Limit analysis of masonry circular buttressed arches under horizontal loads

Abstract

Arches are among the most widely used and characterizing elements in masonry structures, particularly in historical buildings and churches. They are susceptible to severe damage due to seismic actions, excessive vertical loads, or support settlements. While an exhaustive literature is available for the behaviour of arched structures under vertical loads and support settlements, less attention has been devoted to their behaviour under horizontal seismic actions. In this paper, limit analysis approach to circular buttressed arches under horizontal loads is applied through a procedure developed by the authors based on non-linear programming technique. The suggested procedure is automatic, and structural engineers can easily implement it in a computer algebra system to identify the failure mechanism and the position of the hinges that minimises the horizontal load multiplier.

Appendix

The expressions for the evaluation of weights, horizontal and vertical components of displacements of the centroids of rigid bodies involved in the kinematical chains of Fig. 7 are provided in the following subsections. In particular, with reference to the failure mechanisms II and III of Fig. 6, these expressions can be used in Eq. (1) for obtaining the horizontal collapse multipliers λ of the masonry buttressed arch.

1.1 Mechanism II (global mechanism)

From simple geometrical considerations on the rotation angles of the three rigid bodies of the mechanism II shown in Fig. 7c, d, the follow relationships among Ψ _A , Ψ _B and Ψ _C can be found:

$$\psi_{A} = \psi$$

(27)

$$\psi_{B} = \psi_{A} \cdot \frac{{L/2 - (R + t) \cdot \cos \vartheta_{B} }}{{\left( {\frac{L + B}{{1 + \frac{\tan \alpha }{\tan \beta }}}} \right) - \left[ {L/2 - (R + t) \cdot \cos \vartheta_{B} } \right]}}$$

(28)

$$\psi_{C} = \psi_{B} \cdot \frac{{L/2 - \frac{2R + B}{{\left( {1 + \frac{\tan \alpha }{\tan \beta }} \right)}} - R \cdot \cos \vartheta_{C} }}{{B + L/2 + R \cdot \cos \vartheta_{C} }}$$

(29)

where

$$\tan \alpha = \frac{{h_{O} + (R + t) \cdot \sin \vartheta_{B} }}{{L/2 - (R + t) \cdot \cos \vartheta_{B} }}$$

(30)

$$\tan \beta = \frac{{h_{O} + R \cdot \sin \vartheta_{C} }}{{B + L/2 + R \cdot \cos \vartheta_{C} }}$$

(31)

hence, we have (angles in radians):

$$u_{pier,l} = \psi_{A} \cdot \frac{h}{2}$$

(32)

$$v_{pier,r} = \psi_{A} \cdot \frac{B}{2}$$

(33)

$$u_{arch,l} = \psi_{A} \cdot \left[ {h_{O} + \left( {R + \frac{t}{2}} \right) \cdot \sin \left( {\frac{{\vartheta_{B} }}{2}} \right)} \right]$$

(34)

$$v_{arch,l} = - \psi_{A} \cdot \left[ {\frac{L}{2} - \left( {R + \frac{t}{2}} \right) \cdot \cos \left( {\frac{{\vartheta_{B} }}{2}} \right)} \right]$$

(35)

$$u_{arch,c} = \psi_{B} \cdot \left\{ {\left[ {B + L - \left( {\frac{L + B}{{1 + \frac{\tan \alpha }{\tan \beta }}}} \right)} \right] \cdot \tan \beta - \left[ {h_{O} + \left( {R + \frac{t}{2}} \right) \cdot \sin \left( {\frac{{\vartheta_{B} + \vartheta_{C} }}{2}} \right)} \right]} \right\}$$

(36)

$$v_{arch,c} = \psi_{B} \cdot \left[ {\frac{L}{2} - \left( {\frac{L + B}{{1 + \frac{\tan \alpha }{\tan \beta }}}} \right) - \left( {R + \frac{t}{2}} \right) \cdot \cos \left( {\frac{{\vartheta_{B} + \vartheta_{C} }}{2}} \right)} \right]$$

(37)

$$u_{arch,r} = \psi_{C} \cdot \left[ {h_{O} + \left( {R + \frac{t}{2}} \right) \cdot \sin \left( {\frac{{\pi - \vartheta_{C} }}{2}} \right)} \right]$$

(38)

$$v_{arch,r} = \psi_{C} \cdot \left[ {B + \frac{L}{2} - \left( {R + \frac{t}{2}} \right) \cdot \cos \left( {\frac{{\pi - \vartheta_{C} }}{2}} \right)} \right]$$

(39)

$$u_{pier,r} = \psi_{C} \cdot \frac{h}{2}$$

(40)

$$v_{pier,r} = \psi_{C} \cdot \frac{B}{2}$$

(41)

and the corresponding weights can be evaluated as follows:

$$W_{arch,l} = \gamma \cdot s_{A} \cdot \left( {\frac{{\vartheta_{B} }}{2}} \right) \cdot t \cdot (2R + t)$$

(42)

$$W_{arch,c} = \gamma \cdot s_{A} \cdot \left( {\frac{{\vartheta_{C} - \vartheta_{B} }}{2}} \right) \cdot t \cdot (2R + t)$$

(43)

$$W_{arch,r} = \gamma \cdot s_{A} \cdot \left( {\frac{{\pi - \vartheta_{C} }}{2}} \right) \cdot t \cdot (2R + t)$$

(44)

$$W_{pier,l} = W_{pier,r} = \gamma \cdot s_{P} \cdot B \cdot h$$

(45)

where γ is the weight per unit volume of masonry material, s _A is the thickness of the arch, and s _P is the thickness of the pier wall.

