additional deflection w 2

The relation between the core shear strain \(\gamma\)and the additional deflection is defined as, \(\gamma = {{w_{2}^{\prime } d} \mathord{\left/ {\vphantom {{w_{2}^{\prime } d} c}} \right. \kern-0pt} c}\)

τ can now be expressed in terms of \(w_{2}^{\prime }\) as follows, \(\tau = {{Gw_{2}^{\prime } d} \mathord{\left/ {\vphantom {{Gw_{2}^{\prime } d} c}} \right. \kern-0pt} c}\)

where G is the shear modulus of core.

Hence Eq. 9 can now be expressed in the form,

$$- Q_{1}^{{}} = - AGw_{2}^{\prime } + E_{c} (I_{f} )w_{1}^{\prime \prime \prime }$$
(10)

where \(A = {{bd^{2} } \mathord{\left/ {\vphantom {{bd^{2} } c}} \right. \kern-0pt} c}\)

Rearranging terms and noting that \(- Q_{1}^{{}} = Dw_{1}^{\prime \prime \prime }\), one can write, \(w_{2}^{\prime } = {{Q_{1} (1 - {{I_{\text{f}} } \mathord{\left/ {\vphantom {{I_{\text{f}} } I}} \right. \kern-0pt} I})} \mathord{\left/ {\vphantom {{Q_{1} (1 - {{I_{\text{f}} } \mathord{\left/ {\vphantom {{I_{\text{f}} } I}} \right. \kern-0pt} I})} {AG}}} \right. \kern-0pt} {AG}}\)

The term \(a^{2} = {{AG} \mathord{\left/ {\vphantom {{AG} {E_{c} I_{\text{f}} (1 - {{I_{\text{f}} } \mathord{\left/ {\vphantom {{I_{\text{f}} } I}} \right. \kern-0pt} I})}}} \right. \kern-0pt} {E_{c} I_{\text{f}} (1 - {{I_{\text{f}} } \mathord{\left/ {\vphantom {{I_{\text{f}} } I}} \right. \kern-0pt} I})}}\)is introduced.

$$E_{\text{c}} I_{\text{f}} w_{2}^{\prime } = {{Q_{1} } \mathord{\left/ {\vphantom {{Q_{1} } {a^{2} }}} \right. \kern-0pt} {a^{2} }}$$
(11)

w 2 will be resisted by the stiff concrete faces resulting in additional shear force\(Q_{2}^{{}} = - E_{\text{c}} (I_{\text{f}} )w_{2}^{\prime \prime \prime }\)

The total shear force \(Q = Q_{1} + Q_{2} = Q_{1} - E_{c} (I_{f} )w_{2}^{\prime \prime \prime }\)

The term \(w_{2}^{\prime \prime \prime }\) can be expressed in terms of Q 1 using Eq. 11

$$Q_{1}^{\prime \prime } - a^{2} Q_{1} = - a^{2} Q$$
(12)

The solution to the above differential equation is

$$- Q_{1} = C_{1} \cosh (ax) + C_{2} \sinh (ax) + Q$$
(13)

where x is measured along the length of the panel, and C 1 and C 2 are constants.

Now, we will apply the general equation, Eq. 13, to the loading case under study.

Considering the slab loaded under a 4-pt loading where the load is applied at points B and C as shown in Fig. 15.

Fig. 15
figure 15

Schematic for a panel under 4-Point load

Full size image

Since the load is symmetrical, only half the panel will be considered. The equations governing the deflection are as follows:

