A free-standing concrete wall occupies with

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Strain and stress in a deformed solid

Problem

A free-standing concrete wall occupies

0\le z\le H

with

H=20\,\mathrm m

, where

z

is height measured upward from its base, and nothing depends on the horizontal coordinates

x,y

(the wall is uniform and effectively infinite in extent). Because it was cast in layers that grow denser with height, its body force per unit volume is purely vertical and varies linearly with height,

f_z=-\rho_0g(1+kz),\qquad f_x=f_y=0,

with

\rho_0=2400\,\mathrm{kg\,m^{-3}}

k=0.010\,\mathrm{m^{-1}}

and

g=9.8\,\mathrm{m\,s^{-2}}

. The wall carries no other load and its top face

z=H

is free (unloaded). Nothing is assumed about the horizontal normal stresses

\sigma_{xx}

and

\sigma_{yy}

, which may be non-zero. ((a)) Starting from the equilibrium equations

\partial_j\sigma_{ij}+f_i=0

(derived from a force balance on an arbitrary volume with the divergence theorem), write the three Cartesian components for this wall. Use the

x

- and

y

-equations with the free top face to show that

\sigma_{xz}=\sigma_{yz}=0

throughout, and show that the

z

-equation then gives an ordinary differential equation for

\sigma_{zz}(z)

alone, whatever

\sigma_{xx}

and

\sigma_{yy}

are. ((b)) Integrate this equation and use the free-surface condition at

z=H

to find

\sigma_{zz}(z)

. ((c)) Evaluate

\sigma_{zz}

at the base of the wall,

z=0

, and check that the result reduces to the lithostatic value

-\rho_0gH

when

k=0

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