1.2 Mechanism III (mixed mechanism)

The expressions to calculate the rotation angles Ψ _A , Ψ _B and Ψ _C of the rigid bodies involved in the failure mechanism III of Fig. 7e are:

$$\psi_{A} = \psi$$

(46)

$$\psi_{B} = \psi_{A} \cdot \frac{{R \cdot \cos \vartheta_{A} - (R + t) \cdot \cos \vartheta_{B} }}{{d^{II} }}$$

(47)

$$\psi_{C} = \psi_{B} \cdot \frac{{(R + t) \cdot \cos \vartheta_{B} - d^{III} - R \cdot \cos \vartheta_{C} }}{{B + L/2 + R \cdot \cos \vartheta_{C} }}$$

(48)

where d ^III is the horizontal distance between the rotational centres (1,2) = B and (2) reported in Fig. 7e:

$$d^{III} = \frac{{B + \frac{L}{2} + R \cdot \cos \vartheta_{A} - \frac{{h_{O} + R \cdot \sin \vartheta_{A} }}{\tan \beta }}}{{1 + \frac{\tan \alpha }{\tan \beta }}} - [R \cdot \cos \vartheta_{A} - (R + t) \cdot \cos \vartheta_{B} ]$$

(49)

and

$$\tan \alpha = \frac{{ - R \cdot \sin \vartheta_{A} + (R + t) \cdot \sin \vartheta_{B} }}{{R \cdot \cos \vartheta_{A} - (R + t) \cdot \cos \vartheta_{B} }}$$

(50)

$$\tan \beta = \frac{{h_{O} + R \cdot \sin \vartheta_{C} }}{{B + L/2 + R \cdot \cos \vartheta_{C} }}$$

(51)

$$h_{O} = h - (R + t) \cdot \cos \omega$$

(52)

Therefore, the horizontal and vertical components of the displacements of the centroids of the three rigid bodies involved in the mechanism III in Fig. 7f can be evaluated through the following formulas:

$$u_{arch,l} = \psi_{A} \cdot \left[ { - R \cdot \sin \vartheta_{A} + \left( {R + \frac{t}{2}} \right) \cdot \sin \left( {\frac{{\vartheta_{A} + \vartheta_{B} }}{2}} \right)} \right]$$

(53)

$$v_{arch,l} = - \psi_{A} \cdot \left[ {R \cdot \cos \vartheta_{A} - \left( {R + \frac{t}{2}} \right) \cdot \cos \left( {\frac{{\vartheta_{A} + \vartheta_{B} }}{2}} \right)} \right]$$

(54)

$$u_{arch,c} = \psi_{B} \cdot \left\{ {\left[ {B + \frac{L}{2} + (R + t) \cdot \cos \vartheta_{B} + d^{II} } \right] \cdot \tan \beta - \left[ {h_{O} + \left( {R + \frac{t}{2}} \right) \cdot \sin \left( {\frac{{\vartheta_{B} + \vartheta_{C} }}{2}} \right)} \right]} \right\}$$

(55)

$$v_{arch,c} = \psi_{B} \cdot \left[ {(R + t) \cdot \cos \vartheta_{B} - d^{II} - \left( {R + \frac{t}{2}} \right) \cdot \cos \left( {\frac{{\vartheta_{B} + \vartheta_{C} }}{2}} \right)} \right]$$

(56)

$$u_{arch,r} = \psi_{C} \cdot \left[ {h_{O} + \left( {R + \frac{t}{2}} \right) \cdot \sin \left( {\frac{{\pi + \vartheta_{C} }}{2}} \right)} \right]$$

(57)

$$v_{arch,r} = \psi_{C} \cdot \left[ {B + \frac{L}{2} + \left( {R + \frac{t}{2}} \right) \cdot \cos \left( {\frac{{\pi + \vartheta_{C} }}{2}} \right)} \right]$$

(58)

$$u_{pier,r} = \psi_{C} \cdot \frac{h}{2}$$

(59)

$$v_{pier,r} = \psi_{C} \cdot \frac{B}{2}$$

(60)

The weights of the three rigid blocks of buttressed arch to consider for the application of Eq. (1) at the mechanism III, can be calculated by using the following expressions (angles in radians):

$$W_{arch,l} = \gamma \cdot s_{A} \cdot \left( {\frac{{\vartheta_{B} - \vartheta_{A} }}{2}} \right) \cdot t \cdot (2R + t)$$

(61)

$$W_{arch,c} = \gamma \cdot s_{A} \cdot \left( {\frac{{\vartheta_{C} - \vartheta_{B} }}{2}} \right) \cdot t \cdot (2R + t)$$

(62)

$$W_{arch,r} = \gamma \cdot s_{A} \cdot \left( {\frac{{\vartheta_{D} - \vartheta_{C} }}{2}} \right) \cdot t \cdot (2R + t)$$

(63)

$$W_{pier,r} = \gamma \cdot s_{P} \cdot B \cdot h$$

(64)

where γ is the weight per unit volume of masonry material, s _A is the thickness of the arch, and s _P is the thickness of the pier wall.