For Part A-B, with x measured arbitrary from point B,

$$\begin{aligned} - Q_{1} &= C_{{1{\text{AB}}}} \cosh (ax) + C_{{2{\text{AB}}}} \sinh (ax) - W \\ EIw_{{1{\text{AB}}}}^{\prime \prime \prime } &= C_{{1{\text{AB}}}} \cosh (ax) + C_{{2{\text{AB}}}} \sinh (ax) - W \\ EIw_{{1{\text{AB}}}}^{''} &= C_{{1{\text{AB}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} a}} \right. \kern-0pt} a} + C_{{2{\text{AB}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} a}} \right. \kern-0pt} a} - Wx + 2C_{{3{\text{AB}}}}\\ EIw_{{1{\text{AB}}}}^{\prime }& = C_{{1{\text{AB}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} {a^{2} }}} \right. \kern-0pt} {a^{2} }} + C_{{2{\text{AB}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} {a^{2} }}} \right. \kern-0pt} {a^{2} }} - {{Wx^{2} } \mathord{\left/ {\vphantom {{Wx^{2} } 2}} \right. \kern-0pt} 2} + 2C_{{3{\text{AB}}}} x + C_{{4{\text{AB}}}} \\ EIw_{{1{\text{AB}}}}^{{}} &= C_{{1{\text{AB}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} + C_{{2{\text{AB}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} - {{Wx^{3} } \mathord{\left/ {\vphantom {{Wx^{3} } 6}} \right. \kern-0pt} 6} + C_{{3{\text{AB}}}} x^{2} + C_{{4{\text{AB}}}} x + C_{{5{\text{AB}}}}\\ EI_{\text{f}} w_{{2{\text{AB}}}}^{\prime } &= - C_{{1{\text{AB}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} {a^{2} }}} \right. \kern-0pt} {a^{2} }} - C_{{2{\text{AB}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} {a^{2} }}} \right. \kern-0pt} {a^{2} }} + {W \mathord{\left/ {\vphantom {W {a^{2} }}} \right. \kern-0pt} {a^{2} }}\\ EI_{\text{f}} w_{{2{\text{AB}}}}^{{}} &= - C_{{1{\text{AB}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} - C_{{2{\text{AB}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} + {{Wx} \mathord{\left/ {\vphantom {{Wx} {a^{2} }}} \right. \kern-0pt} {a^{2} }} - C_{{6{\text{AB}}}}\\ EI_{\text{f}} w_{{2{\text{AB}}}}^{\prime \prime } &= - C_{{1{\text{AB}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} a}} \right. \kern-0pt} a} - C_{{2{\text{AB}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} a}} \right. \kern-0pt} a} \\ EI_{\text{f}} w_{{2{\text{AB}}}}^{\prime \prime \prime } & = - C_{{1{\text{AB}}}} \cosh (ax) - C_{{2{\text{AB}}}} \sinh (ax) \end{aligned}$$

For Part B-C, with x measured arbitrary from mid-span

$$\begin{aligned} - Q_{1} &= C_{{1{\text{BC}}}} \cosh (ax) + C_{{2{\text{BC}}}} \sinh (ax) \\ EIw_{{1{\text{BC}}}}^{\prime \prime \prime } &= C_{{1{\text{BC}}}} \cosh (ax) + C_{{2{\text{BC}}}} \sinh (ax)\\ EIw_{{1{\text{BC}}}}^{\prime \prime } &= C_{{1{\text{BC}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} a}} \right. \kern-0pt} a} + C_{{2{\text{BC}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} a}} \right. \kern-0pt} a} + 2C_{{3{\text{BC}}}}\\ EIw_{{1{\text{BC}}}}^{\prime } &= C_{{1{\text{BC}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} {a^{2} }}} \right. \kern-0pt} {a^{2} }} + C_{{2{\text{BC}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} {a^{2} }}} \right. \kern-0pt} {a^{2} }} + 2C_{{3{\text{BC}}}} x + C_{{4{\text{BC}}}}\\ EIw_{{1{\text{BC}}}}^{{}} &= C_{{1{\text{BC}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} + C_{{2{\text{BC}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} + C_{{3{\text{BC}}}} x^{2} + C_{{4{\text{BC}}}} x + C_{{5{\text{BC}}}}\\ EI_{\text{f}} w_{{2{\text{BC}}}}^{\prime } &= - C_{{1{\text{BC}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} {a^{2} }}} \right. \kern-0pt} {a^{2} }} - C_{{2{\text{BC}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} {a^{2} }}} \right. \kern-0pt} {a^{2} }}\\ EI_{\text{f}} w_{{2{\text{BC}}}}^{{}} &= - C_{{1{\text{BC}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} - C_{{2{\text{BC}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} - C_{{6{\text{BC}}}}\\ EI_{\text{f}} w_{{2{\text{BC}}}}^{\prime \prime } &= - C_{{1{\text{BC}}}} {{\sinh (ax)} \mathord{\left/ {\vphantom {{\sinh (ax)} a}} \right. \kern-0pt} a} - C_{{2{\text{BC}}}} {{\cosh (ax)} \mathord{\left/ {\vphantom {{\cosh (ax)} a}} \right. \kern-0pt} a} \\ EI_{\text{f}} w_{{2{\text{BC}}}}^{\prime \prime \prime } &= - C_{{1{\text{BC}}}} \cosh (ax) - C_{{2{\text{BC}}}} \sinh (ax) \end{aligned}$$

To determine the constants of integration, the boundary conditions and requirements for continuity at B will be considered.

