Piers, Walls, Buttresses and Towers

Abstract

This chapter is addressed to the structural analysis under vertical loads of walls, piers, buttresses and towers. For them, the non-linear interaction between the destabilizing effects of the axial loads and the masonry no-tension response can be very strong. Instability analysis of the masonry pier under an eccentric axial load is firstly studied in the wake of a significant study of Yokel. The strong sensitivity of the pier strength to the eccentricity of the load is pointed out and comparisons are made with the case of reinforced concrete columns. Statics of buttresses as of retaining walls is also developed in the chapter. For these structural systems the occurrence of inclined lines of cracks has a not negligible influence on their strength. Static analysis of building masonry walls is then examined. For them the presence of offsets of the wall thickness at the various stories plays a relevant role. Instability of towers whose behavior can be strongly influenced by foundation deformability, is analyzed at the end of the section. Special attention has been given to the stability analysis of the Pisa Tower, which recently underwent an outstanding restoration work.

Appendix

Integration of the Yokel differential equation

By means of Eq. (9.15), we obtain

$$\frac{d}{dx}(\frac{dy}{dx})^{2} = \frac{2dy}{dx} \cdot \frac{{d^{2} y}}{{dx^{2} }} = \frac{{2k_{1} }}{{\left( {u_{0} + y} \right)^{2} }} \cdot \frac{dy}{dx}$$

and

$$(\frac{dy}{dx})^{2} = 2k_{1} \int {\frac{dy}{{(u_{0} + y)^{2} }} = - 2k_{1} \frac{1}{{u_{0} + y}} + c_{1} } .$$

$${\text{For}}\,y \to 0,{\text{we}}\,{\text{have}}\,dy/dx \to 0\,{\text{and}}\,{\text{consequently}}\,c_{1} = 2k_{1} /u_{0} ,$$

and

$$(\frac{dy}{dx})^{2} = 2k_{1} (\frac{1}{{u_{0} }} - \frac{1}{{u_{0} + y}}) = \frac{{2k_{1} }}{{u_{0} }} \cdot \frac{y}{{u_{0} + y}},$$

hence

$$\frac{dy}{dx} = \pm \sqrt {\frac{{2k_{1} }}{{u_{0} }}} \cdot (\frac{y}{{u_{0} + y}})^{1/2} .$$

(9.194)

With the position

$$k_{2} = \sqrt {\frac{{2k_{1} }}{{u_{0} }}} = \frac{2}{3}\sqrt {\frac{P}{{Ebu_{0} }}} ,$$

(9.195)

Equation (9.194) becomes:

$$\frac{dy}{dx} = \pm k_{2} (\frac{y}{{u_{0} + y}})^{1/2}$$

or

$$dx = \pm \frac{1}{{k_{2} }}(\frac{{u_{0} + y}}{y})^{1/2} \cdot dy.$$

Integration yields

$$x = \pm \frac{1}{{k_{2} }}\int {(\frac{{u_{0} + y}}{y})^{1/2} dy}$$

and, consequently

$$\int {(\frac{{u_{0} + y}}{y})}^{1/2} dy = \sqrt {y\left( {u_{0} + y} \right)} + \frac{{u_{0} }}{2}\int {\frac{dy}{{\sqrt {y\left( {u_{0} + y} \right)} }}} = \sqrt {y\left( {u_{0} + y} \right)} + u_{0} \ln (\sqrt y + \sqrt {u_{0} + y} ),$$

hence

$$x = \pm \frac{1}{{k_{2} }}\left[ {\sqrt {y\left( {u_{0} + y} \right)} + u_{0} \ln \left( {\sqrt y + \sqrt {u_{0} + y} } \right)} \right] + c_{2} .$$

By accounting that at x = 0, y = 0, we get:

$$c_{2} = - \frac{{u_{0} }}{{k_{2} }}\ln \sqrt {u_{0} }$$

and

$$x = \pm \frac{1}{{k_{2} }}[\sqrt {y\left( {u_{0} + y} \right)} + u_{0} \ln \frac{{\sqrt y + \sqrt {u_{0} + y} }}{{\sqrt {u_{0} } }}].$$

(9.196)

Substituting (9.196) into (9.195) of k ₂ furnishes:

$$x = \pm \frac{3}{2}\sqrt {\frac{{Ebu_{0} }}{P}} \cdot [\sqrt {y\left( {u_{0} + y} \right)} + u_{0} \ln \frac{{\sqrt y + \sqrt {u_{0} + y} }}{{\sqrt {u_{0} } }}].$$

By considering that for x = h/2, y = u ₁  − u ₀ , the solution is expressed in terms of u ₁ :

$$\frac{h}{2} = \pm \frac{3}{2}\sqrt {\frac{{Ebu_{0} }}{P}} \cdot [\sqrt {u_{1} \left( {u_{1} - u_{0} } \right)} + u_{0} \ln \sqrt {\frac{{u_{1} - u_{0} }}{{u_{0} }}} + \sqrt {\frac{{u_{1} }}{{u_{0} }}} ],$$

and hence

$$P = \frac{{9Ebu_{0} }}{{h^{2} }} \cdot [\sqrt {u_{1} \left( {u_{1} - u_{0} } \right)} + u_{0} \ln (\sqrt {\frac{{u_{1} - u_{0} }}{{u_{0} }}} + \sqrt {\frac{{u_{1} }}{{u_{0} }}} )]^{2} .$$

With the position

$$\alpha = u_{0} /u_{1} ,$$

(9.197)

we finally arrive at

$$P = \frac{{9Ebu_{1}^{3} }}{{h^{2} }} \cdot \alpha \cdot [\sqrt {1 - \alpha } + \alpha \,\ln (\sqrt {\frac{1 - \alpha }{\alpha }} + \sqrt {\frac{1}{\alpha }} )]^{2} .$$

(9.198)