Four boundary conditions are available in AB, with x measured arbitrary from point B are,

  1. (1)

    \(w_{{1{\text{AB}}}} = 0\) at \(x = 0\) (Arbitrary)

    $$C_{{2{\text{AB}}}} = - C_{{1{\text{AB}}}} \tanh (aL)$$
  2. (2)

    \(w_{{2{\text{AB}}}} = 0\) at \(x = 0\) (Arbitrary)

    $$C_{{3{\text{AB}}}} = {{WL} \mathord{\left/ {\vphantom {{WL} 2}} \right. \kern-0pt} 2}$$
  3. (3)

    \(M = - WL\) at \(x = 0\)

    $${{C_{{2{\text{AB}}}} } \mathord{\left/ {\vphantom {{C_{{2{\text{AB}}}} } {a^{3} }}} \right. \kern-0pt} {a^{3} }} = - C_{{6{\text{AB}}}}$$
  4. (4)

    \(w^{\prime \prime }_{{1{\text{AB}}}} = 0\) at \(x = L\)

    $${{C_{{2{\text{AB}}}} } \mathord{\left/ {\vphantom {{C_{{2{\text{AB}}}} } {a^{3} }}} \right. \kern-0pt} {a^{3} }} = - C_{{5{\text{AB}}}}$$

In part BC with x measured from midspan, five boundary conditions are available as follows,

  1. (5)

    \(w_{{1{\text{BC}}}} = 0\) at \(x = 0\) (Arbitrary)

    $$C_{3BC} = {{WL} \mathord{\left/ {\vphantom {{WL} 2}} \right. \kern-0pt} 2}$$
  2. (6)

    \(w_{{2{\text{BC}}}} = 0\) at \(x = 0\) (Arbitrary)

    $$C_{4BC} = 0$$
  3. (7)

    \(w^{\prime }_{2BC} = 0\) at \(x = 0\) (Arbitrary)

    $$C_{{1{\text{BC}}}} = 0$$
  4. (8)

    \(w^{\prime }_{{1{\text{BC}}}} = 0\) at \(x = 0\) (Arbitrary)

    $${{C_{{2{\text{BC}}}} } \mathord{\left/ {\vphantom {{C_{{2{\text{BC}}}} } {a^{3} }}} \right. \kern-0pt} {a^{3} }} = - C_{{5{\text{BC}}}}$$
  5. (9)

    \(M = - WL\) at \(x = 0\)

    $${{C_{{2{\text{BC}}}} } \mathord{\left/ {\vphantom {{C_{{2{\text{BC}}}} } {a^{3} }}} \right. \kern-0pt} {a^{3} }} = - C_{{6{\text{BC}}}}$$

Using continuity conditions at point B,

  1. (10)

    \(w^{\prime }_{1}\) is continuous

    $${{C_{{1{\text{AB}}}} } \mathord{\left/ {\vphantom {{C_{{1{\text{AB}}}} } {a^{2} }}} \right. \kern-0pt} {a^{2} }} + C_{{4{\text{AB}}}} = {{C_{{2{\text{BC}}}} \sinh ({{aL} \mathord{\left/ {\vphantom {{aL} 2}} \right. \kern-0pt} 2})} \mathord{\left/ {\vphantom {{C_{{2{\text{BC}}}} \sinh ({{aL} \mathord{\left/ {\vphantom {{aL} 2}} \right. \kern-0pt} 2})} {a^{2} + {{WL^{2} } \mathord{\left/ {\vphantom {{WL^{2} } 2}} \right. \kern-0pt} 2}}}} \right. \kern-0pt} {a^{2} + {{WL^{2} } \mathord{\left/ {\vphantom {{WL^{2} } 2}} \right. \kern-0pt} 2}}}$$
  2. (11)

    \(w^{\prime }_{2}\) is continuous

    $$C_{{4{\text{AB}}}} = {{{{ - W} \mathord{\left/ {\vphantom {{ - W} {a^{2} + }}} \right. \kern-0pt} {a^{2} + }}WL^{2} } \mathord{\left/ {\vphantom {{{{ - W} \mathord{\left/ {\vphantom {{ - W} {a^{2} + }}} \right. \kern-0pt} {a^{2} + }}WL^{2} } 2}} \right. \kern-0pt} 2}$$
  3. (12)

    \(w^{\prime \prime }_{1}\) is continuous

    $$C_{{2{\text{BC}}}} \cosh ({{3aL} \mathord{\left/ {\vphantom {{3aL} 2}} \right. \kern-0pt} 2}) = - W\sinh (aL)$$

The summation of w for part AB can be written as

$$\begin{gathered} w_{\text{AB}} = \left[ {{{\left( { - {{C_{{1{\text{AB}}}} \sinh (ax)} \mathord{\left/ {\vphantom {{C_{{1{\text{AB}}}} \sinh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} - {{C_{{2{\text{AB}}}} \cosh (ax)} \mathord{\left/ {\vphantom {{C_{{2{\text{AB}}}} \cosh (ax)} {a^{3} - C_{{6{\text{AB}}}} + {{Wx} \mathord{\left/ {\vphantom {{Wx} {a^{2} }}} \right. \kern-0pt} {a^{2} }}}}} \right. \kern-0pt} {a^{3} - C_{{6{\text{AB}}}} + {{Wx} \mathord{\left/ {\vphantom {{Wx} {a^{2} }}} \right. \kern-0pt} {a^{2} }}}}} \right)} \mathord{\left/ {\vphantom {{\left( { - {{C_{{1{\text{AB}}}} \sinh (ax)} \mathord{\left/ {\vphantom {{C_{{1{\text{AB}}}} \sinh (ax)} {a^{3} }}} \right. \kern-0pt} {a^{3} }} - {{C_{{2{\text{AB}}}} \cosh (ax)} \mathord{\left/ {\vphantom {{C_{{2{\text{AB}}}} \cosh (ax)} {a^{3} - C_{{6{\text{AB}}}} + {{Wx} \mathord{\left/ {\vphantom {{Wx} {a^{2} }}} \right. \kern-0pt} {a^{2} }}}}} \right. \kern-0pt} {a^{3} - C_{{6{\text{AB}}}} + {{Wx} \mathord{\left/ {\vphantom {{Wx} {a^{2} }}} \right. \kern-0pt} {a^{2} }}}}} \right)} {EI_{\text{f}} }}} \right. \kern-0pt} {EI_{\text{f}} }}} \right]\left[ {1 - \left( {{{I_{\text{f}} } \mathord{\left/ {\vphantom {{I_{\text{f}} } I}} \right. \kern-0pt} I}} \right)} \right] \hfill \\ + \left[ {{{\left( {{{WL^{2} x} \mathord{\left/ {\vphantom {{WL^{2} x} 2}} \right. \kern-0pt} 2} + {{WLx^{2} } \mathord{\left/ {\vphantom {{WLx^{2} } 2}} \right. \kern-0pt} 2} - {{Wx^{3} } \mathord{\left/ {\vphantom {{Wx^{3} } 6}} \right. \kern-0pt} 6}} \right)} \mathord{\left/ {\vphantom {{\left( {{{WL^{2} x} \mathord{\left/ {\vphantom {{WL^{2} x} 2}} \right. \kern-0pt} 2} + {{WLx^{2} } \mathord{\left/ {\vphantom {{WLx^{2} } 2}} \right. \kern-0pt} 2} - {{Wx^{3} } \mathord{\left/ {\vphantom {{Wx^{3} } 6}} \right. \kern-0pt} 6}} \right)} {EI}}} \right. \kern-0pt} {EI}}} \right] \hfill \\ \end{gathered}$$
(14)

The maximum deflection for this part occurs at point x (one third of the panel length) = 93 cm

The summation of w for part BC can be written as

$$w_{\text{BC}} = \left[ {{{\left( {{{C_{{2{\text{BC}}}} \left( {1 - \cosh (ax)} \right)} \mathord{\left/ {\vphantom {{C_{{2{\text{BC}}}} \left( {1 - \cosh (ax)} \right)} {a^{3} }}} \right. \kern-0pt} {a^{3} }}} \right)} \mathord{\left/ {\vphantom {{\left( {{{C_{{2{\text{BC}}}} \left( {1 - \cosh (ax)} \right)} \mathord{\left/ {\vphantom {{C_{{2{\text{BC}}}} \left( {1 - \cosh (ax)} \right)} {a^{3} }}} \right. \kern-0pt} {a^{3} }}} \right)} {EI_{\text{f}} }}} \right. \kern-0pt} {EI_{\text{f}} }}} \right]\left[ {1 - \left( {{{I_{\text{f}} } \mathord{\left/ {\vphantom {{I_{\text{f}} } I}} \right. \kern-0pt} I}} \right)} \right] + \left[ {{{\left( {{{WLx^{2} } \mathord{\left/ {\vphantom {{WLx^{2} } 2}} \right. \kern-0pt} 2}} \right)} \mathord{\left/ {\vphantom {{\left( {{{WLx^{2} } \mathord{\left/ {\vphantom {{WLx^{2} } 2}} \right. \kern-0pt} 2}} \right)} {EI}}} \right. \kern-0pt} {EI}}} \right]$$
(15)

The maximum deflection for this part occurs at x (located at midspan) = 46.5 cm

The maximum deflection at midspan is the summation of the maxima for Eqs. 14 and 15.

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ElKashef, M., AbdelMooty, M. Investigating the use of autoclaved aerated concrete as an infill in reinforced concrete sandwich panels. Mater Struct 48, 2133–2146 (2015). https://doi.org/10.1617/s11527-014-0298-3

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